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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="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> <ul> <li><a href='#adiabatic_switching'>Adiabatic switching</a></li> <li><a href='#adiabatic_limit'>Adiabatic limit</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> <h3 id="adiabatic_switching">Adiabatic switching</h3> <p>In <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, the term <em>adiabatic switching</em> refers to considering a smooth transition between vanishing and non-vanishing <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling</a>: the interaction is slowly, hence (borrowing a term from <a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a>) “<a class="existingWikiWord" href="/nlab/show/adiabatic">adiabatically</a>”, switched on or off. This is mostly a mathematical device, not meant to directly reflect a physical situation of changing coupling, but it does serve to construct physical quantities. This is closely related to the role of <a class="existingWikiWord" href="/nlab/show/operator-valued+distributions">operator-valued distributions</a> which are quantities that give well defined <a class="existingWikiWord" href="/nlab/show/linear+operators">linear operators</a> (hence <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a>) only when evaluated on any <a class="existingWikiWord" href="/nlab/show/bump+function">bump function</a>.</p> <p>Originally adiabatic switching was considered (<a href="#LippmannSchwinger50">Lippmann-Schwinger 50</a>) only in the <a class="existingWikiWord" href="/nlab/show/time">time</a>-direction (for a fixed choice of time on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>) by multiplying the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> term of the <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a>/<a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> by the <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ϵ</mi><mrow><mo stretchy="false">‖</mo><mi>t</mi><mo stretchy="false">‖</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(- \epsilon {\Vert t \Vert})</annotation></semantics></math> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon \in (0,\infty)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/real+number">real number</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>t</mi><mo stretchy="false">‖</mo></mrow></mrow><annotation encoding="application/x-tex">{\Vert t\Vert}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/absolute+value">absolute value</a> of the time <a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a>). Review is for instance in (<a href="#Strocchi13">Strocchi 13, section 6.3</a>).</p> <p>Using this, the <em>Gell-Mann and Low formula</em> (<a href="#GellMannLow51">Gell-Mann & Low 51</a>, see <a href="#Molinari06">Molinari 06</a>) expresses the <a class="existingWikiWord" href="/nlab/show/eigenstates">eigenstates</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>ψ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\vert \psi \rangle</annotation></semantics></math> of an interacting <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><msub><mi>H</mi> <mi>free</mi></msub><mo>+</mo><msub><mi>H</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">H = H_{free} + H_{int}</annotation></semantics></math> in terms of the eigenstates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msub><mi>Ψ</mi> <mi>free</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\vert \Psi_{free} \rangle</annotation></semantics></math> of the free Hamiltonian by the “adiabatic limit”</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><msubsup><mi>Ψ</mi> <mi>int</mi> <mo>±</mo></msubsup><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo>∝</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></munder><msub><mi>S</mi> <mi>ϵ</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mo>±</mo><mn>∞</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mi>Ψ</mi> <mi>free</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \vert \Psi^{\pm}_{int} \rangle \;\propto\; \underset{\epsilon \to 0}{\lim} S_\epsilon(0, \pm \infty) \vert \Psi_{free} \rangle </annotation></semantics></math></div> <p>(if the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> exists) where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>ϵ</mi></msub></mrow><annotation encoding="application/x-tex">S_\epsilon</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> of the adiabatically switched Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ϵ</mi></msub><mo>≔</mo><msub><mi>H</mi> <mi>free</mi></msub><mo>+</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ϵ</mi><mrow><mo stretchy="false">‖</mo><mi>t</mi><mo stretchy="false">‖</mo></mrow></mrow></msup><msub><mi>H</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">H_\epsilon \coloneqq H_{free} + e^{- \epsilon {\Vert t\Vert}}H_{int}</annotation></semantics></math>.</p> <p>More generally, one may consider adiabatic switching taking place not just in time, but in all of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>. This the basis of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> and <a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+quantum+field+theory">locally covariant perturbative quantum field theory</a>:</p> <p>In the construction of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> via the method of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> terms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">L_{int}</annotation></semantics></math> used in the mathematical construction of the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> are multiplied with a “<a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a>” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> which is in fact taken to be a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> of <a class="existingWikiWord" href="/nlab/show/compact+support">compact support</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, hence a <a class="existingWikiWord" href="/nlab/show/bump+function">bump 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>L</mi> <mi>g</mi></msub><mo>=</mo><msub><mi>L</mi> <mi>free</mi></msub><mo>+</mo><mi>g</mi><msub><mi>L</mi> <mi>int</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L_g = L_{free} + g L_{int} \,. </annotation></semantics></math></div> <p>This means that the the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> as modeled by the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>g</mi></msub><mo>≔</mo><mi>T</mi><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><msub><mo>∫</mo> <mi>X</mi></msub><mi>g</mi><mo>:</mo><msub><mi>L</mi> <mi>int</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S_g \coloneqq T \exp( \tfrac{i}{\hbar} \int_{X} g :L_{int}(x): ) </annotation></semantics></math></div> <p>is non-trivial only on a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, towards its boundary it smoothly drops to zero. Hence outside this region the interaction is “switched off”.</p> <p>Since the actual interactions in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> are of course <em>not</em> “switched off” anywhere, the use of an adiabatic switching is just an intermediate mathematical step. Originally in (<a href="#EpsteinGlaser73">Epstein-Glaser 73</a>) the idea was that after having constructed the <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> for any adiabatic switching <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> (“adiabatic limit”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">g \to 1</annotation></semantics></math> had to be taken to remove the switching in the end. Failure of this limit to exist is interpreted as “<a class="existingWikiWord" href="/nlab/show/infrared+divergency">infrared divergency</a>” of the <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> (since the divergency comes from large scales, hence long <a class="existingWikiWord" href="/nlab/show/wavelength">wavelength</a>).</p> <p>But as observed in (<a href="#IlinSlavnov78">Il’in-Slavnov 78</a>) and rediscovered in (<a href="#BrunettiFredenhagen00">Brunetti-Fredenhagen 00</a>), an adiabatic switching map that is unity on a globally hyperbolic sub-spacetime <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">O \subset X</annotation></semantics></math> is sufficient to compute the perturbative <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a>, hence the <a class="existingWikiWord" href="/nlab/show/algebra+of+quantum+observables">algebra of quantum observables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(O)</annotation></semantics></math> on that subspace, and the collection of all of these as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math> ranges forms a <a class="existingWikiWord" href="/nlab/show/causally+local+net+of+observables">causally local net of observables</a> which fully captures the <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> in the sense of the <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> (<a href="S-matrix#PerturbativeQuantumObservablesIsLocalnet">this prop.</a>). This perspective is now known as <em><a class="existingWikiWord" href="/nlab/show/locally+covariant+algebraic+quantum+field+theory">locally covariant algebraic quantum field theory</a></em>.</p> <h3 id="adiabatic_limit">Adiabatic limit</h3> <p>The <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> of the <a class="existingWikiWord" href="/nlab/show/perturbative+S-matrix">perturbative S-matrix</a> as the <a class="existingWikiWord" href="/nlab/show/adiabatic+switching">adiabatic switching</a> is removed (if it exists) is called the <em>adiabatic limit</em> or <em>strong adiabatic limit</em>.</p> <p>If one just asks that the corresponding limit exists for the <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> one speaks of a <em>weak adiabatic limit</em>.</p> <p>Even with the adiabatically switched S-matrix elements (not taking a limit) the <a class="existingWikiWord" href="/nlab/show/local+net+of+quantum+observables">local net of quantum observables</a> is well defined (<a href="S-matrix#PerturbativeQuantumObservablesIsLocalnet">this prop.</a>), this is hence a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo>↦</mo><mi>𝒜</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O} \mapsto \mathcal{A}(\mathcal{O}) </annotation></semantics></math></div> <p>that assigns <a class="existingWikiWord" href="/nlab/show/algebras+of+observables">algebras of observables</a> to <a class="existingWikiWord" href="/nlab/show/causally+closed+subsets">causally closed subsets</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>. The <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> algebra</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>≔</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mi>𝒪</mi></munder></munder><mi>𝒜</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \coloneqq \underset{\underset{\mathcal{O}}{\longrightarrow}}{\lim} \mathcal{A}(\mathcal{O}) </annotation></semantics></math></div> <p>over this <a class="existingWikiWord" href="/nlab/show/functor">functor</a> (in the sense of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>) always exists. This is also called the <em>algebraic adiabatic limit</em>.</p> <p>(See around <a href="#Duch17">Duch 17, section 4</a> for review of strong, weak and algebraic adiabaitc limit; and <a href="#Duch17">Duch 17, chapter II</a> for results on the weak adiabatic limit)</p> <p>Here</p> <ol> <li> <p>the <em>algebraic adiabatic</em> limit defines the <em><a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a></em> in the limit;</p> </li> <li> <p>the weak adiabatic limit may serve to define also the <em><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">states</a></em>, hence the <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> (<a href="#Duch17">Duch 17, p. 113-114</a>).</p> </li> </ol> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/adiabatic">adiabatic</a></li> </ul> <h2 id="references">References</h2> <p>The concept of adiabatic switching in the time direction was introduced in</p> <ul> <li id="LippmannSchwinger50">B. A. Lippmann, <a class="existingWikiWord" href="/nlab/show/Julian+Schwinger">Julian Schwinger</a>, <em>Variational Principles for Scattering Processes. I</em>, Phys. Rev. 79, 469 (1950) (<a href="https://doi.org/10.1103/PhysRev.79.469">doi:10.1103/PhysRev.79.469</a>)</li> </ul> <p>reviewed for instance in</p> <ul> <li id="Strocchi13"><a class="existingWikiWord" href="/nlab/show/Franco+Strocchi">Franco Strocchi</a>, section 6.3 of <em>An Introduction to Non-Perturbative Foundations of Quantum Field Theory</em>, Oxford University Press, 2013</li> </ul> <p>and the corresponding formula for the interacting eigenstates in terms of the free ones is due to</p> <ul> <li id="GellMannLow51"> <p><a class="existingWikiWord" href="/nlab/show/Murray+Gell-Mann">Murray Gell-Mann</a>, F. Low, <em>Bound states in quantum field theory</em> Phys. Rev. 84, 350 (1951)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Murray+Gell-Mann">Murray Gell-Mann</a>, M. L. Goldberger, Phys. Rev. 91 398 (1953)</p> </li> </ul> <p>see</p> <ul> <li id="Molinari06">Luca Guido Molinari, <em>Another proof of Gell-Mann and Low’s theorem</em>, Journal of Mathematical Physics 48, 052113, 2007 (<a href="https://arxiv.org/abs/math-ph/0612030">arXiv:math-ph/0612030</a>)</li> </ul> <p>See also:</p> <ul> <li> <p>C. Lanczos, R. C. Clark, G. H. Derrick (eds.), Ch. 2, Sec 2 in: <em>Mathematical Methods in Solid State and Superfluid Theory</em>, Springer (1986) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/978-1-4899-6435-9">doi:10.1007/978-1-4899-6435-9</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Gell-Mann_and_Low_theorem">Gell-Mann and Low theorem</a></em></p> </li> </ul> <p>The generalization to switching in all space-time directions was considered for the construction of <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> in</p> <ul> <li id="EpsteinGlaser73"><a class="existingWikiWord" href="/nlab/show/Henri+Epstein">Henri Epstein</a> and <a class="existingWikiWord" href="/nlab/show/Vladimir+Glaser">Vladimir Glaser</a>, <em><a class="existingWikiWord" href="/nlab/show/The+Role+of+locality+in+perturbation+theory">The Role of locality in perturbation theory</a></em>, Annales Poincaré Phys. Theor. A 19 (1973) 211.</li> </ul> <p>The observation that this in fact makes causal perturbation theory a tool for constructing <a class="existingWikiWord" href="/nlab/show/local+nets+of+observables">local nets of observables</a> for <a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+quantum+field+theory">locally covariant perturbative quantum field theory</a> is due to</p> <ul> <li id="IlinSlavnov78"> <p>V. A. Il’in and D. S. Slavnov, <em>Observable algebras in the S-matrix approach</em>, Theor. Math. Phys. 36 (1978) 32.</p> </li> <li id="BrunettiFredenhagen00"> <p><a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <em>Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds</em>, Commun. Math. Phys. 208 : 623-661,2000 (<a href="https://arxiv.org/abs/math-ph/9903028">math-ph/9903028</a>)</p> </li> </ul> <p>The term “algebraic adiabatic limit” for the resulting <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> (or its <a class="existingWikiWord" href="/nlab/show/inductive+limit">inductive limit</a>) appears in</p> <ul> <li id="FredenhagenLindner13"><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Falk+Lindner">Falk Lindner</a>, p. 7 of <em>Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics</em>, Communications in Mathematical Physics Volume 332, Issue 3, pp 895-932, 2014 (<a href="https://arxiv.org/abs/1306.6519">arXiv:1306.6519</a>)</li> </ul> <p>The weak adiabatic limit in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> for massive fields was shown to exists in</p> <ul> <li><a href="#EpsteinGlaser73">Epstein-Glaser 73</a></li> </ul> <p>Extension of this result to <a class="existingWikiWord" href="/nlab/show/quantum+electrodynamics">quantum electrodynamics</a> and <a class="existingWikiWord" href="/nlab/show/phi%5E4+theory">phi^4 theory</a> was given in</p> <ul> <li id="BlanchardSeneor75">P. Blanchard and R. Seneor, <em>Green’s functions for theories with massless particles (in perturbation theory)</em>, Ann. Inst. H. Poincaré Sec. A 23 (2), 147–209 (1975) (<a href="http://www.numdam.org/item?id=AIHPA_1975__23_2_147_0">Numdam</a>)</li> </ul> <p>See also</p> <ul> <li id="Scharf95"><a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Scharf">Günter Scharf</a>, section 3.11 of <em><a class="existingWikiWord" href="/nlab/show/Finite+Quantum+Electrodynamics+--+The+Causal+Approach">Finite Quantum Electrodynamics – The Causal Approach</a></em>, Berlin: Springer-Verlag, 1995, 2nd edition</li> </ul> <p>Further extension of the result is due to</p> <ul> <li id="Duch17"><a class="existingWikiWord" href="/nlab/show/Pawe%C5%82+Duch">Paweł Duch</a>, <em>Massless fields and adiabatic limit in quantum field theory</em> (<a href="https://arxiv.org/abs/1709.09907">arXiv:1709.09907</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 12, 2022 at 10:54:57. 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