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advanced and retarded causal propagators in nLab
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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a></p> </li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a></p> </li> <li> <p><a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a></strong></p> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> <p id="models_2"><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>)</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></p> </li> </ul> </div></div> <h4 id="algebraic_quantum_field_theory">Algebraic Quantum Field Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a></p> <h2 id="concepts">Concepts</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a></strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observables">observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+net">field net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> <p><a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a></p> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a></p> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>)</p> </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#existence_and_uniqueness'>Existence and uniqueness</a></li> <li><a href='#Continuity'>Continuity</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#ForKleinGordonOperatorOnMinkowskiSpacetime'>For Klein-Gordon operator on Minkowski spacetime</a></li> <li><a href='#ExampleForDiracOperatorOnMinkowskiSpacetime'>For Dirac operator on Minkowski spacetime</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>What are called <em>advanced</em> and <em>retarded causal Green functions</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>A</mi><mo stretchy="false">/</mo><mi>R</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{A/R}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/Green+functions">Green functions</a> for <a class="existingWikiWord" href="/nlab/show/hyperbolic+differential+operators">hyperbolic differential operators</a> on manifolds with <a class="existingWikiWord" href="/nlab/show/causal+structure">causal structure</a> (e.g. <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a>) whose <a class="existingWikiWord" href="/nlab/show/support">support</a> is in the <a class="existingWikiWord" href="/nlab/show/future+cone">future cone</a> or <a class="existingWikiWord" href="/nlab/show/past+cone">past cone</a>, respectively, of the source excitation. The corresponding <a class="existingWikiWord" href="/nlab/show/integral+kernels">integral kernels</a> hence say how a <a class="existingWikiWord" href="/nlab/show/delta+distribution">point excitation</a> <em>propagates</em> into the future or past, respectively, via the given <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a>, and therefore these are also called the advanced/future <em>propagators</em>.</p> <p>If both advanced and retarded Green functions exist for a differential operator as well as for its <a class="existingWikiWord" href="/nlab/show/formally+adjoint+differential+operator">formal adjoint</a>, then the differental operator is called a <em><a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+operator">Green hyperbolic differential operator</a></em>. The archetypical examples are, on <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetimes">globally hyperbolic spacetimes</a>:</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/normally+hyperbolic+differential+operators">normally hyperbolic differential operators</a> such as the <a class="existingWikiWord" href="/nlab/show/wave+operator">wave operator</a> and the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operators">Dirac operators</a> on <a class="existingWikiWord" href="/nlab/show/spinor+bundles">spinor bundles</a> whose square is a normally hyperbolic differential operator as above.</p> </li> </ol> <p>The advanced/retarded <a class="existingWikiWord" href="/nlab/show/integral+kernel">integral kernel</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>A</mi><mo stretchy="false">/</mo><mi>R</mi></mrow></msub><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta_{A/R} \in \mathcal{D}'(X \times X) </annotation></semantics></math></div> <p>is such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>R</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y) \in supp(\Delta_R)</annotation></semantics></math> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is in the <a class="existingWikiWord" href="/nlab/show/causal+future">causal future</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>supp</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y) \in supp(\Delta_A)</annotation></semantics></math> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is in the <a class="existingWikiWord" href="/nlab/show/causal+past">causal past</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>.</p> </li> </ol> <p>Written as <a class="existingWikiWord" href="/nlab/show/generalized+functions">generalized functions</a> these satisfy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Δ</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_A(x,y) = \Delta_R(y,x) \,. </annotation></semantics></math></div> <p>This implies in particular that</p> <ol> <li>the <em><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></em>, which is the difference of the two</li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub><mo>≔</mo><msub><mi>Δ</mi> <mi>R</mi></msub><mo>−</mo><msub><mi>Δ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> \Delta_S \coloneqq \Delta_R - \Delta_A </annotation></semantics></math></div> <p>is skew-symmetric in its arguments (reflecting the fact that this is the <a class="existingWikiWord" href="/nlab/show/integral+kernel">integral kernel</a> for the <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a> for the <a class="existingWikiWord" href="/nlab/show/free+field">free</a> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> on the given spacetime);</p> <ol> <li>the <em><a class="existingWikiWord" href="/nlab/show/Dirac+propagator">Dirac propagator</a></em>, which is the sum of the two</li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>D</mi></msub><mo>≔</mo><msub><mi>Δ</mi> <mi>R</mi></msub><mo>+</mo><msub><mi>Δ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> \Delta_D \coloneqq \Delta_R + \Delta_A </annotation></semantics></math></div> <p>is symmetric in its arguments, reflecting the fact that this is the integral kernel for <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> away from the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a>.</p> <h2 id="definition">Definition</h2> <div class="num_defn" id="CompactlySourceCausalSupport"> <h6 id="definition_2">Definition</h6> <p><strong>(compactly sourced causal support)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mrow></mrow></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \overset{}{\to} \Sigma</annotation></semantics></math> over a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/conal+causal+structure">causal structure</a></p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_{\Sigma}(-)</annotation></semantics></math> for <a class="existingWikiWord" href="/nlab/show/space+of+sections">spaces of smooth sections</a>, and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Γ</mi> <mi>cp</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mtext>compact support</mtext></mtd></mtr> <mtr><mtd><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mo>±</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mtext>compactly sourced future/past support</mtext></mtd></mtr> <mtr><mtd><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>scp</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mtext>spacelike compact support</mtext></mtd></mtr> <mtr><mtd><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">)</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mtext>future/past compact support</mtext></mtd></mtr> <mtr><mtd><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>tcp</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mtext>timelike compact support</mtext></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Gamma_{cp}(-) & \text{compact support} \\ \Gamma_{\Sigma,\pm cp}(-) & \text{compactly sourced future/past support} \\ \Gamma_{\Sigma,scp}(-) & \text{spacelike compact support} \\ \Gamma_{\Sigma,(f/p)cp}(-) & \text{future/past compact support} \\ \Gamma_{\Sigma,tcp}(-) & \text{timelike compact support} } </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/linear+subspaces">linear subspaces</a> on those smooth sections whose <a class="existingWikiWord" href="/nlab/show/support">support</a> is</p> <ol> <li> <p>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cp</mi></mrow><annotation encoding="application/x-tex">cp</annotation></semantics></math>) inside a <a class="existingWikiWord" href="/nlab/show/compact+subset">compact subset</a></p> </li> <li> <p>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><mi>cp</mi></mrow><annotation encoding="application/x-tex">\pm cp</annotation></semantics></math>) inside the <a class="existingWikiWord" href="/nlab/show/closed+future+cone">closed future cone</a>/<a class="existingWikiWord" href="/nlab/show/closed+past+cone">closed past cone</a>, respectively, of a <a class="existingWikiWord" href="/nlab/show/compact+subset">compact subset</a>,</p> </li> <li> <p>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>scp</mi></mrow><annotation encoding="application/x-tex">scp</annotation></semantics></math>) inside the <a class="existingWikiWord" href="/nlab/show/closed+causal+cone">closed causal cone</a> of a <a class="existingWikiWord" href="/nlab/show/compact+subset">compact subset</a>, which equivalently means that the <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> with every (<a class="existingWikiWord" href="/nlab/show/spacelike">spacelike</a>) <a class="existingWikiWord" href="/nlab/show/Cauchy+surface">Cauchy surface</a> is compact (<a href="#Sanders12">Sanders 13, theorem 2.2</a>),</p> </li> <li> <p>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>fcp</mi></mrow><annotation encoding="application/x-tex">fcp</annotation></semantics></math>) inside the past of a Cauchy surface (<a href="#Sanders12">Sanders 13, def. 3.2</a>),</p> </li> <li> <p>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pcp</mi></mrow><annotation encoding="application/x-tex">pcp</annotation></semantics></math>) inside the future of a Cauchy surface (<a href="#Sanders12">Sanders 13, def. 3.2</a>),</p> </li> <li> <p>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>tcp</mi></mrow><annotation encoding="application/x-tex">tcp</annotation></semantics></math>) inside the future of one Cauchy surface and the past of another (<a href="#Sanders12">Sanders 13, def. 3.2</a>)</p> </li> </ol> </div> <p>(<a href="#Baer14">Bär 14, section 1</a>, <a href="#Khavkine14">Khavkine 14, def. 2.1</a>)</p> <div class="num_defn" id="AdvancedAndRetardedGreenFunctions"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+Green+functions">advanced and retarded Green functions</a>, <a class="existingWikiWord" href="/nlab/show/causal+Green+function">causal Green function</a> and <a class="existingWikiWord" href="/nlab/show/propagators">propagators</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> with <a class="existingWikiWord" href="/nlab/show/causal+structure">causal structure</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \to \Sigma</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+vector+bundle">smooth vector bundle</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P \;\colon\;\Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a> on its <a class="existingWikiWord" href="/nlab/show/space+of+smooth+sections">space of smooth sections</a>.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo>±</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mo>±</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathrm{G}_{P,\pm} \;\colon\; \Gamma_{\Sigma, cp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma, \pm cp}(E) </annotation></semantics></math></div> <p>from spaces of sections of <a class="existingWikiWord" href="/nlab/show/compact+support">compact support</a> to spaces of sections of causally sourced future/past support (def. <a class="maruku-ref" href="#CompactlySourceCausalSupport"></a>) is called an <em><a class="existingWikiWord" href="/nlab/show/advanced+or+retarded+Green+function">advanced or retarded Green function</a></em> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>, respectively, if</p> <ol> <li>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo>∈</mo><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Phi \in \Gamma_{\Sigma,cp}(E_1)</annotation></semantics></math> we have</li> </ol> <div class="maruku-equation" id="eq:AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><mi>P</mi><mo>,</mo><mo>±</mo></mrow></msub><mo>∘</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Φ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Φ</mi></mrow><annotation encoding="application/x-tex"> G_{P,\pm} \circ P(\Phi) = \Phi </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation" id="eq:AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∘</mo><msub><mi>G</mi> <mrow><mi>P</mi><mo>,</mo><mo>±</mo></mrow></msub><mo stretchy="false">(</mo><mi>Φ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Φ</mi></mrow><annotation encoding="application/x-tex"> P \circ G_{P,\pm}(\Phi) = \Phi </annotation></semantics></math></div> <ol> <li>the <a class="existingWikiWord" href="/nlab/show/support">support</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><mi>P</mi><mo>.</mo><mo>±</mo></mrow></msub><mo stretchy="false">(</mo><mi>Φ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_{P.\pm}(\Phi)</annotation></semantics></math> is in the <a class="existingWikiWord" href="/nlab/show/closed+future+cone">closed future cone</a> or <a class="existingWikiWord" href="/nlab/show/closed+past+cone">closed past cone</a> of the support of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math>, respectively.</li> </ol> <p>If the advanced/retarded Green functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mrow><mi>P</mi><mo>±</mo></mrow></msub></mrow><annotation encoding="application/x-tex">G_{P\pm}</annotation></semantics></math> exists, then the difference</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">G</mi> <mi>P</mi></msub><mo>≔</mo><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow></msub><mo>−</mo><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathrm{G}_P \coloneqq \mathrm{G}_{P,+} - \mathrm{G}_{P,-} </annotation></semantics></math></div> <p>is called the <em><a class="existingWikiWord" href="/nlab/show/causal+Green+function">causal Green function</a></em>.</p> <p>If there are <a class="existingWikiWord" href="/nlab/show/integral+kernel">integral kernel</a>, hence <a class="existingWikiWord" href="/nlab/show/distributions+in+two+variables">distributions in two variables</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>P</mi><mo>,</mo><mo>±</mo></mrow></msub><mo>∈</mo><mi>Γ</mi><mo>′</mo><mi>Σ</mi><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><msub><mo>⊠</mo> <mi>Σ</mi></msub><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta_{P,\pm} \in \Gamma'\Sigma( \tilde E^\ast \boxtimes_\Sigma E ) </annotation></semantics></math></div> <p>such that these Green functions are given by the corresponding <a class="existingWikiWord" href="/nlab/show/integral+transform">integral transform</a>, in that (in <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>-notation)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mrow><mi>P</mi><mo>,</mo><mo>±</mo></mrow></msub><mi>Φ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo>∫</mo><mrow><mi>y</mi><mo>∈</mo><mi>Σ</mi></mrow></munder><msub><mi>Δ</mi> <mrow><mi>P</mi><mo>,</mo><mo>±</mo></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Φ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (G_{P,\pm} \Phi)(x) \;=\; \underset{y \in \Sigma}{\int} \Delta_{P, \pm}(x,y) \cdot \Phi(y) </annotation></semantics></math></div> <p>then these integral kernels are called the advanced/retarded <em><a class="existingWikiWord" href="/nlab/show/propagators">propagators</a></em>; similarly then their difference</p> <div class="maruku-equation" id="eq:CausalPropagator"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>P</mi><mo>,</mo><mi>S</mi></mrow></msub><mo>≔</mo><msub><mi>Δ</mi> <mrow><mi>P</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow></msub><mo>−</mo><msub><mi>Δ</mi> <mrow><mi>P</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \Delta_{P,S} \coloneqq \Delta_{P,+} - \Delta_{P,-} </annotation></semantics></math></div> <p>is the corresponding <em><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></em>.</p> </div> <p>(e.g. <a href="#Baer14}">Bär 14, def. 3.2, cor. 3.10</a>)</p> <h2 id="properties">Properties</h2> <h3 id="existence_and_uniqueness">Existence and uniqueness</h3> <div class="num_prop" id="AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+Green+functions">advanced and retarded Green functions</a> of <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+operator">Green hyperbolic differential operator</a> are unique)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+Green+functions">advanced and retarded Green functions</a> (def. <a class="maruku-ref" href="#AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique"></a>) of a <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+operator">Green hyperbolic differential operator</a> are unique.</p> </div> <p>(<a href="#Baer14}">Bär 14, cor. 3.12</a></p> <h3 id="Continuity">Continuity</h3> <div class="num_defn" id="TVSStructureOnSpacesOfSmoothSections"> <h6 id="definition_4">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+space">Fréchet</a> <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a> on <a class="existingWikiWord" href="/nlab/show/spaces+of+smooth+sections">spaces of smooth sections</a> of a <a class="existingWikiWord" href="/nlab/show/smooth+vector+bundle">smooth vector bundle</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>fb</mi></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \overset{fb}{\to} \Sigma</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+vector+bundle">smooth vector bundle</a>. On its <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_\Sigma(E)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/space+of+sections">of smooth sections</a> consider the <a class="existingWikiWord" href="/nlab/show/seminorms">seminorms</a> indexed by a <a class="existingWikiWord" href="/nlab/show/compact+subset">compact subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">K \subset \Sigma</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">N \in \mathbb{N}</annotation></semantics></math> and given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>p</mi> <mi>K</mi> <mi>N</mi></msubsup></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>Φ</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><munder><mi>max</mi><mrow><mi>n</mi><mo>≤</mo><mi>N</mi></mrow></munder><mrow><mo>(</mo><munder><mi>sup</mi><mrow><mi>x</mi><mo>∈</mo><mi>K</mi></mrow></munder><mrow><mo stretchy="false">|</mo><msup><mo>∇</mo> <mi>n</mi></msup><mi>Φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Gamma_\Sigma(E) &\overset{p_K^N}{\longrightarrow}& [0,\infty) \\ \Phi &\mapsto& \underset{n \leq N}{max} \left( \underset{x \in K}{sup} {\vert \nabla^n \Phi(x)\vert}\right) \,, } </annotation></semantics></math></div> <p>where on the right we have the <a class="existingWikiWord" href="/nlab/show/absolute+values">absolute values</a> of the <a class="existingWikiWord" href="/nlab/show/covariant+derivatives">covariant derivatives</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> for any fixed choice of <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/norm">norm</a> on the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product of vector bundles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>Σ</mi><msup><mo stretchy="false">)</mo> <mrow><msubsup><mo>⊗</mo> <mi>Σ</mi> <mi>n</mi></msubsup></mrow></msup><msub><mo>⊗</mo> <mi>Σ</mi></msub><mi>E</mi></mrow><annotation encoding="application/x-tex">(T^\ast \Sigma)^{\otimes_\Sigma^n} \otimes_\Sigma E </annotation></semantics></math>.</p> <p>This makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_\Sigma(E)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+space">Fréchet</a> <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">K \subset \Sigma</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a> then the sub-space of sections</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>K</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E) </annotation></semantics></math></div> <p>of sections whose <a class="existingWikiWord" href="/nlab/show/support">support</a> is inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+space">Fréchet</a> <a class="existingWikiWord" href="/nlab/show/topological+vector+spaces">topological vector spaces</a> with the induced <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, which makes these be <a class="existingWikiWord" href="/nlab/show/closed+subspaces">closed subspaces</a>.</p> </div> <p>(<a href="#Baer14">Bär 14, 2.1, 2.2</a>)</p> <div class="num_defn" id="DistributionalSections"> <h6 id="definition_5">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/distribution">distributional</a> <a class="existingWikiWord" href="/nlab/show/sections">sections</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>fb</mi></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \overset{fb}{\to} \Sigma</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+vector+bundle">smooth vector bundle</a> over a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> with <a class="existingWikiWord" href="/nlab/show/causal+structure">causal structure</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/vector+space">vector</a> <a class="existingWikiWord" href="/nlab/show/spaces+of+smooth+sections">spaces of smooth sections</a> with restricted support from def. <a class="maruku-ref" href="#CompactlySourceCausalSupport"></a> structures of <a class="existingWikiWord" href="/nlab/show/topological+vector+spaces">topological vector spaces</a> via def. <a class="maruku-ref" href="#TVSStructureOnSpacesOfSmoothSections"></a>. We denote the topological <a class="existingWikiWord" href="/nlab/show/dual+spaces">dual spaces</a> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><msub><mo>′</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> \Gamma'_{\Sigma}(\tilde{E}^*) \coloneqq (\Gamma_{\Sigma,cp}(E))^* </annotation></semantics></math></div> <p>etc.</p> <p>This is the space of <em>distributional sections</em> of the bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\tilde{E}^*</annotation></semantics></math>.</p> <p>With this notations, smooth compactly supported sections of the same bundle, regarded as the <a class="existingWikiWord" href="/nlab/show/non-singular+distributions">non-singular distributions</a>, constitute a <a class="existingWikiWord" href="/nlab/show/dense+subset">dense subset</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><munder><mo>↪</mo><mtext>dense</mtext></munder><mi>Γ</mi><msub><mo>′</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_{\Sigma,cp}(\tilde{E}^*) \underset{\text{dense}}{\hookrightarrow} \Gamma'_{\Sigma}(\tilde{E}^*) \,. </annotation></semantics></math></div> <p>Imposing the same restrictions to the <a class="existingWikiWord" href="/nlab/show/supports+of+distributions">supports of distributions</a> as in def. <a class="maruku-ref" href="#CompactlySourceCausalSupport"></a>, we have the following subspaces of distributional sections:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><msub><mo>′</mo> <mrow><mi>Σ</mi><mo>,</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><msub><mo>′</mo> <mrow><mi>Σ</mi><mo>,</mo><mo>±</mo><mi>cp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><msub><mo>′</mo> <mrow><mi>Σ</mi><mo>,</mo><mi>scp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><msub><mo>′</mo> <mrow><mi>Σ</mi><mo>,</mo><mi>fcp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><msub><mo>′</mo> <mrow><mi>Σ</mi><mo>,</mo><mi>pcp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><msub><mo>′</mo> <mrow><mi>Σ</mi><mo>,</mo><mi>tcp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⊂</mo><mi>Γ</mi><msub><mo>′</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma'_{\Sigma,cp}(\tilde E^\ast) , \Gamma'_{\Sigma,\pm cp}(\tilde E^\ast) , \Gamma'_{\Sigma,scp}(\tilde E^\ast) , \Gamma'_{\Sigma,fcp}(\tilde E^\ast) , \Gamma'_{\Sigma,pcp}(\tilde E^\ast) , \Gamma'_{\Sigma,tcp}(\tilde E^\ast) \subset \Gamma'_{\Sigma}(\tilde E^\ast) . </annotation></semantics></math></div></div> <p>(<a href="#Sanders12">Sanders 13</a>, <a href="#Baer14">Bär 14</a>)</p> <div class="num_prop" id="GreenFunctionsAreContinuous"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/causal+Green+functions">causal Green functions</a> of <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+operators">Green hyperbolic differential operators</a> are <a class="existingWikiWord" href="/nlab/show/continuous+linear+maps">continuous linear maps</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+operator">Green hyperbolic differential operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> (def. <a class="maruku-ref" href="#GreenHyperbolicDifferentialOperator"></a>), the advanced, retarded and causal Green functions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> (def. <a class="maruku-ref" href="#AdvancedAndRetardedGreenFunctions"></a>) are <a class="existingWikiWord" href="/nlab/show/continuous+linear+maps">continuous linear maps</a> with respect to the <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a> structure from def. <a class="maruku-ref" href="#TVSStructureOnSpacesOfSmoothSections"></a> and also have a unique continuous extension to the spaces of sections with .larger support (def. <a class="maruku-ref" href="#CompactlySourceCausalSupport"></a>) as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow></msub></mtd> <mtd><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>pcp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>pcp</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></msub></mtd> <mtd><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>fcp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>fcp</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr> <mtr><mtd><msub><mi mathvariant="normal">G</mi> <mi>P</mi></msub></mtd> <mtd><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mrow><mi>Σ</mi><mo>,</mo><mi>tcp</mi></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathrm{G}_{P,+} &\;\colon\; \Gamma_{\Sigma, pcp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma, pcp}(E) , \\ \mathrm{G}_{P,-} &\;\colon\; \Gamma_{\Sigma, fcp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma, fcp}(E) , \\ \mathrm{G}_{P} &\;\colon\; \Gamma_{\Sigma, tcp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma}(E) , \end{aligned} </annotation></semantics></math></div> <p>such that we still have the relation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">G</mi> <mi>P</mi></msub><mo>=</mo><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow></msub><mo>−</mo><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathrm{G}_P = \mathrm{G}_{P,+} - \mathrm{G}_{P,-} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∘</mo><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo>±</mo></mrow></msub><mo>=</mo><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo>±</mo></mrow></msub><mo>∘</mo><mi>P</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex"> P \circ \mathrm{G}_{P,\pm} = \mathrm{G}_{P,\pm} \circ P = id </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>supp</mi><msub><mi mathvariant="normal">G</mi> <mrow><mi>P</mi><mo>,</mo><mo>±</mo></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>α</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⊆</mo><msup><mi>J</mi> <mo>±</mo></msup><mo stretchy="false">(</mo><mi>supp</mi><msup><mover><mi>α</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> supp \mathrm{G}_{P,\pm}(\tilde{\alpha}^*) \subseteq J^\pm(supp \tilde{\alpha}^*) \,. </annotation></semantics></math></div></div> <p>(<a href="#Baer14">Bär 14, thm. 3.8, cor. 3.11</a>)</p> <h2 id="examples">Examples</h2> <h3 id="ForKleinGordonOperatorOnMinkowskiSpacetime">For Klein-Gordon operator on Minkowski spacetime</h3> <p>On <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p,1}</annotation></semantics></math> consider the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mi>Φ</mi><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mi>Φ</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu} \Phi - \left( \tfrac{m c}{\hbar} \right)^2 \Phi \;=\; 0 \,. </annotation></semantics></math></div> <p>Its <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \;=\; (k_0)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/dispersion+relation">dispersion relation</a> of this equation we write</p> <div class="maruku-equation" id="eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo>+</mo><mi>c</mi><msqrt><mrow><msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></msqrt><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \omega(\vec k) \;\coloneqq\; + c \sqrt{ {\vert \vec k \vert}^2 + \left( \tfrac{m c}{\hbar}\right)^2 } \,, </annotation></semantics></math></div> <p>where on the right we choose the <a class="existingWikiWord" href="/nlab/show/non-negative+real+number">non-negative</a> <a class="existingWikiWord" href="/nlab/show/square+root">square root</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>We now discuss</p> <ol> <li> <p><em><a href="#AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime">Advanced and regarded propagators</a></em></p> </li> <li> <p><em><a href="#CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime">Causal propagator</a></em></p> </li> <li> <p><em><a href="#HadamardPropagatorForKleinGordonOnMinkowskiSpacetime">Wightman propagator</a></em></p> </li> <li> <p><em><a href="#FeynmanPropagator">Feynman propagator</a></em></p> </li> <li> <p><em><a href="#WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime">Singular support and Wave front sets</a></em></p> </li> </ol> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p id="AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime"><strong><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagators">advanced and retarded propagators</a> for <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a></strong></p> <div class="num_prop" id="AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime"> <h6 id="proposition_3">Proposition</h6> <p><strong>(mode expansion of <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagators">advanced and retarded propagators</a> for <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+Green+functions">advanced and retarded Green functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>±</mo></msub></mrow><annotation encoding="application/x-tex">G_\pm</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> are given by <a class="existingWikiWord" href="/nlab/show/integral+kernels">integral kernels</a> (“<a class="existingWikiWord" href="/nlab/show/propagators">propagators</a>”)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo>×</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta_\pm \in \mathcal{D}'(\mathbb{R}^{p,1}\times \mathbb{R}^{p,1}) </annotation></semantics></math></div> <p>by (in <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>-notation)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>Φ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow></munder><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>Φ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>dvol</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G_\pm(\Phi) \;=\; \underset{\mathbb{R}^{p,1}}{\int} \Delta_{\pm}(x,y) \Phi(y) \, dvol(y) </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagators">advanced and retarded propagators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta_{\pm}(x,y)</annotation></semantics></math> have the following equivalent expressions:</p> <div class="maruku-equation" id="eq:ModeExpansionForMinkowskiAdvancedRetardedPropagator"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mo>∫</mo><mo>∫</mo><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo>±</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>)</mo></mrow><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><mo>±</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo>∓</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><mo>±</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_\pm(x-y) & = \frac{1}{(2\pi)^{p+1}} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \int \int \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ (k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2 -\left( \tfrac{m c}{\hbar}\right)^2 } \, d k_0 \, d^p \vec k \\ & = \left\{ \array{ \frac{\pm i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{+i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) } \right) d^p \vec k & \vert & \text{if} \, \pm (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \left\{ \array{ \frac{\mp 1}{(2\pi)^{p}} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } d^p \vec k & \vert & \text{if} \, \pm (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \end{aligned} </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega(\vec k)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/dispersion+relation">dispersion relation</a> <a class="maruku-eqref" href="#eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime">(4)</a> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> is a <a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+operator">Green hyperbolic differential operator</a> (<a href="Green+hyperbolic+partial+differential+equation#GreenHyperbolicKleinGordonOperator">this example</a>) therefore its advanced and retarded Green functions exist uniquely (prop. <a class="maruku-ref" href="#AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique"></a>). Moreover, prop. <a class="maruku-ref" href="#GreenFunctionsAreContinuous"></a> says that they are <a class="existingWikiWord" href="/nlab/show/continuous+linear+functionals">continuous linear functionals</a> with respect to the <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a> <a class="existingWikiWord" href="/nlab/show/structures">structures</a> on <a class="existingWikiWord" href="/nlab/show/spaces+of+smooth+sections">spaces of smooth sections</a> (def. <a class="maruku-ref" href="#TVSStructureOnSpacesOfSmoothSections"></a>). In the case of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> this just means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>±</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>⟶</mo><msubsup><mi>C</mi> <mrow><mo>±</mo><mi>cp</mi></mrow> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G_{\pm} \;\colon\; C^\infty_{cp}(\mathbb{R}^{p,1}) \longrightarrow C^\infty_{\pm cp}(\mathbb{R}^{p,1}) </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/continuous+linear+functionals">continuous linear functionals</a> in the standard sense of <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a>. Therefore the <a class="existingWikiWord" href="/nlab/show/Schwartz+kernel+theorem">Schwartz kernel theorem</a> implies the existence of <a class="existingWikiWord" href="/nlab/show/integral+kernels">integral kernels</a> being <a class="existingWikiWord" href="/nlab/show/distributions+in+two+variables">distributions in two variables</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mo>∈</mo><mi>𝒟</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo>×</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta_{\pm} \in \mathcal{D}(\mathbb{R}^{p,1} \times \mathbb{R}^{p,1}) </annotation></semantics></math></div> <p>such that, in the notation of <a class="existingWikiWord" href="/nlab/show/generalized+functions">generalized functions</a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mo>±</mo></msub><mi>α</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow></munder><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>α</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>dvol</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (G_\pm \alpha)(x) \;=\; \underset{\mathbb{R}^{p,1}}{\int} \Delta_{\pm}(x,y) \alpha(y) \, dvol(y) \,. </annotation></semantics></math></div> <p>These integral kernels are the advanced/retarded “<a class="existingWikiWord" href="/nlab/show/propagators">propagators</a>”. We now compute these <a class="existingWikiWord" href="/nlab/show/integral+kernels">integral kernels</a> by making an Ansatz and showing that it has the defining properties, which identifies them by the uniqueness statement of prop. <a class="maruku-ref" href="#AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique"></a>.</p> <p>We make use of the fact that the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> is <a class="existingWikiWord" href="/nlab/show/invariant">invariant</a> under the defnining <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, which is a <a class="existingWikiWord" href="/nlab/show/semidirect+product+group">semidirect product group</a> of the <a class="existingWikiWord" href="/nlab/show/translation+group">translation group</a> and the <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a>.</p> <p>Since the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> is invariant, in particular, under <a class="existingWikiWord" href="/nlab/show/translations">translations</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p,1}</annotation></semantics></math> it is clear that the propagators, as a <a class="existingWikiWord" href="/nlab/show/distribution+in+two+variables">distribution in two variables</a>, depend only on the difference of its two arguments</p> <div class="maruku-equation" id="eq:TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_{\pm}(x,y) = \Delta_{\pm}(x-y) \,. </annotation></semantics></math></div> <p>Since moreover the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> is <a class="existingWikiWord" href="/nlab/show/formally+adjoint+differential+operator">formally self-adjoint</a> (<a href="Klein-Gordon+equation#FormallySelfAdjointKleinGordonOperator">this prop.</a>) this implies that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> the Klein the equation <a class="maruku-eqref" href="#eq:AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator">(2)</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∘</mo><msub><mi>G</mi> <mo>±</mo></msub><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex"> P \circ G_\pm = id </annotation></semantics></math></div> <p>is equivalent to the equation <a class="maruku-eqref" href="#eq:AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator">(1)</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>±</mo></msub><mo>∘</mo><mi>P</mi><mo>=</mo><mi>id</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G_\pm \circ P = id \,. </annotation></semantics></math></div> <p>Therefore it is sufficient to solve for the first of these two equation, subject to the defining support conditions. In terms of the <a class="existingWikiWord" href="/nlab/show/propagator">propagator</a> <a class="existingWikiWord" href="/nlab/show/integral+kernels">integral kernels</a> this means that we have to solve the <a class="existingWikiWord" href="/nlab/show/distribution">distributional</a> equation</p> <div class="maruku-equation" id="eq:KleinGordonEquationOnAdvacedRetardedPropagator"><span class="maruku-eq-number">(7)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \left( \eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu} - \left( \tfrac{m c}{\hbar} \right)^2 \right) \Delta_\pm(x-y) \;=\; \delta(x-y) </annotation></semantics></math></div> <p>subject to the condition that the <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">distributional support</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mrow><mo>(</mo><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⊂</mo><mrow><mo>{</mo><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><msubsup><mo stretchy="false">|</mo> <mi>η</mi> <mn>2</mn></msubsup></mrow><mo><</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mo>±</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> supp\left( \Delta_{\pm}(x-y) \right) \subset \left\{ {\vert x-y\vert^2_\eta}\lt 0 \;\,,\; \pm(x^0 - y^ 0) \gt 0 \right\} \,. </annotation></semantics></math></div> <p>We make the <em>Ansatz</em> that we assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_{\pm}</annotation></semantics></math>, as a distribution in a single variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x-y</annotation></semantics></math>, is a <a class="existingWikiWord" href="/nlab/show/tempered+distribution">tempered distribution</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mo>∈</mo><mi>𝒮</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Delta_\pm \in \mathcal{S}'(\mathbb{R}^{p,1}) \,, </annotation></semantics></math></div> <p>hence amenable to <a class="existingWikiWord" href="/nlab/show/Fourier+transform+of+distributions">Fourier transform of distributions</a>. If we do find a solution this way, it is guaranteed to be the unique solution by prop. <a class="maruku-ref" href="#AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique"></a>.</p> <p>By <a href="Fourier+transform#BasicPropertiesOfFourierTransformOverCartesianSpaces">this prop.</a> the <a class="existingWikiWord" href="/nlab/show/Fourier+transform+of+distributions">distributional Fourier transform</a> of equation <a class="maruku-eqref" href="#eq:KleinGordonEquationOnAdvacedRetardedPropagator">(7)</a> is</p> <div class="maruku-equation" id="eq:FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator"><span class="maruku-eq-number">(8)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mi>k</mi> <mi>μ</mi></msub><msub><mi>k</mi> <mi>ν</mi></msub><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mover><mrow><msub><mi>Δ</mi> <mo>±</mo></msub></mrow><mo>^</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mover><mi>δ</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left( - \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) \widehat{\Delta_{\pm}}(k) & = \widehat{\delta}(k) \\ & = 1 \end{aligned} \,, </annotation></semantics></math></div> <p>where in the second line we used the <a class="existingWikiWord" href="/nlab/show/Fourier+transform+of+distributions">Fourier transform</a> of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> from <a href="Dirac+distribution#FourierTransformOfDeltaDistribution">this example</a>.</p> <p>Notice that this implies that the <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a> of the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub><mo>≔</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex"> \Delta_S \coloneqq \Delta_+ - \Delta_- </annotation></semantics></math></div> <p>satisfies the homogeneous equation:</p> <div class="maruku-equation" id="eq:FourierVersionOfPDEForKleinGordonCausalPropagator"><span class="maruku-eq-number">(9)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mi>k</mi> <mi>μ</mi></msub><msub><mi>k</mi> <mi>ν</mi></msub><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mover><mrow><msub><mi>Δ</mi> <mi>S</mi></msub></mrow><mo>^</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \left( - \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) \widehat{\Delta_S}(k) \;=\; 0 \,, </annotation></semantics></math></div> <p>Hence we are now reduced to finding solutions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><msub><mi>Δ</mi> <mo>±</mo></msub></mrow><mo>^</mo></mover><mo>∈</mo><mi>𝒮</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat{\Delta_\pm} \in \mathcal{S}'(\mathbb{R}^{p,1})</annotation></semantics></math> to <a class="maruku-eqref" href="#eq:FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator">(8)</a> such that their <a class="existingWikiWord" href="/nlab/show/Fourier+inversion+theorem">Fourier inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_\pm</annotation></semantics></math> has the required <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">support</a> properties.</p> <p>We discuss this by a variant of the <a class="existingWikiWord" href="/nlab/show/Cauchy+principal+value">Cauchy principal value</a>:</p> <p>Suppose the following <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> of <a class="existingWikiWord" href="/nlab/show/non-singular+distributions">non-singular distributions</a> in the <a class="existingWikiWord" href="/nlab/show/variable">variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">k \in \mathbb{R}^{p,1}</annotation></semantics></math> exists in the space of <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a></p> <div class="maruku-equation" id="eq:LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator"><span class="maruku-eq-number">(10)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><msup><mo stretchy="false">|</mo> <mn>2</mn></msup></mrow><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim} \frac{1}{ (k_0 \mp i \epsilon)^2 - {\vert \vec k\vert^2} - \left( \tfrac{m c}{\hbar} \right)^2 } \;\in\; \mathcal{D}'(\mathbb{R}^{p,1}) </annotation></semantics></math></div> <p>meaning that for each <a class="existingWikiWord" href="/nlab/show/bump+function">bump function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b \in C^\infty_{cp}(\mathbb{R}^{p,1})</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow></munder><mfrac><mrow><mi>b</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><msup><mi>d</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>k</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim} \underset{\mathbb{R}^{p,1}}{\int} \frac{b(k)}{ (k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 } d^{p+1}k \;\in\; \mathbb{C} </annotation></semantics></math></div> <p>exists. Then this limit is clearly a solution to the distributional equation <a class="maruku-eqref" href="#eq:FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator">(8)</a> because on those bump functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b(k)</annotation></semantics></math> which happen to be products with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mi>k</mi> <mi>μ</mi></msub><mi>k</mi><mo>−</mo><mi>ν</mi><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(-\eta^{\mu \nu}k_\mu k-\nu - \left( \tfrac{m c}{\hbar}\right)^2\right)</annotation></semantics></math> we clearly have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow></munder><mfrac><mrow><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mi>k</mi> <mi>μ</mi></msub><msub><mi>k</mi> <mi>ν</mi></msub><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mi>b</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><msup><mi>d</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>k</mi></mtd> <mtd><mo>=</mo><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow></munder><munder><munder><mrow><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mi>k</mi> <mi>μ</mi></msub><msub><mi>k</mi> <mi>ν</mi></msub><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>1</mn></mrow></munder><mi>b</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>k</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">⟨</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim} \underset{\mathbb{R}^{p,1}}{\int} \frac{ \left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) b(k) }{ (k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 } d^{p+1}k & = \underset{\mathbb{R}^{p,1}}{\int} \underset{= 1}{ \underbrace{ \underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim} \frac{ \left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) }{ (k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 } } } b(k)\, d^{p+1}k \\ & = \langle 1, b\rangle \,. \end{aligned} </annotation></semantics></math></div> <p>Moreover, if the limiting distribution <a class="maruku-eqref" href="#eq:LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator">(10)</a> exists, then it is clearly a <a class="existingWikiWord" href="/nlab/show/tempered+distribution">tempered distribution</a>, hence we may apply <a class="existingWikiWord" href="/nlab/show/Fourier+inversion+theorem">Fourier inversion</a> to obtain <a class="existingWikiWord" href="/nlab/show/Green+functions">Green functions</a></p> <div class="maruku-equation" id="eq:AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets"><span class="maruku-eq-number">(11)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><munder><mo>∫</mo><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow></munder><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_{\pm}(x,y) \;\coloneqq\; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{1}{(2\pi)^{p+1}} \underset{\mathbb{R}^{p,1}}{\int} \frac{e^{i k_\mu (x-y)^\mu}}{ (k_0 \mp i \epsilon )^2 - {\vert \vec k\vert}^2 - \left(\tfrac{m c}{\hbar}\right)^2 } d k_0 d^p \vec k \,. </annotation></semantics></math></div> <p>To see that this is the correct answer, we need to check the defining support property.</p> <p>Finally, by the <a class="existingWikiWord" href="/nlab/show/Fourier+inversion+theorem">Fourier inversion theorem</a>, to show that the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> <a class="maruku-eqref" href="#eq:LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator">(10)</a> indeed exists it is sufficient to show that the limit in <a class="maruku-eqref" href="#eq:AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets">(11)</a> exists.</p> <p>We compute as follows</p> <div class="maruku-equation" id="eq:TheSupportOfTheCandidateAdvancedRetardedPropagatorIsinTheFutureOrPastRespectively"><span class="maruku-eq-number">(12)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mo>∫</mo><mo>∫</mo><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mo>∫</mo><mo>∫</mo><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mo>∫</mo><mo>∫</mo><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><mo stretchy="false">)</mo><mo>−</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>∓</mo><mi>i</mi><mi>ϵ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo>±</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>)</mo></mrow><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><mo>±</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo>∓</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><mo>±</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_\pm(x-y) & = \frac{1}{(2\pi)^{p+1}} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \int \int \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ (k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2 -\left( \tfrac{m c}{\hbar}\right)^2 } \, d k_0 \, d^p \vec k \\ & = \frac{1}{(2\pi)^{p+1}} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \int \int \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ (k_0 \mp i \epsilon)^2 - \left(\omega(\vec k)/c\right)^2 } \, d k_0 \, d^p \vec k \\ &= \frac{1}{(2\pi)^{p+1}} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \int \int \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ \left( (k_0 \mp i\epsilon) - \omega(\vec k)/c \right) \left( (k_0 \mp i \epsilon) + \omega(\vec k)/c \right) } \, d k_0 \, d^p \vec k \\ & = \left\{ \array{ \frac{\pm i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k & \vert & \text{if} \, \pm (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \left\{ \array{ \frac{\mp 1}{(2\pi)^{p}} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } d^p \vec k & \vert & \text{if} \, \pm (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \end{aligned} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega(\vec k)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/dispersion+relation">dispersion relation</a> <a class="maruku-eqref" href="#eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime">(4)</a> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a>. The last step is simply the application of <a class="existingWikiWord" href="/nlab/show/Euler%27s+formula">Euler's formula</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mn>2</mn><mi>i</mi></mrow></mfrac></mstyle><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>α</mi></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>α</mi></mrow></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\sin(\alpha) = \tfrac{1}{2 i }\left( e^{i \alpha} - e^{- i \alpha}\right)</annotation></semantics></math>.</p> <p>Here the key step is the application of <a class="existingWikiWord" href="/nlab/show/Cauchy%27s+integral+formula">Cauchy's integral formula</a> in the fourth step. We spell this out now for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_+</annotation></semantics></math>, the discussion for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_-</annotation></semantics></math> is the same, just with the appropriate signs reversed.</p> <ol> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(x^0 - y^0) \gt 0</annotation></semantics></math> thn the expression <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>e</mi> <mrow><msub><mi>ik</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">e^{ik_0 (x^0 - y^0)}</annotation></semantics></math> decays with <em><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary part</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">k_0</annotation></semantics></math>, so that we may expand the <a class="existingWikiWord" href="/nlab/show/integration">integration</a> <a class="existingWikiWord" href="/nlab/show/domain">domain</a> into the <a class="existingWikiWord" href="/nlab/show/upper+half+plane">upper half plane</a> as</li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub></mtd> <mtd><mo>=</mo><mphantom><mo lspace="verythinmathspace" rspace="0em">+</mo></mphantom><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>0</mn></msubsup><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mo>+</mo><msubsup><mo>∫</mo> <mn>0</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mn>∞</mn></mrow></msubsup><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mn>∞</mn></mrow> <mn>0</mn></msubsup><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mo>+</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>∞</mn></msubsup><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>;</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \int_{-\infty}^\infty d k_0 & = \phantom{+} \int_{-\infty}^0 d k_0 + \int_{0}^{+ i \infty} d k_0 \\ & = + \int_{+i \infty}^0 d k_0 + \int_0^\infty d k_0 \,; \end{aligned} </annotation></semantics></math></div> <p>Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(x^0 - y^0) \lt 0</annotation></semantics></math> then we may analogously expand into the <a class="existingWikiWord" href="/nlab/show/lower+half+plane">lower half plane</a>.</p> <ol> <li>This integration domain may then further be completed to two <a class="existingWikiWord" href="/nlab/show/contour+integrations">contour integrations</a>. For the expansion into the <a class="existingWikiWord" href="/nlab/show/upper+half+plane">upper half plane</a> these encircle counter-clockwise the <a class="existingWikiWord" href="/nlab/show/poles">poles</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mi>ϵ</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\pm \omega(\vec k)+ i\epsilon \in \mathbb{C}</annotation></semantics></math>, while for expansion into the <a class="existingWikiWord" href="/nlab/show/lower+half+plane">lower half plane</a> no poles are being encircled.</li> </ol> <p><img src="https://ncatlab.org/nlab/files/ContourForAdvancedPropagator.png" height="280" /></p> <ol> <li> <p>Apply <a class="existingWikiWord" href="/nlab/show/Cauchy%27s+integral+formula">Cauchy's integral formula</a> to find in the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(x^0 - y^0)\gt 0</annotation></semantics></math> the sum of the <a class="existingWikiWord" href="/nlab/show/residues">residues</a> at these two <a class="existingWikiWord" href="/nlab/show/poles">poles</a> times <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">2\pi i</annotation></semantics></math>, zero in the other case. (For the retarded propagator we get <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><mi>π</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">- 2 \pi i</annotation></semantics></math> times the residues, because now the contours encircling non-trivial poles go clockwise).</p> </li> <li> <p>The result is now non-singular at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon = 0</annotation></semantics></math> and therefore the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon \to 0</annotation></semantics></math> is now computed by evaluating at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon = 0</annotation></semantics></math>.</p> </li> </ol> <p>This computation shows a) that the limiting distribution indeed exists, and b) that the <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">support</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_+</annotation></semantics></math> is in the future, and that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_-</annotation></semantics></math> is in the past.</p> <p>Hence it only remains to see now that the support of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_\pm</annotation></semantics></math> is inside the <a class="existingWikiWord" href="/nlab/show/causal+cone">causal cone</a>. But this follows from the previous argument, by using that the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> is invariant under <a class="existingWikiWord" href="/nlab/show/Lorentz+transformations">Lorentz transformations</a>: This implies that the support is in fact in the <a class="existingWikiWord" href="/nlab/show/future">future</a> of <em>every</em> spacelike slice through the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p,1}</annotation></semantics></math>, hence in the <a class="existingWikiWord" href="/nlab/show/closed+future+cone">closed future cone</a> of the origin.</p> </div> <div class="num_cor" id="CausalPropagatorIsSkewSymmetric"> <h6 id="corollary">Corollary</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> is skew-symmetric)</strong></p> <p>Under reversal of arguments the <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+causal+propagators">advanced and retarded causal propagators</a> from prop. <a class="maruku-ref" href="#AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime"></a> are related by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Δ</mi> <mo>∓</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_{\pm}(y-x) = \Delta_\mp(x-y) \,. </annotation></semantics></math></div> <p>It follows that the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>≔</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\Delta \coloneqq \Delta_+ - \Delta_-</annotation></semantics></math> is skew-symmetric in its arguments:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_S(x-y) = - \Delta_S(y-x) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By prop. <a class="maruku-ref" href="#AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime"></a> we have with <a class="maruku-eqref" href="#eq:ModeExpansionForMinkowskiAdvancedRetardedPropagator">(5)</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mo>±</mo></msub><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo>±</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>)</mo></mrow><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><mo>∓</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo>±</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>)</mo></mrow><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><mo>∓</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo>∓</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>)</mo></mrow><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><mo>∓</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Δ</mi> <mo>∓</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_\pm(y-x) & = \left\{ \array{ \frac{\pm i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{-i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x -\vec y)} - e^{+i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) } \right) d^p \vec k & \vert & \text{if} \, \mp (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \left\{ \array{ \frac{\pm i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{+i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) } \right) d^p \vec k & \vert & \text{if} \, \mp (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \left\{ \array{ \frac{\mp i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{+i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) } \right) d^p \vec k & \vert & \text{if} \, \mp (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \Delta_\mp(x-y) \end{aligned} </annotation></semantics></math></div> <p>Here in the second step we applied <a class="existingWikiWord" href="/nlab/show/change+of+integration+variables">change of integration variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec k \mapsto - \vec k</annotation></semantics></math> (which introduces <em>no</em> sign because in addition to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>d</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">d \vec k \mapsto - d \vec k</annotation></semantics></math> the integration domain reverses <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>).</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p id="CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime"><strong><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></strong></p> <div class="num_prop" id="ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski"> <h6 id="proposition_4">Proposition</h6> <p><strong>(mode expansion of <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> for <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> <a class="maruku-eqref" href="#eq:CausalPropagator">(3)</a> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> for <a class="existingWikiWord" href="/nlab/show/mass">mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p,1}</annotation></semantics></math> is given, in <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a> notation, by</p> <div class="maruku-equation" id="eq:CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime"><span class="maruku-eq-number">(13)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>)</mo></mrow><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_S(x,y) & = \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k \\ & = \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x -\vec y)} d^p \vec k \,, \end{aligned} </annotation></semantics></math></div> <p>where in the second line we used <a class="existingWikiWord" href="/nlab/show/Euler%27s+formula">Euler's formula</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mn>2</mn><mi>i</mi></mrow></mfrac></mstyle><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>α</mi></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>α</mi></mrow></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">sin(\alpha)= \tfrac{1}{2i}\left( e^{i \alpha} - e^{-i \alpha} \right)</annotation></semantics></math>.</p> <p>In particular this shows that the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> is <a class="existingWikiWord" href="/nlab/show/real+part">real</a>, in that it is equal to its <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugate</a></p> <div class="maruku-equation" id="eq:CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal"><span class="maruku-eq-number">(14)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow><mo>(</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mo>*</mo></msup><mo>=</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left(\Delta_S(x,y)\right)^\ast = \Delta_S(x,y) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By definition and using the expression from prop. <a class="maruku-ref" href="#AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime"></a> for the <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+causal+propagators">advanced and retarded causal propagators</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>)</mo></mrow><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><mo>+</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>)</mo></mrow><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><mo>−</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mo>)</mo></mrow><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_S(x,y) & \coloneqq \Delta_+(x,y) - \Delta_-(x,y) \\ & = \left\{ \array{ \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k & \vert & \text{if} \, + (x^0 - y^0) \gt 0 \\ \frac{(-1) (-1) i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k & \vert & \text{if} \, - (x^0 - y^0) \gt 0 } \right. \\ & = \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k \\ & = \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x -\vec y)} d^p \vec k \end{aligned} </annotation></semantics></math></div> <p>For the reality, notice from the last line that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msup><mrow><mo>(</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mo>*</mo></msup></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left(\Delta_S(x,y)\right)^\ast & = \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{-i \vec k \cdot (\vec x -\vec y)} d^p \vec k \\ & = \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{+i \vec k \cdot (\vec x -\vec y)} d^p \vec k \\ & = \Delta_S(x,y) \,, \end{aligned} </annotation></semantics></math></div> <p>where in the last step we used the <a class="existingWikiWord" href="/nlab/show/change+of+integration+variables">change of integration variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec k \mapsto - \vec k</annotation></semantics></math> (whih introduces no sign, since on top of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>d</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">d \vec k \mapsto - d \vec k</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> of the integration <a class="existingWikiWord" href="/nlab/show/domain">domain</a> changes).</p> </div> <p>We consider a couple of equivalent expressions for the causal propagator which are useful for computations:</p> <div class="num_prop" id="CausalPropagatorForKleinGordonOnMinkowskiAsContourIntegral"> <h6 id="proposition_5">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> for <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> as a <a class="existingWikiWord" href="/nlab/show/contour+integral">contour integral</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> at <a class="existingWikiWord" href="/nlab/show/mass">mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> has the following equivalent expression, as a <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>, given as a <a class="existingWikiWord" href="/nlab/show/contour+integral">contour integral</a> along a curve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\vec k)</annotation></semantics></math> going counter-clockwise around the two <a class="existingWikiWord" href="/nlab/show/poles">poles</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub><mo>=</mo><mo>±</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">k_0 = \pm \omega(\vec k)/c</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>∫</mo><munder><mo>∮</mo><mrow><mi>C</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></munder><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mi>g</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mi>k</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_S(x,y) \;=\; (2\pi)^{-(p+1)} \int \underset{C(\vec k)}{\oint} \frac{e^{i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2g } \,d k_0 \,d^{p} k \,. </annotation></semantics></math></div></div> <p><img src="https://ncatlab.org/nlab/files/ContourForCausalPropagator.png" height="160" /></p> <blockquote> <p>graphics grabbed from <a href="#Kocic16">Kocic 16</a></p> </blockquote> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By <a class="existingWikiWord" href="/nlab/show/Cauchy%27s+integral+formula">Cauchy's integral formula</a> we compute as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>∫</mo><munder><mo>∮</mo><mrow><mi>C</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></munder><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>−</mo><msup><mi>y</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mi>k</mi></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>∫</mo><munder><mo>∮</mo><mrow><mi>C</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></munder><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><msup><mi>x</mi> <mn>0</mn></msup></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msubsup><mi>k</mi> <mn>0</mn> <mn>2</mn></msubsup><mo>−</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>2</mn></msup></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>∫</mo><munder><mo>∮</mo><mrow><mi>C</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></munder><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>+</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>−</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mn>2</mn><mi>π</mi><mi>i</mi><mo>∫</mo><mrow><mo>(</mo><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} (2\pi)^{-(p+1)} \int \underset{C(\vec k)}{\oint} \frac{e^{i k_\mu (x^\mu - y^\mu)}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 } \,d k_0 \,d^{p} k & = (2\pi)^{-(p+1)} \int \underset{C(\vec k)}{\oint} \frac{ e^{i k_0 x^0} e^{ i \vec k \cdot (\vec x - \vec y)} }{ k_0^2 - \omega(\vec k)^2/c^2 } \,d k_0 \,d^p \vec k \\ & = (2\pi)^{-(p+1)} \int \underset{C(\vec k)}{\oint} \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ ( k_0 + \omega(\vec k)/c ) ( k_0 - \omega(\vec k)/c ) } \,d k_0 \,d^p \vec k \\ & = (2\pi)^{-(p+1)} 2\pi i \int \left( \frac{ e^{i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} } { 2 \omega(\vec k)/c } - \frac{ e^{ - i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} }{ 2 \omega(\vec k)/c } \right) \,d^p \vec k \\ & = i (2\pi)^{-p} \int \frac{1}{\omega(\vec k)/c} sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y)} \,d^p \vec k \,. \end{aligned} </annotation></semantics></math></div> <p>The last line is the expression for the causal propagator from prop. <a class="maruku-ref" href="#ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski"></a></p> </div> <div class="num_prop" id="CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator"> <h6 id="proposition_6">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> as <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a> of <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> on the <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transformed</a> <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> at <a class="existingWikiWord" href="/nlab/show/mass">mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> has the following equivalent expression, as a <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><mi>δ</mi><mrow><mo>(</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>+</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup></mrow></msup><msup><mi>d</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>k</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Delta_S(x,y) \;=\; i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k \,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/integrand">integrand</a> is the product of the <a class="existingWikiWord" href="/nlab/show/sign+function">sign function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">k_0</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> of the <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> and a <a class="existingWikiWord" href="/nlab/show/plane+wave">plane wave</a> factor.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>By decomposing the integral over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">k_0</annotation></semantics></math> into its negative and its positive half, and applying the <a class="existingWikiWord" href="/nlab/show/change+of+integration+variables">change of integration variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub><mo>=</mo><mo>±</mo><msqrt><mi>h</mi></msqrt></mrow><annotation encoding="application/x-tex">k_0 = \pm\sqrt{h}</annotation></semantics></math> we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><mi>δ</mi><mrow><mo>(</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>+</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup></mrow></msup><msup><mi>d</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>k</mi></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>∞</mn></msubsup><mi>δ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mi>k</mi> <mn>0</mn> <mn>2</mn></msubsup><mo>+</mo><msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>0</mn></msubsup><mi>δ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mi>k</mi> <mn>0</mn> <mn>2</mn></msubsup><mo>+</mo><msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover> <mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>∞</mn></msubsup><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mi>h</mi></msqrt></mrow></mfrac><mi>δ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>h</mi><mo>+</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><msqrt><mi>h</mi></msqrt><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow></msup><mi>d</mi><mi>h</mi><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>∞</mn></msubsup><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mi>h</mi></msqrt></mrow></mfrac><mi>δ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>h</mi><mo>+</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msqrt><mi>h</mi></msqrt><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow></msup><mi>d</mi><mi>h</mi><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mn>0</mn></msup><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow></msup><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mn>0</mn></msup><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow></msup><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mn>0</mn></msup><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k & = + i (2\pi)^{-p} \int \int_0^\infty \delta\left( -k_0^2 + \vec k^2 + \left( \tfrac{m c}{\hbar}\right)^2 \right) e^{ i k_0 (x^0 - y^0) + i \vec k \cdot (\vec x - \vec y)} d k_0 \, d^p \vec k \\ & \phantom{=} - i (2\pi)^{-p} \int \int_{-\infty}^0 \delta\left( -k_0^2 + \vec k^2 + \left(\tfrac{m c}{\hbar}\right)^2 \right) e^{ i k_0 (x^0 - y^0)+ i \vec k \cdot (\vec x - \vec y) } d k_0 \, d^{p} \vec k \\ & = +i (2\pi)^{-p} \int \int_0^\infty \frac{1}{2 \sqrt{h}} \delta\left( -h + \omega(\vec k)^2/c^2 \right) e^{ + i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x } d h \, d^{p} \vec k \\ & \phantom{=} - i (2\pi)^{-p} \int \int_0^\infty \frac{1}{2 \sqrt{h}} \delta\left( - h + \omega(\vec k)^2/c^2 \right) e^{ - i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x } d h \, d^{p} \vec k \\ & = +i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x} d^{p} \vec k \\ & \phantom{=} - i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ - i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x } d^{p} \vec k \\ & = -(2 \pi)^{-p} \int \frac{1}{\omega(\vec k)/c} sin\left( \omega(\vec k)(x-y)^0/c \right) e^{i \vec k \cdot (\vec x - \vec y)} \end{aligned} </annotation></semantics></math></div> <p>The last line is the expression for the causal propagator from prop. <a class="maruku-ref" href="#ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski"></a>.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p id="HadamardPropagatorForKleinGordonOnMinkowskiSpacetime"><strong><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></strong></p> <p>Prop. <a class="maruku-ref" href="#CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator"></a> exhibits the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> as the difference of a contribution for <a class="existingWikiWord" href="/nlab/show/positive+real+number">positive</a> temporal <a class="existingWikiWord" href="/nlab/show/angular+frequency">angular frequency</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub><mo>∝</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k_0 \propto \omega(\vec k)</annotation></semantics></math> (hence positive <a class="existingWikiWord" href="/nlab/show/energy">energy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hbar \omega(\vec k)</annotation></semantics></math> and a contribution of negative temporal <a class="existingWikiWord" href="/nlab/show/angular+frequency">angular frequency</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/positive+real+number">positive</a> <a class="existingWikiWord" href="/nlab/show/frequency">frequency</a> contribution to the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> is called the <em><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></em> (def. <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a> below), also known as the the <em><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> of the <a class="existingWikiWord" href="/nlab/show/free+field">free</a> <a class="existingWikiWord" href="/nlab/show/real+scalar+field">real scalar field</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a></em>. Notice that the temporal component of the <a class="existingWikiWord" href="/nlab/show/wave+vector">wave vector</a> is proportional to the <em>negative</em> <a class="existingWikiWord" href="/nlab/show/angular+frequency">angular frequency</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ω</mi><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex"> k_0 = -\omega/c </annotation></semantics></math></div> <p>(see at <em><a class="existingWikiWord" href="/nlab/show/plane+wave">plane wave</a></em>), therefore the appearance of the <a class="existingWikiWord" href="/nlab/show/step+function">step function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(-k_0)</annotation></semantics></math> in <a class="maruku-eqref" href="#eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime">(15)</a> below:</p> <div class="num_defn" id="StandardHadamardDistributionOnMinkowskiSpacetime"> <h6 id="definition_6">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> or <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> for <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <em><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></em> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> at <a class="existingWikiWord" href="/nlab/show/mass">mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is the <a class="existingWikiWord" href="/nlab/show/tempered+distribution">tempered</a> <a class="existingWikiWord" href="/nlab/show/distribution+in+two+variables">distribution in two variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub><mo>∈</mo><mi>𝒮</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta_H \in \mathcal{S}'(\mathbb{R}^{p,1})</annotation></semantics></math> which as a <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a> is given by the expression</p> <div class="maruku-equation" id="eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime"><span class="maruku-eq-number">(15)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mi>δ</mi><mrow><mo>(</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup><mo>)</mo></mrow><mi>Θ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>−</mo><msup><mi>y</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>k</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_H(x,y) & \coloneqq \frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k \\ & = \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k \,, \end{aligned} </annotation></semantics></math></div> <p>Here in the first line we have in the <a class="existingWikiWord" href="/nlab/show/integrand">integrand</a> the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> of the <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> times a <a class="existingWikiWord" href="/nlab/show/plane+wave">plane wave</a> and times the <a class="existingWikiWord" href="/nlab/show/step+function">step function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math> of the temporal component of the <a class="existingWikiWord" href="/nlab/show/wave+vector">wave vector</a>. In the second line we used the <a class="existingWikiWord" href="/nlab/show/change+of+integration+variables">change of integration variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub><mo>=</mo><msqrt><mi>h</mi></msqrt></mrow><annotation encoding="application/x-tex">k_0 = \sqrt{h}</annotation></semantics></math>, then the definition of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> and the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega(\vec k)</annotation></semantics></math> is by definition the <a class="existingWikiWord" href="/nlab/show/non-negative+real+number">non-negative</a> solution to the Klein-Gordon <a class="existingWikiWord" href="/nlab/show/dispersion+relation">dispersion relation</a>.</p> </div> <p>(e.g. <a href="Hadamard+distribution#KhavineMoretti14">Khavkine-Moretti 14, equation (38) and section 3.4</a>)</p> <div class="num_prop" id="ContourIntegralForStandardHadamardPropagatorOnMinkowskiSpacetime"> <h6 id="proposition_7">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/contour+integral">contour integral</a> representation of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> from def. <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a> is equivalently given by the <a class="existingWikiWord" href="/nlab/show/contour+integral">contour integral</a></p> <div class="maruku-equation" id="eq:StandardHadamardPropagatorOnMinkowskiSpacetimeInTermsOfContourIntegral"><span class="maruku-eq-number">(16)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>∫</mo><munder><mo>∮</mo><mrow><msub><mi>C</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></munder><mfrac><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><msup><mi>d</mi> <mi>p</mi></msup><mi>k</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Delta_H(x,y) \;=\; -i(2\pi)^{-(p+1)} \int \underset{C_+(\vec k)}{\oint} \frac{e^{-i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 } d k_0 d^{p} k \,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/Jordan+curve">Jordan curve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>⊂</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">C_+(\vec k) \subset \mathbb{C}</annotation></semantics></math> runs counter-clockwise, enclosing the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>∈</mo><mi>ℝ</mi><mo>⊂</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}</annotation></semantics></math>, but not enclosing the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>∈</mo><mi>ℝ</mi><mo>⊂</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">- \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}</annotation></semantics></math>.</p> <p><img src="https://ncatlab.org/nlab/files/ContourForHadamardPropagator.png" height="200" /></p> <blockquote> <p>graphics grabbed from <a href="#Kocic16">Kocic 16</a></p> </blockquote> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>We compute as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo>−</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>∫</mo><munder><mo>∮</mo><mrow><msub><mi>C</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></munder><mfrac><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></mfrac><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><msup><mi>d</mi> <mi>p</mi></msup><mi>k</mi></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>∫</mo><msub><mo>∮</mo> <mrow><msub><mi>C</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msub><mfrac><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><msup><mi>x</mi> <mn>0</mn></msup></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><msubsup><mi>k</mi> <mn>0</mn> <mn>2</mn></msubsup><mo>−</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>2</mn></msup></mrow></mfrac><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>∫</mo><munder><mo>∮</mo><mrow><msub><mi>C</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></munder><mfrac><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>−</mo><msub><mi>ω</mi> <mi>ϵ</mi></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>ω</mi> <mi>ϵ</mi></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msup><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} -i(2\pi)^{-(p+1)} \int \underset{C_+(\vec k)}{\oint} \frac{e^{ - i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 } d k_0 d^{p} k & = -i(2\pi)^{-(p+1)} \int \oint_{C_+(\vec k)} \frac{ e^{ -i k_0 x^0} e^{i \vec k \cdot (\vec x - \vec y)} }{ k_0^2 - \omega(\vec k)^2/c^2 } d k_0 d^p \vec k \\ & = -i(2\pi)^{-(p+1)} \int \underset{C_+(\vec k)}{\oint} \frac{ e^{ - i k_0 (x^0-y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ ( k_0 - \omega_\epsilon(\vec k) ) ( k_0 + \omega_\epsilon(\vec k) ) } d k_0 d^p \vec k \\ & = (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)} e^{-i \omega(\vec k) (x^0-y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} d^p \vec k \,. \end{aligned} </annotation></semantics></math></div> <p>The last step is application of <a class="existingWikiWord" href="/nlab/show/Cauchy%27s+integral+formula">Cauchy's integral formula</a>, which says that the <a class="existingWikiWord" href="/nlab/show/contour+integral">contour integral</a> picks up the <a class="existingWikiWord" href="/nlab/show/residue">residue</a> of the <a class="existingWikiWord" href="/nlab/show/pole">pole</a> of the <a class="existingWikiWord" href="/nlab/show/integrand">integrand</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>∈</mo><mi>ℝ</mi><mo>⊂</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}</annotation></semantics></math>. The last line is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta_H(x,y)</annotation></semantics></math>, by definition <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a>.</p> </div> <div class="num_prop" id="SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime"> <h6 id="proposition_8">Proposition</h6> <p><strong>(skew-symmetric part of <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> is the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> (def. <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a>) is of the form</p> <div class="maruku-equation" id="eq:DeompositionOfHadamardPropagatorOnMinkowkski"><span class="maruku-eq-number">(17)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mi>S</mi></msub><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo>)</mo></mrow><mo>+</mo><mi>H</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_H & = \tfrac{i}{2} \Delta_S + H \\ & = \tfrac{i}{2} \left( \Delta_+ - \Delta_- \right) + H \end{aligned} \,, </annotation></semantics></math></div> <p>where</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_S</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> (prop. <a class="maruku-ref" href="#AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime"></a>), which is real <a class="maruku-eqref" href="#eq:CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal">(14)</a> and skew-symmetric (prop. <a class="maruku-ref" href="#CausalPropagatorIsSkewSymmetric"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AA</mi></mphantom><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\Delta_S(x,y))^\ast = \Delta_S(x,y) \phantom{AA} \,, \phantom{AA} \Delta_S(y,x) = - \Delta_S(x,y) </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is real and symmetric</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AA</mi></mphantom><mi>H</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (H(x,y))^\ast = H(x,y) \phantom{AA} \,, \phantom{AA} H(y,x) = H(x,y) </annotation></semantics></math></div></li> </ol> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>By applying <a class="existingWikiWord" href="/nlab/show/Euler%27s+formula">Euler's formula</a> to <a class="maruku-eqref" href="#eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime">(15)</a> we obtain</p> <div class="maruku-equation" id="eq:SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime"><span class="maruku-eq-number">(18)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><munder><munder><mrow><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>cos</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>≔</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_H(x,y) & = \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k \\ & = \tfrac{i}{2} \underset{= \Delta_S(x,y)}{ \underbrace{ \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k }} \;+\; \underset{ \coloneqq H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k }} \end{aligned} </annotation></semantics></math></div> <p>On the left this identifies the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> by <a class="maruku-eqref" href="#eq:CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime">(13)</a>, prop. <a class="maruku-ref" href="#ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski"></a>.</p> <p>The second summand changes, both under complex conjugation as well as under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x-y) \mapsto (y-x)</annotation></semantics></math>, via <a class="existingWikiWord" href="/nlab/show/change+of+integration+variables">change of integration variables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec k \mapsto - \vec k</annotation></semantics></math> (because the <a class="existingWikiWord" href="/nlab/show/cosine">cosine</a> is an even function). This does not change the integral, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is symmetric.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p id="FeynmanPropagator"><strong><a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></strong></p> <p>We have seen that the <a class="existingWikiWord" href="/nlab/show/positive+real+number">positive</a> <a class="existingWikiWord" href="/nlab/show/frequency">frequency</a> component of the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_S</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> (prop. <a class="maruku-ref" href="#AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime"></a>) is the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> (def. <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a>) given, according to prop. <a class="maruku-ref" href="#SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime"></a>, by <a class="maruku-eqref" href="#eq:DeompositionOfHadamardPropagatorOnMinkowkski">(17)</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mi>S</mi></msub><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo>)</mo></mrow><mo>+</mo><mi>H</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_H & = \tfrac{i}{2} \Delta_S + H \\ & = \tfrac{i}{2} \left( \Delta_+ - \Delta_- \right) + H \end{aligned} \,. </annotation></semantics></math></div> <p>There is an evident variant of this combination, which will be of interest:</p> <div class="num_defn" id="FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime"> <h6 id="definition_7">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> for <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <em><a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></em> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is the <a class="existingWikiWord" href="/nlab/show/linear+combination">linear combination</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub><mo>≔</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo>)</mo></mrow><mo>+</mo><mi>H</mi></mrow><annotation encoding="application/x-tex"> \Delta_F \coloneqq \tfrac{i}{2} \left( \Delta_+ + \Delta_- \right) + H </annotation></semantics></math></div> <p>where the first term is proportional to the sum of the <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagators">advanced and retarded propagators</a> (prop. <a class="maruku-ref" href="#AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime"></a>) and the second is the symmetric part of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> according to prop. <a class="maruku-ref" href="#SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime"></a>.</p> <p>Similarly the <em><a class="existingWikiWord" href="/nlab/show/anti-Feynman+propagator">anti-Feynman propagator</a></em> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mover><mi>F</mi><mo>¯</mo></mover></msub><mo>≔</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo>)</mo></mrow><mo>−</mo><mi>H</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_{\overline{F}} \coloneqq \tfrac{i}{2} \left( \Delta_+ + \Delta_- \right) - H \,. </annotation></semantics></math></div></div> <div class="num_prop" id="ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime"> <h6 id="proposition_9">Proposition</h6> <p><strong>(mode expansion for <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> of <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> (def. <a class="maruku-ref" href="#FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime"></a>) for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is given by the following equivalent expressions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_F(x,y) & = \left\{ \array{ \frac{1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)c} e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \array{ \Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0 \\ \Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0 } \right. \end{aligned} </annotation></semantics></math></div> <p>Similarly the <a class="existingWikiWord" href="/nlab/show/anti-Feynman+propagator">anti-Feynman propagator</a> is equivalently given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mover><mi>F</mi><mo>¯</mo></mover></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mo lspace="verythinmathspace" rspace="0em">−</mo><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mfrac><mo lspace="verythinmathspace" rspace="0em">−</mo><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_{\overline{F}}(x,y) & = \left\{ \array{ \frac{-}{(2\pi)^p} \int \frac{1}{\omega(\vec k)c} e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{-}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \array{ -\Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0 \\ -\Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0 } \right. \end{aligned} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>By the mode expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_{\pm}</annotation></semantics></math> from <a class="maruku-eqref" href="#eq:ModeExpansionForMinkowskiAdvancedRetardedPropagator">(5)</a> and the mode expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> from <a class="maruku-eqref" href="#eq:SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime">(18)</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><munder><munder><mrow><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mtext>for</mtext><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow></munder><mo>+</mo><munder><munder><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>cos</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><munder><munder><mrow><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn><mo>+</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mtext>for</mtext><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow></munder><mo>+</mo><munder><munder><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>cos</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_F(x,y) & = \left\{ \array{ \underset{ = \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0 }{ \underbrace{ \frac{- i}{(2\pi)^{p}} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } + \underset{ = H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } &\vert& (x^0 - y^0) \gt 0 \\ \underset{ = 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0 }{ \underbrace{ \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } + \underset{ = H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \array{ \frac{1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \array{ \Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0 \\ \Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0 } \right. \end{aligned} </annotation></semantics></math></div> <p>where in the second line we used <a class="existingWikiWord" href="/nlab/show/Euler%27s+formula">Euler's formula</a>. The last line follows by comparison with <a class="maruku-eqref" href="#eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime">(15)</a> and using that the integral over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec k</annotation></semantics></math> is invariant under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec k \mapsto - \vec k</annotation></semantics></math>.</p> <p>The computation for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mover><mi>F</mi><mo>¯</mo></mover></msub></mrow><annotation encoding="application/x-tex">\Delta_{\overline{F}}</annotation></semantics></math> is the same, only now with a minus sign in front of the <a class="existingWikiWord" href="/nlab/show/cosine">cosine</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mover><mi>F</mi><mo>¯</mo></mover></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><munder><munder><mrow><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mtext>for</mtext><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow></munder><mo>−</mo><munder><munder><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>cos</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><munder><munder><mrow><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn><mo>+</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mtext>for</mtext><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow></munder><mo>−</mo><munder><munder><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>cos</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_{\overline{F}}(x,y) & = \left\{ \array{ \underset{ = \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0 }{ \underbrace{ \frac{- i}{(2\pi)^{p}} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } - \underset{ = H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } &\vert& (x^0 - y^0) \gt 0 \\ \underset{ = 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0 }{ \underbrace{ \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } - \underset{ = H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \array{ \frac{-1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{-1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{-1i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \array{ - \Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0 \\ - \Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0 } \right. \end{aligned} </annotation></semantics></math></div></div> <p>As before for the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a>, there are equivalent reformulations of the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a>, which are useful for computations:</p> <div class="num_prop" id="FeynmanPropagatorAsACauchyPrincipalvalue"> <h6 id="proposition_10">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> as a <a class="existingWikiWord" href="/nlab/show/Cauchy+principal+value">Cauchy principal value</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> and <a class="existingWikiWord" href="/nlab/show/anti-Feynman+propagator">anti-Feynman propagator</a> (def. <a class="maruku-ref" href="#FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime"></a>) for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is equivalently given by the following expressions, respectively:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mrow><mtable><mtr><mtd><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>Δ</mi> <mover><mi>F</mi><mo>¯</mo></mover></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>−</mo><msup><mi>y</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>±</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left. \array{ \Delta_F(x,y) \\ \Delta_{\overline{F}}(x,y) } \right\} & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu (x^\mu - y^\mu)} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \end{aligned} </annotation></semantics></math></div> <p>where we have a <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> of <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a> as for the <a class="existingWikiWord" href="/nlab/show/Cauchy+principal+value">Cauchy principal value</a> (<a href="Cauchy+principal+vlue#CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta">this prop</a>).</p> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>We compute as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mi>i</mi><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>−</mo><msup><mi>y</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>±</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mi>i</mi><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>−</mo><msup><mi>y</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><munder><munder><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>±</mo><mi>i</mi><mi>ϵ</mi><mo>)</mo></mrow><mo>⏟</mo></munder><mrow><mo>≔</mo><msub><mi>ω</mi> <mrow><mo>±</mo><mi>ϵ</mi></mrow></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>2</mn></msup></mrow></munder></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mi>i</mi><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>−</mo><msup><mi>y</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mrow><mo>(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>−</mo><msub><mi>ω</mi> <mrow><mo>±</mo><mi>ϵ</mi></mrow></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>ω</mi> <mrow><mo>±</mo><mi>ϵ</mi></mrow></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mfrac><mrow><mo>∓</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo>±</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mtd></mtr> <mtr><mtd><mfrac><mrow><mo>∓</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo>∓</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></msup><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>Δ</mi> <mover><mi>F</mi><mo>¯</mo></mover></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu (x^\mu - y^\mu)} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu (x^\mu - y^\mu)} }{ (k_0)^2 - \underset{ \coloneqq \omega_{\pm\epsilon}(\vec k)^2/c^2 }{\underbrace{ \left( \omega(\vec k)^2/c^2 \pm i \epsilon \right) }} } \, d k_0 \, d^p \vec k \\ & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu (x^\mu - y^\mu)} }{ \left( k_0 - \omega_{\pm \epsilon}(\vec k)/c \right) \left( k_0 + \omega_{\pm \epsilon}(\vec k)/c \right) } \, d k_0 \, d^p \vec k \\ & = \left\{ \array{ \frac{\mp 1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{\pm i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{\mp 1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{\mp i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \array{ \Delta_F(x,y) \\ \Delta_{\overline{F}}(x,y) } \right. \end{aligned} </annotation></semantics></math></div> <p>Here</p> <ol> <li> <p>In the first step we introduced the <a class="existingWikiWord" href="/nlab/show/complex+number">complex</a> <a class="existingWikiWord" href="/nlab/show/square+root">square root</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mrow><mo>±</mo><mi>ϵ</mi></mrow></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega_{\pm \epsilon}(\vec k)</annotation></semantics></math>. For this to be compatible with the choice of <em>non-negative</em> square root for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon = 0</annotation></semantics></math> in <a class="maruku-eqref" href="#eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime">(4)</a> we need to choose that complex square root whose <a class="existingWikiWord" href="/nlab/show/complex+phase">complex phase</a> is one half that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>−</mo><mi>i</mi><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\omega(\vec k)^2 - i \epsilon</annotation></semantics></math> (instead of that plus <a class="existingWikiWord" href="/nlab/show/%CF%80">π</a>). This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>ϵ</mi></mrow></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega_{+ \epsilon}(\vec k)</annotation></semantics></math> is in the <em><a class="existingWikiWord" href="/nlab/show/upper+half+plane">upper half plane</a></em> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mo>−</mo></msub><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega_-(\vec k)</annotation></semantics></math> is in the <a class="existingWikiWord" href="/nlab/show/lower+half+plane">lower half plane</a>.</p> </li> <li> <p>In the third step we observe that</p> <ol> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(x^0 - y^0) \gt 0</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/integrand">integrand</a> decays for <a class="existingWikiWord" href="/nlab/show/positive+real+number">positive</a> <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary part</a> and hence the integration over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">k_0</annotation></semantics></math> may be deformed to a <a class="existingWikiWord" href="/nlab/show/Jordan+curve">contour</a> which encircles the <a class="existingWikiWord" href="/nlab/show/pole">pole</a> in the <a class="existingWikiWord" href="/nlab/show/upper+half+plane">upper half plane</a>;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(x^0 - y^0) \lt 0</annotation></semantics></math> the integrand decays for <a class="existingWikiWord" href="/nlab/show/negative+real+number">negative</a> <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary part</a> and hence the integration over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">k_0</annotation></semantics></math> may be deformed to a <a class="existingWikiWord" href="/nlab/show/Jordan+curve">contour</a> which encircles the <a class="existingWikiWord" href="/nlab/show/pole">pole</a> in the <a class="existingWikiWord" href="/nlab/show/lower+half+plane">lower half plane</a></p> </li> </ol> <p>and then apply <a class="existingWikiWord" href="/nlab/show/Cauchy%27s+integral+formula">Cauchy's integral formula</a> which picks out <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">2\pi i</annotation></semantics></math> times the <a class="existingWikiWord" href="/nlab/show/residue">residue</a> a these poles.</p> <p><img src="https://ncatlab.org/nlab/files/ContourForFeynmanPropagator.png" height="300" /></p> <p>Notice that when completing to a contour in the <a class="existingWikiWord" href="/nlab/show/lower+half+plane">lower half plane</a> we pick up a minus signs from the fact that now the contour runs clockwise.</p> </li> <li> <p>In the fourth step we used prop. <a class="maruku-ref" href="#ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime"></a>.</p> </li> </ol> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p id="WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime"><strong><a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> and <a class="existingWikiWord" href="/nlab/show/wave+front+sets">wave front sets</a></strong></p> <p>We now discuss the <a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> and the <a class="existingWikiWord" href="/nlab/show/wave+front+sets">wave front sets</a> of the various <a class="existingWikiWord" href="/nlab/show/propagators">propagators</a> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>.</p> <div class="num_prop" id="SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone"> <h6 id="proposition_11">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> of the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> of the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_S</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, regarded via <a class="existingWikiWord" href="/nlab/show/translation">translation</a> <a class="existingWikiWord" href="/nlab/show/invariant">invariance</a> as a <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a> in a single variable <a class="maruku-eqref" href="#eq:TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime">(6)</a> is the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a> of the origin:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>supp</mi> <mi>sing</mi></msub><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> supp_{sing}(\Delta_S) \;=\; \left\{ x \in \mathbb{R}^{p,1} \,\vert\, {\vert x\vert}^2_\eta = 0 \right\} \,. </annotation></semantics></math></div></div> <div class="proof" id="ProofThatSingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone"> <h6 id="proof_10">Proof</h6> <p>By prop. <a class="maruku-ref" href="#CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator"></a> the causal propagator is equivalently the <a class="existingWikiWord" href="/nlab/show/Fourier+transform+of+distributions">Fourier transform of distributions</a> of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> of the <a class="existingWikiWord" href="/nlab/show/mass+shell">mass shell</a> times the <a class="existingWikiWord" href="/nlab/show/sign+function">sign function</a> of the <a class="existingWikiWord" href="/nlab/show/angular+frequency">angular frequency</a>; and by basic properties of the Fourier transform this is the <a class="existingWikiWord" href="/nlab/show/convolution+of+distributions">convolution of distributions</a> of the separate Fourier transforms:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>∝</mo><mover><mrow><mi>δ</mi><mrow><mo>(</mo><msup><mi>η</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>^</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>∝</mo><mover><mrow><mi>δ</mi><mrow><mo>(</mo><msup><mi>η</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>^</mo></mover><mo>⋆</mo><mover><mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>^</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_S(x) & \propto \widehat{ \delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) } \\ &\propto \widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)} \star \widehat{sgn( k_0 )} \end{aligned} </annotation></semantics></math></div> <p>By (<a href="#GelfandShilov66">Gel’fand-Shilov 66, III 2.11 (7), p 294</a>), see <a href="Cauchy+principal+value#FourierTransformOfDeltaDistributionappliedToMassShell">this prop.</a>, the <a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> of the first convolution factor is the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a>.</p> <p>The second factor is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mover><mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>^</mo></mover></mtd> <mtd><mo>∝</mo><mrow><mo>(</mo><mn>2</mn><mover><mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>−</mo><mover><mn>1</mn><mo>^</mo></mover><mo>)</mo></mrow><mi>δ</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>∝</mo><mrow><mo>(</mo><mn>2</mn><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><msup><mi>x</mi> <mn>0</mn></msup><mo>+</mo><msup><mn>0</mn> <mo>+</mo></msup></mrow></mfrac></mstyle><mo>−</mo><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>)</mo></mrow><mi>δ</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \widehat{sgn(k_0)} & \propto \left(2\widehat{\Theta(k_0)} - \widehat{1}\right) \delta(\vec k) \\ & \propto \left(2\tfrac{1}{i x^0 + 0^+} - \delta(x^0)\right) \delta(\vec k) \end{aligned} </annotation></semantics></math></div> <p>(by <a href="Dirac+distribution#FourierTransformOfDeltaDistribution">this example</a> and <a href="Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution">this example</a>) and hence the <a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a> of the second factor is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mrow><mo>(</mo><mover><mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>)</mo></mrow><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>k</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\} </annotation></semantics></math></div> <p>(by <a href="wavefront+set#WaveFrontOfDeltaDistribution">this example</a> and <a href="Cauchy+principal+value#PrincipalValueOfInverseFunctionCharacteristicEquation">this example</a>).</p> <p>With this the statement follows, via a <a class="existingWikiWord" href="/nlab/show/partition+of+unity">partition of unity</a>, from <a href="convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions">this prop.</a>.</p> <p>For illustration we now make this general argument more explicit in the special case of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> p + 1 = 3 + 1 </annotation></semantics></math></div> <p>by computing an explicit form for the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> in terms of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a>, the <a class="existingWikiWord" href="/nlab/show/Heaviside+distribution">Heaviside distribution</a> and <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth</a> <a class="existingWikiWord" href="/nlab/show/Bessel+functions">Bessel functions</a>.</p> <p>We follow (<a href="causal+perturbation+theory#Scharf95">Scharf 95 (2.3.18)</a>).</p> <p>Consider the formula for the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> in terms of the mode expansion <a class="maruku-eqref" href="#eq:CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime">(13)</a>. Since the <a class="existingWikiWord" href="/nlab/show/integrand">integrand</a> here depends on the <a class="existingWikiWord" href="/nlab/show/wave+vector">wave vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec k</annotation></semantics></math> only via its <a class="existingWikiWord" href="/nlab/show/norm">norm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert \vec k\vert}</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/angle">angle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> it makes with the given <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/vector">vector</a> via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mspace width="thinmathspace"></mspace><mi>cos</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \vec k \cdot (\vec x - \vec y) \;=\; {\vert \vec k\vert} \, {\vert \vec x\vert} \, \cos(\theta) </annotation></semantics></math></div> <p>we may express the <a class="existingWikiWord" href="/nlab/show/integration">integration</a> in terms of <a class="existingWikiWord" href="/nlab/show/polar+coordinates">polar coordinates</a> as follws:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>vol</mi> <mrow><msup><mi>S</mi> <mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><munder><mo>∫</mo><mrow><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow></munder><munder><mo>∫</mo><mrow><mi>θ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo stretchy="false">]</mo></mrow></munder><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mspace width="thinmathspace"></mspace><mi>d</mi><mi>θ</mi><mo>∧</mo><mi>d</mi><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_S(x - y) & = \frac{-1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k \\ & = \frac{- vol_{S^{p-2}}}{(2\pi)^p} \underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int} \underset{ \theta \in [0,\pi] }{\int} \frac{ 1 }{ \omega(\vec k)/c } \sin\left( \omega(\vec k) (x^0 - y^0) /c \right) e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) } {\vert \vec k\vert} ({\vert \vec k\vert} \sin(\theta))^{p-2} \, d \theta \wedge d {\vert \vec k\vert} \end{aligned} </annotation></semantics></math></div> <p>In the special case of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p + 1 = 3 + 1</annotation></semantics></math> this becomes</p> <div class="maruku-equation" id="eq:StepsInComputingCausalPropagatorIn3plus1Dimension"><span class="maruku-eq-number">(19)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><mi>π</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mn>3</mn></msup></mrow></mfrac><munder><mo>∫</mo><mrow><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow></munder><mfrac><mrow><msup><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup></mrow><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><munder><munder><mrow><munder><mo>∫</mo><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></munder><msup><mi>e</mi> <mrow><mi>i</mi><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></msup><mi>d</mi><mi>cos</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></mfrac></mstyle><mrow><mo>(</mo><msup><mi>e</mi> <mrow><mi>i</mi><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></msup><mo>−</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></msup><mo>)</mo></mrow></mrow></munder><mo>∧</mo><mi>d</mi><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></mfrac><munder><mo>∫</mo><mrow><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow></munder><mfrac><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></mfrac><munder><mo>∫</mo><mrow><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow></munder><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></mfrac><munder><mo>∫</mo><mrow><mi>κ</mi><mo>∈</mo><mi>ℝ</mi></mrow></munder><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mi>κ</mi><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>κ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">|</mo></mrow></mfrac><mrow><mo>(</mo><munder><munder><mrow><munder><mo>∫</mo><mrow><mi>κ</mi><mo>∈</mo><mi>ℝ</mi></mrow></munder><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>κ</mi><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>)</mo></mrow><mi>d</mi><mi>κ</mi></mrow><mo>⏟</mo></munder><mrow><mo>≔</mo><msub><mi>I</mi> <mo>+</mo></msub></mrow></munder><mo>+</mo><munder><munder><mrow><munder><mo>∫</mo><mrow><mi>κ</mi><mo>∈</mo><mi>ℝ</mi></mrow></munder><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>−</mo><mi>κ</mi><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>κ</mi></mrow><mo>⏟</mo></munder><mrow><mo>≔</mo><msub><mi>I</mi> <mo>−</mo></msub></mrow></munder><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_S(x - y) & = \frac{- 2\pi}{(2\pi)^{3}} \underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int} \frac{ {\vert \vec k \vert}^2 }{ \omega(\vec k)/c } \sin\left( \omega(\vec k) (x^0 - y^0) /c \right) \underset{ = \tfrac{1}{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} } \left( e^{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}} - e^{-i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}} \right) }{ \underbrace{ \underset{ \cos(\theta) \in [-1,1] }{\int} e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) } d \cos(\theta) } } \wedge d {\vert \vec k \vert} \\ & = \frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}} \underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int} \frac{ {\vert \vec k \vert} }{ \omega(\vec k)/c } \sin\left( \omega(\vec k) (x^0 - y^0) /c \right) \sin\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right) \, d {\vert \vec k\vert} \\ & = \frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}} \frac{d}{d {\vert \vec x - \vec y \vert } } \underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int} \frac{ 1 }{ \omega(\vec k)/c } \sin\left( \omega(\vec k) (x^0 - y^0) /c \right) \cos\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right) \, d {\vert \vec k\vert} \\ & = \frac{- 1}{(2\pi)^{2} {\vert \vec x - \vec y\vert}} \frac{d}{d {\vert \vec x - \vec y \vert } } \underset{ \kappa \in \mathbb{R} }{\int} \frac{ 1 }{ \omega(\kappa)/c } \sin\left( \omega(\kappa) (x^0 - y^0) /c \right) \cos\left( \kappa\, {\vert \vec x - \vec y\vert} \right) \, d \kappa \\ & = \frac{- 1}{2(2\pi)^{2} {\vert \vec x - \vec y\vert}} \frac{d}{d {\vert \vec x - \vec y} \vert } \left( \underset{\coloneqq I_+}{ \underbrace{ \underset{ \kappa \in \mathbb{R} }{\int} \frac{ 1 }{ \omega(\kappa)/c } \sin\left( \omega(\kappa) (x^0 - y^0) /c + \kappa\, {\vert \vec x - \vec y\vert} \right) d\kappa } } + \underset{ \coloneqq I_- }{ \underbrace{ \underset{ \kappa \in \mathbb{R} }{\int} \frac{ 1 }{ \omega(\kappa)/c } \sin\left( \omega(\kappa) (x^0 - y^0) /c - \kappa\, {\vert \vec x - \vec y\vert} \right) \, d \kappa } } \right) \,. \end{aligned} </annotation></semantics></math></div> <p>Here in the second but last step we renamed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo>≔</mo><mrow><mo stretchy="false">|</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">\kappa \coloneqq {\vert \vec k\vert}</annotation></semantics></math> and doubled the integration domain for convenience, and in the last step we used the <a class="existingWikiWord" href="/nlab/show/trigonometric+identity">trigonometric identity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\sin(\alpha) \cos(\beta)\;=\; \tfrac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)</annotation></semantics></math>.</p> <p>In order to further evaluate this, we parameterize the remaining components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">/</mo><mi>c</mi><mo>,</mo><mi>κ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\omega/c, \kappa)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/wave+vector">wave vector</a> by the dual <a class="existingWikiWord" href="/nlab/show/rapidity">rapidity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math>, via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow><mo>(</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo>(</mo><mi>sinh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> \left(\cosh(z)\right)^2 - \left( \sinh(z)\right)^2 = 1 </annotation></semantics></math></div> <p>as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><mi>cosh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AA</mi></mphantom><mi>κ</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><mi>sinh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \omega(\kappa)/c \;=\; \left( \tfrac{m c}{\hbar} \right) \cosh(z) \phantom{AA} \,, \phantom{AA} \kappa \;=\; \left( \tfrac{m c}{\hbar} \right) \sinh(z) \,, </annotation></semantics></math></div> <p>which makes use of the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega(\kappa)</annotation></semantics></math> is non-negative, by construction. This <a class="existingWikiWord" href="/nlab/show/change+of+integration+variables">change of integration variables</a> makes the integrals under the braces above become</p> <div class="maruku-equation" id="eq:TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski"><span class="maruku-eq-number">(20)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mo>±</mo></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>sin</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>±</mo><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mi>sinh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> I_\pm \;=\; \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \left( (x^0 - y^0) \cosh(z) \pm {\vert \vec x - \vec y\vert} \sinh(z) \right) \right) \, d z \,. </annotation></semantics></math></div> <p>Next we similarly parameterize the vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x-y</annotation></semantics></math> by its <a class="existingWikiWord" href="/nlab/show/rapidity">rapidity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>. That parameterization depends on whether <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x-y</annotation></semantics></math> is spacelike or not, and if not, whether it is future or past directed.</p> <p>First, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x-y</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/spacelike">spacelike</a> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\vert x-y\vert}^2_\eta \gt 0</annotation></semantics></math> then we may parameterize as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mi>sinh</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AA</mi></mphantom><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>=</mo><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mi>cosh</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (x^0 - y^0) = \sqrt{{\vert x-y\vert}^2_\eta} \sinh(\tau) \phantom{AA} \,, \phantom{AA} {\vert \vec x - \vec y\vert} = \sqrt{ {\vert x-y\vert}^2_\eta} \cosh(\tau) </annotation></semantics></math></div> <p>which yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>I</mi> <mo>±</mo></msub></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>sin</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mrow><mo>(</mo><mi>sinh</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>±</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mi>sinh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>sin</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mrow><mo>(</mo><mi>sinh</mi><mrow><mo>(</mo><mi>τ</mi><mo>±</mo><mi>z</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>sin</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mrow><mo>(</mo><mi>sinh</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} I_{\pm} & = \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \left( \sinh(\tau) \cosh(z) \pm \cosh(\tau) \sinh(z) \right) \right) \, d z \\ & = \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta} \left( \sinh\left( \tau \pm z\right) \right) \right) \, d z \\ & = \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \left( \sinh\left( z \right) \right) \right) \, d z \\ & = 0 \,, \end{aligned} </annotation></semantics></math></div> <p>where in the last line we observe that the integrand is a skew-symmetric function of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math>.</p> <p>Second, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x-y</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/timelike">timelike</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(x^0 - y^0) \gt 0</annotation></semantics></math> then we may parameterize as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mi>cosh</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AA</mi></mphantom><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mo>=</mo><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mi>sinh</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (x^0 - y^0) = \sqrt{ -{\vert x-y\vert}^2_\eta} \cosh(\tau) \phantom{AA} \,, \phantom{AA} {\vert \vec x - \vec y\vert} = \sqrt{ -{\vert x - y\vert}^2_\eta } \sinh(\tau) </annotation></semantics></math></div> <p>which yields</p> <div class="maruku-equation" id="eq:IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski"><span class="maruku-eq-number">(21)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>I</mi> <mo>±</mo></msub></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>sin</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>±</mo><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mi>sinh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>sin</mi><mrow><mo>(</mo><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mrow><mo>(</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>±</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mi>sinh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>sin</mi><mrow><mo>(</mo><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mrow><mo>(</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>z</mi><mo>±</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>π</mi><msub><mi>J</mi> <mn>0</mn></msub><mrow><mo>(</mo><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} I_\pm & = \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \left( (x^0 - y^0) \cosh(z) \pm {\vert \vec x - \vec y\vert} \sinh(z) \right) \right) \, d z \\ & = \int_{-\infty}^\infty \sin\left( \sqrt{ - {\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \left( \cosh(\tau)\cosh(z) \pm \cosh(\tau) \sinh(z) \right) \right) \, d z \\ & = \int_{-\infty}^\infty \sin\left( \sqrt{ - {\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \left( \cosh(z \pm \tau) \right) \right) \, d z \\ & = \pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right) \end{aligned} \,. </annotation></semantics></math></div> <p>Here in the last line we identified the integral representation of the <a class="existingWikiWord" href="/nlab/show/Bessel+function">Bessel function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">J_0</annotation></semantics></math> of order 0 (see <a href="Bessel+function#eq:J0AsIntSinOfxCoshtdt">here</a>). The important point here is that this is a smooth function.</p> <p>Similarly, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x-y</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/timelike">timelike</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(x^0 - y^0) \lt 0</annotation></semantics></math> then the same argument yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mo>±</mo></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>π</mi><msub><mi>J</mi> <mn>0</mn></msub><mrow><mo>(</mo><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> I_\pm = - \pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right) </annotation></semantics></math></div> <p>In conclusion, the general form of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mo>±</mo></msub></mrow><annotation encoding="application/x-tex">I_\pm</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mo>±</mo></msub><mo>=</mo><mi>π</mi><mi>sgn</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mi>Θ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><msub><mi>J</mi> <mn>0</mn></msub><mrow><mo>(</mo><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> I_\pm = \pi sgn(x^0 - y^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right) J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \right) \,. </annotation></semantics></math></div> <p>Therefore we end up with</p> <div class="maruku-equation" id="eq:FinalResultOfComputationOf3Plus1dCausalPropagator"><span class="maruku-eq-number">(22)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow></mrow></mfrac><mi>sgn</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mi>Θ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><msub><mi>J</mi> <mn>0</mn></msub><mrow><mo>(</mo><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac><mi>sgn</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mi>Θ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><msub><mi>J</mi> <mn>0</mn></msub><mrow><mo>(</mo><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mfrac><mi>d</mi><mrow><mi>d</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><msubsup><mo stretchy="false">|</mo> <mi>η</mi> <mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac><mi>sgn</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mi>Θ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><msub><mi>J</mi> <mn>0</mn></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mi>sgn</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mrow><mo>(</mo><mi>δ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>−</mo><mspace width="thickmathspace"></mspace><mi>Θ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mrow><mo>(</mo><msubsup><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow></mrow></mfrac><msub><mi>J</mi> <mn>0</mn></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mo>)</mo></mrow><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_S(x,y) & = \frac{1}{4 \pi {\vert \vec x - \vec y\vert}} \frac{d}{d {\vert \vec x - \vec y\vert}} sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right) J_0\left( \sqrt{ -{\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \right) \\ & = \frac{-1}{2 \pi } \frac{d}{d (-{\vert x-y\vert}^2_\eta)} sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right) J_0\left( \sqrt{-{\vert x-y \vert}^2_\eta} \tfrac{m c}{\hbar} \right) \\ & = -\frac{1}{2 \pi } \frac{d}{d (- \vert x-y\vert^2_{\eta})} sgn(x^0) \Theta\left( - {\vert x - y\vert}^2_\eta \right) J_0\left( \tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta } \right) \\ & = \frac{-1}{2\pi} sgn(x^0) \left( \delta\left( -{\vert x-y\vert}^2_\eta \right) \;-\; \Theta\left( -{\vert x-y\vert}^2_\eta \right) \frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) } J_0\left( \tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta } \right) \right) \end{aligned} </annotation></semantics></math></div></div> <div class="num_prop" id="SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone"> <h6 id="proposition_12">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> (def. <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a>) for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, regarded via <a class="existingWikiWord" href="/nlab/show/translation">translation</a> <a class="existingWikiWord" href="/nlab/show/invariant">invariance</a> as a <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> in a single variable, is the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a> of the origin:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>supp</mi> <mi>sing</mi></msub><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> supp_{sing}(\Delta_H) = \left\{ x \in \mathbb{R}^{p,1} \;\vert\; {\vert x\vert}^2_\eta = 0 \right\} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>By prop. <a class="maruku-ref" href="#CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator"></a> the causal propagator is equivalently the <a class="existingWikiWord" href="/nlab/show/Fourier+transform+of+distributions">Fourier transform of distributions</a> of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> of the <a class="existingWikiWord" href="/nlab/show/mass+shell">mass shell</a> times the <a class="existingWikiWord" href="/nlab/show/sign+function">sign function</a> of the <a class="existingWikiWord" href="/nlab/show/angular+frequency">angular frequency</a>; and by basic properties of the Fourier transform this is the <a class="existingWikiWord" href="/nlab/show/convolution+of+distributions">convolution of distributions</a> of the separate Fourier transforms:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>∝</mo><mover><mrow><mi>δ</mi><mrow><mo>(</mo><msup><mi>η</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>^</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>∝</mo><mover><mrow><mi>δ</mi><mrow><mo>(</mo><msup><mi>η</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow></mrow><mo>^</mo></mover><mo>⋆</mo><mover><mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>^</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_S(x) & \propto \widehat{ \delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) } \\ &\propto \widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)} \star \widehat{sgn( k_0 )} \end{aligned} </annotation></semantics></math></div> <p>By (<a href="#GelfandShilov66">Gel’fand-Shilov 66, III 2.11 (7), p 294</a>), see <a href="Cauchy+principal+value#FourierTransformOfDeltaDistributionappliedToMassShell">this prop.</a>, the <a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> of the first convolution factor is the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a>.</p> <p>The second factor is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>∝</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mi>i</mi><msup><mi>x</mi> <mn>0</mn></msup><mo>+</mo><msup><mn>0</mn> <mo>+</mo></msup></mrow></mfrac></mstyle><mi>δ</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \widehat{\Theta(k_0)} \propto \tfrac{1}{i x^0 + 0^+} \delta(\vec k) </annotation></semantics></math></div> <p>(by <a href="Dirac+distribution#FourierTransformOfDeltaDistribution">this example</a> and <a href="Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution">this example</a>) and hence the <a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a> of the second factor is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mrow><mo>(</mo><mover><mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>)</mo></mrow><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>k</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\} </annotation></semantics></math></div> <p>(by <a href="wavefront+set#WaveFrontOfDeltaDistribution">this example</a> and <a href="Cauchy+principal+value#PrincipalValueOfInverseFunctionCharacteristicEquation">this example</a>).</p> <p>With this the statement follows, via a <a class="existingWikiWord" href="/nlab/show/partition+of+unity">partition of unity</a>, from <a href="convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions">this prop.</a>.</p> <p>For illustration, we now make this general statement fully explicit in the special case of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> p + 1 = 3 + 1 </annotation></semantics></math></div> <p>by computing an explicit form for the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> in terms of the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a>, the <a class="existingWikiWord" href="/nlab/show/Heaviside+distribution">Heaviside distribution</a> and <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth</a> <a class="existingWikiWord" href="/nlab/show/Bessel+functions">Bessel functions</a>.</p> <p>We follow (<a href="causal+perturbation+theory#Scharf95">Scharf 95 (2.3.36)</a>).</p> <p>By <a class="maruku-eqref" href="#eq:SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime">(18)</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><munder><munder><mrow><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>sin</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><mi>cos</mi><mrow><mo>(</mo><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>⏟</mo></munder><mrow><mo>≔</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_H(x,y) & = \tfrac{i}{2} \underset{= \Delta_S(x,y)}{ \underbrace{ \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k }} \;+\; \underset{ \coloneqq H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k }} \end{aligned} </annotation></semantics></math></div> <p>The first summand, proportional to the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a>, which we computed as <a class="maruku-eqref" href="#eq:FinalResultOfComputationOf3Plus1dCausalPropagator">(22)</a> in prop. <a class="maruku-ref" href="#SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone"></a> to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi></mrow><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mi>sgn</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mrow><mo>(</mo><mi>δ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>−</mo><mspace width="thickmathspace"></mspace><mi>Θ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mrow><mo>(</mo><msubsup><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow></mrow></mfrac><msub><mi>J</mi> <mn>0</mn></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tfrac{i}{2}\Delta_S(x,y) \;=\; \frac{-i}{4\pi} sgn(x^0) \left( \delta\left( -{\vert x-y\vert}^2_\eta \right) \;-\; \Theta\left( -{\vert x-y\vert}^2_\eta \right) \frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) } J_0\left( \tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta } \right) \right) \,. </annotation></semantics></math></div> <p>The second term is computed in a directly analogous fashion: The integrals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mo>±</mo></msub></mrow><annotation encoding="application/x-tex">I_\pm</annotation></semantics></math> from <a class="maruku-eqref" href="#eq:TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski">(20)</a> are now</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mo>±</mo></msub><mo>≔</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>cos</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mi>cosh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>±</mo><mrow><mo stretchy="false">|</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">|</mo></mrow><mi>sinh</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi></mrow><annotation encoding="application/x-tex"> I_\pm \coloneqq \int_{-\infty}^\infty \cos\left( \tfrac{m c}{\hbar} \left( (x^0 - y^0) \cosh(z) \pm {\vert \vec x - \vec y\vert} \sinh(z) \right) \right) \, d z </annotation></semantics></math></div> <p>Parameterizing by <a class="existingWikiWord" href="/nlab/show/rapidity">rapidity</a>, as in the proof of prop. <a class="maruku-ref" href="#SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone"></a>, one finds that for <a class="existingWikiWord" href="/nlab/show/timelike">timelike</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x-y</annotation></semantics></math> this is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>I</mi> <mo>±</mo></msub></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>cos</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mrow><mo>(</mo><mi>cosh</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>π</mi><msub><mi>N</mi> <mn>0</mn></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} I_\pm & = \int_{-\infty}^\infty \cos\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \left( \cosh\left( z \right) \right) \right) \, d z \\ & = - \pi N_0 \left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \right) \end{aligned} </annotation></semantics></math></div> <p>while for <a class="existingWikiWord" href="/nlab/show/spacelike">spacelike</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x-y</annotation></semantics></math> it is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>I</mi> <mo>±</mo></msub></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mi>cos</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mrow><mo>(</mo><mi>sinh</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>2</mn><msub><mi>K</mi> <mn>0</mn></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} I_\pm & = \int_{-\infty}^\infty \cos\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \left( \sinh\left( z \right) \right) \right) \, d z \\ & = 2 K_0 \left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \right) \,, \end{aligned} </annotation></semantics></math></div> <p>where we identified the integral representations of the <a class="existingWikiWord" href="/nlab/show/Neumann+function">Neumann function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">N_0</annotation></semantics></math> (see <a href="Bessel+function#N0AsIntSinOfxCoshtdt">here</a>) and of the <a class="existingWikiWord" href="/nlab/show/modified+Bessel+function">modified Bessel function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">K_0</annotation></semantics></math> (see <a href="Bessel+function#eq:K0AsIntSinOfxCoshtdt">here</a>).</p> <p>As for the <a class="existingWikiWord" href="/nlab/show/Bessel+function">Bessel function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">J_0</annotation></semantics></math> in <a class="maruku-eqref" href="#eq:IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski">(21)</a> the key point is that these are <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>. Hence we conclude that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∝</mo><mspace width="thickmathspace"></mspace><mfrac><mi>d</mi><mrow><mi>d</mi><mrow><mo>(</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow></mrow></mfrac><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>Θ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><msub><mi>N</mi> <mn>0</mn></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mo>)</mo></mrow><mo>+</mo><mi>Θ</mi><mrow><mo>(</mo><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>)</mo></mrow><mstyle displaystyle="false"><mfrac><mn>2</mn><mi>π</mi></mfrac></mstyle><msub><mi>K</mi> <mn>0</mn></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><msqrt><mrow><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup></mrow></msqrt><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H(x,y) \;\propto\; \frac{d}{d \left( {\vert x-y\vert}^2_\eta \right)} \left( -\Theta\left( -{\vert x-y\vert}^2_\eta \right) N_0 \left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \right) + \Theta\left( {\vert x-y\vert}^2_\eta \right) \tfrac{2}{\pi} K_0 \left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \right) \right) \,. </annotation></semantics></math></div> <p>This expression has singularities on the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a> due to the <a class="existingWikiWord" href="/nlab/show/step+functions">step functions</a>. In fact the expression being differentiated is continuous at the light cone (<a href="#Scharf95">Scharf 95 (2.3.34)</a>), so that the singularity on the light cone is not a <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a> singularity from the derivative of the step functions. Accordingly it does not cancel the singularity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tfrac{i}{2}\Delta_S(x,y)</annotation></semantics></math> as above, and hence the singular support of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> is still the whole light cone.</p> </div> <div class="num_prop" id="SingularSupportOfFeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime"> <h6 id="proposition_13">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> of <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> for <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> of the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> and of the <a class="existingWikiWord" href="/nlab/show/anti-Feynman+propagator">anti-Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mover><mi>F</mi><mo>¯</mo></mover></msub></mrow><annotation encoding="application/x-tex">\Delta_{\overline{F}}</annotation></semantics></math> (def. <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a>) for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, regarded via <a class="existingWikiWord" href="/nlab/show/translation">translation</a> <a class="existingWikiWord" href="/nlab/show/invariant">invariance</a> as a <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> in a single variable, is the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a> of the origin:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mrow><mtable><mtr><mtd><msub><mi>supp</mi> <mi>sing</mi></msub><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>supp</mi> <mi>sing</mi></msub><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mover><mi>F</mi><mo>¯</mo></mover></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left. \array{ supp_{sing}(\Delta_F) \\ supp_{sing}(\Delta_{\overline{F}}) } \right\} = \left\{ x \in \mathbb{R}^{p,1} \;\vert\; {\vert x\vert}^2_\eta = 0 \right\} \,. </annotation></semantics></math></div></div> <p>(e.g <a href="Feynman+propagator#DeWitt03">DeWitt 03 (27.85)</a>)</p> <div class="proof"> <h6 id="proof_12">Proof</h6> <p>By prop. <a class="maruku-ref" href="#FeynmanPropagatorAsACauchyPrincipalvalue"></a> the Feynman propagator is equivalently the <a class="existingWikiWord" href="/nlab/show/Cauchy+principal+value">Cauchy principal value</a> of the inverse of the Fourier transformed Klein-Gordon operator:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub><mspace width="thickmathspace"></mspace><mo>∝</mo><mspace width="thickmathspace"></mspace><mover><mfrac><mn>1</mn><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>+</mo><mi>i</mi><msup><mn>0</mn> <mo>+</mo></msup></mrow></mfrac><mo>^</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_F \;\propto\; \widehat{ \frac{1}{-k_\mu k^\mu - \left(\tfrac{m c}{\hbar}\right)^2 + i 0^+} } \,. </annotation></semantics></math></div> <p>With this the statement follows immediately from the result (<a href="#GelfandShilov66">Gel’fand-Shilov 66, III 2.8 (8) and (9), p 289</a>), see <a href="Cauchy+principal+value#FourierTransformOfPrincipalValueOfPowerOfQuadraticForm">this prop.</a>.</p> </div> <div class="num_prop" id="WaveFronSetsForKGPropagatorsOnMinkowski"> <h6 id="proposition_14">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/wave+front+sets">wave front sets</a> of <a class="existingWikiWord" href="/nlab/show/propagators">propagators</a> of <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a> of the various <a class="existingWikiWord" href="/nlab/show/propagators">propagators</a> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, regarded, via <a class="existingWikiWord" href="/nlab/show/translation">translation</a> <a class="existingWikiWord" href="/nlab/show/invariant">invariance</a>, as <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a> in a single variable, are as follows:</p> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_S</annotation></semantics></math> (prop. <a class="maruku-ref" href="#ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski"></a>) has wave front set all pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,k)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> both on the lightcone:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><msubsup><mrow><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thinmathspace"></mspace><mi>k</mi><mo>≠</mo><mn>0</mn><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> WF(\Delta_S) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \, k \neq 0 \right\} </annotation></semantics></math></div><center> <img src="https://ncatlab.org/nlab/files/RetGreenFunction.png" width="60" /> <br /> - <br /> <img src="https://ncatlab.org/nlab/files/AdvancedGreenFunction.png" width="60" /> </center> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>H</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_H</annotation></semantics></math> (def. <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a>) has wave front set all pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,k)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> both on the light cone and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mn>0</mn></msup><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k^0 \gt 0</annotation></semantics></math>:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><msubsup><mrow><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><msup><mi>k</mi> <mn>0</mn></msup><mo>></mo><mn>0</mn><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \; k^0 \gt 0 \right\} </annotation></semantics></math></div><center> <img src="https://ncatlab.org/nlab/files/HadamardPropagator.png" width="60" /> </center> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_S</annotation></semantics></math> (def. <a class="maruku-ref" href="#FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime"></a>) has wave front set all pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,k)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> both on the light cone and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>></mo><mn>0</mn><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mo>±</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0</annotation></semantics></math></li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><msubsup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><msubsup><mrow><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">|</mo></mrow> <mi>η</mi> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo>±</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>></mo><mn>0</mn><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mo>±</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>></mo><mn>0</mn><mo>)</mo></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \; \left( \pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0 \right) \right\} </annotation></semantics></math></div><center> <img src="https://ncatlab.org/nlab/files/FeynmanPropagator.png" width="60" /> </center></div> <p>(<a href="Hadamard+distribution#Radzikowski96">Radzikowski 96, (16)</a>)</p> <div class="proof"> <h6 id="proof_13">Proof</h6> <p>First regarding the causal propagator:</p> <p>By prop. <a class="maruku-ref" href="#SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone"></a> the <a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_S</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/light+cone">light cone</a>.</p> <p>Since the causal propagator is a solution to the homogeneous Klein-Gordon equation, the <a class="existingWikiWord" href="/nlab/show/propagation+of+singularities+theorem">propagation of singularities theorem</a> says that also all <a class="existingWikiWord" href="/nlab/show/wave+vectors">wave vectors</a> in the wave front set are lightlike. Hence it just remains to show that all non-vanishing lightlike wave vectors based on the lightcone in spacetime indeed do appear in the wave front set.</p> <p>To that end, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msubsup><mi>C</mi> <mi>cp</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b \in C^\infty_{cp}(\mathbb{R}^{p,1})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/bump+function">bump function</a> whose <a class="existingWikiWord" href="/nlab/show/compact+support">compact support</a> includes the origin.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a \in \mathbb{R}^{p,1}</annotation></semantics></math> a point on the light cone, we need to determine the decay property of the Fourier transform of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \mapsto b(x-a)\Delta_S(x)</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/convolution+of+distributions">convolution of distributions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>a</mi> <mi>μ</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\hat b(k)e^{i k_\mu a^\mu}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>Δ</mi><mo>^</mo></mover> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat \Delta_S(k)</annotation></semantics></math>. By prop. <a class="maruku-ref" href="#CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator"></a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>Δ</mi><mo>^</mo></mover> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∝</mo><mspace width="thickmathspace"></mspace><mi>δ</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>)</mo></mrow><mi>sgn</mi><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \widehat \Delta_{S}(k) \;\propto\; \delta\left( -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \right) sgn(k_0) \,. </annotation></semantics></math></div> <p>This means that the convolution product is the smearing of the mass shell by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mo lspace="0em" rspace="thinmathspace">u</mo></msub><msup><mi>a</mi> <mi>μ</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\widehat b(k)e^{i k_\u a^\mu}</annotation></semantics></math>.</p> <p>Since the mass shell asymptotes to the light cone, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>a</mi> <mi>μ</mi></msup></mrow></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">e^{i k_\mu a^\mu} = 1</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> on the light cone (given that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> is on the light cone), this implies the claim.</p> <p>Now for the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a>:</p> <p>By def. <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a> its Fourier transform is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>Δ</mi><mo>^</mo></mover> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∝</mo><mspace width="thickmathspace"></mspace><mi>δ</mi><mrow><mo>(</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup><mo>)</mo></mrow><mi>Θ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \widehat \Delta_H(k) \;\propto\; \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) </annotation></semantics></math></div> <p>Moreover, its <a class="existingWikiWord" href="/nlab/show/singular+support">singular support</a> is also the light cone (prop. <a class="maruku-ref" href="#SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone"></a>).</p> <p>Therefore now same argument as before says that the wave front set consists of wave vectors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> on the light cone, but now due to the <a class="existingWikiWord" href="/nlab/show/step+function">step function</a> factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(-k_0)</annotation></semantics></math> it must satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mn>0</mn></msub><mo>=</mo><msup><mi>k</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">0 \leq - k_0 = k^0</annotation></semantics></math>.</p> <p>Finally regarding the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a>:</p> <p>By prop. <a class="maruku-ref" href="#ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime"></a> the Feynman propagator coincides with the positive frequency Wightman propagator for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mn>0</mn></msup><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^0 \gt 0</annotation></semantics></math> and with the “negative frequency Hadamard operator” for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mn>0</mn></msup><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^0 \lt 0</annotation></semantics></math>. Therefore the form of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">WF(\Delta_F)</annotation></semantics></math> now follows directly with that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">WF(\Delta_H)</annotation></semantics></math> above.</p> </div> <h3 id="ExampleForDiracOperatorOnMinkowskiSpacetime">For Dirac operator on Minkowski spacetime</h3> <p>Finally we observe that the <a class="existingWikiWord" href="/nlab/show/propagators">propagators</a> for the <a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> follow immediately from the propagators for the <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>:</p> <div class="num_prop" id="DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators"> <h6 id="proposition_15">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a> for <a class="existingWikiWord" href="/nlab/show/Dirac+equation">Dirac equation</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, which in <a class="existingWikiWord" href="/nlab/show/Feynman+slash+notation">Feynman slash notation</a> reads</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>D</mi></mtd> <mtd><mo>≔</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mrow><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msup><mi>γ</mi> <mi>μ</mi></msup><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} D & \coloneqq -i {\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \\ & = -i \gamma^\mu \frac{\partial}{\partial x^\mu} + \tfrac{m c}{\hbar} \end{aligned} \,. </annotation></semantics></math></div> <p>Its <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagators">advanced and retarded propagators</a> (def. <a class="maruku-ref" href="#AdvancedAndRetardedGreenFunctions"></a>) are the <a class="existingWikiWord" href="/nlab/show/derivatives+of+distributions">derivatives of distributions</a> of the advanced and retarded propagators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>±</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_\pm</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> (prop. <a class="maruku-ref" href="#AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime"></a>) by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>+</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">{\partial\!\!\!/\,} + m</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>D</mi><mo>,</mo><mo>±</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mrow><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>−</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><msub><mi>Δ</mi> <mo>±</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_{D, \pm} \;=\; \left( -i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar} \right) \Delta_{\pm} \,. </annotation></semantics></math></div> <p>Hence the same is true for the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>D</mi><mo>,</mo><mi>S</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mrow><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>−</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><msub><mi>Δ</mi> <mi>S</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_{D, S} \;=\; \left( -i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar} \right) \Delta_{S} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_14">Proof</h6> <p>Applying a <a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a> does not change the <a class="existingWikiWord" href="/nlab/show/support">support</a> of a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>, hence also not the <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">support of a distribution</a>. Therefore the uniqueness of the advanced and retarded propagators (prop. <a class="maruku-ref" href="#AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique"></a>) together with the translation-invariance and the anti-<a class="existingWikiWord" href="/nlab/show/formally+self-adjoint+differential+operator">formally self-adjointness</a> of the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> (as for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> <a class="maruku-eqref" href="#eq:TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime">(6)</a> implies that it is sufficent to check that applying the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>D</mi><mo>,</mo><mo>±</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{D, \pm}</annotation></semantics></math> yields the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a>. This follows since the Dirac operator squares to the Klein-Gordon operator:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mrow><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><msub><mi>Δ</mi> <mrow><mi>D</mi><mo>,</mo><mo>±</mo></mrow></msub></mtd> <mtd><mo>=</mo><munder><munder><mrow><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mrow><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mrow><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>−</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mo>□</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup></mrow></munder><msub><mi>Δ</mi> <mo>±</mo></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>δ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left( -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \right) \Delta_{D, \pm} & = \underset{ = \Box - \left(\tfrac{m c}{\hbar}\right)^2}{ \underbrace{ \left( -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \right) \left( -i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar} \right) } } \Delta_{\pm} \\ & = \delta \end{aligned} \,. </annotation></semantics></math></div></div> <p>Similarly we obtain the other <a class="existingWikiWord" href="/nlab/show/propagators">propagators</a> for the <a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a> from those of the <a class="existingWikiWord" href="/nlab/show/real+scalar+field">real scalar field</a>:</p> <div class="num_defn" id="HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime"> <h6 id="definition_8">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> for <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <em><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></em> for the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is the <a class="existingWikiWord" href="/nlab/show/positive+real+number">positive</a> <a class="existingWikiWord" href="/nlab/show/frequency">frequency</a> part of the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> (prop. <a class="maruku-ref" href="#DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators"></a>), hence the <a class="existingWikiWord" href="/nlab/show/derivative+of+distributions">derivative of distributions</a> of the Wightman propagator for the Klein-Gordon field (def. <a class="maruku-ref" href="#StandardHadamardDistributionOnMinkowskiSpacetime"></a>) by the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mrow><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mi>δ</mi><mrow><mo>(</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>+</mo><msup><mi>m</mi> <mn>2</mn></msup><mo>)</mo></mrow><mi>Θ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow><mi>k</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo stretchy="false">)</mo><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>−</mo><msup><mi>y</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>k</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mi>p</mi></msup></mrow></mfrac><mo>∫</mo><mfrac><mrow><msup><mi>γ</mi> <mn>0</mn></msup><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mover><mi>γ</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle></mrow><mrow><mn>2</mn><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi></mrow></mfrac><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mi>ω</mi><mo stretchy="false">(</mo><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>0</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>c</mi><mo>+</mo><mi>i</mi><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>y</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left( -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \right)\Delta_{H}(x,y) & = \frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) ( {k\!\!\!/\,} + \tfrac{m c}{\hbar}) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k \\ & = \frac{1}{(2\pi)^p} \int \frac{ \gamma^0 \omega(\vec k)/c + \vec \gamma \cdot \vec k + \tfrac{m c}{\hbar} }{2 \omega(\vec k)/c} e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k \,. \end{aligned} </annotation></semantics></math></div> <p>Here we used the expression <a class="maruku-eqref" href="#eq:StandardHadamardDistributionOnMinkowskiSpacetime">(?)</a> for the Wightman propagator of the Klein-Gordon equation.</p> </div> <div class="num_defn" id="FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime"> <h6 id="definition_9">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> for <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>The <em><a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></em> for the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is the linear combination</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>D</mi><mo>,</mo><mi>F</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>Δ</mi> <mrow><mi>D</mi><mo>,</mo><mi>H</mi></mrow></msub><mo>+</mo><mi>i</mi><msub><mi>Δ</mi> <mrow><mi>D</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \Delta_{D, F} \;\coloneqq\; \Delta_{D,H} + i \Delta_{D, -} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> (def. <a class="maruku-ref" href="#HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime"></a>) and the retarded propagator (prop. <a class="maruku-ref" href="#DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators"></a>). By prop. <a class="maruku-ref" href="#FeynmanPropagatorAsACauchyPrincipalvalue"></a> this means that it is the <a class="existingWikiWord" href="/nlab/show/derivative+of+distributions">derivative of distributions</a> of the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+equation">Klein-Gordon equation</a> (def. <a class="maruku-ref" href="#FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime"></a>) by the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Δ</mi> <mrow><mi>D</mi><mo>,</mo><mi>F</mi></mrow></msub></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mrow><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><munder><mi>lim</mi><mfrac linethickness="0"><mrow><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow></mrow><mrow><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></mrow></mfrac></munder><mfrac><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>∫</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow> <mn>∞</mn></msubsup><mfrac><mrow><mrow><mo>(</mo><mrow><mi>k</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mspace width="thinmathspace"></mspace></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow><msup><mi>e</mi> <mrow><mi>i</mi><msub><mi>k</mi> <mi>μ</mi></msub><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>−</mo><msup><mi>y</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>k</mi> <mi>μ</mi></msub><msup><mi>k</mi> <mi>μ</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle><mo>)</mo></mrow> <mn>2</mn></msup><mo>+</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>k</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><msup><mi>d</mi> <mi>p</mi></msup><mover><mi>k</mi><mo stretchy="false">→</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Delta_{D, F} & = \left( -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \right)\Delta_{F}(x,y) & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{-i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ \left( {k\!\!\!/\,} + \tfrac{m c}{\hbar} \right) e^{i k_\mu (x^\mu - y^\mu)} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k \,. \end{aligned} </annotation></semantics></math></div></div> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/propagators">propagators</a> (i.e. <a class="existingWikiWord" href="/nlab/show/integral+kernels">integral kernels</a> of <a class="existingWikiWord" href="/nlab/show/Green+functions">Green functions</a>)</strong> <br /> <strong>for the <a class="existingWikiWord" href="/nlab/show/wave+operator">wave operator</a> and <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a></strong> <br /> <strong>on a <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetime">globally hyperbolic spacetime</a> such as <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>:</strong></p> <table><thead><tr><th>name</th><th>symbol</th><th><a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a></th><th>as <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum exp. value</a> <br /> of <a class="existingWikiWord" href="/nlab/show/operator-valued+distribution">field operators</a></th><th>as a <a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">product</a> of <br /> <a class="existingWikiWord" href="/nlab/show/operator-valued+distribution">field operators</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_1"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd><msub><mi>Δ</mi> <mi>S</mi></msub></mtd> <mtd><mo>=</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned}\Delta_S & = \Delta_+ - \Delta_- \end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/RetGreenFunction.png" width="60" /> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_2"><semantics><mrow><mphantom><mi>A</mi></mphantom><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>−</mo></mrow><annotation encoding="application/x-tex">\phantom{A}\,\,\,-</annotation></semantics></math> <br /> <img src="https://ncatlab.org/nlab/files/AdvancedGreenFunction.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_3"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>i</mi><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & i \hbar \, \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned} </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Peierls+bracket">Peierls-Poisson bracket</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/advanced+propagator">advanced propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_4"><semantics><mrow><msub><mi>Δ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_+</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/RetGreenFunction.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_5"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>i</mi><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>x</mi><mo>≥</mo><mi>y</mi></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>y</mi><mo>≥</mo><mi>x</mi></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & i \hbar \, \Delta_+(x,y) = \\ & \left\{ \array{ \left\langle \; \left[ \mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& x \geq y \\ 0 &\vert& y \geq x } \right. \end{aligned} </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/future">future</a> part of <br /> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/retarded+propagator">retarded propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_6"><semantics><mrow><msub><mi>Δ</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_-</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/AdvancedGreenFunction.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_7"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>i</mi><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mo>−</mo></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>y</mi><mo>≥</mo><mi>x</mi></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>x</mi><mo>≥</mo><mi>y</mi></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & i \hbar \, \Delta_-(x,y) = \\ & \left\{ \array{ \left\langle \; \left[\mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& y \geq x \\ 0 &\vert& x \geq y } \right. \end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/past">past</a> part of <br /> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_8"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd><msub><mi>Δ</mi> <mi>H</mi></msub></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo>)</mo></mrow><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><msub><mi>Δ</mi> <mi>S</mi></msub><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Δ</mi> <mi>F</mi></msub><mo>−</mo><mi>i</mi><msub><mi>Δ</mi> <mo>−</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} \Delta_H &= \tfrac{i}{2}\left( \Delta_+ - \Delta_-\right) + H\\ & = \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_- \end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/HadamardPropagator.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_9"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><munder><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mo>:</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & \hbar \, \Delta_H(x,y) \\ & = \left\langle \; \mathbf{\Phi}(x) \mathbf{\Phi}(y) \; \right\rangle \\ & = \underset{ = 0 }{\underbrace{\left\langle \; : \mathbf{\Phi}(x) \mathbf{\Phi}(y) : \; \right\rangle}} \\ & \phantom{=} + \left\langle \; \left[ \mathbf{\Phi}^{(-)}(x), \mathbf{\Phi}^{(+)}(y) \right] \; \right\rangle \end{aligned} </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/positive+real+number">positive</a> <a class="existingWikiWord" href="/nlab/show/frequency">frequency</a> of <br /> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a>, <br /> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>-product, <br /> <a class="existingWikiWord" href="/nlab/show/2-point+function">2-point function</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_10"><semantics><mrow><mphantom><mo>=</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{=}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_11"><semantics><mrow><mphantom><mo>=</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{=}</annotation></semantics></math> or generally of <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_12"><semantics><mrow><mphantom><mo>=</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{=}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_13"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd><msub><mi>Δ</mi> <mi>F</mi></msub></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mrow><mo>(</mo><msub><mi>Δ</mi> <mo>+</mo></msub><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>Δ</mi> <mo>−</mo></msub><mo>)</mo></mrow><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>i</mi><msub><mi>Δ</mi> <mi>D</mi></msub><mo>+</mo><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Δ</mi> <mi>H</mi></msub><mo>+</mo><mi>i</mi><msub><mi>Δ</mi> <mo>−</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned}\Delta_F & = \tfrac{i}{2}\left( \Delta_+ + \Delta_- \right) + H \\ & = i \Delta_D + H \\ & = \Delta_H + i \Delta_- \end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><img src="https://ncatlab.org/nlab/files/FeynmanPropagator.png" width="60" /></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_8ad2292c16e949c06e1578ff9cf7d277b6dcdf97_14"><semantics><mrow><mrow><mtable columnalign="right left right left right left right left right left" columnspacing="0em" displaystyle="true"><mtr><mtd></mtd> <mtd><mi>ℏ</mi><mspace width="thinmathspace"></mspace><msub><mi>Δ</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mi>T</mi><mrow><mo>(</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>x</mi><mo>≥</mo><mi>y</mi></mtd></mtr> <mtr><mtd><mrow><mo>⟨</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⟩</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>y</mi><mo>≥</mo><mi>x</mi></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\begin{aligned} & \hbar \, \Delta_F(x,y) \\ & = \left\langle \; T\left( \; \mathbf{\Phi}(x)\mathbf{\Phi}(y) \;\right) \; \right\rangle \\ & = \left\{ \array{ \left\langle \; \mathbf{\Phi}(x)\mathbf{\Phi}(x) \; \right\rangle &\vert& x \geq y \\ \left\langle \; \mathbf{\Phi}(y) \mathbf{\Phi}(x) \; \right\rangle &\vert& y \geq x } \right.\end{aligned}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a></td></tr> </tbody></table> <p>(see also <a class="existingWikiWord" href="/nlab/show/Mikica+Kocic">Kocic</a>‘s overview: <a class="existingWikiWord" href="/nlab/files/KGPropagatorsOnMinkowskiTable.pdf" title="pdf">pdf</a>)</p></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green+hyperbolic+differential+operator">Green hyperbolic differential operator</a></li> </ul> <h2 id="references">References</h2> <p>General discussion includes</p> <ul> <li id="Baer14"> <p><a class="existingWikiWord" href="/nlab/show/Christian+B%C3%A4r">Christian Bär</a>, <em>Green-hyperbolic operators on globally hyperbolic spacetimes</em>, Communications in Mathematical Physics 333, 1585-1615 (2014) (<a href="http://dx.doi.org/10.1007/s00220-014-2097-7">doi</a>, <a href="https://arxiv.org/abs/1310.0738">arXiv:1310.0738</a>)</p> </li> <li id="Khavkine14"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Khavkine">Igor Khavkine</a>, <em>Covariant phase space, constraints, gauge and the Peierls formula</em>, Int. J. Mod. Phys. A, 29, 1430009 (2014) (<a href="https://arxiv.org/abs/1402.1282">arXiv:1402.1282</a>)</p> </li> </ul> <p>based on</p> <ul> <li id="Sanders12"><a class="existingWikiWord" href="/nlab/show/Ko+Sanders">Ko Sanders</a>, <em>A note on spacelike and timelike compactness</em>, Classical and Quantum Gravity 30, 115014 (2012) (<a href="http://dx.doi.org/10.1088/0264-9381/30/11/115014">doi</a>, <a href="https://arxiv.org/abs/1211.2469">arXiv:1211.2469</a>)</li> </ul> <p>Textbook discussion for <a class="existingWikiWord" href="/nlab/show/free+fields">free fields</a> in <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> is in</p> <ul> <li id="Scharf95"> <p><a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Scharf">Günter Scharf</a>, section 2.3 of <em><a class="existingWikiWord" href="/nlab/show/Finite+Quantum+Electrodynamics+--+The+Causal+Approach">Finite Quantum Electrodynamics – The Causal Approach</a></em>, Springer 1995</p> </li> <li id="Scharf01"> <p><a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Scharf">Günter Scharf</a>, section 1 of <em><a class="existingWikiWord" href="/nlab/show/Quantum+Gauge+Theories+--+A+True+Ghost+Story">Quantum Gauge Theories – A True Ghost Story</a></em>, Wiley 2001</p> </li> </ul> <p>An overview of the <a class="existingWikiWord" href="/nlab/show/Green+functions">Green functions</a> of the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a>, hence of the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a>, <a class="existingWikiWord" href="/nlab/show/advanced+propagator">advanced propagator</a>, <a class="existingWikiWord" href="/nlab/show/retarded+propagator">retarded propagator</a>, <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> etc. is given in</p> <ul> <li id="Kocic16"><a class="existingWikiWord" href="/nlab/show/Mikica+Kocic">Mikica Kocic</a>, <em>Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions</em>, 2016 (<a class="existingWikiWord" href="/nlab/files/KGPropagatorsOnMinkowskiTable.pdf" title="pdf">pdf</a>)</li> </ul> <p>Discussion on general <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetimes">globally hyperbolic spacetimes</a> includes</p> <ul> <li> <p>F. Friedlander, <em>The Wave Equation on a Curved Space-Time</em>, Cambridge: Cambridge University Press, 1975</p> </li> <li id="BaerGinouxPfaeffle07"> <p><a class="existingWikiWord" href="/nlab/show/Christian+B%C3%A4r">Christian Bär</a>, <a class="existingWikiWord" href="/nlab/show/Nicolas+Ginoux">Nicolas Ginoux</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Pf%C3%A4ffle">Frank Pfäffle</a>, <em>Wave Equations on Lorentzian Manifolds and Quantization</em>, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (<a href="https://arxiv.org/abs/0806.1036">arXiv:0806.1036</a>)</p> </li> <li id="Ginoux08"> <p><a class="existingWikiWord" href="/nlab/show/Nicolas+Ginoux">Nicolas Ginoux</a>, <em>Linear wave equations</em>, Ch. 3 in <a class="existingWikiWord" href="/nlab/show/Christian+B%C3%A4r">Christian Bär</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <em>Quantum Field Theory on Curved Spacetimes: Concepts and Methods</em>, Lecture Notes in Physics, Vol. 786, Springer, 2009</p> </li> </ul> <p>Review in the context of <a class="existingWikiWord" href="/nlab/show/perturbative+algebraic+quantum+field+theory">perturbative algebraic quantum field theory</a> includes</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, sections 4.1 and 6.2.3 of <em>Perturbative Algebraic Quantum Field Theory</em>, Mathematical Physics Studies, Springer 2016 (<a href="https://link.springer.com/book/10.1007%2F978-3-319-25901-7">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 5, 2018 at 06:25:13. 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