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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*{Green hyperbolic partial differential equation} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - 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{formally_adjoint_green_hyperbolic_operators}{Formally adjoint Green hyperbolic operators}\dotfill \pageref*{formally_adjoint_green_hyperbolic_operators} \linebreak \noindent\hyperlink{ContinousLinearFunctionalsOnTheSolutionSpace}{Continuous linear functionals on the solution space}\dotfill \pageref*{ContinousLinearFunctionalsOnTheSolutionSpace} \linebreak \noindent\hyperlink{PSymplecticAndPPeierlsBracket}{The $P$-Symplectic and $P$-Peierls brackets}\dotfill \pageref*{PSymplecticAndPPeierlsBracket} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{[[partial differential equation]]} (PDE) is \emph{Green hyperbolic} (\hyperlink{Baer14}{B\"a{}r 14, def. 3.2}, \hyperlink{Khavkine14}{Khavkine 14, def. 2.2}) if it behaves like a [[normally hyperbolic differential equation]] on a [[globally hyperbolic spacetime]] in that it has unique [[advanced and retarded Green functions]]. \emph{Duhamel's principle} essentially establishes the equivalence between [[hyperbolic differential equations]] with a well-posed [[Cauchy problem]] and Green hyperbolic systems. (\hyperlink{Khavkine14}{Khavkine 14, p. 12}) \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Let $\Sigma$ be a [[time orientation|time-oriented]] [[Lorentzian manifold]] of [[dimension]] $p+1$, or more generally a [[conal manifold]] with [[conal causal structure]]. \begin{displaymath} \itexarray{ E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma } \end{displaymath} be a [[smooth vector bundle]]. Write \begin{displaymath} \tilde E^\ast \;\coloneqq\; E^\ast \otimes_\Sigma \wedge_\Sigma^{p+1} T^\ast \Sigma \end{displaymath} for the \emph{[[density|densitized]] [[dual vector bundle]]}, hence the [[tensor product of vector bundles]] of the [[dual vector bundle]] with the [[differential n-form]]-bundle. \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{FormallyAdjointDifferentialOperators}\hypertarget{FormallyAdjointDifferentialOperators}{} \textbf{([[formally adjoint differential operators]])} Two [[differential operators]] \begin{displaymath} P, P^\ast \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(\tilde E^\ast) \end{displaymath} are called \emph{[[formally adjoint differential operators]]} via a [[bilinear map|bilinear]] [[differential operator]] \begin{equation} K \;\colon\; \Gamma_\Sigma(E) \otimes \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(\wedge^{p} T^\ast \Sigma) \label{FormallyAdjointDifferentialOperatorWitness}\end{equation} such that for all $\Phi_1, \Phi_2 \in \Gamma_\Sigma(E)$ we have \begin{displaymath} P(\Phi_1) \cdot \Phi_2 - \Phi_1 \cdot P^\ast(\Phi_2) \;=\; d K(\Phi_1, \Phi_2) \end{displaymath} \end{defn} (\hyperlink{Baer14}{B\"a{}r 14, 3. (1)} \hyperlink{Khavkine14}{Khavkine 14, def. 2.4}) \begin{defn} \label{AdvancedAndRetardedGreenFunctions}\hypertarget{AdvancedAndRetardedGreenFunctions}{} \textbf{([[advanced and retarded Green functions]] and [[causal Green function]])} 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 \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} \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]]}. \end{defn} (\hyperlink{Baer14}}{B\"a{}r 14, def. 3.2, cor. 3.10}) \begin{defn} \label{GreenHyperbolicDifferentialOperator}\hypertarget{GreenHyperbolicDifferentialOperator}{} \textbf{([[Green hyperbolic differential operator]])} Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] over a [[smooth manifold]] $\Sigma$ with [[causal structure]]. A [[linear differential operator|linear]] [[hyperbolic differential operator]]. \begin{displaymath} P \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_{\Sigma}(\tilde E^\ast) \end{displaymath} is called \emph{Green hyperbolic} with respect to the given [[causal structure]] if $P$ as well as its [[formal adjoint differential operator]] $P^\ast$ (def. \ref{FormallyAdjointDifferentialOperators}) admit [[advanced and retarded Green functions]] (def. \ref{AdvancedAndRetardedGreenFunctions}). \end{defn} (\hyperlink{Baer14}}{B\"a{}r 14, def. 3.2}, \hyperlink{Khavkine14}{Khavkine 14, def. 2.2}) \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{AdvancedAndRetardedGreenFunctions}) of a [[Green hyperbolic differential operator]] (def. \ref{GreenHyperbolicDifferentialOperator}) are unique. \end{prop} (\hyperlink{Baer14}}{B\"a{}r 14, cor. 3.12} \hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties} \hypertarget{formally_adjoint_green_hyperbolic_operators}{}\subsubsection*{{Formally adjoint Green hyperbolic operators}}\label{formally_adjoint_green_hyperbolic_operators} \begin{prop} \label{}\hypertarget{}{} \textbf{([[causal Green functions]] of [[formally adjoint differential operator|formally adjoint]] [[Green hyperbolic differential operators]] are [[formally adjoint differential operator|formally adjoint]])} Let \begin{displaymath} P, P^\ast \;\colon\;\Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast) \end{displaymath} be a pair of Green hyperbolic differential operators (def. \ref{GreenHyperbolicDifferentialOperator}) which are [[formally adjoint differential operator|formally adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}). Then also their [[causal Green functions]] $\mathrm{G}_P$ and $G_{P^\ast}$ (def. \ref{AdvancedAndRetardedGreenFunctions}) are [[formally adjoint differential operators]], up to a sign: \begin{displaymath} \left( \mathrm{G}_P \right)^\ast \;=\; - \mathrm{G}_{P^\ast} \,. \end{displaymath} \end{prop} (\hyperlink{Khavkine14}{Khavkine 14, (24), (25)}) \hypertarget{ContinousLinearFunctionalsOnTheSolutionSpace}{}\subsubsection*{{Continuous linear functionals on the solution space}}\label{ContinousLinearFunctionalsOnTheSolutionSpace} \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]]. Finally, the [[vector spaces]] of smooth sections with prescribed causal support (def. \ref{CompactlySourceCausalSupport}) are [[inductive limits]] of vector spaces $\Gamma_{\Sigma,K}(E)$ as above, and hence they inherit [[topological vector space]] [[structure]] by forming the corresponding [[inductive limit]] in the [[category]] of [[locally convex topological vector spaces]]. For instance \begin{displaymath} \Gamma_{\Sigma,cp}(E) \;\coloneqq\; \underset{\underset{ {K \subset \Sigma} \atop {K\, \text{compact}} }{\longrightarrow}}{\lim} \Gamma_{\Sigma,K}(E) \end{displaymath} etc. \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}) \begin{prop} \label{DistributionsWithCausalSupports}\hypertarget{DistributionsWithCausalSupports}{} \textbf{(topological duality with causally restricted supports)} Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] over a [[smooth manifold]] with [[causal structure]]. Then there are the following [[isomorphisms]] of [[topological vector spaces]] between [[dual spaces]] of [[spaces of sections]] and spaces of distributional sections, with restricted supports (def. \ref{DistributionalSections}): \begin{displaymath} \begin{aligned} \Gamma_{\Sigma,cp}(E)^* &\simeq \Gamma'_{\Sigma}(\tilde E^\ast) , \\ \Gamma_{\Sigma,+cp}(E)^* &\simeq \Gamma'_{\Sigma,fcp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,-cp}(E)^* &\simeq \Gamma'_{\Sigma,pcp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,scp}(E)^* &\simeq \Gamma'_{\Sigma,tcp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,fcp}(E)^* &\simeq \Gamma'_{\Sigma,+cp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,pcp}(E)^* &\simeq \Gamma'_{\Sigma,-cp}(\tilde E^\ast) , \\ \Gamma_{\Sigma,tcp}(E)^* &\simeq \Gamma'_{\Sigma,scp}(\tilde E^\ast) , \\ \Gamma_{\Sigma}(E)^* &\simeq \Gamma'_{\Sigma,cp}(\tilde E^\ast) . \end{aligned} \end{displaymath} \end{prop} (\hyperlink{Sanders12}{Sanders 13, thm. 4.3}, \hyperlink{Baer14}{B\"a{}r 14, lem. 2.14}) \begin{prop} \label{ExactSequenceOfGreenHyperbolicSystem}\hypertarget{ExactSequenceOfGreenHyperbolicSystem}{} \textbf{([[exact sequence]] of [[Green hyperbolic differential operator]])} Let $\Gamma_\Sigma(E) \overset{P}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast)$ be a [[Green hyperbolic differential operator]] (def. \ref{GreenHyperbolicDifferentialOperator}) with [[causal Green function]] $\mathrm{G}$ (def. \ref{GreenHyperbolicDifferentialOperator}). Then the sequences \begin{displaymath} 0 \to \Gamma_{\Sigma,cp}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma,cp}(\tilde E^\ast) \overset{\mathrm{G}_P}{\longrightarrow} \Gamma_{\Sigma,scp}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma,scp}(\tilde E^\ast) \to 0 , \end{displaymath} \begin{displaymath} 0 \to \Gamma_{\Sigma,tcp}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma,tcp}(\tilde E^\ast) \overset{\mathrm{G}_P}{\longrightarrow} \Gamma_{\Sigma}(E) \overset{P}{\longrightarrow} \Gamma_{\Sigma}(\tilde E^\ast) \to 0 \end{displaymath} of these operators restricted to functions with causally restricted supports as indicated (def. \ref{CompactlySourceCausalSupport}) are [[exact sequence]]s of [[topological vector spaces]] and continuous [[linear maps]] between them. Under passing to [[dual spaces]] and using the isomorphisms of spaces of distributional sections (def. \ref{DistributionalSections}) from prop. \ref{DistributionsWithCausalSupports} this yields the following dual [[exact sequence]] of [[topological vector spaces]] and continuous [[linear maps]] between them: \begin{equation} 0 \to \Gamma'_{\Sigma,tcp}(E) \overset{P^*}{\longrightarrow} \Gamma'_{\Sigma,tcp}(\tilde E^\ast) \overset{-\mathrm{G}_{P^*}}{\longrightarrow} \Gamma'_{\Sigma}(E) \overset{P^*}{\longrightarrow} \Gamma'_{\Sigma}(\tilde E^\ast) \to 0 , \label{GreenHyperbolicOperatorDualExactSequence}\end{equation} \begin{equation} 0 \to \Gamma'_{\Sigma,cp}(E) \overset{P^*}{\longrightarrow} \Gamma'_{\Sigma,cp}(\tilde E^\ast) \overset{-\mathrm{G}_{P^*}}{\longrightarrow} \Gamma'_{\Sigma,scp}(E) \overset{P^*}{\longrightarrow} \Gamma'_{\Sigma,scp}(\tilde E^\ast) \to 0 \label{GreenHyperbolicOperatorDualExactSequence2}\end{equation} In particular this means that there is a [[linear isomorphism]] between the space $ker_{scp}(P)$ of spatially compact solutions to the differential equation and the [[quotient space]] of the [[compact support|compactly supported]] dual sections by the [[image]] of $P$: \begin{equation} \Gamma_{\Sigma,cp}(\tilde E^\ast)/im(P) \underoverset{\simeq}{\phantom{A}\mathrm{G}_P\phantom{A}}{\longrightarrow} ker_{scp}(P) \,. \label{SolutionSpaceIsomorphicToQuotientByImP}\end{equation} \end{prop} The following proof is a slight refinement of (\hyperlink{Khavkine14}{Khavkine 14, prop. 2.1}). The refinement consists of reducing the check of exactness to the construction of a contracting homotopy of the complex into itself (cf. the answer to \href{https://mathoverflow.net/a/209024}{MO208985}, where this refinement has appeared in a simplified context). \begin{proof} Let $\Sigma_p^-, \Sigma_p^+ \subset \Sigma$ be two Cauchy surfaces, with $\Sigma_p^-$ in the past of $\Sigma_p^+$. Let also $\{\chi_+,\chi_-\}$ be a [[partition of unity]] subordinate to the cover $\{J^+(\Sigma_p^-), J^-(\Sigma_p^+)\}$ of $\Sigma$, that is, smooth functions such $\chi_+ + \chi_- = 1$, while $\chi_+ = 0$ on the past of $\Sigma_p^-$ and $\chi_- = 0$ on the future of $\Sigma_p^+$. We can use these functions to define the following [[contracting homotopy]] of our complex into itself: \begin{displaymath} \begin{array}{ccccccccccc} 0 & \to & \Gamma_{\Sigma,cp}(E) & \overset{P}{\longrightarrow} & \Gamma_{\Sigma,cp}(\tilde E^\ast) & \overset{\mathrm{G}}{\longrightarrow} & \Gamma_{\Sigma,scp}(E) & \overset{P}{\longrightarrow} & \Gamma_{\Sigma,scp}(\tilde E^\ast) & \to & 0 \\ & & {\downarrow} id & {\swarrow} {}_\chi\mathrm{G} & {\downarrow} id & {\swarrow} P_\chi & {\downarrow} id & {\swarrow} \mathrm{G}_\chi & {\downarrow} id \\ 0 & \to & \Gamma_{\Sigma,cp}(E) & \overset{P}{\longrightarrow} & \Gamma_{\Sigma,cp}(\tilde E^\ast) & \overset{\mathrm{G}}{\longrightarrow} & \Gamma_{\Sigma,scp}(E) & \overset{P}{\longrightarrow} & \Gamma_{\Sigma,scp}(\tilde E^\ast) & \to & 0 \end{array} \end{displaymath} The homotopy maps are defined as follows: \begin{displaymath} \begin{aligned} {}_\chi\mathrm{G}[\tilde{\alpha}^*] &= \chi_+ \mathrm{G}_-[\tilde{\alpha}^*] + \chi_- \mathrm{G}_+[\tilde{\alpha}^*] , \\ P_\chi[\psi] &= P[\chi_+\psi] - \chi_+ P[\psi] = -P[\chi_-\psi] + \chi_- P[\psi] , \\ \mathrm{G}_\chi[\tilde{\alpha}^*] &= \mathrm{G}_+[\chi_+\tilde{\alpha}^*] + \mathrm{G}_-[\chi_-\tilde{\alpha}^*] . \end{aligned} \end{displaymath} The contracting identities \begin{displaymath} \begin{aligned} {}_\chi\mathrm{G} \circ P &= id , \\ P\circ {}_\chi\mathrm{G} + P_\chi \circ \mathrm{G} &= id , \\ \mathrm{G}\circ P_\chi + \mathrm{G}_\chi \circ P &= id , \\ P \circ \mathrm{G}_\chi &= id , \end{aligned} \end{displaymath} are simply a matter of direct calculation. The identity morphism of our complex to itself induces an isomorphism on its cohomology. On the other hand, since this morphism itself is induced by a homotopy, it must be in fact be the zero map on cohomology. This is only possible when all cohomologies vanish and our complex is exact. The continuity of the differential operators $P$ and $P^*$ is standard. The continuity of the Green function acting on smooth functions was already noted in prop. \ref{GreenFunctionsAreContinuous}. To see the second exact sequences, observe that differential operators extend continuously to distributions in a standard way. The only nontrivial check is on the Green functions. Their continuity is discussed in (\hyperlink{Sanders12}{Sanders 13, sec. 5}) and (\hyperlink{Baer14}{Baer 14, lem. 4.1}). The exactness follows from the same argument as in the previous argument (since a contracting homotopy dualizes to a contracting homotopy). The exactness of a sequence similar to the one above also appears as (\hyperlink{Baer14}{B\"a{}r 14, thm. 4.3}). The continuity of the extensions of Green functions to distribution follows from standard arguments, which was checked for instance in (\hyperlink{Baer14}{B\"a{}r 14, lem. 4.1}). The standard argument consists of noting that the adjoint of a continuous linear map is also continuous, both for the weak and strong topologies on distributions (\hyperlink{Treves67}{Treves 67, prop.-cor. 19.5}). \end{proof} Putting the above results together, it follows: \begin{prop} \label{DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions}\hypertarget{DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions}{} \textbf{([[distributions]] on Green hyperbolic PDE solution space are the [[generalized PDE solutions]])} Let $P, P^\ast \;\colon\; \Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast)$ be a pair of [[Green hyperbolic differential operators]] (def. \ref{GreenHyperbolicDifferentialOperator}) which are [[formally adjoint differential operator|formally adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}). Then 1) the canonical pairing (from prop. \ref{DistributionsWithCausalSupports}) \begin{displaymath} \itexarray{ \Gamma'_{\Sigma,cp}(\tilde E^\ast) &\otimes& \Gamma_\Sigma(E) &\overset{}{\longrightarrow}& \mathbb{C} \\ \alpha^\ast &,& \Phi &\mapsto& \int \alpha^\ast_a(x) \Phi^a(x)\, dvol_\Sigma(x) } \end{displaymath} induces a [[continuous linear map|continuous]] [[linear isomorphism]] \begin{displaymath} (ker(P))^\ast \;\simeq\; \Gamma'_{\Sigma,cp}(\tilde E^\ast)/im_{cp}(P^\ast) \end{displaymath} 2) a [[continuous linear functional]] on the solution space \begin{displaymath} u_{sol} \in \left(ker_{scp}(P)\right)^\ast \end{displaymath} is equivalently a [[distribution|distributional section]] (def. \ref{DistributionalSections}) \begin{displaymath} u \in \Gamma'_{\Sigma}(E) \end{displaymath} which is a [[generalized solution]] to the differential equation \begin{displaymath} P^\ast u = 0 \,, \end{displaymath} and this is a [[continuous linear map|continuous]] [[linear isomorphism]] given by pullback along the [[causal Green function]] $\mathrm{G}_P$ (def. \ref{AdvancedAndRetardedGreenFunctions}): \begin{equation} \left(ker_{scp}(P)\right)^\ast \underoverset{\simeq}{-\mathrm{G}_{P^\ast} = (-)\circ \mathrm{G}_P}{\longrightarrow} \left\{ u \in \Gamma'_{\Sigma}(E) \,\vert\, P^* u = 0 \right\} \,. \label{ContinuousLinearDualOfSpacelikeCompactSolutionSpaceIsUnconstrainedDistributionsThatAreGeneralizedSolutions}\end{equation} Similarly: \begin{equation} \left(ker(P)\right)^\ast \underoverset{\simeq}{-\mathrm{G}_{P^\ast} = (-)\circ \mathrm{G}_P}{\longrightarrow} \left\{ u \in \Gamma'_{\Sigma,scp}(E) \,\vert\, P^* u = 0 \right\} \,. \label{ContinuousLinearDualOfUnconstrainedSolutionSpaceIsSpacelikeCompactlySupportedDistributionsThatAreGeneralizedSolutions}\end{equation} \end{prop} \begin{proof} Observe that both $\ker_{scp}(P) \subset \Gamma_{\Sigma,scp}(E)$ and $im_{tcp}(P^*) \subset \Gamma'_{\Sigma,tcp}(\tilde{E}^*)$ are [[closed subspaces]]: the first by [[continuous function|continuity]] of $P$ and the second by exactness of the sequence \eqref{GreenHyperbolicOperatorDualExactSequence} in prop. \ref{ExactSequenceOfGreenHyperbolicSystem}, which implies that this, too, is the kernel of a continuous linear functional. This implies (\hyperlink{Treves67}{Treves 67, props. 35.5, 35.6}) that the linear isomorphisms \begin{displaymath} (ker(P))^\ast \;\simeq\; \Gamma'_{\Sigma,cp}(\tilde E^\ast)/im_{cp}(P^\ast) \end{displaymath} and \begin{equation} (\ker_{scp}(P))^* \;\simeq\; \Gamma'_{\Sigma,tcp}(\tilde{E}^*) / im_{tcp}(P^*) \label{ContinuousLinearDualsOnGreenHyperbolicSolutionSpace}\end{equation} obtained from the underlying exact sequences of vector spaces in prop. \ref{ExactSequenceOfGreenHyperbolicSystem} are also [[continuous linear map|continuous]] linear isomorphisms for the [[dual space]] topology on the left. The first of these is the statement 1) to be proven. From the second, by once again exploiting the exactness of the sequence \eqref{GreenHyperbolicOperatorDualExactSequence} in prop. \ref{ExactSequenceOfGreenHyperbolicSystem}, we also have the chain of isomorphisms \begin{equation} \Gamma'_{\Sigma,tcp}(\tilde{E}^*) / im_{tcp}(P^*) \simeq \Gamma'_{\Sigma,tcp}(\tilde{E}^*) / \ker_{tcp}(G_{P^*}) \underoverset{\simeq}{G_{P^\ast}}{\to} \ker(P^*) \subset \Gamma'_{\Sigma}({E}). \label{ChainOfIsomorphisms}\end{equation} Combining this with \eqref{ContinuousLinearDualsOnGreenHyperbolicSolutionSpace} yields the desired isomorphism in 2) The same argument applied to the exact sequence \eqref{GreenHyperbolicOperatorDualExactSequence2} yields the second statement in 2) \end{proof} \hypertarget{PSymplecticAndPPeierlsBracket}{}\subsubsection*{{The $P$-Symplectic and $P$-Peierls brackets}}\label{PSymplecticAndPPeierlsBracket} \begin{defn} \label{BracketPSymplecticAndPPeierls}\hypertarget{BracketPSymplecticAndPPeierls}{} \textbf{($P$-Symplectic and $P$-Peierls bracket)} Let \begin{displaymath} P, P^\ast \;\colon\;\Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast) \end{displaymath} be a pair of [[Green hyperbolic differential operators]] (def. \ref{GreenHyperbolicDifferentialOperator}) which are [[formally adjoint differential operator|formally adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}) via a differential operator $K$ \eqref{FormallyAdjointDifferentialOperatorWitness}. Then: \begin{enumerate}% \item Consider $\ker_{scp}(P) \subset \Gamma_{\Sigma,scp}(E)$ and $\ker'(P^*) \subset \Gamma'_{\Sigma}(E)$, where $\ker'$ simply means that we are considering distributional solutions. For any [[Cauchy surface]] $\Sigma_p \overset{\iota_{\Sigma_p}}{\hookrightarrow} \Sigma$ the \emph{$P$-Symplectic bracket} is the [[bilinear map]] \begin{displaymath} \itexarray{ \ker_{scp}(P) \otimes \ker'(P^*) &\overset{ \left\{-,- \right\}_{\Sigma_p,K} }{\longrightarrow}& \mathbb{R} \\ (\mathbf{\Phi}_1, \mathbf{\Phi}_2) &\mapsto& \underset{\Sigma_p}{\int} (\iota_{\Sigma_p})^\ast K(\mathbf{\Phi}_1, \mathbf{\Phi}_2) } \end{displaymath} Note that, even though $\Phi_2\in \ker'(P^*)$ is a distribution on $\Sigma$, we are integrating it over a codimension-$1$ surface $\Sigma_p \hookrightarrow \Sigma$. In this case, the restriction to $\Sigma_p$ is well-defined by the theorem on the [[pullback of a distribution]] (\href{pullback+of+a+distribution#PullbackOfDistributionsWhoseWaveFrontDoesNotIntersectNormalBundle}{this prop.} there). The only condition to check is that the [[conormal bundle]] of the embedding $\Sigma_p \hookrightarrow \Sigma$ does not intersect $WF(\Phi_2)$. But, since the $\Phi_2$ is annihilated by $P^*$, by the theorem on the [[propagation of singularities theorem|propagation of singularities]] (\href{propagation+of+singularities+theorem#PropagationOfSingularitiesTheorem}{this prop.} there), $WF(\Phi_2)$ contains only covectors that are characteristic with respect to $P^*$ (those on which the [[principal symbol]] of $P^*$ fails to be invertible). But by its definition, a Cauchy surface $\Sigma_p$ must be nowhere characteristic, meaning that its conormal bundle does not intersect $WF(\Phi_2)$.\newline Because the arguments $\Phi_1$ and $\Phi_2$ are annihilated respectively by $P$ and $P^*$, equation \eqref{FormallyAdjointDifferentialOperatorWitness} ensures that the definition of $\{-,-\}_{\Sigma_p,K}$ is actually independent of the choice of $\Sigma_p$. \item The \emph{$P$-Peierls bracket} is the [[bilinear map]] \begin{displaymath} \itexarray{ \Gamma_{\Sigma,cp}(\tilde E^\ast) \otimes \Gamma'_{\Sigma,tcp}(\tilde E^\ast) & \overset{ \left\{ -,- \right\}_{\Sigma, \mathrm{G}} }{ \longrightarrow } & \mathbb{R} \\ (\tilde \alpha^\ast_1, \tilde \alpha^\ast_2) &\mapsto& \underset{\Sigma}{\int} \tilde \alpha^\ast_1 \cdot \mathrm{G}_{P^*}(\tilde \alpha^\ast_2) = -\underset{\Sigma}{\int} \tilde \mathrm{G}_{P}(\alpha^\ast_1) \cdot \tilde \alpha^\ast_2 } \end{displaymath} Since $\mathrm{G}_P$ annihilates $im_{cp}(P)$ and the image of $\mathrm{G}_P$ is annihilated by any distribution in $im'_{tcp}(P^*)$ (again, $im'$ refers to the fact that we are considering distributional sections), the $P$-Peierls bracket descends to a bilinear map on \begin{displaymath} \itexarray{ \Gamma_{\Sigma,cp}(\tilde E^\ast) / im_{cp}(P^*) \otimes \Gamma'_{\Sigma,tcp}(\tilde E^\ast) / im'_{tcp}(P) & \overset{ \left\{ -,- \right\}_{\Sigma, \mathrm{G}} }{ \longrightarrow } & \mathbb{R} } \end{displaymath} \end{enumerate} \end{defn} \begin{prop} \label{PSymplecticToPPeierls}\hypertarget{PSymplecticToPPeierls}{} \textbf{([[causal Green function]] transforms $P$-Peierls bracket to $P$-symplectic pairing)} Let \begin{displaymath} P, P^\ast \;\colon\;\Gamma_\Sigma(E) \overset{}{\longrightarrow} \Gamma_\Sigma(\tilde E^\ast) \end{displaymath} be a pair of [[Green hyperbolic differential operators]] (def. \ref{GreenHyperbolicDifferentialOperator}) which are [[formally adjoint differential operator|formally adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}) via a differential operator $K$ \eqref{FormallyAdjointDifferentialOperatorWitness}. Then the [[causal Green function]] intertwines the $P$-Symplectic bracket with the $P$-Peierls bracket (def. \ref{BracketPSymplecticAndPPeierls}) in that for every [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ and all \begin{displaymath} \tilde \alpha_1^\ast \;\in\; \Gamma_{\Sigma,cp}(\tilde E^\ast) , \quad \tilde \alpha_2^\ast \;\in\; \Gamma'_{\Sigma,tcp}(\tilde E^\ast) \end{displaymath} we have \begin{equation} \left\{ \tilde \alpha^\ast_1, \tilde \alpha^\ast_2 \right\}_{\Sigma,\mathrm{G}} \;=\; -\left\{ \mathrm{G}_P(\tilde \alpha^\ast_1), \mathrm{G}_{P^*}(\tilde \alpha^\ast_2) \right\}_{\Sigma_p,K} \,. \label{RelationBetweenSymplecticAndPPeierlsBracket}\end{equation} \end{prop} (\hyperlink{Khavkine14}{Khavkine 14, lemma 2.5}) As currently defined, the pairings $\{-,-\}_{\Sigma_p,K}$ and $\{-,-\}_{\Sigma,G}$ are far from being anti-symmetric. In particular, the two arguments may come from very different spaces. In the self-adjoint case $P=P^*$, we may ask whether the domains of the two arguments may be respectively enlarged or shrunk to be equal and such that the pairing becomes anti-symmetric. When that is possible, we get an honest symplectic or Poisson bracket. The simplest such choices are \begin{displaymath} \itexarray{ \ker_{scp}(P) \otimes \ker_{scp}(P) & \overset{ \left\{ -,- \right\}_{\Sigma_p, K} }{ \longrightarrow } & \mathbb{R} \\ \Gamma_{\Sigma,cp}(\tilde E^\ast) \otimes \Gamma_{\Sigma,cp}(\tilde E^\ast) & \overset{ \left\{ -,- \right\}_{\Sigma, \mathrm{G}} }{ \longrightarrow } & \mathbb{R} } \end{displaymath} This problem is analogous to studying (anti-)self-adjoint extensions of a symmetric unbounded operator on Hilbert space. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{example} \label{GreenHyperbolicKleinGordonOperator}\hypertarget{GreenHyperbolicKleinGordonOperator}{} \textbf{([[Klein-Gordon operator]] is [[Green hyperbolic differential operator|Green hyperbolic]])} For $\Sigma$ a [[globally hyperbolic spacetimes]] then the [[Klein-Gordon operator]] $P = \Box - m^2$ (i.e. the [[wave operator]] for $m = 0$) is [[Green hyperbolic differential operator|Green hyperbolic]] according to def. \ref{GreenHyperbolicDifferentialOperator} (e. g. \hyperlink{BaerGinouxPfaeffle07}{B\"a{}r-Ginoux-Pfaeffle 07}) and formally self-adjoint (\href{formal+adjoint+differential+operator#FormallySelfAdjointKleinGordonOperator}{this example}). The corresponding $P$-Peierls bracket (def. \ref{BracketPSymplecticAndPPeierls}) is the original [[Peierls bracket]]. \end{example} \begin{example} \label{GreenHyperbolicDiracOperator}\hypertarget{GreenHyperbolicDiracOperator}{} \textbf{([[Dirac operator]] is [[Green hyperbolic differential operator|Green hyperbolic]])} The [[Dirac operator]] $D$ squaring to a Green hyperbolic operator is itself Green hyperbolic. \end{example} (\hyperlink{Baer14}{B\"a{}r 14, corollary 3.15, example 3.16}) \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[François Trèves]], \emph{Topological Vector Spaces, Distributions and Kernels} (Academic Press, New York, 1967) \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 [[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}) \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} [[!redirects Green hyperbolic partial differential equations]] [[!redirects Green hyperbolic differential equation]] [[!redirects Green hyperbolic differential equations]] [[!redirects Duhamel's principle]] [[!redirects Duhamel principle]] [[!redirects Green hyperbolic PDE]] [[!redirects Green hyperbolic PDEs]] [[!redirects Green hyperbolic differential operator]] [[!redirects Green hyperbolic differential operators]] \end{document}