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manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/metric">metric</a>, <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/isometry+group">isometry group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space+of+Riemannian+metrics">moduli space of Riemannian metrics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo-Riemannian manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian manifold</a>, <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic">geodesic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic+convexity">geodesic convexity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geodesic+flow">geodesic flow</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Levi-Civita+connection">Levi-Civita connection</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+curvature">Riemann curvature</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+metric+connection">torsion of a metric connection</a></li> </ul> </li> </ul> <h2 id="further_concepts">Further concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+inner+product">Hodge inner product</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gradient">gradient</a>, <a class="existingWikiWord" href="/nlab/show/gradient+flow">gradient flow</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a>-theorem</li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a>, <a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/Riemannian+geometry+-+contents">Edit this sidebar</a> </p> </div></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='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#formal_selfadjointness'>Formal self-adjointness</a></li> <li><a href='#bicharacteristic_flow_and_propagation_of_singularities'>Bicharacteristic flow and propagation of singularities</a></li> <li><a href='#FundamentalSolutions'>Fundamental solutions</a></li> <li><a href='#relation_to_schrdinger_equation'>Relation to Schrödinger equation</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>The <em>Klein-Gordon equation</em> is the <a class="existingWikiWord" href="/nlab/show/linear+differential+equation">linear</a> <a class="existingWikiWord" href="/nlab/show/partial+differential+equation">partial differential equation</a> which is the <a class="existingWikiWord" href="/nlab/show/equation+of+motion">equation of motion</a> of a <a class="existingWikiWord" href="/nlab/show/free+field">free</a> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of possibly non-vanishing <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 some (possibly <a class="existingWikiWord" href="/nlab/show/curved+spacetime">curved</a>) <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> (<a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentzian manifold</a>): it is the relativistic <a class="existingWikiWord" href="/nlab/show/wave+equation">wave equation</a> with inhomogeneity the mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>m</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">m^2</annotation></semantics></math>.</p> <p>The structure of the Klein-Gordon equation appears also in the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> of richer <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> than just <a class="existingWikiWord" href="/nlab/show/scalar+fields">scalar fields</a>, where now the underlying <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> may more generally be some <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>. Therefore the <a class="existingWikiWord" href="/nlab/show/fundamental+solutions">fundamental solutions</a> of the Klein-Gordon equation, called the <em><a class="existingWikiWord" href="/nlab/show/propagators">propagators</a></em> (see <a href="#FundamentalSolutions">below</a>) pervades all of relativistic <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>.</p> <h2 id="definition">Definition</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <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>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> (a <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo-Riemannian manifold</a>) and a <a class="existingWikiWord" href="/nlab/show/real+number">real number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">m \in \mathbb{R}_{\geq 0}</annotation></semantics></math>, then the <em>Klein-Gordon equation</em> is the <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> on <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\phi \colon X \to \mathbb{R}</annotation></semantics></math> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mo>□</mo> <mi>g</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>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"> \left( \Box_g - \left( \tfrac{m c}{\hbar} \right)^2 \right) \phi \;=\; 0 \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>□</mo> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">\Box_g</annotation></semantics></math> denotes the <em><a class="existingWikiWord" href="/nlab/show/wave+operator">wave operator</a></em> on <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>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> (the analog of the <a class="existingWikiWord" href="/nlab/show/Laplace+operator">Laplace operator</a> in <a class="existingWikiWord" href="/nlab/show/Lorentzian+geometry">Lorentzian geometry</a>) and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mrow><mi>m</mi><mi>c</mi></mrow><mi>ℏ</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\tfrac{m c }{\hbar}</annotation></semantics></math> is for the purpose of pure <a class="existingWikiWord" href="/nlab/show/PDE">PDE</a> theory just a <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, while regarded as equipped with <a class="existingWikiWord" href="/nlab/show/physical+units">physical units</a> it is the inverse <em><a class="existingWikiWord" href="/nlab/show/Compton+wavelength">Compton wavelength</a></em> 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>.</p> <p>This is the <a class="existingWikiWord" href="/nlab/show/equation+of+motion">equation of motion</a> of the <a class="existingWikiWord" href="/nlab/show/free+field">free</a> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> on <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>, of <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> and subject to a <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> as encoded in the metric <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>.</p> <h2 id="examples">Examples</h2> <p>If <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>g</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">(X,g) = \mathbb{R}^{p,1}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> equipped with its canonical <a class="existingWikiWord" href="/nlab/show/coordinate+functions">coordinate functions</a> <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><mi>c</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">x^0 = c t</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>x</mi> <mi>i</mi></msup><msubsup><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>p</mi></msubsup></mrow><annotation encoding="application/x-tex">\{x^i\}_{i = 1}^p</annotation></semantics></math>, then the Klein-Gordon equation reads as follows (using <a class="existingWikiWord" href="/nlab/show/Einstein+summation+convention">Einstein summation convention</a>)</p> <div class="maruku-equation"><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><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn></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) \phi \;=\; 0 </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><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><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><msup><mi>c</mi> <mn>2</mn></msup></mrow></mfrac></mstyle><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup></mrow><mrow><mo>∂</mo><msup><mi>t</mi> <mn>2</mn></msup></mrow></mfrac><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>x</mi> <mi>i</mi></msup></mrow></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>x</mi> <mi>i</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><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn></mrow><annotation encoding="application/x-tex"> \left( -\tfrac{1}{c^2} \frac{\partial^2}{\partial t^2} + \underoverset{i = 1}{p}{\sum}\frac{\partial}{\partial x^i} \frac{\partial}{\partial x^i} - \left( \tfrac{m c}{\hbar} \right)^2 \right) \phi \;=\; 0 </annotation></semantics></math></div> <h2 id="properties">Properties</h2> <h3 id="formal_selfadjointness">Formal self-adjointness</h3> <div class="num_example" id="FormallySelfAdjointKleinGordonOperator"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> is <a class="existingWikiWord" href="/nlab/show/formal+adjoint+differential+operator">formally self-adjoint</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Sigma = \mathbb{R}^{p,1}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> with <a class="existingWikiWord" href="/nlab/show/Minkowski+metric">Minkowski metric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> and 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><mo>×</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">E \coloneqq \Sigma \times \mathbb{R}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/trivial+line+bundle">trivial line bundle</a>. The canonical <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dvol</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">dvol_\Sigma</annotation></semantics></math> induces an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <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><mo>≃</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\tilde E^\ast \simeq E</annotation></semantics></math>.</p> <p>Consider then 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><mo stretchy="false">(</mo><mo>□</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>m</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><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>Σ</mi><mo>×</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo>×</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">⟨</mo><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Box - m^2) \;\colon\; \Gamma_\Sigma(\Sigma \times \mathbb{R}) \longrightarrow \Gamma_\Sigma(\Sigma \times \mathbb{R}) \otimes \langle dvol_\Sigma\rangle \,. </annotation></semantics></math></div> <p>This is its own <a class="existingWikiWord" href="/nlab/show/formal+adjoint+differential+operator">formal adjoint</a> witnessed by the bilinear differential operator given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>Φ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>Φ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><msub><mi>Φ</mi> <mn>1</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><msub><mi>Φ</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>Φ</mi> <mn>1</mn></msub><mfrac><mrow><mo>∂</mo><msub><mi>Φ</mi> <mn>2</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>)</mo></mrow><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>ν</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K(\Phi_1, \Phi_2) \;\coloneqq\; \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <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><mi>K</mi><mo stretchy="false">(</mo><msub><mi>Φ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>Φ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>d</mi><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><msub><mi>Φ</mi> <mn>1</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><msub><mi>Φ</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>Φ</mi> <mn>1</mn></msub><mfrac><mrow><mo>∂</mo><msub><mi>Φ</mi> <mn>2</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>)</mo></mrow><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>ν</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mrow><mo>(</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>Φ</mi> <mn>1</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>∂</mo><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><msub><mi>Φ</mi> <mn>2</mn></msub><mo>+</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mfrac><mrow><mo>∂</mo><msub><mi>Φ</mi> <mn>1</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>Φ</mi> <mn>2</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mfrac><mrow><mo>∂</mo><msub><mi>Φ</mi> <mn>1</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>Φ</mi> <mn>2</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>+</mo><msub><mi>Φ</mi> <mn>1</mn></msub><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>Φ</mi> <mn>2</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>ν</mi></msup><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>)</mo></mrow><mo>)</mo></mrow><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>Φ</mi> <mn>1</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>∂</mo><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><msub><mi>Φ</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>Φ</mi> <mn>1</mn></msub><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>Φ</mi> <mn>2</mn></msub></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>ν</mi></msup><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>)</mo></mrow><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo>□</mo><mo stretchy="false">(</mo><msub><mi>Φ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>Φ</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>Φ</mi> <mn>1</mn></msub><mo>□</mo><mo stretchy="false">(</mo><msub><mi>Φ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} d K(\Phi_1, \Phi_2) & = d \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \\ &= \left( \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 + \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu} \right) - \left( \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu} + \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) \right) dvol_\Sigma \\ & = \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 - \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) dvol_\Sigma \\ & = \Box(\Phi_1) \Phi_2 - \Phi_1 \Box (\Phi_2) \end{aligned} </annotation></semantics></math></div></div> <h3 id="bicharacteristic_flow_and_propagation_of_singularities">Bicharacteristic flow and propagation of singularities</h3> <p>The <a class="existingWikiWord" href="/nlab/show/bicharacteristic+strips">bicharacteristic strips</a> of the Klein-Gordon operator are <a class="existingWikiWord" href="/nlab/show/cotangent+vectors">cotangent vectors</a> along <a class="existingWikiWord" href="/nlab/show/lightlike">lightlike</a> <a class="existingWikiWord" href="/nlab/show/geodesics">geodesics</a> (<a href="bicharacteristic+flow#BicharachteristicFlowOfKleinGordonOperator">this example</a>).</p> <h3 id="FundamentalSolutions">Fundamental solutions</h3> <p>On a <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetime">globally hyperbolic spacetime</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> the Klein-Gordon equation has unique advanced and retarded <a class="existingWikiWord" href="/nlab/show/Green+functions">Green functions</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>R</mi></msub><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>M</mi><mo>×</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta_R \in \mathcal{D}'(M\times M)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>A</mi></msub><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>M</mi><mo>×</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta_A \in \mathcal{D}'(M\times M)</annotation></semantics></math> respectively.</p> <p>The <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+Green+functions">advanced and retarded Green functions</a> are uniquely distinguished by their <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">support</a> properties. Namely, <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 lspace="0em" rspace="thinmathspace">supp</mo><msub><mi>Δ</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">(x,y) \in \operatorname{supp} \Delta_R</annotation></semantics></math> only 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>, while <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 lspace="0em" rspace="thinmathspace">supp</mo><msub><mi>Δ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">(x,y) \in \operatorname{supp} \Delta_A</annotation></semantics></math> only 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> <p>Their difference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" 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 = \Delta_R - \Delta_A</annotation></semantics></math> is a bisolution known as the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a>, which is the <a class="existingWikiWord" href="/nlab/show/Peierls+bracket">Peierls bracket</a> which gives the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> on the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> of the <a class="existingWikiWord" href="/nlab/show/free+field">free</a> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>. This in turn defines the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> of the free scalar field, which yields the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of the free scalar field to a <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>.</p> <p>Other important Green functions or bisolutions include any (anti-)<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> <mover><mi>F</mi><mo stretchy="false">¯</mo></mover></msub></mrow><annotation encoding="application/x-tex">\Delta_{\bar{F}}</annotation></semantics></math>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_F</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/Hadamard+propagator">Hadamard propagator</a>. Unfortunately, it is not possible to identify them by a simple support condition. On Minkowski space, they are identified by the support of their Fourier transform. On curved spacetimes, there are two possibilities. One specifies the asymptotic expansion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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">\Delta(x,y)</annotation></semantics></math> in a geodesically convex neighborhood of the diagonal <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> to be of a special <em>Hadamard form</em>. The other specifies constraints on the <a class="existingWikiWord" href="/nlab/show/wavefront+set">wavefront set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">WF(\Delta)</annotation></semantics></math>. The possibilities were proven to be equivalent in <a href="#Radzikowski96">Radzikowski 96</a>, which made essential use of the relevant notions of microlocal analysis and of <em>distinguished parametrices</em> introduced in <a href="#DuistermaatHoermander72">DuistermaatHörmander 72</a>.</p> <p>According to <a href="#Radzikowski96">Radzikowski 96</a>, the constraints on the wavefront sets of important Green functions and bisolutions can be diagrammatically illustrated as follows below. Note that the primed <a class="existingWikiWord" href="/nlab/show/wavefront+set">wavefront set</a> of a distribution on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>×</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">M\times M</annotation></semantics></math> is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo>′</mo><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>;</mo><mi>p</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>q</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>T</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>M</mi><mo>×</mo><mi>M</mi><mo stretchy="false">)</mo><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">WF'(\Delta) = \{ (x,y;p,-q) \in T^*(M\times M) \mid (x,y;p,q) \in WF(\Delta) \}</annotation></semantics></math>. The diagrams illustrate tuples <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>;</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>T</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>M</mi><mo>×</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y; p,q) \in T^*(M\times M)</annotation></semantics></math>, where the vertex of a cone corresponds to <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> and the cone illustrates all points <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> linked to <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> by null geodesics; the arrows illustrate the allowed directions 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>, with <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> linked by parallel transport and both tangent to null geodesic linking <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>.</p> <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>These propagators govern the construction of the <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a> of 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 <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetime">globally hyperbolic spacetime</a>, as well as the further <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> to <a class="existingWikiWord" href="/nlab/show/interaction">interacting</a> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory+on+curved+spacetimes">on curved spacetimes</a> via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>. See at <em><a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+quantum+field+theory">locally covariant perturbative quantum field theory</a></em> for more on this.</p> <h3 id="relation_to_schrdinger_equation">Relation to Schrödinger equation</h3> <p>Sometimes the Klein-Gordon equation is thought of as a <a class="existingWikiWord" href="/nlab/show/general+relativity">relativistic</a> refinement of the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+equation">Schrödinger equation</a> (as one passes from the non-relativistic to the <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>). But this requires some care. A priori the Klein-Gordon equation takes as arguments a <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, whereas the Schrödinger equation takes as argument a <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> on <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+equation">wave equation</a>, <a class="existingWikiWord" href="/nlab/show/wave">wave</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/plane+wave">plane wave</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+vector">wave vector</a>, <a class="existingWikiWord" href="/nlab/show/wavelength">wavelength</a>, <a class="existingWikiWord" href="/nlab/show/frequency">frequency</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fourier+analysis">Fourier analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heat+equation">heat equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+equation">wave equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+equation">Schrödinger equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+equation">Dirac equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sine-Gordon+equation">sine-Gordon equation</a></p> </li> </ul> <h2 id="references">References</h2> <p>The Klein-Gordon equation is named after <a class="existingWikiWord" href="/nlab/show/Oskar+Klein">Oskar Klein</a> and <a class="existingWikiWord" href="/nlab/show/Walter+Gordon">Walter Gordon</a>.</p> <p>Textbook account:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/James+D.+Bjorken">James D. Bjorken</a>, <a class="existingWikiWord" href="/nlab/show/Sidney+D.+Drell">Sidney D. Drell</a>: <em>Relativistic Quantum Mechanics</em>, McGrawHill (1964) [<a href="https://archive.org/details/relativisticquan0000bjor/page/n1/mode/2up">ark:/13960/t5fc2v05h</a>, <a href="https://emineter.wordpress.com/wp-content/uploads/2018/10/james-d-bjorken-sidney-d-drell-relativistic-quantum-mechanics-1964.pdf">pdf</a>, <a href="http://www.mmmut.ac.in/News_content/14331tpnews_11122020.pdf">pdf</a>]</li> </ul> <p>An overview over the KG <a class="existingWikiWord" href="/nlab/show/propagators">propagators</a> on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> 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>The <a class="existingWikiWord" href="/nlab/show/Hadamard+propagator">Hadamard propagator</a> for the Klein-Gordon equation on general <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetimes">globally hyperbolic spacetimes</a> was found in</p> <ul> <li id="Radzikowski96"><a class="existingWikiWord" href="/nlab/show/Marek+Radzikowski">Marek Radzikowski</a>, <em>Micro-local approach to the Hadamard condition in quantum field theory on curved space-time</em>, Commun. Math. Phys. 179 (1996), 529–553 (<a href="http://projecteuclid.org/euclid.cmp/1104287114">Euclid</a>)</li> </ul> <p>The original reference on the relevant notions of microlocal analysis and distinguished parametrices of the Klein-Gordon equation is</p> <ul> <li id="DuistermaatHoermander72"><a class="existingWikiWord" href="/nlab/show/Johann+Duistermaat">Johann Duistermaat</a>, <a class="existingWikiWord" href="/nlab/show/Lars+H%C3%B6rmander">Lars Hörmander</a>, <em>Fourier integral operators. II</em>, Acta Mathematica 128 (1972), 183-269 (<a href="http://dx.doi.org/10.1007/bf02392165">doi</a>)</li> </ul> <p>Textbook accounts include</p> <ul> <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>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation">Klein-Gordon equation</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 16, 2024 at 06:44:18. 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