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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{advanced and retarded causal propagators} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory} [[!include AQFT and operator algebra contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{existence_and_uniqueness}{Existence and uniqueness}\dotfill \pageref*{existence_and_uniqueness} \linebreak \noindent\hyperlink{Continuity}{Continuity}\dotfill \pageref*{Continuity} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{ForKleinGordonOperatorOnMinkowskiSpacetime}{For Klein-Gordon operator on Minkowski spacetime}\dotfill \pageref*{ForKleinGordonOperatorOnMinkowskiSpacetime} \linebreak \noindent\hyperlink{ExampleForDiracOperatorOnMinkowskiSpacetime}{For Dirac operator on Minkowski spacetime}\dotfill \pageref*{ExampleForDiracOperatorOnMinkowskiSpacetime} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} What are called \emph{advanced} and \emph{retarded causal Green functions} $\Delta_{A/R}$ are [[Green functions]] for [[hyperbolic differential operators]] on manifolds with [[causal structure]] (e.g. [[spacetimes]]) whose [[support]] is in the [[future cone]] or [[past cone]], respectively, of the source excitation. The corresponding [[integral kernels]] hence say how a [[delta distribution|point excitation]] \emph{propagates} into the future or past, respectively, via the given [[differential equation]], and therefore these are also called the advanced/future \emph{propagators}. If both advanced and retarded Green functions exist for a differential operator as well as for its [[formally adjoint differential operator|formal adjoint]], then the differental operator is called a \emph{[[Green hyperbolic differential operator]]}. The archetypical examples are, on [[globally hyperbolic spacetimes]]: \begin{enumerate}% \item [[normally hyperbolic differential operators]] such as the [[wave operator]] and the [[Klein-Gordon operator]]; \item [[Dirac operators]] on [[spinor bundles]] whose square is a normally hyperbolic differential operator as above. \end{enumerate} The advanced/retarded [[integral kernel]] \begin{displaymath} \Delta_{A/R} \in \mathcal{D}'(X \times X) \end{displaymath} is such that \begin{enumerate}% \item $(x,y) \in supp(\Delta_R)$ precisely if $x$ is in the [[causal future]] of $y$; \item $(x,y) \in supp(\Delta_A)$ precisely if $x$ is in the [[causal past]] of $y$. \end{enumerate} Written as [[generalized functions]] these satisfy \begin{displaymath} \Delta_A(x,y) = \Delta_R(y,x) \,. \end{displaymath} This implies in particular that \begin{enumerate}% \item the \emph{[[causal propagator]]}, which is the difference of the two \end{enumerate} \begin{displaymath} \Delta_S \coloneqq \Delta_R - \Delta_A \end{displaymath} is skew-symmetric in its arguments (reflecting the fact that this is the [[integral kernel]] for the [[Peierls-Poisson bracket]] for the [[free field|free]] [[scalar field]] on the given spacetime); \begin{enumerate}% \item the \emph{[[Dirac propagator]]}, which is the sum of the two \end{enumerate} \begin{displaymath} \Delta_D \coloneqq \Delta_R + \Delta_A \end{displaymath} is symmetric in its arguments, reflecting the fact that this is the integral kernel for [[time-ordered products]] away from the [[diagonal]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{CompactlySourceCausalSupport}\hypertarget{CompactlySourceCausalSupport}{} \textbf{(compactly sourced causal support)} Given a [[vector bundle]] $E \overset{}{\to} \Sigma$ over a manifold $\Sigma$ with [[conal causal structure|causal structure]] Write $\Gamma_{\Sigma}(-)$ for [[space of sections|spaces of smooth sections]], and write \begin{displaymath} \itexarray{ \Gamma_{cp}(-) & \text{compact support} \\ \Gamma_{\Sigma,\pm cp}(-) & \text{compactly sourced future/past support} \\ \Gamma_{\Sigma,scp}(-) & \text{spacelike compact support} \\ \Gamma_{\Sigma,(f/p)cp}(-) & \text{future/past compact support} \\ \Gamma_{\Sigma,tcp}(-) & \text{timelike compact support} } \end{displaymath} for the [[linear subspaces]] on those smooth sections whose [[support]] is \begin{enumerate}% \item ($cp$) inside a [[compact subset]] \item ($\pm cp$) inside the [[closed future cone]]/[[closed past cone]], respectively, of a [[compact subset]], \item ($scp$) inside the [[closed causal cone]] of a [[compact subset]], which equivalently means that the [[intersection]] with every ([[spacelike]]) [[Cauchy surface]] is compact (\hyperlink{Sanders12}{Sanders 13, theorem 2.2}), \item ($fcp$) inside the past of a Cauchy surface (\hyperlink{Sanders12}{Sanders 13, def. 3.2}), \item ($pcp$) inside the future of a Cauchy surface (\hyperlink{Sanders12}{Sanders 13, def. 3.2}), \item ($tcp$) inside the future of one Cauchy surface and the past of another (\hyperlink{Sanders12}{Sanders 13, def. 3.2}) \end{enumerate} \end{defn} (\hyperlink{Baer14}{B\"a{}r 14, section 1}, \hyperlink{Khavkine14}{Khavkine 14, def. 2.1}) \begin{defn} \label{AdvancedAndRetardedGreenFunctions}\hypertarget{AdvancedAndRetardedGreenFunctions}{} \textbf{([[advanced and retarded Green functions]], [[causal Green function]] and [[propagators]])} Let $\Sigma$ be a [[smooth manifold]] with [[causal structure]], let $E \to \Sigma$ be a [[smooth vector bundle]] and let $P \;\colon\;\Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)$ be a [[differential operator]] on its [[space of smooth sections]]. Then a [[linear map]] \begin{displaymath} \mathrm{G}_{P,\pm} \;\colon\; \Gamma_{\Sigma, cp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma, \pm cp}(E) \end{displaymath} from spaces of sections of [[compact support]] to spaces of sections of causally sourced future/past support (def. \ref{CompactlySourceCausalSupport}) is called an \emph{[[advanced or retarded Green function]]} for $P$, respectively, if \begin{enumerate}% \item for all $\Phi \in \Gamma_{\Sigma,cp}(E_1)$ we have \end{enumerate} \begin{equation} G_{P,\pm} \circ P(\Phi) = \Phi \label{AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator}\end{equation} and \begin{equation} P \circ G_{P,\pm}(\Phi) = \Phi \label{AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator}\end{equation} \begin{enumerate}% \item the [[support]] of $G_{P.\pm}(\Phi)$ is in the [[closed future cone]] or [[closed past cone]] of the support of $\Phi$, respectively. \end{enumerate} If the advanced/retarded Green functions $G_{P\pm}$ exists, then the difference \begin{displaymath} \mathrm{G}_P \coloneqq \mathrm{G}_{P,+} - \mathrm{G}_{P,-} \end{displaymath} is called the \emph{[[causal Green function]]}. If there are [[integral kernel]], hence [[distributions in two variables]] \begin{displaymath} \Delta_{P,\pm} \in \Gamma'\Sigma( \tilde E^\ast \boxtimes_\Sigma E ) \end{displaymath} such that these Green functions are given by the corresponding [[integral transform]], in that (in [[generalized function]]-notation) \begin{displaymath} (G_{P,\pm} \Phi)(x) \;=\; \underset{y \in \Sigma}{\int} \Delta_{P, \pm}(x,y) \cdot \Phi(y) \end{displaymath} then these integral kernels are called the advanced/retarded \emph{[[propagators]]}; similarly then their difference \begin{equation} \Delta_{P,S} \coloneqq \Delta_{P,+} - \Delta_{P,-} \label{CausalPropagator}\end{equation} is the corresponding \emph{[[causal propagator]]}. \end{defn} (e.g. \hyperlink{Baer14}}{B\"a{}r 14, def. 3.2, cor. 3.10}) \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{existence_and_uniqueness}{}\subsubsection*{{Existence and uniqueness}}\label{existence_and_uniqueness} \begin{prop} \label{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}\hypertarget{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}{} \textbf{([[advanced and retarded Green functions]] of [[Green hyperbolic differential operator]] are unique)} The [[advanced and retarded Green functions]] (def. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}) of a [[Green hyperbolic differential operator]] are unique. \end{prop} (\hyperlink{Baer14}}{B\"a{}r 14, cor. 3.12} \hypertarget{Continuity}{}\subsubsection*{{Continuity}}\label{Continuity} \begin{defn} \label{TVSStructureOnSpacesOfSmoothSections}\hypertarget{TVSStructureOnSpacesOfSmoothSections}{} \textbf{([[Fréchet space|Fréchet]] [[topological vector space]] on [[spaces of smooth sections]] of a [[smooth vector bundle]])} Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]]. On its [[real vector space]] $\Gamma_\Sigma(E)$ [[space of sections|of smooth sections]] consider the [[seminorms]] indexed by a [[compact subset]] $K \subset \Sigma$ and a [[natural number]] $N \in \mathbb{N}$ and given by \begin{displaymath} \itexarray{ \Gamma_\Sigma(E) &\overset{p_K^N}{\longrightarrow}& [0,\infty) \\ \Phi &\mapsto& \underset{n \leq N}{max} \left( \underset{x \in K}{sup} {\vert \nabla^n \Phi(x)\vert}\right) \,, } \end{displaymath} where on the right we have the [[absolute values]] of the [[covariant derivatives]] of $\Phi$ for any fixed choice of [[connection on a bundle|connection]] on $E$ and [[norm]] on the [[tensor product of vector bundles]] $(T^\ast \Sigma)^{\otimes_\Sigma^n} \otimes_\Sigma E$. This makes $\Gamma_\Sigma(E)$ a [[Fréchet space|Fréchet]] [[topological vector space]]. For $K \subset \Sigma$ any [[closed subset]] then the sub-space of sections \begin{displaymath} \Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E) \end{displaymath} of sections whose [[support]] is inside $K$ becomes a [[Fréchet space|Fréchet]] [[topological vector spaces]] with the induced [[subspace topology]], which makes these be [[closed subspaces]]. \end{defn} (\hyperlink{Baer14}{B\"a{}r 14, 2.1, 2.2}) \begin{defn} \label{DistributionalSections}\hypertarget{DistributionalSections}{} \textbf{([[distribution|distributional]] [[sections]])} Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] over a [[smooth manifold]] with [[causal structure]]. The [[vector space|vector]] [[spaces of smooth sections]] with restricted support from def. \ref{CompactlySourceCausalSupport} structures of [[topological vector spaces]] via def. \ref{TVSStructureOnSpacesOfSmoothSections}. We denote the topological [[dual spaces]] by \begin{displaymath} \Gamma'_{\Sigma}(\tilde{E}^*) \coloneqq (\Gamma_{\Sigma,cp}(E))^* \end{displaymath} etc. This is the space of \emph{distributional sections} of the bundle $\tilde{E}^*$. With this notations, smooth compactly supported sections of the same bundle, regarded as the [[non-singular distributions]], constitute a [[dense subset]] \begin{displaymath} \Gamma_{\Sigma,cp}(\tilde{E}^*) \underset{\text{dense}}{\hookrightarrow} \Gamma'_{\Sigma}(\tilde{E}^*) \,. \end{displaymath} Imposing the same restrictions to the [[supports of distributions]] as in def. \ref{CompactlySourceCausalSupport}, we have the following subspaces of distributional sections: \begin{displaymath} \Gamma'_{\Sigma,cp}(\tilde E^\ast) , \Gamma'_{\Sigma,\pm cp}(\tilde E^\ast) , \Gamma'_{\Sigma,scp}(\tilde E^\ast) , \Gamma'_{\Sigma,fcp}(\tilde E^\ast) , \Gamma'_{\Sigma,pcp}(\tilde E^\ast) , \Gamma'_{\Sigma,tcp}(\tilde E^\ast) \subset \Gamma'_{\Sigma}(\tilde E^\ast) . \end{displaymath} \end{defn} (\hyperlink{Sanders12}{Sanders 13}, \hyperlink{Baer14}{B\"a{}r 14}) \begin{prop} \label{GreenFunctionsAreContinuous}\hypertarget{GreenFunctionsAreContinuous}{} \textbf{([[causal Green functions]] of [[Green hyperbolic differential operators]] are [[continuous linear maps]])} Given a [[Green hyperbolic differential operator]] $P$ (def. \ref{GreenHyperbolicDifferentialOperator}), the advanced, retarded and causal Green functions of $P$ (def. \ref{AdvancedAndRetardedGreenFunctions}) are [[continuous linear maps]] with respect to the [[topological vector space]] structure from def. \ref{TVSStructureOnSpacesOfSmoothSections} and also have a unique continuous extension to the spaces of sections with .larger support (def. \ref{CompactlySourceCausalSupport}) as follows: \begin{displaymath} \begin{aligned} \mathrm{G}_{P,+} &\;\colon\; \Gamma_{\Sigma, pcp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma, pcp}(E) , \\ \mathrm{G}_{P,-} &\;\colon\; \Gamma_{\Sigma, fcp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma, fcp}(E) , \\ \mathrm{G}_{P} &\;\colon\; \Gamma_{\Sigma, tcp}(\tilde E^\ast) \longrightarrow \Gamma_{\Sigma}(E) , \end{aligned} \end{displaymath} such that we still have the relation \begin{displaymath} \mathrm{G}_P = \mathrm{G}_{P,+} - \mathrm{G}_{P,-} \end{displaymath} and \begin{displaymath} P \circ \mathrm{G}_{P,\pm} = \mathrm{G}_{P,\pm} \circ P = id \end{displaymath} and \begin{displaymath} supp \mathrm{G}_{P,\pm}(\tilde{\alpha}^*) \subseteq J^\pm(supp \tilde{\alpha}^*) \,. \end{displaymath} \end{prop} (\hyperlink{Baer14}{B\"a{}r 14, thm. 3.8, cor. 3.11}) \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{ForKleinGordonOperatorOnMinkowskiSpacetime}{}\subsubsection*{{For Klein-Gordon operator on Minkowski spacetime}}\label{ForKleinGordonOperatorOnMinkowskiSpacetime} On [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ consider the [[Klein-Gordon operator]] \begin{displaymath} \eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu} \Phi - \left( \tfrac{m c}{\hbar} \right)^2 \Phi \;=\; 0 \,. \end{displaymath} Its [[Fourier transform]] is \begin{displaymath} - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \;=\; (k_0)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 \,. \end{displaymath} The [[dispersion relation]] of this equation we write \begin{equation} \omega(\vec k) \;\coloneqq\; + c \sqrt{ {\vert \vec k \vert}^2 + \left( \tfrac{m c}{\hbar}\right)^2 } \,, \label{DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime}\end{equation} where on the right we choose the [[non-negative real number|non-negative]] [[square root]]. $\,$ We now discuss \begin{enumerate}% \item \emph{\hyperlink{AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime}{Advanced and regarded propagators}} \item \emph{\hyperlink{CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}{Causal propagator}} \item \emph{\hyperlink{HadamardPropagatorForKleinGordonOnMinkowskiSpacetime}{Wightman propagator}} \item \emph{\hyperlink{FeynmanPropagator}{Feynman propagator}} \item \emph{\hyperlink{WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime}{Singular support and Wave front sets}} \end{enumerate} $\,$ \textbf{[[advanced and retarded propagators]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]]} \begin{prop} \label{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}\hypertarget{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}{} \textbf{(mode expansion of [[advanced and retarded propagators]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]])} The [[advanced and retarded Green functions]] $G_\pm$ of the [[Klein-Gordon operator]] on [[Minkowski spacetime]] are given by [[integral kernels]] (``[[propagators]]'') \begin{displaymath} \Delta_\pm \in \mathcal{D}'(\mathbb{R}^{p,1}\times \mathbb{R}^{p,1}) \end{displaymath} by (in [[generalized function]]-notation) \begin{displaymath} G_\pm(\Phi) \;=\; \underset{\mathbb{R}^{p,1}}{\int} \Delta_{\pm}(x,y) \Phi(y) \, dvol(y) \end{displaymath} where the [[advanced and retarded propagators]] $\Delta_{\pm}(x,y)$ have the following equivalent expressions: \begin{equation} \begin{aligned} \Delta_\pm(x-y) & = \frac{1}{(2\pi)^{p+1}} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \int \int \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ (k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2 -\left( \tfrac{m c}{\hbar}\right)^2 } \, d k_0 \, d^p \vec k \\ & = \left\{ \itexarray{ \frac{\pm i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{+i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) } \right) d^p \vec k & \vert & \text{if} \, \pm (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \left\{ \itexarray{ \frac{\mp 1}{(2\pi)^{p}} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } d^p \vec k & \vert & \text{if} \, \pm (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \end{aligned} \label{ModeExpansionForMinkowskiAdvancedRetardedPropagator}\end{equation} Here $\omega(\vec k)$ denotes the [[dispersion relation]] \eqref{DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime} of the [[Klein-Gordon equation]]. \end{prop} \begin{proof} The [[Klein-Gordon operator]] is a [[Green hyperbolic differential operator]] (\href{Green+hyperbolic+partial+differential+equation#GreenHyperbolicKleinGordonOperator}{this example}) therefore its advanced and retarded Green functions exist uniquely (prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}). Moreover, prop. \ref{GreenFunctionsAreContinuous} says that they are [[continuous linear functionals]] with respect to the [[topological vector space]] [[structures]] on [[spaces of smooth sections]] (def. \ref{TVSStructureOnSpacesOfSmoothSections}). In the case of the [[Klein-Gordon operator]] this just means that \begin{displaymath} G_{\pm} \;\colon\; C^\infty_{cp}(\mathbb{R}^{p,1}) \longrightarrow C^\infty_{\pm cp}(\mathbb{R}^{p,1}) \end{displaymath} are [[continuous linear functionals]] in the standard sense of [[distributions]]. Therefore the [[Schwartz kernel theorem]] implies the existence of [[integral kernels]] being [[distributions in two variables]] \begin{displaymath} \Delta_{\pm} \in \mathcal{D}(\mathbb{R}^{p,1} \times \mathbb{R}^{p,1}) \end{displaymath} such that, in the notation of [[generalized functions]], \begin{displaymath} (G_\pm \alpha)(x) \;=\; \underset{\mathbb{R}^{p,1}}{\int} \Delta_{\pm}(x,y) \alpha(y) \, dvol(y) \,. \end{displaymath} These integral kernels are the advanced/retarded ``[[propagators]]''. We now compute these [[integral kernels]] by making an Ansatz and showing that it has the defining properties, which identifies them by the uniqueness statement of prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}. We make use of the fact that the [[Klein-Gordon equation]] is [[invariant]] under the defnining [[action]] of the [[Poincaré group]] on [[Minkowski spacetime]], which is a [[semidirect product group]] of the [[translation group]] and the [[Lorentz group]]. Since the [[Klein-Gordon operator]] is invariant, in particular, under [[translations]] in $\mathbb{R}^{p,1}$ it is clear that the propagators, as a [[distribution in two variables]], depend only on the difference of its two arguments \begin{equation} \Delta_{\pm}(x,y) = \Delta_{\pm}(x-y) \,. \label{TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime}\end{equation} Since moreover the [[Klein-Gordon operator]] is [[formally adjoint differential operator|formally self-adjoint]] (\href{Klein-Gordon+equation#FormallySelfAdjointKleinGordonOperator}{this prop.}) this implies that for $P$ the Klein the equation \eqref{AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator} \begin{displaymath} P \circ G_\pm = id \end{displaymath} is equivalent to the equation \eqref{AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator} \begin{displaymath} G_\pm \circ P = id \,. \end{displaymath} Therefore it is sufficient to solve for the first of these two equation, subject to the defining support conditions. In terms of the [[propagator]] [[integral kernels]] this means that we have to solve the [[distribution|distributional]] equation \begin{equation} \left( \eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu} - \left( \tfrac{m c}{\hbar} \right)^2 \right) \Delta_\pm(x-y) \;=\; \delta(x-y) \label{KleinGordonEquationOnAdvacedRetardedPropagator}\end{equation} subject to the condition that the [[support of a distribution|distributional support]] is \begin{displaymath} supp\left( \Delta_{\pm}(x-y) \right) \subset \left\{ {\vert x-y\vert^2_\eta}\lt 0 \;\,,\; \pm(x^0 - y^ 0) \gt 0 \right\} \,. \end{displaymath} We make the \emph{Ansatz} that we assume that $\Delta_{\pm}$, as a distribution in a single variable $x-y$, is a [[tempered distribution]] \begin{displaymath} \Delta_\pm \in \mathcal{S}'(\mathbb{R}^{p,1}) \,, \end{displaymath} hence amenable to [[Fourier transform of distributions]]. If we do find a solution this way, it is guaranteed to be the unique solution by prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}. By \href{Fourier+transform#BasicPropertiesOfFourierTransformOverCartesianSpaces}{this prop.} the [[Fourier transform of distributions|distributional Fourier transform]] of equation \eqref{KleinGordonEquationOnAdvacedRetardedPropagator} is \begin{equation} \begin{aligned} \left( - \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) \widehat{\Delta_{\pm}}(k) & = \widehat{\delta}(k) \\ & = 1 \end{aligned} \,, \label{FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator}\end{equation} where in the second line we used the [[Fourier transform of distributions|Fourier transform]] of the [[delta distribution]] from \href{Dirac+distribution#FourierTransformOfDeltaDistribution}{this example}. Notice that this implies that the [[Fourier transform]] of the [[causal propagator]] \begin{displaymath} \Delta_S \coloneqq \Delta_+ - \Delta_- \end{displaymath} satisfies the homogeneous equation: \begin{equation} \left( - \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) \widehat{\Delta_S}(k) \;=\; 0 \,, \label{FourierVersionOfPDEForKleinGordonCausalPropagator}\end{equation} Hence we are now reduced to finding solutions $\widehat{\Delta_\pm} \in \mathcal{S}'(\mathbb{R}^{p,1})$ to \eqref{FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator} such that their [[Fourier inversion theorem|Fourier inverse]] $\Delta_\pm$ has the required [[support of a distribution|support]] properties. We discuss this by a variant of the [[Cauchy principal value]]: Suppose the following [[limit of a sequence|limit]] of [[non-singular distributions]] in the [[variable]] $k \in \mathbb{R}^{p,1}$ exists in the space of [[distributions]] \begin{equation} \underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim} \frac{1}{ (k_0 \mp i \epsilon)^2 - {\vert \vec k\vert^2} - \left( \tfrac{m c}{\hbar} \right)^2 } \;\in\; \mathcal{D}'(\mathbb{R}^{p,1}) \label{LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator}\end{equation} meaning that for each [[bump function]] $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ the [[limit of a sequence|limit]] in $\mathbb{C}$ \begin{displaymath} \underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim} \underset{\mathbb{R}^{p,1}}{\int} \frac{b(k)}{ (k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 } d^{p+1}k \;\in\; \mathbb{C} \end{displaymath} exists. Then this limit is clearly a solution to the distributional equation \eqref{FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator} because on those bump functions $b(k)$ which happen to be products with $\left(-\eta^{\mu \nu}k_\mu k-\nu - \left( \tfrac{m c}{\hbar}\right)^2\right)$ we clearly have \begin{displaymath} \begin{aligned} \underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim} \underset{\mathbb{R}^{p,1}}{\int} \frac{ \left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) b(k) }{ (k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 } d^{p+1}k & = \underset{\mathbb{R}^{p,1}}{\int} \underset{= 1}{ \underbrace{ \underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim} \frac{ \left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) }{ (k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2 } } } b(k)\, d^{p+1}k \\ & = \langle 1, b\rangle \,. \end{aligned} \end{displaymath} Moreover, if the limiting distribution \eqref{LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator} exists, then it is clearly a [[tempered distribution]], hence we may apply [[Fourier inversion theorem|Fourier inversion]] to obtain [[Green functions]] \begin{equation} \Delta_{\pm}(x,y) \;\coloneqq\; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{1}{(2\pi)^{p+1}} \underset{\mathbb{R}^{p,1}}{\int} \frac{e^{i k_\mu (x-y)^\mu}}{ (k_0 \mp i \epsilon )^2 - {\vert \vec k\vert}^2 - \left(\tfrac{m c}{\hbar}\right)^2 } d k_0 d^p \vec k \,. \label{AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets}\end{equation} To see that this is the correct answer, we need to check the defining support property. Finally, by the [[Fourier inversion theorem]], to show that the [[limit of a sequence|limit]] \eqref{LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator} indeed exists it is sufficient to show that the limit in \eqref{AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets} exists. We compute as follows \begin{equation} \begin{aligned} \Delta_\pm(x-y) & = \frac{1}{(2\pi)^{p+1}} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \int \int \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ (k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2 -\left( \tfrac{m c}{\hbar}\right)^2 } \, d k_0 \, d^p \vec k \\ & = \frac{1}{(2\pi)^{p+1}} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \int \int \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ (k_0 \mp i \epsilon)^2 - \left(\omega(\vec k)/c\right)^2 } \, d k_0 \, d^p \vec k \\ &= \frac{1}{(2\pi)^{p+1}} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \int \int \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ \left( (k_0 \mp i\epsilon) - \omega(\vec k)/c \right) \left( (k_0 \mp i \epsilon) + \omega(\vec k)/c \right) } \, d k_0 \, d^p \vec k \\ & = \left\{ \itexarray{ \frac{\pm i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k & \vert & \text{if} \, \pm (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \left\{ \itexarray{ \frac{\mp 1}{(2\pi)^{p}} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } d^p \vec k & \vert & \text{if} \, \pm (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \end{aligned} \label{TheSupportOfTheCandidateAdvancedRetardedPropagatorIsinTheFutureOrPastRespectively}\end{equation} where $\omega(\vec k)$ denotes the [[dispersion relation]] \eqref{DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime} of the [[Klein-Gordon equation]]. The last step is simply the application of [[Euler's formula]] $\sin(\alpha) = \tfrac{1}{2 i }\left( e^{i \alpha} - e^{- i \alpha}\right)$. Here the key step is the application of [[Cauchy's integral formula]] in the fourth step. We spell this out now for $\Delta_+$, the discussion for $\Delta_-$ is the same, just with the appropriate signs reversed. \begin{enumerate}% \item If $(x^0 - y^0) \gt 0$ thn the expression $e^{ik_0 (x^0 - y^0)}$ decays with \emph{[[positive number|positive]] [[imaginary part]]} of $k_0$, so that we may expand the [[integration]] [[domain]] into the [[upper half plane]] as \end{enumerate} \begin{displaymath} \begin{aligned} \int_{-\infty}^\infty d k_0 & = \phantom{+} \int_{-\infty}^0 d k_0 + \int_{0}^{+ i \infty} d k_0 \\ & = + \int_{+i \infty}^0 d k_0 + \int_0^\infty d k_0 \,; \end{aligned} \end{displaymath} Conversely, if $(x^0 - y^0) \lt 0$ then we may analogously expand into the [[lower half plane]]. \begin{enumerate}% \item This integration domain may then further be completed to two [[contour integrations]]. For the expansion into the [[upper half plane]] these encircle counter-clockwise the [[poles]] at $\pm \omega(\vec k)+ i\epsilon \in \mathbb{C}$, while for expansion into the [[lower half plane]] no poles are being encircled. \end{enumerate} \begin{enumerate}% \item Apply [[Cauchy's integral formula]] to find in the case $(x^0 - y^0)\gt 0$ the sum of the [[residues]] at these two [[poles]] times $2\pi i$, zero in the other case. (For the retarded propagator we get $- 2 \pi i$ times the residues, because now the contours encircling non-trivial poles go clockwise). \item The result is now non-singular at $\epsilon = 0$ and therefore the [[limit of a sequence|limit]] $\epsilon \to 0$ is now computed by evaluating at $\epsilon = 0$. \end{enumerate} This computation shows a) that the limiting distribution indeed exists, and b) that the [[support of a distribution|support]] of $\Delta_+$ is in the future, and that of $\Delta_-$ is in the past. Hence it only remains to see now that the support of $\Delta_\pm$ is inside the [[causal cone]]. But this follows from the previous argument, by using that the [[Klein-Gordon equation]] is invariant under [[Lorentz transformations]]: This implies that the support is in fact in the [[future]] of \emph{every} spacelike slice through the origin in $\mathbb{R}^{p,1}$, hence in the [[closed future cone]] of the origin. \end{proof} \begin{cor} \label{CausalPropagatorIsSkewSymmetric}\hypertarget{CausalPropagatorIsSkewSymmetric}{} \textbf{([[causal propagator]] is skew-symmetric)} Under reversal of arguments the [[advanced and retarded causal propagators]] from prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} are related by \begin{displaymath} \Delta_{\pm}(y-x) = \Delta_\mp(x-y) \,. \end{displaymath} It follows that the [[causal propagator]] $\Delta \coloneqq \Delta_+ - \Delta_-$ is skew-symmetric in its arguments: \begin{displaymath} \Delta_S(x-y) = - \Delta_S(y-x) \,. \end{displaymath} \end{cor} \begin{proof} By prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} we have with \eqref{ModeExpansionForMinkowskiAdvancedRetardedPropagator} \begin{displaymath} \begin{aligned} \Delta_\pm(y-x) & = \left\{ \itexarray{ \frac{\pm i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{-i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x -\vec y)} - e^{+i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) } \right) d^p \vec k & \vert & \text{if} \, \mp (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \left\{ \itexarray{ \frac{\pm i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{+i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) } \right) d^p \vec k & \vert & \text{if} \, \mp (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \left\{ \itexarray{ \frac{\mp i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{+i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c - i \vec k \cdot (\vec x - \vec y) } \right) d^p \vec k & \vert & \text{if} \, \mp (x^0 - y^0) \gt 0 \\ 0 & \vert & \text{otherwise} } \right. \\ & = \Delta_\mp(x-y) \end{aligned} \end{displaymath} Here in the second step we applied [[change of integration variables]] $\vec k \mapsto - \vec k$ (which introduces \emph{no} sign because in addition to $d \vec k \mapsto - d \vec k$ the integration domain reverses [[orientation]]). \end{proof} $\,$ \textbf{[[causal propagator]]} \begin{prop} \label{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}\hypertarget{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}{} \textbf{(mode expansion of [[causal propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])} The [[causal propagator]] \eqref{CausalPropagator} for the [[Klein-Gordon equation]] for [[mass]] $m$ on [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ is given, in [[generalized function]] notation, by \begin{equation} \begin{aligned} \Delta_S(x,y) & = \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k \\ & = \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x -\vec y)} d^p \vec k \,, \end{aligned} \label{CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime}\end{equation} where in the second line we used [[Euler's formula]] $sin(\alpha)= \tfrac{1}{2i}\left( e^{i \alpha} - e^{-i \alpha} \right)$. In particular this shows that the [[causal propagator]] is [[real part|real]], in that it is equal to its [[complex conjugation|complex conjugate]] \begin{equation} \left(\Delta_S(x,y)\right)^\ast = \Delta_S(x,y) \,. \label{CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal}\end{equation} \end{prop} \begin{proof} By definition and using the expression from prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} for the [[advanced and retarded causal propagators]] we have \begin{displaymath} \begin{aligned} \Delta_S(x,y) & \coloneqq \Delta_+(x,y) - \Delta_-(x,y) \\ & = \left\{ \itexarray{ \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k & \vert & \text{if} \, + (x^0 - y^0) \gt 0 \\ \frac{(-1) (-1) i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k & \vert & \text{if} \, - (x^0 - y^0) \gt 0 } \right. \\ & = \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2\omega(\vec k)/c} \left( e^{i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x -\vec y)} - e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k \\ & = \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x -\vec y)} d^p \vec k \end{aligned} \end{displaymath} For the reality, notice from the last line that \begin{displaymath} \begin{aligned} \left(\Delta_S(x,y)\right)^\ast & = \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{-i \vec k \cdot (\vec x -\vec y)} d^p \vec k \\ & = \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{+i \vec k \cdot (\vec x -\vec y)} d^p \vec k \\ & = \Delta_S(x,y) \,, \end{aligned} \end{displaymath} where in the last step we used the [[change of integration variables]] $\vec k \mapsto - \vec k$ (whih introduces no sign, since on top of $d \vec k \mapsto - d \vec k$ the [[orientation]] of the integration [[domain]] changes). \end{proof} We consider a couple of equivalent expressions for the causal propagator which are useful for computations: \begin{prop} \label{CausalPropagatorForKleinGordonOnMinkowskiAsContourIntegral}\hypertarget{CausalPropagatorForKleinGordonOnMinkowskiAsContourIntegral}{} \textbf{([[causal propagator]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]] as a [[contour integral]])} The [[causal propagator]] for the [[Klein-Gordon equation]] at [[mass]] $m$ on [[Minkowski spacetime]] has the following equivalent expression, as a [[generalized function]], given as a [[contour integral]] along a curve $C(\vec k)$ going counter-clockwise around the two [[poles]] at $k_0 = \pm \omega(\vec k)/c$: \begin{displaymath} \Delta_S(x,y) \;=\; (2\pi)^{-(p+1)} \int \underset{C(\vec k)}{\oint} \frac{e^{i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2g } \,d k_0 \,d^{p} k \,. \end{displaymath} \end{prop} \begin{quote}% graphics grabbed from \hyperlink{Kocic16}{Kocic 16} \end{quote} \begin{proof} By [[Cauchy's integral formula]] we compute as follows: \begin{displaymath} \begin{aligned} (2\pi)^{-(p+1)} \int \underset{C(\vec k)}{\oint} \frac{e^{i k_\mu (x^\mu - y^\mu)}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 } \,d k_0 \,d^{p} k & = (2\pi)^{-(p+1)} \int \underset{C(\vec k)}{\oint} \frac{ e^{i k_0 x^0} e^{ i \vec k \cdot (\vec x - \vec y)} }{ k_0^2 - \omega(\vec k)^2/c^2 } \,d k_0 \,d^p \vec k \\ & = (2\pi)^{-(p+1)} \int \underset{C(\vec k)}{\oint} \frac{ e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ ( k_0 + \omega(\vec k)/c ) ( k_0 - \omega(\vec k)/c ) } \,d k_0 \,d^p \vec k \\ & = (2\pi)^{-(p+1)} 2\pi i \int \left( \frac{ e^{i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} } { 2 \omega(\vec k)/c } - \frac{ e^{ - i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} }{ 2 \omega(\vec k)/c } \right) \,d^p \vec k \\ & = i (2\pi)^{-p} \int \frac{1}{\omega(\vec k)/c} sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y)} \,d^p \vec k \,. \end{aligned} \end{displaymath} The last line is the expression for the causal propagator from prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski} \end{proof} \begin{prop} \label{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator}\hypertarget{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator}{} \textbf{([[causal propagator]] as [[Fourier transform]] of [[delta distribution]] on the [[Fourier transform|Fourier transformed]] [[Klein-Gordon operator]])} The [[causal propagator]] for the [[Klein-Gordon equation]] at [[mass]] $m$ on [[Minkowski spacetime]] has the following equivalent expression, as a [[generalized function]]: \begin{displaymath} \Delta_S(x,y) \;=\; i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k \,, \end{displaymath} where the [[integrand]] is the product of the [[sign function]] of $k_0$ with the [[delta distribution]] of the [[Fourier transform]] of the [[Klein-Gordon operator]] and a [[plane wave]] factor. \end{prop} \begin{proof} By decomposing the integral over $k_0$ into its negative and its positive half, and applying the [[change of integration variables]] $k_0 = \pm\sqrt{h}$ we get \begin{displaymath} \begin{aligned} i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k & = + i (2\pi)^{-p} \int \int_0^\infty \delta\left( -k_0^2 + \vec k^2 + \left( \tfrac{m c}{\hbar}\right)^2 \right) e^{ i k_0 (x^0 - y^0) + i \vec k \cdot (\vec x - \vec y)} d k_0 \, d^p \vec k \\ & \phantom{=} - i (2\pi)^{-p} \int \int_{-\infty}^0 \delta\left( -k_0^2 + \vec k^2 + \left(\tfrac{m c}{\hbar}\right)^2 \right) e^{ i k_0 (x^0 - y^0)+ i \vec k \cdot (\vec x - \vec y) } d k_0 \, d^{p} \vec k \\ & = +i (2\pi)^{-p} \int \int_0^\infty \frac{1}{2 \sqrt{h}} \delta\left( -h + \omega(\vec k)^2/c^2 \right) e^{ + i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x } d h \, d^{p} \vec k \\ & \phantom{=} - i (2\pi)^{-p} \int \int_0^\infty \frac{1}{2 \sqrt{h}} \delta\left( - h + \omega(\vec k)^2/c^2 \right) e^{ - i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x } d h \, d^{p} \vec k \\ & = +i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x} d^{p} \vec k \\ & \phantom{=} - i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ - i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x } d^{p} \vec k \\ & = -(2 \pi)^{-p} \int \frac{1}{\omega(\vec k)/c} sin\left( \omega(\vec k)(x-y)^0/c \right) e^{i \vec k \cdot (\vec x - \vec y)} \end{aligned} \end{displaymath} The last line is the expression for the causal propagator from prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}. \end{proof} $\,$ \textbf{[[Wightman propagator]]} Prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} exhibits the [[causal propagator]] of the [[Klein-Gordon operator]] on [[Minkowski spacetime]] as the difference of a contribution for [[positive real number|positive]] temporal [[angular frequency]] $k_0 \propto \omega(\vec k)$ (hence positive [[energy]] $\hbar \omega(\vec k)$ and a contribution of negative temporal [[angular frequency]]. The [[positive real number|positive]] [[frequency]] contribution to the [[causal propagator]] is called the \emph{[[Wightman propagator]]} (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} below), also known as the the \emph{[[vacuum state]] [[2-point function]] of the [[free field|free]] [[real scalar field]] on [[Minkowski spacetime]]}. Notice that the temporal component of the [[wave vector]] is proportional to the \emph{negative} [[angular frequency]] \begin{displaymath} k_0 = -\omega/c \end{displaymath} (see at \emph{[[plane wave]]}), therefore the appearance of the [[step function]] $\Theta(-k_0)$ in \eqref{HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime} below: \begin{defn} \label{StandardHadamardDistributionOnMinkowskiSpacetime}\hypertarget{StandardHadamardDistributionOnMinkowskiSpacetime}{} \textbf{([[Wightman propagator]] or [[vacuum state]] [[2-point function]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]])} The \emph{[[Wightman propagator]]} for the [[Klein-Gordon operator]] at [[mass]] $m$ on [[Minkowski spacetime]] is the [[tempered distribution|tempered]] [[distribution in two variables]] $\Delta_H \in \mathcal{S}'(\mathbb{R}^{p,1})$ which as a [[generalized function]] is given by the expression \begin{equation} \begin{aligned} \Delta_H(x,y) & \coloneqq \frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k \\ & = \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k \,, \end{aligned} \label{HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime}\end{equation} Here in the first line we have in the [[integrand]] the [[delta distribution]] of the [[Fourier transform]] of the [[Klein-Gordon operator]] times a [[plane wave]] and times the [[step function]] $\Theta$ of the temporal component of the [[wave vector]]. In the second line we used the [[change of integration variables]] $k_0 = \sqrt{h}$, then the definition of the [[delta distribution]] and the fact that $\omega(\vec k)$ is by definition the [[non-negative real number|non-negative]] solution to the Klein-Gordon [[dispersion relation]]. \end{defn} (e.g. \href{Hadamard+distribution#KhavineMoretti14}{Khavkine-Moretti 14, equation (38) and section 3.4}) \begin{prop} \label{ContourIntegralForStandardHadamardPropagatorOnMinkowskiSpacetime}\hypertarget{ContourIntegralForStandardHadamardPropagatorOnMinkowskiSpacetime}{} \textbf{([[contour integral]] representation of the [[Wightman propagator]] for the [[Klein-Gordon operator]] on [[Minkowski spacetime]])} The [[Wightman propagator]] from def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} is equivalently given by the [[contour integral]] \begin{equation} \Delta_H(x,y) \;=\; -i(2\pi)^{-(p+1)} \int \underset{C_+(\vec k)}{\oint} \frac{e^{-i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 } d k_0 d^{p} k \,, \label{StandardHadamardPropagatorOnMinkowskiSpacetimeInTermsOfContourIntegral}\end{equation} where the [[Jordan curve]] $C_+(\vec k) \subset \mathbb{C}$ runs counter-clockwise, enclosing the point $+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$, but not enclosing the point $- \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$. \begin{quote}% graphics grabbed from \hyperlink{Kocic16}{Kocic 16} \end{quote} \end{prop} \begin{proof} We compute as follows: \begin{displaymath} \begin{aligned} -i(2\pi)^{-(p+1)} \int \underset{C_+(\vec k)}{\oint} \frac{e^{ - i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 } d k_0 d^{p} k & = -i(2\pi)^{-(p+1)} \int \oint_{C_+(\vec k)} \frac{ e^{ -i k_0 x^0} e^{i \vec k \cdot (\vec x - \vec y)} }{ k_0^2 - \omega(\vec k)^2/c^2 } d k_0 d^p \vec k \\ & = -i(2\pi)^{-(p+1)} \int \underset{C_+(\vec k)}{\oint} \frac{ e^{ - i k_0 (x^0-y^0)} e^{i \vec k \cdot (\vec x - \vec y)} }{ ( k_0 - \omega_\epsilon(\vec k) ) ( k_0 + \omega_\epsilon(\vec k) ) } d k_0 d^p \vec k \\ & = (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)} e^{-i \omega(\vec k) (x^0-y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} d^p \vec k \,. \end{aligned} \end{displaymath} The last step is application of [[Cauchy's integral formula]], which says that the [[contour integral]] picks up the [[residue]] of the [[pole]] of the [[integrand]] at $+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$. The last line is $\Delta_H(x,y)$, by definition \ref{StandardHadamardDistributionOnMinkowskiSpacetime}. \end{proof} \begin{prop} \label{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}\hypertarget{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}{} \textbf{(skew-symmetric part of [[Wightman propagator]] is the [[causal propagator]])} The [[Wightman propagator]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) is of the form \begin{equation} \begin{aligned} \Delta_H & = \tfrac{i}{2} \Delta_S + H \\ & = \tfrac{i}{2} \left( \Delta_+ - \Delta_- \right) + H \end{aligned} \,, \label{DeompositionOfHadamardPropagatorOnMinkowkski}\end{equation} where \begin{enumerate}% \item $\Delta_S$ is the [[causal propagator]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}), which is real \eqref{CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal} and skew-symmetric (prop. \ref{CausalPropagatorIsSkewSymmetric}) \begin{displaymath} (\Delta_S(x,y))^\ast = \Delta_S(x,y) \phantom{AA} \,, \phantom{AA} \Delta_S(y,x) = - \Delta_S(x,y) \end{displaymath} \item $H$ is real and symmetric \begin{displaymath} (H(x,y))^\ast = H(x,y) \phantom{AA} \,, \phantom{AA} H(y,x) = H(x,y) \end{displaymath} \end{enumerate} \end{prop} \begin{proof} By applying [[Euler's formula]] to \eqref{HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime} we obtain \begin{equation} \begin{aligned} \Delta_H(x,y) & = \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k \\ & = \tfrac{i}{2} \underset{= \Delta_S(x,y)}{ \underbrace{ \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k }} \;+\; \underset{ \coloneqq H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k }} \end{aligned} \label{SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime}\end{equation} On the left this identifies the [[causal propagator]] by \eqref{CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime}, prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}. The second summand changes, both under complex conjugation as well as under $(x-y) \mapsto (y-x)$, via [[change of integration variables]] $\vec k \mapsto - \vec k$ (because the [[cosine]] is an even function). This does not change the integral, and hence $H$ is symmetric. \end{proof} $\,$ \textbf{[[Feynman propagator]]} We have seen that the [[positive real number|positive]] [[frequency]] component of the [[causal propagator]] $\Delta_S$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) is the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) given, according to prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}, by \eqref{DeompositionOfHadamardPropagatorOnMinkowkski} \begin{displaymath} \begin{aligned} \Delta_H & = \tfrac{i}{2} \Delta_S + H \\ & = \tfrac{i}{2} \left( \Delta_+ - \Delta_- \right) + H \end{aligned} \,. \end{displaymath} There is an evident variant of this combination, which will be of interest: \begin{defn} \label{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}\hypertarget{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}{} \textbf{([[Feynman propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])} The \emph{[[Feynman propagator]]} for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[linear combination]] \begin{displaymath} \Delta_F \coloneqq \tfrac{i}{2} \left( \Delta_+ + \Delta_- \right) + H \end{displaymath} where the first term is proportional to the sum of the [[advanced and retarded propagators]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) and the second is the symmetric part of the [[Wightman propagator]] according to prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}. Similarly the \emph{[[anti-Feynman propagator]]} is \begin{displaymath} \Delta_{\overline{F}} \coloneqq \tfrac{i}{2} \left( \Delta_+ + \Delta_- \right) - H \,. \end{displaymath} \end{defn} \begin{prop} \label{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime}\hypertarget{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime}{} \textbf{(mode expansion for [[Feynman propagator]] of [[Klein-Gordon equation]] on [[Minkowski spacetime]])} The [[Feynman propagator]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is given by the following equivalent expressions \begin{displaymath} \begin{aligned} \Delta_F(x,y) & = \left\{ \itexarray{ \frac{1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)c} e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \itexarray{ \Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0 \\ \Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0 } \right. \end{aligned} \end{displaymath} Similarly the [[anti-Feynman propagator]] is equivalently given by \begin{displaymath} \begin{aligned} \Delta_{\overline{F}}(x,y) & = \left\{ \itexarray{ \frac{-}{(2\pi)^p} \int \frac{1}{\omega(\vec k)c} e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{-}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \itexarray{ -\Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0 \\ -\Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0 } \right. \end{aligned} \end{displaymath} \end{prop} \begin{proof} By the mode expansion of $\Delta_{\pm}$ from \eqref{ModeExpansionForMinkowskiAdvancedRetardedPropagator} and the mode expansion of $H$ from \eqref{SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime} we have \begin{displaymath} \begin{aligned} \Delta_F(x,y) & = \left\{ \itexarray{ \underset{ = \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0 }{ \underbrace{ \frac{- i}{(2\pi)^{p}} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } + \underset{ = H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } &\vert& (x^0 - y^0) \gt 0 \\ \underset{ = 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0 }{ \underbrace{ \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } + \underset{ = H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \itexarray{ \frac{1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \itexarray{ \Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0 \\ \Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0 } \right. \end{aligned} \end{displaymath} where in the second line we used [[Euler's formula]]. The last line follows by comparison with \eqref{HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime} and using that the integral over $\vec k$ is invariant under $\vec k \mapsto - \vec k$. The computation for $\Delta_{\overline{F}}$ is the same, only now with a minus sign in front of the [[cosine]]: \begin{displaymath} \begin{aligned} \Delta_{\overline{F}}(x,y) & = \left\{ \itexarray{ \underset{ = \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0 }{ \underbrace{ \frac{- i}{(2\pi)^{p}} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } - \underset{ = H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } &\vert& (x^0 - y^0) \gt 0 \\ \underset{ = 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0 }{ \underbrace{ \frac{+ i}{(2\pi)^{p}} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } - \underset{ = H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k } } &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \itexarray{ \frac{-1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{-1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{-1i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \itexarray{ - \Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0 \\ - \Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0 } \right. \end{aligned} \end{displaymath} \end{proof} As before for the [[causal propagator]], there are equivalent reformulations of the [[Feynman propagator]], which are useful for computations: \begin{prop} \label{FeynmanPropagatorAsACauchyPrincipalvalue}\hypertarget{FeynmanPropagatorAsACauchyPrincipalvalue}{} \textbf{([[Feynman propagator]] as a [[Cauchy principal value]])} The [[Feynman propagator]] and [[anti-Feynman propagator]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is equivalently given by the following expressions, respectively: \begin{displaymath} \begin{aligned} \left. \itexarray{ \Delta_F(x,y) \\ \Delta_{\overline{F}}(x,y) } \right\} & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu (x^\mu - y^\mu)} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \end{aligned} \end{displaymath} where we have a [[limit of a sequence|limit]] of [[distributions]] as for the [[Cauchy principal value]] (\href{Cauchy+principal+vlue#CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta}{this prop}). \end{prop} \begin{proof} We compute as follows: \begin{displaymath} \begin{aligned} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu (x^\mu - y^\mu)} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu (x^\mu - y^\mu)} }{ (k_0)^2 - \underset{ \coloneqq \omega_{\pm\epsilon}(\vec k)^2/c^2 }{\underbrace{ \left( \omega(\vec k)^2/c^2 \pm i \epsilon \right) }} } \, d k_0 \, d^p \vec k \\ & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu (x^\mu - y^\mu)} }{ \left( k_0 - \omega_{\pm \epsilon}(\vec k)/c \right) \left( k_0 + \omega_{\pm \epsilon}(\vec k)/c \right) } \, d k_0 \, d^p \vec k \\ & = \left\{ \itexarray{ \frac{\mp 1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{\pm i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \gt 0 \\ \frac{\mp 1}{(2\pi)^p} \int \frac{1}{2\omega(\vec k)c} e^{\mp i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k &\vert& (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \itexarray{ \Delta_F(x,y) \\ \Delta_{\overline{F}}(x,y) } \right. \end{aligned} \end{displaymath} Here \begin{enumerate}% \item In the first step we introduced the [[complex number|complex]] [[square root]] $\omega_{\pm \epsilon}(\vec k)$. For this to be compatible with the choice of \emph{non-negative} square root for $\epsilon = 0$ in \eqref{DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime} we need to choose that complex square root whose [[complex phase]] is one half that of $\omega(\vec k)^2 - i \epsilon$ (instead of that plus [[π]]). This means that $\omega_{+ \epsilon}(\vec k)$ is in the \emph{[[upper half plane]]} and $\omega_-(\vec k)$ is in the [[lower half plane]]. \item In the third step we observe that \begin{enumerate}% \item for $(x^0 - y^0) \gt 0$ the [[integrand]] decays for [[positive real number|positive]] [[imaginary part]] and hence the integration over $k_0$ may be deformed to a [[Jordan curve|contour]] which encircles the [[pole]] in the [[upper half plane]]; \item for $(x^0 - y^0) \lt 0$ the integrand decays for [[negative real number|negative]] [[imaginary part]] and hence the integration over $k_0$ may be deformed to a [[Jordan curve|contour]] which encircles the [[pole]] in the [[lower half plane]] \end{enumerate} and then apply [[Cauchy's integral formula]] which picks out $2\pi i$ times the [[residue]] a these poles. Notice that when completing to a contour in the [[lower half plane]] we pick up a minus signs from the fact that now the contour runs clockwise. \item In the fourth step we used prop. \ref{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime}. \end{enumerate} \end{proof} $\,$ $\,$ \textbf{[[singular support]] and [[wave front sets]]} We now discuss the [[singular support]] and the [[wave front sets]] of the various [[propagators]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]]. \begin{prop} \label{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}\hypertarget{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}{} \textbf{([[singular support]] of the [[causal propagator]] of the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[light cone]])} The [[singular support]] of the [[causal propagator]] $\Delta_S$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded via [[translation]] [[invariant|invariance]] as a [[generalized function]] in a single variable \eqref{TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime} is the [[light cone]] of the origin: \begin{displaymath} supp_{sing}(\Delta_S) \;=\; \left\{ x \in \mathbb{R}^{p,1} \,\vert\, {\vert x\vert}^2_\eta = 0 \right\} \,. \end{displaymath} \end{prop} \begin{proof} By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} the causal propagator is equivalently the [[Fourier transform of distributions]] of the [[delta distribution]] of the [[mass shell]] times the [[sign function]] of the [[angular frequency]]; and by basic properties of the Fourier transform this is the [[convolution of distributions]] of the separate Fourier transforms: \begin{displaymath} \begin{aligned} \Delta_S(x) & \propto \widehat{ \delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) } \\ &\propto \widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)} \star \widehat{sgn( k_0 )} \end{aligned} \end{displaymath} By (\hyperlink{GelfandShilov66}{Gel'fand-Shilov 66, III 2.11 (7), p 294}), see \href{Cauchy+principal+value#FourierTransformOfDeltaDistributionappliedToMassShell}{this prop.}, the [[singular support]] of the first convolution factor is the [[light cone]]. The second factor is \begin{displaymath} \begin{aligned} \widehat{sgn(k_0)} & \propto \left(2\widehat{\Theta(k_0)} - \widehat{1}\right) \delta(\vec k) \\ & \propto \left(2\tfrac{1}{i x^0 + 0^+} - \delta(x^0)\right) \delta(\vec k) \end{aligned} \end{displaymath} (by \href{Dirac+distribution#FourierTransformOfDeltaDistribution}{this example} and \href{Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution}{this example}) and hence the [[wave front set]] of the second factor is \begin{displaymath} WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\} \end{displaymath} (by \href{wavefront+set#WaveFrontOfDeltaDistribution}{this example} and \href{Cauchy+principal+value#PrincipalValueOfInverseFunctionCharacteristicEquation}{this example}). With this the statement follows, via a [[partition of unity]], from \href{convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions}{this prop.}. For illustration we now make this general argument more explicit in the special case of [[spacetime]] [[dimension]] \begin{displaymath} p + 1 = 3 + 1 \end{displaymath} by computing an explicit form for the [[causal propagator]] in terms of the [[delta distribution]], the [[Heaviside distribution]] and [[smooth function|smooth]] [[Bessel functions]]. We follow (\href{causal+perturbation+theory#Scharf95}{Scharf 95 (2.3.18)}). Consider the formula for the [[causal propagator]] in terms of the mode expansion \eqref{CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime}. Since the [[integrand]] here depends on the [[wave vector]] $\vec k$ only via its [[norm]] ${\vert \vec k\vert}$ and the [[angle]] $\theta$ it makes with the given [[spacetime]] [[vector]] via \begin{displaymath} \vec k \cdot (\vec x - \vec y) \;=\; {\vert \vec k\vert} \, {\vert \vec x\vert} \, \cos(\theta) \end{displaymath} we may express the [[integration]] in terms of [[polar coordinates]] as follws: \begin{displaymath} \begin{aligned} \Delta_S(x - y) & = \frac{-1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y)} \, d^p \vec k \\ & = \frac{- vol_{S^{p-2}}}{(2\pi)^p} \underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int} \underset{ \theta \in [0,\pi] }{\int} \frac{ 1 }{ \omega(\vec k)/c } \sin\left( \omega(\vec k) (x^0 - y^0) /c \right) e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) } {\vert \vec k\vert} ({\vert \vec k\vert} \sin(\theta))^{p-2} \, d \theta \wedge d {\vert \vec k\vert} \end{aligned} \end{displaymath} In the special case of [[spacetime]] [[dimension]] $p + 1 = 3 + 1$ this becomes \begin{equation} \begin{aligned} \Delta_S(x - y) & = \frac{- 2\pi}{(2\pi)^{3}} \underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int} \frac{ {\vert \vec k \vert}^2 }{ \omega(\vec k)/c } \sin\left( \omega(\vec k) (x^0 - y^0) /c \right) \underset{ = \tfrac{1}{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} } \left( e^{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}} - e^{-i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}} \right) }{ \underbrace{ \underset{ \cos(\theta) \in [-1,1] }{\int} e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) } d \cos(\theta) } } \wedge d {\vert \vec k \vert} \\ & = \frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}} \underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int} \frac{ {\vert \vec k \vert} }{ \omega(\vec k)/c } \sin\left( \omega(\vec k) (x^0 - y^0) /c \right) \sin\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right) \, d {\vert \vec k\vert} \\ & = \frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}} \frac{d}{d {\vert \vec x - \vec y \vert } } \underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int} \frac{ 1 }{ \omega(\vec k)/c } \sin\left( \omega(\vec k) (x^0 - y^0) /c \right) \cos\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right) \, d {\vert \vec k\vert} \\ & = \frac{- 1}{(2\pi)^{2} {\vert \vec x - \vec y\vert}} \frac{d}{d {\vert \vec x - \vec y \vert } } \underset{ \kappa \in \mathbb{R} }{\int} \frac{ 1 }{ \omega(\kappa)/c } \sin\left( \omega(\kappa) (x^0 - y^0) /c \right) \cos\left( \kappa\, {\vert \vec x - \vec y\vert} \right) \, d \kappa \\ & = \frac{- 1}{2(2\pi)^{2} {\vert \vec x - \vec y\vert}} \frac{d}{d {\vert \vec x - \vec y} \vert } \left( \underset{\coloneqq I_+}{ \underbrace{ \underset{ \kappa \in \mathbb{R} }{\int} \frac{ 1 }{ \omega(\kappa)/c } \sin\left( \omega(\kappa) (x^0 - y^0) /c + \kappa\, {\vert \vec x - \vec y\vert} \right) d\kappa } } + \underset{ \coloneqq I_- }{ \underbrace{ \underset{ \kappa \in \mathbb{R} }{\int} \frac{ 1 }{ \omega(\kappa)/c } \sin\left( \omega(\kappa) (x^0 - y^0) /c - \kappa\, {\vert \vec x - \vec y\vert} \right) \, d \kappa } } \right) \,. \end{aligned} \label{StepsInComputingCausalPropagatorIn3plus1Dimension}\end{equation} Here in the second but last step we renamed $\kappa \coloneqq {\vert \vec k\vert}$ and doubled the integration domain for convenience, and in the last step we used the [[trigonometric identity]] $\sin(\alpha) \cos(\beta)\;=\; \tfrac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)$. In order to further evaluate this, we parameterize the remaining components $(\omega/c, \kappa)$ of the [[wave vector]] by the dual [[rapidity]] $z$, via \begin{displaymath} \left(\cosh(z)\right)^2 - \left( \sinh(z)\right)^2 = 1 \end{displaymath} as \begin{displaymath} \omega(\kappa)/c \;=\; \left( \tfrac{m c}{\hbar} \right) \cosh(z) \phantom{AA} \,, \phantom{AA} \kappa \;=\; \left( \tfrac{m c}{\hbar} \right) \sinh(z) \,, \end{displaymath} which makes use of the fact that $\omega(\kappa)$ is non-negative, by construction. This [[change of integration variables]] makes the integrals under the braces above become \begin{equation} I_\pm \;=\; \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \left( (x^0 - y^0) \cosh(z) \pm {\vert \vec x - \vec y\vert} \sinh(z) \right) \right) \, d z \,. \label{TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski}\end{equation} Next we similarly parameterize the vector $x-y$ by its [[rapidity]] $\tau$. That parameterization depends on whether $x-y$ is spacelike or not, and if not, whether it is future or past directed. First, if $x-y$ is [[spacelike]] in that ${\vert x-y\vert}^2_\eta \gt 0$ then we may parameterize as \begin{displaymath} (x^0 - y^0) = \sqrt{{\vert x-y\vert}^2_\eta} \sinh(\tau) \phantom{AA} \,, \phantom{AA} {\vert \vec x - \vec y\vert} = \sqrt{ {\vert x-y\vert}^2_\eta} \cosh(\tau) \end{displaymath} which yields \begin{displaymath} \begin{aligned} I_{\pm} & = \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \left( \sinh(\tau) \cosh(z) \pm \cosh(\tau) \sinh(z) \right) \right) \, d z \\ & = \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta} \left( \sinh\left( \tau \pm z\right) \right) \right) \, d z \\ & = \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \left( \sinh\left( z \right) \right) \right) \, d z \\ & = 0 \,, \end{aligned} \end{displaymath} where in the last line we observe that the integrand is a skew-symmetric function of $z$. Second, if $x-y$ is [[timelike]] with $(x^0 - y^0) \gt 0$ then we may parameterize as \begin{displaymath} (x^0 - y^0) = \sqrt{ -{\vert x-y\vert}^2_\eta} \cosh(\tau) \phantom{AA} \,, \phantom{AA} {\vert \vec x - \vec y\vert} = \sqrt{ -{\vert x - y\vert}^2_\eta } \sinh(\tau) \end{displaymath} which yields \begin{equation} \begin{aligned} I_\pm & = \int_{-\infty}^\infty \sin\left( \tfrac{m c}{\hbar} \left( (x^0 - y^0) \cosh(z) \pm {\vert \vec x - \vec y\vert} \sinh(z) \right) \right) \, d z \\ & = \int_{-\infty}^\infty \sin\left( \sqrt{ - {\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \left( \cosh(\tau)\cosh(z) \pm \cosh(\tau) \sinh(z) \right) \right) \, d z \\ & = \int_{-\infty}^\infty \sin\left( \sqrt{ - {\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \left( \cosh(z \pm \tau) \right) \right) \, d z \\ & = \pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right) \end{aligned} \,. \label{IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski}\end{equation} Here in the last line we identified the integral representation of the [[Bessel function]] $J_0$ of order 0 (see \href{Bessel+function#eq:J0AsIntSinOfxCoshtdt}{here}). The important point here is that this is a smooth function. Similarly, if $x-y$ is [[timelike]] with $(x^0 - y^0) \lt 0$ then the same argument yields \begin{displaymath} I_\pm = - \pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right) \end{displaymath} In conclusion, the general form of $I_\pm$ is \begin{displaymath} I_\pm = \pi sgn(x^0 - y^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right) J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \right) \,. \end{displaymath} Therefore we end up with \begin{equation} \begin{aligned} \Delta_S(x,y) & = \frac{1}{4 \pi {\vert \vec x - \vec y\vert}} \frac{d}{d {\vert \vec x - \vec y\vert}} sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right) J_0\left( \sqrt{ -{\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \right) \\ & = \frac{-1}{2 \pi } \frac{d}{d (-{\vert x-y\vert}^2_\eta)} sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right) J_0\left( \sqrt{-{\vert x-y \vert}^2_\eta} \tfrac{m c}{\hbar} \right) \\ & = -\frac{1}{2 \pi } \frac{d}{d (- \vert x-y\vert^2_{\eta})} sgn(x^0) \Theta\left( - {\vert x - y\vert}^2_\eta \right) J_0\left( \tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta } \right) \\ & = \frac{-1}{2\pi} sgn(x^0) \left( \delta\left( -{\vert x-y\vert}^2_\eta \right) \;-\; \Theta\left( -{\vert x-y\vert}^2_\eta \right) \frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) } J_0\left( \tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta } \right) \right) \end{aligned} \label{FinalResultOfComputationOf3Plus1dCausalPropagator}\end{equation} \end{proof} \begin{prop} \label{SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}\hypertarget{SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}{} \textbf{([[singular support]] of the [[Wightman propagator]] of the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[light cone]])} The [[singular support]] of the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded via [[translation]] [[invariant|invariance]] as a [[distribution]] in a single variable, is the [[light cone]] of the origin: \begin{displaymath} supp_{sing}(\Delta_H) = \left\{ x \in \mathbb{R}^{p,1} \;\vert\; {\vert x\vert}^2_\eta = 0 \right\} \,. \end{displaymath} \end{prop} \begin{proof} By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} the causal propagator is equivalently the [[Fourier transform of distributions]] of the [[delta distribution]] of the [[mass shell]] times the [[sign function]] of the [[angular frequency]]; and by basic properties of the Fourier transform this is the [[convolution of distributions]] of the separate Fourier transforms: \begin{displaymath} \begin{aligned} \Delta_S(x) & \propto \widehat{ \delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) } \\ &\propto \widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)} \star \widehat{sgn( k_0 )} \end{aligned} \end{displaymath} By (\hyperlink{GelfandShilov66}{Gel'fand-Shilov 66, III 2.11 (7), p 294}), see \href{Cauchy+principal+value#FourierTransformOfDeltaDistributionappliedToMassShell}{this prop.}, the [[singular support]] of the first convolution factor is the [[light cone]]. The second factor is \begin{displaymath} \widehat{\Theta(k_0)} \propto \tfrac{1}{i x^0 + 0^+} \delta(\vec k) \end{displaymath} (by \href{Dirac+distribution#FourierTransformOfDeltaDistribution}{this example} and \href{Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution}{this example}) and hence the [[wave front set]] of the second factor is \begin{displaymath} WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\} \end{displaymath} (by \href{wavefront+set#WaveFrontOfDeltaDistribution}{this example} and \href{Cauchy+principal+value#PrincipalValueOfInverseFunctionCharacteristicEquation}{this example}). With this the statement follows, via a [[partition of unity]], from \href{convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions}{this prop.}. For illustration, we now make this general statement fully explicit in the special case of [[spacetime]] [[dimension]] \begin{displaymath} p + 1 = 3 + 1 \end{displaymath} by computing an explicit form for the [[causal propagator]] in terms of the [[delta distribution]], the [[Heaviside distribution]] and [[smooth function|smooth]] [[Bessel functions]]. We follow (\href{causal+perturbation+theory#Scharf95}{Scharf 95 (2.3.36)}). By \eqref{SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime} we have \begin{displaymath} \begin{aligned} \Delta_H(x,y) & = \tfrac{i}{2} \underset{= \Delta_S(x,y)}{ \underbrace{ \frac{-1}{(2\pi)^p} \int \frac{1}{\omega(\vec k)/c} \sin\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k }} \;+\; \underset{ \coloneqq H(x,y) }{ \underbrace{ \frac{1}{(2\pi)^p} \int \frac{1}{2 \omega(\vec k)/c} \cos\left( \omega(\vec k)(x^0 - y^0)/c \right) e^{i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k }} \end{aligned} \end{displaymath} The first summand, proportional to the [[causal propagator]], which we computed as \eqref{FinalResultOfComputationOf3Plus1dCausalPropagator} in prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} to be \begin{displaymath} \tfrac{i}{2}\Delta_S(x,y) \;=\; \frac{-i}{4\pi} sgn(x^0) \left( \delta\left( -{\vert x-y\vert}^2_\eta \right) \;-\; \Theta\left( -{\vert x-y\vert}^2_\eta \right) \frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) } J_0\left( \tfrac{m c}{\hbar} \sqrt{ -{\vert x-y\vert}^2_\eta } \right) \right) \,. \end{displaymath} The second term is computed in a directly analogous fashion: The integrals $I_\pm$ from \eqref{TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski} are now \begin{displaymath} I_\pm \coloneqq \int_{-\infty}^\infty \cos\left( \tfrac{m c}{\hbar} \left( (x^0 - y^0) \cosh(z) \pm {\vert \vec x - \vec y\vert} \sinh(z) \right) \right) \, d z \end{displaymath} Parameterizing by [[rapidity]], as in the proof of prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}, one finds that for [[timelike]] $x-y$ this is \begin{displaymath} \begin{aligned} I_\pm & = \int_{-\infty}^\infty \cos\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \left( \cosh\left( z \right) \right) \right) \, d z \\ & = - \pi N_0 \left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \right) \end{aligned} \end{displaymath} while for [[spacelike]] $x-y$ it is \begin{displaymath} \begin{aligned} I_\pm & = \int_{-\infty}^\infty \cos\left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \left( \sinh\left( z \right) \right) \right) \, d z \\ & = 2 K_0 \left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \right) \,, \end{aligned} \end{displaymath} where we identified the integral representations of the [[Neumann function]] $N_0$ (see \href{Bessel+function#N0AsIntSinOfxCoshtdt}{here}) and of the [[modified Bessel function]] $K_0$ (see \href{Bessel+function#eq:K0AsIntSinOfxCoshtdt}{here}). As for the [[Bessel function]] $J_0$ in \eqref{IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski} the key point is that these are [[smooth functions]]. Hence we conclude that \begin{displaymath} H(x,y) \;\propto\; \frac{d}{d \left( {\vert x-y\vert}^2_\eta \right)} \left( -\Theta\left( -{\vert x-y\vert}^2_\eta \right) N_0 \left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \right) + \Theta\left( {\vert x-y\vert}^2_\eta \right) \tfrac{2}{\pi} K_0 \left( \tfrac{m c}{\hbar} \sqrt{ {\vert x-y\vert}^2_\eta } \right) \right) \,. \end{displaymath} This expression has singularities on the [[light cone]] due to the [[step functions]]. In fact the expression being differentiated is continuous at the light cone (\hyperlink{Scharf95}{Scharf 95 (2.3.34)}), so that the singularity on the light cone is not a [[delta distribution]] singularity from the derivative of the step functions. Accordingly it does not cancel the singularity of $\tfrac{i}{2}\Delta_S(x,y)$ as above, and hence the singular support of $\Delta_H$ is still the whole light cone. \end{proof} \begin{prop} \label{SingularSupportOfFeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}\hypertarget{SingularSupportOfFeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}{} \textbf{([[singular support]] of [[Feynman propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])} The [[singular support]] of the [[Feynman propagator]] $\Delta_H$ and of the [[anti-Feynman propagator]] $\Delta_{\overline{F}}$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded via [[translation]] [[invariant|invariance]] as a [[distribution]] in a single variable, is the [[light cone]] of the origin: \begin{displaymath} \left. \itexarray{ supp_{sing}(\Delta_F) \\ supp_{sing}(\Delta_{\overline{F}}) } \right\} = \left\{ x \in \mathbb{R}^{p,1} \;\vert\; {\vert x\vert}^2_\eta = 0 \right\} \,. \end{displaymath} \end{prop} (e.g \href{Feynman+propagator#DeWitt03}{DeWitt 03 (27.85)}) \begin{proof} By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} the Feynman propagator is equivalently the [[Cauchy principal value]] of the inverse of the Fourier transformed Klein-Gordon operator: \begin{displaymath} \Delta_F \;\propto\; \widehat{ \frac{1}{-k_\mu k^\mu - \left(\tfrac{m c}{\hbar}\right)^2 + i 0^+} } \,. \end{displaymath} With this the statement follows immediately from the result (\hyperlink{GelfandShilov66}{Gel'fand-Shilov 66, III 2.8 (8) and (9), p 289}), see \href{Cauchy+principal+value#FourierTransformOfPrincipalValueOfPowerOfQuadraticForm}{this prop.}. \end{proof} \begin{prop} \label{WaveFronSetsForKGPropagatorsOnMinkowski}\hypertarget{WaveFronSetsForKGPropagatorsOnMinkowski}{} \textbf{([[wave front sets]] of [[propagators]] of [[Klein-Gordon equation]] on [[Minkowski spacetime]])} The [[wave front set]] of the various [[propagators]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded, via [[translation]] [[invariant|invariance]], as [[distributions]] in a single variable, are as follows: \begin{itemize}% \item the [[causal propagator]] $\Delta_S$ (prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the lightcone: \end{itemize} \begin{displaymath} WF(\Delta_S) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \, k \neq 0 \right\} \end{displaymath} \begin{itemize}% \item the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the light cone and $k^0 \gt 0$: \end{itemize} \begin{displaymath} WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \; k^0 \gt 0 \right\} \end{displaymath} \begin{itemize}% \item the [[Feynman propagator]] $\Delta_S$ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the light cone and $\pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0$ \end{itemize} \begin{displaymath} WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \; \left( \pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0 \right) \right\} \end{displaymath} \end{prop} (\href{Hadamard+distribution#Radzikowski96}{Radzikowski 96, (16)}) \begin{proof} First regarding the causal propagator: By prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} the [[singular support]] of $\Delta_S$ is the [[light cone]]. Since the causal propagator is a solution to the homogeneous Klein-Gordon equation, the [[propagation of singularities theorem]] says that also all [[wave vectors]] in the wave front set are lightlike. Hence it just remains to show that all non-vanishing lightlike wave vectors based on the lightcone in spacetime indeed do appear in the wave front set. To that end, let $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ be a [[bump function]] whose [[compact support]] includes the origin. For $a \in \mathbb{R}^{p,1}$ a point on the light cone, we need to determine the decay property of the Fourier transform of $x \mapsto b(x-a)\Delta_S(x)$. This is the [[convolution of distributions]] of $\hat b(k)e^{i k_\mu a^\mu}$ with $\widehat \Delta_S(k)$. By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} we have \begin{displaymath} \widehat \Delta_{S}(k) \;\propto\; \delta\left( -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \right) sgn(k_0) \,. \end{displaymath} This means that the convolution product is the smearing of the mass shell by $\widehat b(k)e^{i k_\u a^\mu}$. Since the mass shell asymptotes to the light cone, and since $e^{i k_\mu a^\mu} = 1$ for $k$ on the light cone (given that $a$ is on the light cone), this implies the claim. Now for the [[Wightman propagator]]: By def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} its Fourier transform is of the form \begin{displaymath} \widehat \Delta_H(k) \;\propto\; \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) \end{displaymath} Moreover, its [[singular support]] is also the light cone (prop. \ref{SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}). Therefore now same argument as before says that the wave front set consists of wave vectors $k$ on the light cone, but now due to the [[step function]] factor $\Theta(-k_0)$ it must satisfy $0 \leq - k_0 = k^0$. Finally regarding the [[Feynman propagator]]: By prop. \ref{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime} the Feynman propagator coincides with the positive frequency Wightman propagator for $x^0 \gt 0$ and with the ``negative frequency Hadamard operator'' for $x^0 \lt 0$. Therefore the form of $WF(\Delta_F)$ now follows directly with that of $WF(\Delta_H)$ above. \end{proof} \hypertarget{ExampleForDiracOperatorOnMinkowskiSpacetime}{}\subsubsection*{{For Dirac operator on Minkowski spacetime}}\label{ExampleForDiracOperatorOnMinkowskiSpacetime} Finally we observe that the [[propagators]] for the [[Dirac field]] on [[Minkowski spacetime]] follow immediately from the propagators for the [[scalar field]]: \begin{prop} \label{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}\hypertarget{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}{} \textbf{([[advanced and retarded propagator]] for [[Dirac equation]] on [[Minkowski spacetime]])} Consider the [[Dirac operator]] on [[Minkowski spacetime]], which in [[Feynman slash notation]] reads \begin{displaymath} \begin{aligned} D & \coloneqq -i {\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \\ & = -i \gamma^\mu \frac{\partial}{\partial x^\mu} + \tfrac{m c}{\hbar} \end{aligned} \,. \end{displaymath} Its [[advanced and retarded propagators]] (def. \ref{AdvancedAndRetardedGreenFunctions}) are the [[derivatives of distributions]] of the advanced and retarded propagators $\Delta_\pm$ for the [[Klein-Gordon equation]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) by ${\partial\!\!\!/\,} + m$: \begin{displaymath} \Delta_{D, \pm} \;=\; \left( -i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar} \right) \Delta_{\pm} \,. \end{displaymath} Hence the same is true for the [[causal propagator]]: \begin{displaymath} \Delta_{D, S} \;=\; \left( -i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar} \right) \Delta_{S} \,. \end{displaymath} \end{prop} \begin{proof} Applying a [[differential operator]] does not change the [[support]] of a [[smooth function]], hence also not the [[support of a distribution]]. Therefore the uniqueness of the advanced and retarded propagators (prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}) together with the translation-invariance and the anti-[[formally self-adjoint differential operator|formally self-adjointness]] of the [[Dirac operator]] (as for the [[Klein-Gordon operator]] \eqref{TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime} implies that it is sufficent to check that applying the [[Dirac operator]] to the $\Delta_{D, \pm}$ yields the [[delta distribution]]. This follows since the Dirac operator squares to the Klein-Gordon operator: \begin{displaymath} \begin{aligned} \left( -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \right) \Delta_{D, \pm} & = \underset{ = \Box - \left(\tfrac{m c}{\hbar}\right)^2}{ \underbrace{ \left( -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \right) \left( -i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar} \right) } } \Delta_{\pm} \\ & = \delta \end{aligned} \,. \end{displaymath} \end{proof} Similarly we obtain the other [[propagators]] for the [[Dirac field]] from those of the [[real scalar field]]: \begin{defn} \label{HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}\hypertarget{HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}{} \textbf{([[Wightman propagator]] for [[Dirac operator]] on [[Minkowski spacetime]])} The \emph{[[Wightman propagator]]} for the [[Dirac operator]] on [[Minkowski spacetime]] is the [[positive real number|positive]] [[frequency]] part of the [[causal propagator]] (prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}), hence the [[derivative of distributions]] of the Wightman propagator for the Klein-Gordon field (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) by the [[Dirac operator]]: \begin{displaymath} \begin{aligned} \left( -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \right)\Delta_{H}(x,y) & = \frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) ( {k\!\!\!/\,} + \tfrac{m c}{\hbar}) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k \\ & = \frac{1}{(2\pi)^p} \int \frac{ \gamma^0 \omega(\vec k)/c + \vec \gamma \cdot \vec k + \tfrac{m c}{\hbar} }{2 \omega(\vec k)/c} e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) } \, d^p \vec k \,. \end{aligned} \end{displaymath} Here we used the expression \eqref{StandardHadamardDistributionOnMinkowskiSpacetime} for the Wightman propagator of the Klein-Gordon equation. \end{defn} \begin{defn} \label{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime}\hypertarget{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime}{} \textbf{([[Feynman propagator]] for [[Dirac operator]] on [[Minkowski spacetime]])} The \emph{[[Feynman propagator]]} for the [[Dirac operator]] on [[Minkowski spacetime]] is the linear combination \begin{displaymath} \Delta_{D, F} \;\coloneqq\; \Delta_{D,H} + i \Delta_{D, -} \end{displaymath} of the [[Wightman propagator]] (def. \ref{HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}) and the retarded propagator (prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}). By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} this means that it is the [[derivative of distributions]] of the [[Feynman propagator]] of the [[Klein-Gordon equation]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) by the [[Dirac operator]] \begin{displaymath} \begin{aligned} \Delta_{D, F} & = \left( -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar} \right)\Delta_{F}(x,y) & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{-i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ \left( {k\!\!\!/\,} + \tfrac{m c}{\hbar} \right) e^{i k_\mu (x^\mu - y^\mu)} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k \,. \end{aligned} \end{displaymath} \end{defn} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} [[!include propagators - table]] $\,$ \begin{itemize}% \item [[Green hyperbolic differential operator]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} General discussion includes \begin{itemize}% \item [[Christian Bär]], \emph{Green-hyperbolic operators on globally hyperbolic spacetimes}, Communications in Mathematical Physics 333, 1585-1615 (2014) (\href{http://dx.doi.org/10.1007/s00220-014-2097-7}{doi}, \href{https://arxiv.org/abs/1310.0738}{arXiv:1310.0738}) \item [[Igor Khavkine]], \emph{Covariant phase space, constraints, gauge and the Peierls formula}, Int. J. Mod. Phys. A, 29, 1430009 (2014) (\href{https://arxiv.org/abs/1402.1282}{arXiv:1402.1282}) \end{itemize} based on \begin{itemize}% \item [[Ko Sanders]], \emph{A note on spacelike and timelike compactness}, Classical and Quantum Gravity 30, 115014 (2012) (\href{http://dx.doi.org/10.1088/0264-9381/30/11/115014}{doi}, \href{https://arxiv.org/abs/1211.2469}{arXiv:1211.2469}) \end{itemize} Textbook discussion for [[free fields]] in [[Minkowski spacetime]] is in \begin{itemize}% \item [[Günter Scharf]], section 2.3 of \emph{[[Finite Quantum Electrodynamics -- The Causal Approach]]}, Springer 1995 \item [[Günter Scharf]], section 1 of \emph{[[Quantum Gauge Theories -- A True Ghost Story]]}, Wiley 2001 \end{itemize} An overview of the [[Green functions]] of the [[Klein-Gordon operator]], hence of the [[Feynman propagator]], [[advanced propagator]], [[retarded propagator]], [[causal propagator]] etc. is given in \begin{itemize}% \item [[Mikica Kocic]], \emph{Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions}, 2016 ([[KGPropagatorsOnMinkowskiTable.pdf:file]]) \end{itemize} Discussion on general [[globally hyperbolic spacetimes]] includes \begin{itemize}% \item F. Friedlander, \emph{The Wave Equation on a Curved Space-Time}, Cambridge: Cambridge University Press, 1975 \item [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], \emph{Wave Equations on Lorentzian Manifolds and Quantization}, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (\href{https://arxiv.org/abs/0806.1036}{arXiv:0806.1036}) \item [[Nicolas Ginoux]], \emph{Linear wave equations}, Ch. 3 in [[Christian Bär]], [[Klaus Fredenhagen]], \emph{Quantum Field Theory on Curved Spacetimes: Concepts and Methods}, Lecture Notes in Physics, Vol. 786, Springer, 2009 \end{itemize} Review in the context of [[perturbative algebraic quantum field theory]] includes \begin{itemize}% \item [[Katarzyna Rejzner]], sections 4.1 and 6.2.3 of \emph{Perturbative Algebraic Quantum Field Theory}, Mathematical Physics Studies, Springer 2016 (\href{https://link.springer.com/book/10.1007%2F978-3-319-25901-7}{pdf}) \end{itemize} [[!redirects retarded and advanced causal propagators]] [[!redirects advanced and retarded propagators]] [[!redirects retarded and advanced propagators]] [[!redirects advanced and retarded propagator]] [[!redirects retarded and advanced propagator]] [[!redirects advanced propagator]] [[!redirects advanced propagators]] [[!redirects retarded propagator]] [[!redirects retarded propagators]] [[!redirects advanced causal propagator]] [[!redirects advanced causal propagators]] [[!redirects retarded causal propagator]] [[!redirects retarded causal propagators]] [[!redirects advanced Green function]] [[!redirects advanced Green functions]] [[!redirects retarded Green function]] [[!redirects retarded Green functions]] [[!redirects retarded and advanced Green functions]] [[!redirects advanced and retarded Green functions]] [[!redirects advanced Green's function]] [[!redirects advanced Green's functions]] [[!redirects retarded Green's function]] [[!redirects retarded Green's functions]] [[!redirects advanced or retarded Green function]] [[!redirects advanced or retarded Green functions]] [[!redirects retarded and advanced Green's functions]] [[!redirects advanced and retarded Green's functions]] \end{document}