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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*{A first idea of quantum field theory -- Phase space} \hypertarget{PhaseSpace}{}\subsection*{{Phase space}}\label{PhaseSpace} In this chapter we discuss these topics: \begin{itemize}% \item \emph{\hyperlink{PreymplecticPhaseSpace}{Covariant phase space}} \item \emph{\hyperlink{BVResolutionOfTheCovariantPhaseSpace}{BV-Resolution of the covariant phase space}} \item \emph{\hyperlink{HamiltonianLocalObservablesOnACauchySurface}{Hamiltonian local observables}} \end{itemize} $\,$ It might seem that with the construction of the [[local observables]] (def. \ref{LocalObservables}) on the [[on-shell]] [[space of field histories]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) the [[field theory]] defined by a [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) has been completely analyzed: This data specifies, in principle, which [[field histories]] are realized, and which [[observable]] properties these have. In particular, if the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) admit [[Cauchy surfaces]] (def. \ref{CauchySurface} below), i.e. spatial [[codimension]] 1 slices of [[spacetimes]] such that a [[field history]] is uniquely specified already by its restriction to the [[infinitesimal neighbourhood]] of that spatial slice, then a sufficiently complete collection of [[local observables]] whose spacetime support (def. \ref{SpacetimeSupport}) [[covering|covers]] that Cauchy surface allows to \emph{predict} the evolution of the field histories through time from that Cauchy surface. This is all what one might think a theory of physical fields should accomplish, and in fact this is essentially all that was thought to be required of a theory of nature from about [[Isaac Newton]]`s time to about [[Max Planck]]'s time. But we have seen that a remarkable aspect of [[Lagrangian field theory]] is that the [[de Rham differential]] of the [[local Lagrangian density]] $\mathbf{L}$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) decomposes into \emph{two} kinds of [[variational differential forms]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}), one of which is the [[Euler-Lagrange form]] which determines the [[equations of motion]] \eqref{EulerLagrangeEquationGeneral}. However, there is a second contribution: The \emph{[[presymplectic current]]} $\Omega_{BFV} \in \Omega^{p,2}_{\Sigma}(E)$ \eqref{PresymplecticCurrent}. Since this is of horizontal degree $p$, its [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) implies a further structure on the [[space of field histories]] restricted to [[spacetime]] [[submanifolds]] of dimension $p$ (i.e. of spacetime ``[[codimension]] 1''). There may be such submanifolds such that this restriction to their [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) does not actually change the [[on-shell]] [[space of field histories]], these are called the \emph{[[Cauchy surfaces]]} (def. \ref{CauchySurface} below). By the [[Hamiltonian Noether theorem]] (prop. \ref{HamiltonianDifferentialForms}) the [[presymplectic current]] induces [[infinitesimal symmetries]] acting on [[field histories]] and [[local observables]], given by the [[Poisson bracket Lie n-algebra|local Poisson bracket]] (prop. \ref{LocalPoissonBracket}). The [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) of the [[presymplectic current]] to these [[Cauchy surfaces]] yields the corresponding [[infinitesimal symmetry]] group acting on the [[on-shell]] [[field histories]], whose [[Lie bracket]] is the \emph{[[Poisson bracket]]} pairing on [[on-shell]] [[observables]] (example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} below). This data, the [[on-shell]] [[space of field histories]] on the [[infinitesimal neighbourhood]] of a [[Cauchy surface]] equipped with [[infinitesimal symmetry]] exhibited by the [[Poisson bracket]] is called the \emph{[[phase space]]} of the theory (def. \ref{PhaseSpaceAssociatedWithCauchySurface}) below. In fact if enough [[Cauchy surfaces]] exist, then the [[presymplectic forms]] associated with any one choice turn out do agree after [[pullback of differential forms|pullback]] to the full [[on-shell]] [[space of field histories]], exhibiting this as the \emph{[[covariant phase space]]} of the theory (prop. \ref{CovariantPhaseSpace} below) which is hence manifestly independent of aa choice of space/time splitting. Accordingly, also the [[Poisson bracket]] on [[on-shell]] [[observables]] exists in a covariant form; for [[free field theories]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] (def. \ref{GreenHyperbolicDifferentialOperator}) this is called the \emph{[[Peierls-Poisson bracket]]} (theorem \ref{PPeierlsBracket} below). The [[integral kernel]] for this [[Peierls-Poisson bracket]] is called the \emph{[[causal propagator]]} (prop. \ref{GreenFunctionsAreContinuous}). Its ``[[normal ordered product|normal ordered]]'' or ``[[positive real number|positive]] [[frequency]] component'', called the \emph{[[Wightman propagator]]} (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} below) as well as the corresponding [[time-ordered product|time-ordered]] variant, called the \emph{[[Feynman propagator]]} (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime} below), which we discuss in detail in \emph{\hyperlink{Propagators}{Propagators}} below, control the [[causal perturbation theory]] for constructing [[perturbative quantum field theory]] by [[deformation quantization|deforming]] the commutative pointwise product of [[on-shell]] [[observables]] to a [[non-commutative algebra|non-commutative product]] governed to first order by the [[Peierls-Poisson bracket]]. To see how such a [[deformation quantization]] comes about conceptually from the [[phase space]] strucure, notice from the basic principles of [[homotopy theory]] that given any [[structure]] on a [[space]] which is [[invariant]] with respect to a [[symmetry group]] [[action|acting]] on the space (here: the [[presymplectic current]]) then the true structure at hand is the [[homotopy quotient]] of that [[space]] by that [[symmetry group]]. We will explain this further below. This here just to point out that the [[homotopy quotient]] of the [[phase space]] by the [[Hamiltonian vector field|infinitesimal symmetries of the presymplectic current]] is called the \emph{[[symplectic groupoid]]} and that the \emph{true} [[algebra of observables]] is hence the ([[polarization|polarized]]) [[groupoid convolution algebra|convolution algebra of functions]] on this groupoid. This turns out to the ``[[algebra of quantum observables]]'' and the passage from the naive [[local observables]] on [[presymplectic manifold|presymplectic]] [[phase space]] to this non-commutative algebra of functions on its [[homotopy quotient]] to the [[symplectic groupoid]] is called \emph{[[quantization]]}. This we discuss in much detail \hyperlink{Quantization}{below}; for the moment this is just to motivate why the [[covariant phase space]] is the crucial construction to be extracted from a [[Lagrangian field theory]]. $\,$ \begin{displaymath} \itexarray{ \left\{ \itexarray{ \text{on-shell space} \\ \text{ of field histories} \\ \text{restricted to} \\ \text{Cauchy surface} } \right\} &\overset{\itexarray{ \text{homotopy} \\ \text{quotient} \\ \text{by} \\ \text{infinitesimal} \\ \text{symmetries} }}{\longrightarrow} & \left\{ \itexarray{ \text{covariant} \\ \text{phase space} } \right\} &\overset{ \itexarray{\text{Lie algebra} \\ \text{of functions} } }{\longrightarrow}& \left\{ \itexarray{ \text{Poisson algebra} \\ \text{of observables} } \right\} \\ & \searrow & \Big\downarrow{}^\mathrlap{{\text{Lie integration}}} && {}^{\mathllap{quantization}}\Big\downarrow \\ && \left\{ \itexarray{ \text{symplectic} \\ \text{groupoid} } \right\} & \overset{ \itexarray{ \text{polarized} \\ \text{convolution} \\ \text{algebra} } }{\longrightarrow}& \left\{ \itexarray{ \text{quantum algebra} \\ \text{of observables} } \right\} } \end{displaymath} $\,$ \textbf{Covariant phase space} \begin{defn} \label{CauchySurface}\hypertarget{CauchySurface}{} \textbf{([[Cauchy surface]])} Given a [[Lagrangian field theory]] $(E, \mathbf{L})$ on a [[spacetime]] $\Sigma$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), then a \emph{[[Cauchy surface]]} is a [[submanifold]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{SmoothManifoldInsideDiffeologicalSpaces}) such that the restriction map from the [[on-shell]] [[space of field histories]] $\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}$ \eqref{OnShellFieldHistories} to the space $\Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0}$ \eqref{OnShellFieldHistoriesInHigherCodimension} of on-shell field histories restricted to the [[infinitesimal neighbourhood]] of $\Sigma_p$ (example \ref{InfinitesimalNeighbourhood}) is an [[isomorphism]]: \begin{equation} \Gamma_\Sigma(E)_{\delta_{EL} \mathbf{L} = 0 } \underoverset{\simeq}{(-)\vert_{N_\Sigma \Sigma_p}}{\longrightarrow} \Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0} \,. \label{CauchySurfaceIsomorphismOnHistorySpace}\end{equation} \end{defn} \begin{example} \label{NormallyHyperbolicOperatorsHaveCauchySurfaces}\hypertarget{NormallyHyperbolicOperatorsHaveCauchySurfaces}{} \textbf{([[normally hyperbolic differential operators]] have [[Cauchy surfaces]])} Given a [[Lagrangian field theory]] $(E, \mathbf{L})$ on a [[spacetime]] $\Sigma$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) whose [[equations of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) are given by a [[normally hyperbolic differential operator]] (def. \ref{NormallyHyperbolicDifferentialOperator}), then it admits [[Cauchy surfaces]] in the sense of Def. \ref{CauchySurface}. \end{example} (e.g. \href{hyperbolic+differential+operator#BaerGinouxPfaeffle07}{Bär-Ginoux-Pfäffle 07, section 3.2}) \begin{defn} \label{PhaseSpaceAssociatedWithCauchySurface}\hypertarget{PhaseSpaceAssociatedWithCauchySurface}{} \textbf{([[phase space]] associated with a [[Cauchy surface]])} Given a [[Lagrangian field theory]] $(E, \mathbf{L})$ on a [[spacetime]] $\Sigma$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) and given a [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{CauchySurface}) then the corresponding \emph{[[phase space]]} is \begin{enumerate}% \item the [[super smooth set]] $\Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0}$ \eqref{OnShellFieldHistoriesInHigherCodimension} of [[on-shell]] [[field histories]] restricted to the [[infinitesimal neighbourhood]] of $\Sigma_p$; \item equipped with the [[differential 2-form]] (as in def. \ref{DifferentialFormsOnDiffeologicalSpaces}) \begin{equation} \omega_{\Sigma_p} \;\coloneqq\; \tau_{\Sigma_p}\left(\Omega_{BFV}\right) \;\in\; \Omega^2\left( \Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0} \right) \label{TransgressionOfPresymplecticCurrentToCauchySurface}\end{equation} which is the distributional [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) of the [[presymplectic current]] $\Omega_{BFV}$ (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) to $\Sigma_p$. This $\omega_{\Sigma_p}$ is a [[closed differential form]] in the sense of def. \ref{DifferentialFormsOnDiffeologicalSpaces}, due to prop. \ref{TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative} and using that $\Omega_{BFV} = \delta \Theta_{BFV}$ is closed by definition \eqref{PresymplecticCurrent}. As such this is called the \emph{[[presymplectic form]]} on the phase space. \end{enumerate} \end{defn} \begin{example} \label{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory}\hypertarget{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory}{} \textbf{(evaluation of [[transgression of variational differential forms|transgressed variational form]] on [[tangent vectors]] for [[free field theory]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) which is [[free field theory|free]] (def. \ref{FreeFieldTheory}) hence whose [[field bundle]] is a some [[smooth vector bundle|smooth]] [[super vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) and whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] is [[linear differential equation|linear]]. Then the [[synthetic differential geometry|synthetic]] [[tangent bundle]] (def. \ref{TangentBundleSynthetic}) of the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ \eqref{OnShellFieldHistories} with spacelike compact support (def \ref{CompactlySourceCausalSupport}) is canonically identified with the [[Cartesian product]] of this [[super smooth set]] with itself \begin{displaymath} T\left( \Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0} \right) \;\simeq\; \left(\Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}\right) \times \left(\Gamma_{\Sigma,scp}(E)_{\delta_{EL} \mathbf{L} = 0}\right) \,. \end{displaymath} With field coordinates as in example \ref{TrivialVectorBundleAsAFieldBundle}, we may expand the [[presymplectic current]] as \begin{displaymath} \Omega_{BFV} = \left(\Omega_{BFV}\right)^{\mu_1, \cdots, \mu_{k_1}, \nu_1, \cdots, \nu_{k_2}, \kappa}_{a_1 a_2} \delta \phi^{a_1}_{\mu_1 \cdots \mu_k} \wedge \delta \phi^{a_2}_{\nu_1 \cdots \nu_{k_2}} \wedge \iota_{\partial_\kappa} dvol_\Sigma \,, \end{displaymath} where the components $(\Omega_{BFV})_{a_1 a_2}^{\mu_1, \cdots, \mu_{k_1}, \nu_1, \cdots, \nu_{k_2}, \kappa}$ are smooth functions on the [[jet bundle]]. Under these identifications the value of the [[presymplectic form]] $\omega_{\Sigma_p}$ \eqref{TransgressionOfPresymplecticCurrentToCauchySurface} on two [[tangent vectors]] $\vec \Phi_1, \vec \Phi_2 \in \Gamma_{\Sigma,scp}(E)$ at a point $\Phi \in \Gamma_{\Sigma,scp}(E)$ is \begin{displaymath} \omega_{\Sigma_p}(\vec \Phi_1, \vec \Phi_2) \;=\; \underset{\Sigma_p}{\int} \left(\Omega_{BFV}\right)^{\mu_1, \cdots, \mu_{k_1}, \nu_1, \cdots, \nu_{k_2}, \kappa}_{a_1 a_2}(\Phi(x)) \left( \frac{\partial}{\partial x^{\mu_1}} \cdots \frac{\partial}{\partial x^{\mu_{k_1}}} \vec \Phi_1(x) \right) \left( \frac{\partial}{\partial x^{\nu_1}} \cdots \frac{\partial}{\partial x^{\nu_{k_2}}} \vec \Phi_2(x) \right) \, \iota_{\partial_\kappa} dvol_\Sigma(x) \,. \end{displaymath} \end{example} \begin{example} \label{PresymplecticFormForFreeRealScalarField}\hypertarget{PresymplecticFormForFreeRealScalarField}{} \textbf{([[presymplectic form]] for [[free field|free]] [[real scalar field]])} Consider the [[Lagrangian field theory]] for the [[free field|free]] [[real scalar field]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}. Under the identification of example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} the [[presymplectic form]] on the [[phase space]] (def. \ref{PhaseSpaceAssociatedWithCauchySurface}) associated with a [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ is given by \begin{displaymath} \begin{aligned} \omega_{\Sigma_p}(\vec \Phi_1, \vec\Phi_2) & = \int_{\Sigma_{p}} \left( \frac{\partial \vec \Phi_1}{\partial x^\mu}(x) \vec \Phi_2(x) - \vec \Phi_1(x) \frac{\partial \vec \Phi_2}{\partial x^\mu}(x) \right) \eta^{\mu \nu} \iota_{\partial_\mu} dvol_{\Sigma_{p}}(x) \\ & = \underset{\Sigma_p}{\int} K(\vec \Phi_1, \vec \Phi_2) \,. \end{aligned} \end{displaymath} Here the first equation follows via example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} from the form of $\Omega_{BFV}$ from example \ref{FreeScalarFieldEOM}, while the second equation identifies the integrand as the witness $K$ for the [[formally adjoint differential operator|formally self-adjointness]] of the [[Klein-Gordon equation]] from example \ref{FormallySelfAdjointKleinGordonOperator}. \end{example} \begin{example} \label{PresymplecticFormForFreeDiracField}\hypertarget{PresymplecticFormForFreeDiracField}{} \textbf{([[presymplectic form]] for [[free field theory|free]] [[Dirac field]])} Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] (example \ref{LagrangianDensityForDiracField}). Under the identification of example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} the [[presymplectic form]] on the [[phase space]] (def. \ref{PhaseSpaceAssociatedWithCauchySurface}) associated with a [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ is given by \begin{displaymath} \begin{aligned} \omega_{\Sigma_p}(\theta_1 \vec \Psi_1, \theta_2 \vec\Psi_2) & = \int_{\Sigma_{p}} \left( \overline{\theta_1 \vec \psi_1}\gamma^\mu \left( \theta_2 \vec \Psi_2 \right) \right) \iota_{\partial_\mu} dvol_{\Sigma_{p}}(x) \\ & = \underset{\Sigma_p}{\int} K(\vec \Phi_1, \vec \Phi_2) \,. \end{aligned} \end{displaymath} Here the first equation follows via example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory} from the form of $\Omega_{BFV}$ from example \ref{PresymplecticCurrentDiracField}, while the second equation identifies the integrand as the witness $K$ for the [[formally adjoint differential operator|formally self-adjointness]] of the [[Dirac equation]] from example \ref{DiracOperatorOnDiracSpinorsIsFormallySelfAdjointDifferentialOperator}. \end{example} \begin{prop} \label{CovariantPhaseSpace}\hypertarget{CovariantPhaseSpace}{} \textbf{([[covariant phase space]])} Consider $(E, \mathbf{L})$ a [[Lagrangian field theory]] on a [[spacetime]] $\Sigma$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). Let \begin{displaymath} \Sigma_{tra} \overset{tra}{\hookrightarrow} \Sigma \end{displaymath} be a [[submanifold]] [[manifold with boundary|with two boundary components]] $\partial \Sigma_{tra} = \Sigma_{in} \sqcup \Sigma_{out}$ , both of which are [[Cauchy surfaces]] (def. \ref{CauchySurface}). Then the corresponding inclusion diagram \begin{displaymath} \itexarray{ && \Sigma_{tra} \\ & {}^{\mathllap{in}}\nearrow && \nwarrow^{\mathrm{out}} \\ \Sigma_{in} && && \Sigma_{out} } \end{displaymath} induces a [[Lagrangian correspondence]] between the associated [[phase spaces]] (def. \ref{PhaseSpaceAssociatedWithCauchySurface}) \begin{displaymath} \itexarray{ && \Gamma_{\Sigma_{tra}}(E)_{\delta_{EL} \mathbf{L} = 0} \\ & {}^{\mathllap{ (-)\vert_{in} }}\swarrow && \searrow^{\mathrlap{ (-)\vert_{out} }} \\ \Gamma_{\Sigma^{(in)}}(E)_{\delta_{EL}\mathbf{L}= 0} && && \Gamma_{\Sigma^{(out)}}(E)_{\delta_{EL}\mathbf{L}= 0} \\ & {}_{\mathllap{\omega_{in}}}\searrow && \swarrow_{\mathrlap{\omega_{out}}} \\ && \mathbf{\Omega}^{2} } \end{displaymath} in that the [[pullback of differential forms|pullback]] of the two [[presymplectic forms]] \eqref{TransgressionOfPresymplecticCurrentToCauchySurface} coincides on the space of field histories: \begin{displaymath} \left( (-)\vert_{in}\right)^\ast\left( \omega_{in}\right) \;=\; \left( (-)\vert_{out} \right)^\ast \left( \omega_{out} \right) \phantom{AAAA} \in \Omega^2 \left( \Gamma_{\Sigma_{tra}}(E)_{\delta_{EL} \mathbf{L} = 0} \right) \,. \end{displaymath} Hence there is a well defined [[presymplectic form]] \begin{displaymath} \omega \in \Omega^2\left( \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L}} = 0 \right) \end{displaymath} on the genuine [[space of field histories]], given by $\omega \coloneqq i^\ast \omega_{\Sigma_p}$ for any Cauchy surface $\Sigma_p \overset{i}{\hookrightarrow} \Sigma$. This [[presymplectic smooth space]] \begin{displaymath} \left( \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L}} \,,\, \omega \right) \end{displaymath} is therefore called the \emph{[[covariant phase space]]} of the [[Lagrangian field theory]] $(E,\mathbf{L})$. \end{prop} \begin{proof} By prop. \ref{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell} the total spacetime derivative $d \Omega_{BFV}$ of the [[presymplectic current]] vanishes [[on-shell]]: \begin{displaymath} d \Omega_{BFV} = - \delta \delta_{EL} \mathbf{L} \end{displaymath} in that the [[pullback of differential forms|pullback]] (def. \ref{PullbackOfDifferential1FormsOnCartesianSpaces}) along the [[shell]] inclusion $\mathcal{E} \overset{i_{\mathcal{E}}}{\hookrightarrow} J^\infty_\Sigma(E)$ \eqref{ShellInJetBundle} vanishes: \begin{displaymath} \begin{aligned} (i_{\mathcal{E}})^\ast \left( d \Omega_{BFV} \right) & = - (i_{\mathcal{E}})^\ast \left( \delta \delta_{EL} \mathcal{L} \right) \\ & = - \delta \underset{ = 0 }{ \underbrace{ (i_{\mathcal{E}})^\ast \left( \delta_{EL} \mathbf{L} \right) } } \\ & = 0 \end{aligned} \end{displaymath} This implies that the transgression of $d \Omega_{BFV}$ to the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma_{tra}}(E)_{\delta_{EL}\mathbf{L} = 0}$ vanishes (since by definition \eqref{EquationOfMotionEL} that involves pulling back through the shell inclusion) \begin{displaymath} \tau_{\Sigma_{tra}}(d \Omega_{BFV}) = 0 \,. \end{displaymath} But then the claim follows with prop. \ref{TransgressionOfVariationaldifferentialFormsCompatibleWithVariationalDerivative}: \begin{displaymath} \begin{aligned} 0 & = \tau_{\Sigma_{tra}}(d \Omega_{BFV}) \\ & = ((-)\vert_{\Sigma_{tra}})^\ast \tau_{\partial \Sigma_{tra}} \Omega_{BFV} \,. \end{aligned} \end{displaymath} \end{proof} \begin{theorem} \label{PPeierlsBracket}\hypertarget{PPeierlsBracket}{} \textbf{([[polynomial Poisson algebra|polynomial Poisson bracket]] on [[covariant phase space]] -- the [[Peierls bracket]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) such that \begin{enumerate}% \item it is a [[free field theory]] (def. \ref{FreeFieldTheory}) \item whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] $P \Phi = 0$ (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) is \begin{enumerate}% \item [[formally adjoint differential operator|formally self-adjoint]] or [[formally adjoint differential operator|formally anti self-adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}) such that \begin{itemize}% \item the integral over the witness $K$ \eqref{FormallyAdjointDifferentialOperatorWitness} is the [[presymplectic form]] \eqref{TransgressionOfPresymplecticCurrentToCauchySurface}: $\omega_{\Sigma_p} = \underset{\Sigma_p}{\int} K$; \end{itemize} \item [[Green hyperbolic differential operator|Green hyperbolic]] (def. \ref{GreenHyperbolicDifferentialOperator}). \end{enumerate} \end{enumerate} Write \begin{displaymath} \mathrm{G}_P \;\colon\; LinObs(E_{scp},\mathbf{L})^{reg} \overset{\mathrm{G}_P}{\longrightarrow} \Gamma_{\Sigma,scp}(E)_{\delta_{EL}\mathbf{L} = 0} \end{displaymath} for the linear map from regular linear field observables (def. \ref{RegularLinearFieldObservables}) to on-shell [[field histories]] with spatially compact support (def. \ref{CompactlySourceCausalSupport}) given under the identification \eqref{RegularLinearObservablesAreCompactlySupportedSectionsModuloImageOfP} by the [[causal Green function]] $\mathrm{G}_P$ (def. \ref{AdvancedAndRetardedGreenFunctions}). Then for every [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{CauchySurface}) this map is an inverse to the [[presymplectic form]] $\omega_{\Sigma_p}$ (def. \ref{PhaseSpaceAssociatedWithCauchySurface}) in that, under the identification of tangent vectors to field histories from example \ref{EvaluationOfTransgressedVariationalFormsOnTangentVectorsForFreeFieldTheory}, we have that the composite \begin{equation} \itexarray{ \omega_{\Sigma_p}(\mathrm{G}_P(-),(-)) \;=\; ev &\colon& LinObs(E_{scp},\mathbf{L})^{reg} &\otimes& \Gamma_{\Sigma,scp}(E) &\longrightarrow& \mathbb{C} \\ && (A &,& \Phi) &\mapsto& A(\Phi) } \label{ForGreenHyperbolicFreeFieldTheoryCausalGreenFunctionIsInverseToPresymplecticFormOnRegularLinearObservables}\end{equation} equals the [[evaluation map]] of observables on field histories. This means that for every [[Cauchy surface]] $\Sigma_p$ the [[presymplectic form]] $\omega_{\Sigma_p}$ restricts to a \emph{[[symplectic form]]} on regular linear observables. The corresponding \emph{[[Poisson bracket]]} is \begin{displaymath} \left\{ -,- \right\}_{\Sigma_p} \;\coloneqq\; \omega_{\Sigma_p}(\mathrm{G}_P(-), \mathrm{G}_P(-)) \;\;\colon\;\; LinObs(E_{scp},\mathbf{L})^{reg} \otimes LinObs(E_{scp},\mathbf{L})^{reg} \longrightarrow \mathbb{R} \,. \end{displaymath} Moreover, equation \eqref{ForGreenHyperbolicFreeFieldTheoryCausalGreenFunctionIsInverseToPresymplecticFormOnRegularLinearObservables} implies that this is the \emph{covariant [[Poisson bracket]]} in the sense of the [[covariant phase space]] (def. \ref{CovariantPhaseSpace}) in that it does not actually depend on the choice of [[Cauchy surface]]. An equivalent expression for the Poisson bracket that makes its independence from the choice of Cauchy surface manifest is the \emph{$P$-[[Peierls bracket]]} given by \begin{equation} \itexarray{ LinObs(E_{scp},\mathbf{L})^{reg} \otimes LinObs(E_{scp},\mathbf{L})^{reg} &\overset{\{-,-\}}{\longrightarrow}& \mathbb{R} \\ (\alpha^\ast, \beta^\ast) &\mapsto& \underset{\Sigma}{\int} \mathrm{G}(\alpha^\ast) \cdot \beta^\ast \, dvol_\Sigma } \label{ThePPeierlsBracket}\end{equation} where on the left $\alpha^\ast, \beta^\ast \in \Gamma_{\Sigma,cp}(E^\ast) \simeq LinObs(E_{scp},\mathbf{L})^{reg}$ Hence under the given assumptions, for every Cauchy surface the [[Poisson bracket]] associated with that Cauchy surface equals the invariantly (``covariantly'') defined [[Peierls bracket]] \begin{displaymath} \{-,-\}_{\Sigma_p} = \{-,-\} \,. \end{displaymath} Finally this means that in terms of the [[causal propagator]] $\Delta$ \eqref{CausalPropagator} the covariant [[Peierls-Poisson bracket]] is given in [[generalized function]]-notation by \begin{equation} \{\alpha^\ast, \beta^\ast\} \;=\; \underset{\Sigma}{\int} \underset{\Sigma}{\int} \alpha^\ast(x) \cdot \Delta(x,y) \cdot \beta^\ast(y) \, dvol_\Sigma(x)\, dvol_\Sigma(y) \label{CausalPropagatorPPeierlsBracket}\end{equation} Therefore, while the point-evaluation field observables $\mathbf{\Phi}^a(x)$ (def. \ref{PointEvaluationObservables}) are not themselves regular observables (def. \ref{RegularLinearFieldObservables}), the [[Peierls-Poisson bracket]] \eqref{CausalPropagatorPPeierlsBracket} is induced from the following distributional bracket between them \begin{displaymath} \left\{ \mathbf{\Phi}^a(x) , \mathbf{\Phi}^b(y) \right\} \;=\; \Delta^{a b}(x,y) \end{displaymath} with the [[causal propagator]] \eqref{CausalPropagator} on the right, in that with the identification \eqref{AverageOfFieldObservableIsRegularLinearObservables} the [[Peierls-Poisson bracket]] on regular linear observables arises as follows: \begin{displaymath} \begin{aligned} \left\{ \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x) \,,\, \underset{\Sigma}{\int} \beta^\ast_b(y) \mathbf{\Phi}^b(y) \, dvol_\Sigma(y) \right\} & = \underset{\Sigma}{\int} \underset{\Sigma}{\int} \alpha^\ast_a(x) \underset{= \Delta^{a b}(x,y)}{ \underbrace{ \left\{ \mathbf{\Phi}^a(x), \mathbf{\Phi}^b(y) \right\} } } \beta^\ast_b(y) \, dvol_\Sigma(x)\, dvol_\Sigma(y) \\ & = \underset{\Sigma}{\int} \underset{\Sigma}{\int} \alpha^\ast_a(x) \Delta^{a b}(x,y) \beta^\ast_b(y) \, dvol_\Sigma(x)\, dvol_\Sigma(y) \end{aligned} \end{displaymath} \end{theorem} (\href{Green+hyperbolic+partial+differential+equation#Khavkine14}{Khavkine 14, lemma 2.5}) \begin{proof} Consider two more Cauchy surfaces $\Sigma_p^\pm \hookrightarrow I^\pm(\Sigma) \hookrightarrow \Sigma$, in the [[future]] $I^+$ and in the [[past]] $I^-$ of $\Sigma$, respectively. Choose a [[partition of unity]] on $\Sigma$ consisting of two elements $\chi^\pm \in C^\infty(\Sigma)$ with [[support]] bounded by these Cauchy surfaces: $supp(\chi_\pm) \subset I^\pm(\Sigma^{\mp})$. Then define \begin{equation} P_\chi \;\colon\; \Gamma_{\Sigma,scp}(E) \longrightarrow \Gamma_{\Sigma,cp}(E^\ast) \label{SplittingOfGreenExactSequenceType}\end{equation} by \begin{equation} \begin{aligned} P_\chi(\Phi) & \coloneqq \phantom{-} P(\chi_+ \Phi) \\ & = - P(\chi_- \Phi) \,. \end{aligned} \label{SplittingOfGreenExactSequence}\end{equation} Notice that the [[support]] of the partitioned field history is in the compactly sourced future/past cone \begin{equation} \chi_\pm \Phi \;\in\; \Gamma_{\Sigma,\pm cp}(E) \label{ChipmPhiIsSupportedInPastFuture}\end{equation} since $\Phi$ is supported in the compactly sourced causal cone, but that $P(\chi_\pm \Phi)$ indeed has [[compact support]] as required by \eqref{SplittingOfGreenExactSequenceType}: Since $P(\Phi) = 0$, by assumption, the support is the intersection of that of $\Phi$ with that of $d \chi_\pm$, and the first is spacelike compact by assumption, while the latter is timelike compact, by definition of partition of unity. Similarly, the equality in \eqref{SplittingOfGreenExactSequence} holds because by [[partition of unity]] $P(\chi_+ \Phi) + P(\chi_-\Phi) = P((\chi_+ + \chi_-)\Phi ) = P(\Phi) = 0$. It follows that \begin{equation} \begin{aligned} \mathrm{G}_P \circ P_\chi (\Phi) & = \left( \mathrm{G}_{P,+} - \mathrm{G}_{P,-} \right) P_\chi (\Phi) \\ & = \underset{ = \chi_+ \Phi}{\underbrace{\mathrm{G}_{P,+} P(\chi_+ \Phi)}} + \underset{ = \chi_- \Phi }{\underbrace{\mathrm{G}_{P,-} P(\chi_- \Phi)}} \\ & = (\chi_+ + \chi_-)\Phi \\ & = \Phi \,, \end{aligned} \label{PchiIsRightInverseToGP}\end{equation} where in the second line we chose from the two equivalent expressions \eqref{SplittingOfGreenExactSequence} such that via \eqref{ChipmPhiIsSupportedInPastFuture} the defining property of the [[advanced and retarded Green functions|advanced or retarded Green function]], respectively, may be applied, as shown under the braces. (\href{Green+hyperbolic+differential+equation#Khavkine14}{Khavkine 14, lemma 2.1}) Now we apply this to the computation of $\omega_{\Sigma_p}(\mathrm{G}_P(-),-)$: \begin{displaymath} \begin{aligned} \omega_{\Sigma_P}(\mathrm{G}_P(\alpha^\ast),\vec \Phi) & = \underset{\Sigma_P}{\int} K(\mathrm{G}_P(\alpha^\ast), \vec \Phi) \\ & = \underset{\Sigma_P}{\int} K(\mathrm{G}_P(\alpha^\ast), \chi_+\vec \Phi) + \underset{\Sigma_P}{\int} K(\mathrm{G}_P(\alpha^\ast), \chi_-\vec \Phi) \\ & = \underset{I^-(\Sigma_P)}{\int} d K(\mathrm{G}_P(\alpha^\ast), \chi_+\vec \Phi) - \underset{I^+(\Sigma_P)}{\int} d K(\mathrm{G}_P(\alpha^\ast), \chi_-\vec \Phi) \\ & = \underset{I^-(\Sigma_P)}{\int} \left( \underset{= 0}{ \underbrace{ P(\mathrm{G}_P(\alpha^\ast))}} \cdot \chi_+\vec \Phi \mp \mathrm{G}_P(\alpha^\ast) \cdot P(\chi_+ \vec \Phi) \right) dvol_\Sigma - \underset{I^+(\Sigma_P)}{\int} \left( \underset{= 0}{ \underbrace{ P(\mathrm{G}_P(\alpha^\ast))}} \cdot \chi_-\vec \Phi \mp \mathrm{G}_P(\alpha^\ast) \cdot P(\chi_- \vec \Phi) \right) dvol_\Sigma \\ & = \mp \left( \underset{I^-(\Sigma_P)}{\int} \mathrm{G}_P(\alpha^\ast) \cdot P(\chi_+ \vec \Phi) dvol_\Sigma + \underset{I^+(\Sigma_P)}{\int} \mathrm{G}_P(\alpha^\ast) \cdot P(\chi_+ \vec \Phi) dvol_\Sigma \right) \\ & = \underset{\Sigma}{\int} \mathrm{G}_P(\alpha^\ast) \cdot P(\chi_+ \vec \Phi) dvol_\Sigma \\ & = \underset{\Sigma}{\int} \alpha^\ast \cdot \mathrm{G}_{P} (P (\chi_+ \vec \Phi)) \\ & = \underset{\Sigma}{\int} \alpha^\ast \cdot \vec \Phi \end{aligned} \end{displaymath} Here we computed as follows: \begin{enumerate}% \item applied the assumption that $\omega_{\Sigma_p}(-,-) = \underset{\Sigma_p}{\int} K(-,-)$; \item applied the above partition of unity; \item used the [[Stokes theorem]] (prop. \ref{StokesTheorem}) for the past and the future of $\Sigma_p$, respectively; \item applied the definition of $d K$ as the witness of the formal (anti-) self-adjointness of $P$ (def. \ref{FormallyAdjointDifferentialOperators}); \item used $P\circ \mathrm{G}_p = 0$ on $\Gamma_{\Sigma,cp}(E^\ast)$ (def. \ref{AdvancedAndRetardedGreenFunctions}) and used \eqref{SplittingOfGreenExactSequence}; \item unified the two integration domains, now that the integrands are the same; \item used the formally (anti-)self adjointness of the Green functions (example \ref{CausalGreenFunctionOfFormallyAdjointDifferentialOperatorAreFormallyAdjoint}); \item used \eqref{PchiIsRightInverseToGP}. \end{enumerate} \end{proof} \begin{example} \label{PeierlsBracketEistsForScalarFieldAndDiracField}\hypertarget{PeierlsBracketEistsForScalarFieldAndDiracField}{} \textbf{([[scalar field]] and [[Dirac field]] have [[covariant phase space|covariant]] [[Peierls-Poisson bracket]])} Examples of [[free field theory|free]] [[Lagrangian field theories]] for which the assumptions of theorem \ref{PPeierlsBracket} are satisfied, so that the covariant [[Poisson bracket]] exists in the form of the [[Peierls bracket]] include \begin{itemize}% \item the [[free field theory|free]] [[real scalar field]] (example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}); \item the [[free field theory|free]] [[Dirac field]] (example \ref{LagrangianDensityForDiracField}). \end{itemize} For the [[free field theory|free]] [[scalar field]] this is the statement of example \ref{GreenHyperbolicKleinGordonEquation} with example \ref{PresymplecticFormForFreeRealScalarField}, while for the [[Dirac field]] this is the statement of example \ref{GreenHyperbolicDiracOperator} with example \ref{PresymplecticFormForFreeDiracField}. \end{example} For the [[free field theory|free]] [[electromagnetic field]] (example \ref{ElectromagnetismLagrangianDensity}) the assumptions of theorem \ref{PPeierlsBracket} are violated, the [[covariant phase space]] does not exist. But in the discussion of \emph{\hyperlink{GaugeFixing}{Gauge fixing}}, below, we will find that for an equivalent re-incarnation of the electromagnetic field, they are met after all. $\,$ \textbf{BV-resolution of the covariant phase space} So far we have discussed the [[covariant phase space]] (prop. \ref{CovariantPhaseSpace}) in terms of explicit restriction to the [[shell]]. We now turn to the more flexible perspective where a [[homological resolution]] of the [[shell]] in terms of ``[[antifields]]'' is used (def. \ref{BVComplexOfOrdinaryLagrangianDensity}). \begin{example} \label{BVPresymplecticCurrent}\hypertarget{BVPresymplecticCurrent}{} \textbf{(BV-presymplectic current)} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) whose [[field bundle]] $E$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) and whose [[Lagrangian density]] $\mathbf{L}$ is spacetime-independent (example \ref{ShellForSpacetimeIndependentLagrangians}). Let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}$ be a constant section of the shell \eqref{ConstantSectionOfTrivialShellBundle}. Then in the BV-variational bicomplex \eqref{ComparisonMorphismFromOrdinaryBVComplexToLocalObservables} there exists the \emph{BV-presymplectic potential} \begin{equation} \Theta_{BV} \;\coloneqq\; \phi^{\ddagger}_a \delta \phi^a \, dvol_\Sigma \;\in\; \Omega^{p,1}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}} \label{BVPresymplecticPotential}\end{equation} and the corresponding \emph{BV-presymplectic current} \begin{displaymath} \Omega_{BV} ;\in\; \Omega^{p,2}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}} \end{displaymath} defined by \begin{displaymath} \begin{aligned} \Omega_{BV} & \coloneqq \delta \Theta_{BV} \\ & = \delta \phi^{\ddagger}_a \wedge \delta \phi^a \wedge dvol_{\Sigma} \end{aligned} \,, \end{displaymath} where $(\phi^a)$ are the given [[field (physics)|field]] [[coordinates]], $\phi^{\ddagger}_a$ the corresponding [[antifield]] coordinates \eqref{AntifieldCoordinates} and $\frac{\delta_{EL} \mathbf{L}}{\delta \phi^a}$ the corresponding components of the [[Euler-Lagrange form]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}). \end{example} \begin{prop} \label{ResolutionOfCovariantPhaseSpaceCorrespondence}\hypertarget{ResolutionOfCovariantPhaseSpaceCorrespondence}{} \textbf{(local BV-BFV relation)} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) whose [[field bundle]] $E$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) and whose [[Lagrangian density]] $\mathbf{L}$ is spacetime-independent (example \ref{ShellForSpacetimeIndependentLagrangians}). Let $\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}$ be a constant section of the shell \eqref{ConstantSectionOfTrivialShellBundle}. Then the BV-presymplectic current $\Omega_{BV}$ (def. \ref{BVPresymplecticCurrent}) witnesses the [[on-shell]] vanishing (prop. \ref{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell}) of the [[total derivative|total spacetime derivative]] of the genuine [[presymplectic current]] $\Omega_{BFV}$ (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) in that the [[total derivative|total spacetime derivative]] of $\Omega_{BFV}$ equals the BV-differential $s_{BV}$ of $\Omega_{BV}$: \begin{displaymath} d \Omega_{BFV} = s \Omega_{BV} \,. \end{displaymath} Hence if $\Sigma_{tra} \hookrightarrow \Sigma$ is a [[submanifold]] of [[spacetime]] of full dimension $p+1$ [[manifold with boundary|with boundary]] $\partial \Sigma_{tra} = \Sigma_{in} \sqcup \Sigma_{out}$ \begin{displaymath} \itexarray{ && \Sigma_{tra} \\ & {}^{\mathllap{in}}\nearrow && \nwarrow^{\mathrm{out}} \\ \Sigma_{in} && && \Sigma_{out} } \end{displaymath} then the [[pullback of differential forms|pullback]] of the two [[presymplectic forms]] \eqref{TransgressionOfPresymplecticCurrentToCauchySurface} on the incoming and outgoing [[spaces of field histories]], respectively, differ by the BV-differential of the transgression of the BV-presymplectic current: \begin{displaymath} \left( (-)\vert_{in}\right)^\ast\left( \omega_{in}\right) \;-\; \left( (-)\vert_{out} \right)^\ast \left( \omega_{out} \right) = \tau_{\mathbb{D} \times \Sigma_{tra}} ( s \Omega_{BV} ) \phantom{AAAA} \in \Omega^2 \left( \Gamma_{\Sigma_{tra}}(E)_{\delta_{EL} \mathbf{L} = 0} \right) \,. \end{displaymath} This [[homological resolution]] of the [[Lagrangian correspondence]] that exhibits the ``covariance'' of the [[covariant phase space]] (prop. \ref{CovariantPhaseSpace}) is known as the \emph{BV-BFV relation} (\href{BV-BRST+formalism#CattaneoMnevReshetikhin12}{Cattaneo-Mnev-Reshetikhin 12 (9)}). \end{prop} \begin{proof} For the first statement we compute as follows: \begin{displaymath} \begin{aligned} s \Omega_{BV} & = - \delta (s \phi^{\ddagger}_a) \delta \phi^a \wedge dvol_{\Sigma} \\ & = - \delta \frac{\delta_{EL}L }{\delta \phi^a} \delta \phi^a dvol_{\Sigma} \\ & = - \delta \delta_{EL}\mathbf{L} \\ & = d \Omega_{BFV} \,, \end{aligned} \end{displaymath} where the first steps simply unwind the definitions, and where the last step is prop. \ref{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell}. With this the second statement follows by immediate generalization of the proof of prop. \ref{CovariantPhaseSpace}. \end{proof} \begin{example} \label{DerivedPresymplecticCurrentOfRealScalarField}\hypertarget{DerivedPresymplecticCurrentOfRealScalarField}{} \textbf{(derived [[presymplectic current]] of [[real scalar field]])} Consider a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) without any non-trivial implicit [[infinitesimal gauge transformations]] (def. \ref{ImplicitInfinitesimalGaugeSymmetry}); for instance the [[real scalar field]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}. Inside its [[local BV-complex]] (def. \ref{BVComplexOfOrdinaryLagrangianDensity}) we may form the linear combination of \begin{enumerate}% \item the [[presymplectic current]] $\Omega_{BFV}$ (example \ref{FreeScalarFieldEOM}) \item the BF-presymplectic current $\Omega_{BV}$ (example \ref{BVPresymplecticCurrent}). \end{enumerate} This yields a vertical 2-form \begin{displaymath} \Omega \;\coloneqq\; \Omega_{BV} + \Omega_{BFV} \;\; \in \Omega^{p,2}_\Sigma(E)\vert_{\mathcal{E}_{BV}} \end{displaymath} which might be called the \emph{derived presymplectic current}. Similarly we may form the linear combination of 1. the presymplectic potential current $\Theta_{BFV}$ \eqref{dLDecomposition} \begin{enumerate}% \item the BF-presymplectic potential current $\Theta_{BV}$ \eqref{BVPresymplecticPotential} \item the [[Lagrangian density]] $\mathbf{L}$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) \end{enumerate} hence \begin{displaymath} \Theta \;\coloneqq\; \Theta_{BV} + \underset{Lepage}{\underbrace{ \Theta_{BFV} + \mathbf{L} }} \end{displaymath} (where the sum of the two terms on the right is the [[Lepage form]] \eqref{TheLepage}). This might be called the \emph{derived presymplectic potental current}. We then have that \begin{displaymath} (\delta + (d-s))\Omega \;=\; 0 \end{displaymath} and in fact \begin{displaymath} (\delta + (d-s))\Theta \;=\; \Omega \,. \end{displaymath} \end{example} \begin{proof} Of course the first statement follows from the second, but in fact the two contributions of the first statement even vanish separately: \begin{displaymath} \delta \Omega = 0 \,, \phantom{AAAA} (d-s)\Omega = 0 \,. \end{displaymath} The statement on the left is immediate from the definitions, since $\Omega = \delta \Theta$. For the statement on the right we compute \begin{displaymath} \begin{aligned} (d - s) (\Omega_{BV} + \Omega_{BFV}) & = \underset{= 0}{\underbrace{d \Omega_{BFV} - \underset{ = 0 }{\underbrace{ s \Omega_{BV}}} }} + \underset{ = 0}{\underbrace{ d \Omega_{BV} - s \Omega_{BFV} }} \\ & = 0 \end{aligned} \end{displaymath} Here the first term vanishes via the local BV-BFV relation (prop. \ref{ResolutionOfCovariantPhaseSpaceCorrespondence}) while the other two terms vanish simply by degree reasons. Similarly for the second statement we compute as follows: \begin{displaymath} \begin{aligned} (\delta + (d - s) ) \Theta & = \underset{ = \Omega_{BV} + \Omega_{BFV}}{\underbrace{ \delta (\Theta_{BV} + \Theta_{BFV}) }} + \underset{ = \delta \mathbf{L}}{\underbrace{\mathbf{d} \mathbf{L}}} + \underset{ = 0 }{\underbrace{ (d-s) \mathbf{L} }} + (d-s)(\Theta_{BV} + \Theta_{BFV}) \\ & = \Omega_{BV} + \Omega_{BFV} + \delta \mathbf{L} + \underset{ = 0}{\underbrace{d \Theta_{BV}}} - \underset{ = \delta_{EL} \mathbf{L} }{\underbrace{ s \Theta_{BV}}} + \underset{ = \delta_{EL}\mathbf{L} - \delta \mathbf{L} }{\underbrace{ d \Theta_{BFV} } } - \underset{ = 0 }{\underbrace{ s \Theta_{BFV} }} \\ & = \Omega_{BV} + \Omega_{BFV} \end{aligned} \,. \end{displaymath} Here the direct vanishing of various terms is again by simple degree reasons, and otherwise we used the definition of $\Omega$ and, crucially, the variational identity $\delta \mathbf{L} = \delta_{EL}\mathbf{L} - d \Theta_{BFV}$ \eqref{dLDecomposition}. \end{proof} $\,$ \textbf{Hamiltonian local observables} We have defined the \emph{[[local observables]]} (def. \ref{LocalObservables}) as the [[transgression of variational differential forms|transgressions]] of horizontal $p+1$-forms (with compact spacetime support) to the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ over all of [[spacetime]] $\Sigma$. More explicitly, these could be called the \emph{spacetime local observables}. But with every choice of [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{CauchySurface}) comes another notion of local observables: those that are [[transgression of variational differential forms|transgressions]] of horizontal $p$-forms (instead of $p+1$-forms) to the [[on-shell]] [[space of field histories]] restricted to the [[infinitesimal neighbourhood]] of that Cauchy surface (def. \ref{FieldHistoriesOnInfinitesimalNeighbourhoodOfSubmanifoldOfSpacetime}): $\Gamma_{\Sigma_p}(E)_{\delta_{EL} \mathbf{L} = 0}$. These are \emph{spatially local observables}, with respect to the given choice of [[Cauchy surface]]. Among these spatially local observables are the \emph{Hamiltonian local observables} (def. \ref{HamiltonianLocalObservables} below) which are [[transgression of variational differential forms|transgressions]] specifically of the [[Hamiltonian differential forms|Hamiltonian forms]] (def. \ref{HamiltonianForms}). These inherit a transgression of the [[Poisson bracket Lie n-algebra|local Poisson bracket]] (prop. \ref{LocalPoissonBracket}) to a [[Poisson bracket]] on Hamiltonian local observables (def. \ref{PoissonBracketOnHamiltonianLocalObservables} below). This is known as the \emph{[[Peierls bracket]]} (example \ref{PoissonBracketForRealScalarField} below). \begin{defn} \label{HamiltonianLocalObservables}\hypertarget{HamiltonianLocalObservables}{} \textbf{(Hamiltonian local observables)} Let $(E, \mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). Consider a [[local observable]] (def. \ref{LocalObservables}) \begin{displaymath} \tau_\Sigma(A) \;\colon\; \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \longrightarrow \mathbb{C} \,, \end{displaymath} hence the [[transgression of variational differential forms|transgression]] of a variational horizontal $p+1$-form $A \in \Omega^{p+1,0}_{\Sigma,cp}(E)$ of compact spacetime support. Given a [[Cauchy surface]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{CauchySurface}) we say that $\tau_\Sigma (A)$ is \emph{[[Hamiltonian]]} if it is also the transgression of a [[Hamiltonian differential form]] (def. \ref{HamiltonianForms}), hence if there exists \begin{displaymath} (H,v) \in \Omega^{p,0}_{\Sigma, Ham}(E) \end{displaymath} whose transgression over the Cauchy surface $\Sigma_p$ equals the transgression of $A$ over all of spacetime $\Sigma$, under the isomorphism \eqref{CauchySurfaceIsomorphismOnHistorySpace} \begin{displaymath} \itexarray{ \Gamma_\Sigma(E)_{\delta_{EL} \mathbf{L} = 0 } && \underoverset{\simeq}{(-)\vert_{N_\Sigma \Sigma_p}}{\longrightarrow} && \Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0} \\ & {}_{\mathllap{\tau_\Sigma}(A)}\searrow && \swarrow_{\mathrlap{ \tau_{\Sigma_p}(H) }} \\ && \mathbf{\Omega}^2 } \end{displaymath} \end{defn} Beware that the [[local observable]] $\tau_{\Sigma_p}(H)$ defined by a [[Hamiltonian differential form]] $H \in \Omega^{p,0}_{\Sigma,Ham}(E)$ as in def. \ref{HamiltonianLocalObservables} does in general depend not just on the choice of $H$, but also on the choice $\Sigma_p$ of the Cauchy surface. The exception are those Hamiltonian forms which are \emph{[[conserved currents]]}: \begin{prop} \label{ConservedCharge}\hypertarget{ConservedCharge}{} \textbf{([[conserved charges]] -- [[transgression of variational differential forms|transgression]] of [[conserved currents]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). If a [[Hamiltonian differential form]] $J \in \Omega^{p,0}_{\Sigma,Ham}(E)$ (def. \ref{HamiltonianForms}) happens to be a [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}) in that its [[total derivative|total spacetime derivative]] vanishes [[on-shell]] \begin{displaymath} d J \vert_{\mathcal{E}} \;= \; 0 \end{displaymath} then the induced Hamiltonian [[local observable]] $\tau_{\Sigma_p}(J)$ (def. \ref{HamiltonianLocalObservables}) is independent of the choice of [[Cauchy surface]] $\Sigma_p$ (def \ref{CauchySurface}) in that for $\Sigma_p, \Sigma'_p \hookrightarrow \Sigma$ any two Cauchy surfaces which are [[cobordism|cobordant]], then \begin{displaymath} \tau_{\Sigma_p}(J) = \tau_{\Sigma'_p}(J) \,. \end{displaymath} The resulting [[constant function|constant]] is called the \emph{[[conserved charge]]} of the conserved current, traditionally denoted \begin{displaymath} Q \;\coloneqq\; \tau_{\Sigma_p}(J) \,. \end{displaymath} \end{prop} \begin{proof} By definition the [[transgression of variational differential forms|transgression]] of $d J$ vanishes on the [[on-shell]] [[space of field histories]]. Therefore the result is given by [[Stokes' theorem]] (prop. \ref{StokesTheorem}). \end{proof} \begin{defn} \label{PoissonBracketOnHamiltonianLocalObservables}\hypertarget{PoissonBracketOnHamiltonianLocalObservables}{} \textbf{([[Poisson bracket]] of [[Hamiltonian differential form|Hamiltonian]] [[local observables]] on [[covariant phase space]])} Let $(E, \mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) where the [[field bundle]] $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}). We say that the \emph{[[Poisson bracket]]} on Hamiltonian local observables (def. \ref{HamiltonianLocalObservables}) is the [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) of the [[Poisson bracket Lie n-algebra|local Poisson bracket]] (def. \ref{LocalPoissonBracket}) of the corresponding [[Hamiltonian differential forms]] (def. \ref{LocalPoissonBracket}) to the [[covariant phase space]] (def. \ref{CovariantPhaseSpace}). Explicitly: for $\Sigma_p \hookrightarrow \Sigma$ a choice of [[Cauchy surface]] (def. \ref{CauchySurface}) then the Poisson bracket between two local Hamiltonian observables $\tau_{\Sigma_p}((H_i, v_i))$ is \begin{equation} \left\{ \tau_{\Sigma_p}((H_1, v_1)) \,,\, \tau_{\Sigma_p}( (H_2, v_2) ) \right\} \;\coloneqq\; \tau_{\Sigma_p}( \, \{ (H_1, v_1), (H_2, v_2) \} \, ) \,, \label{PoissonBracketTransgressedToCauchySurface}\end{equation} where on the right we have the transgression of the [[Poisson bracket Lie n-algebra|local Poisson bracket]] $\{(H_1, v_1), (H_2, v_2)\}$ of [[Hamiltonian differential forms]] on the [[jet bundle]] from prop. \ref{LocalPoissonBracket}. \end{defn} \begin{proof} We need to see that equation \eqref{PoissonBracketTransgressedToCauchySurface} is well defined, in that it does not depend on the choice of Hamiltonian form $(H_i, v_i)$ representing the local Hamiltonian observable $\tau_{\Sigma_p}(H_i)$. It is clear that all the transgressions involved depend only on the restriction of the Hamiltonian forms to the pullback of the jet bundle to the [[infinitesimal neighbourhood]] $N_\Sigma \Sigma_p$. Moreover, the Poisson bracket on the jet bundle \eqref{LocalPoissonLieBracket} clearly respects this restriction. If a Hamiltonian differential form $H$ is in the [[kernel]] of the transgression map relative to $\Sigma_p$, in that for every smooth collection $\Phi_{(-)} \colon U \to \Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0}$ of field histories (according to def. \ref{DifferentialFormsOnDiffeologicalSpaces}) we have (by def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) \begin{displaymath} \int_{\Sigma_p} j^\infty_\Sigma(\Phi_{(-)})^\ast H \;= \;0 \;\;\; \in \Omega^p(U) \end{displaymath} then the fact that the \emph{[[kernel of integration is the exact differential forms]]} says that $j^\infty_\Sigma(\Phi_{(-)})^\ast H \in \Omega^p(U \times \Sigma)$ is $d_\Sigma$-[[exact differential form|exact]] and hence in particular $d_\Sigma$-[[closed differential form|closed]] for all $\Phi_{(-)}$: \begin{displaymath} d_\Sigma j^\infty(\Phi_{(-)})^\ast H \;=\; 0 \,. \end{displaymath} By prop. \ref{PullbackAlongJetProlongationIntertwinesHorizontalDerivative} this means that \begin{displaymath} j^\infty(\Phi_{(-)})^\ast ( d H ) \;= \; 0 \end{displaymath} for all $\Phi_{(-)}$. Since $H \in \Omega^{p,0}_\Sigma(E)$ is horizontal, the same proposition (see also example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}) implies that in fact $H$ is horizontally closed: \begin{displaymath} d H \;=\; 0 \,. \end{displaymath} Now since the field bundle $E \overset{fb}{\to} \Sigma$ is [[trivial bundle|trivial]] by assumption, prop. \ref{HorizontalVariationalComplexOfTrivialFieldBundleIsExact} applies and says that this horizontally closed form on the jet bundle is in fact horizontally exact. In conclusion this shows that the [[kernel]] of the [[transgression of variational differential forms|transgression]] map $\tau_{\Sigma_p} \;\colon\; \Omega^{p,0}_\Sigma(E) \to C^\infty\left( \Gamma_{\Sigma_p}(E)\right)$ is precisely the space of horizontally exact horizontal $p$-forms. Therefore the claim now follows with the statement that horizontally exact [[Hamiltonian differential forms]] constitute a [[Lie ideal]] for the local Poisson bracket on the jet bundle; this is lemma \ref{HorizontallyExactFormsDropOutOfLocalLieBracket}. \end{proof} \begin{example} \label{PoissonBracketForRealScalarField}\hypertarget{PoissonBracketForRealScalarField}{} \textbf{([[Poisson bracket]] of the [[real scalar field]])} Consider the [[Lagrangian field theory]] of the [[free field|free]] [[scalar field]] (example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}), and consider the [[Cauchy surface]] defined by $x^0 = 0$. By example \ref{LocalPoissonBracketForRealScalarField} the [[Poisson bracket Lie n-algebra|local Poisson bracket]] of the [[Hamiltonian forms]] \begin{displaymath} Q \coloneqq \phi \iota_{\partial_0} dvol_\Sigma \in \Omega^{p,0}(E) \end{displaymath} and \begin{displaymath} P \coloneqq \eta^{\mu \nu} \phi_{,\mu} \iota_{\partial_\nu} dvol_{\Sigma} \in \Omega^{p,0}(E) \,. \end{displaymath} is \begin{displaymath} \{Q,P\} = \iota_{v_Q} \iota_{v_P} \omega = \iota_{\partial_0} dvol_\Sigma \,. \end{displaymath} Upon [[transgression of variational differential forms|transgression]] according to def. \ref{PoissonBracketOnHamiltonianLocalObservables} this yields the following [[Poisson bracket]] \begin{displaymath} \left\{ \int_{\Sigma_p} b_1(\vec x) \phi(t,\vec x) \iota_{\partial_0} dvol_\Sigma(x) d^p \vec x \;,\; \int_{\Sigma_p} b_2(\vec x) \partial_0 \phi(t,\vec x) \iota_{\partial_0} dvol_\Sigma(\vec x) \right\} \;=\; \int_{\Sigma_p} b_1(\vec x) b_2(\vec x) \iota_{\partial_0} dvol_\Sigma(\vec x) d^p \vec x \,, \end{displaymath} where \begin{displaymath} \mathbf{\Phi}(x), \partial_0 \mathbf{\Phi}(x) \;:\; PhaseSpace(\Sigma_p^t) \to \mathbb{R} \end{displaymath} denote the point-evaluation observables (example \ref{PointEvaluationObservables}), which act on a field history $\Phi \in \Gamma_\Sigma(E) = C^\infty(\Sigma)$ as \begin{displaymath} \mathbf{\Phi}(x) \;\colon\; \Phi \mapsto \Phi(x) \phantom{AAAAAAAA} \partial_0 \mathbf{\Phi}(x) \;\colon\; \Phi \mapsto \partial_0 \Phi(x) \,. \end{displaymath} Notice that these point-evaluation functions themselves do not arise as the transgression of elements in $\Omega^{p,0}(E)$; only their smearings such as $\int_{\Sigma_p} b_1 \phi dvol_{\Sigma_p}$ do. Nevertheless we may express the above Poisson bracket conveniently via the [[integral kernel]] \begin{equation} \left\{ \mathbf{\Phi}(t,\vec x), \partial_0\mathbf{\Phi}(t,\vec y) \right\} \;=\; \delta(\vec x - \vec y) \,. \label{PoissonBracketOfScalarFieldPointEvaluationOnMinkowskiSpacetime}\end{equation} \end{example} \begin{prop} \label{PoissonBracketForDiracField}\hypertarget{PoissonBracketForDiracField}{} \textbf{([[super Lie algebra|super]]-[[Poisson bracket]] of the [[Dirac field]])} Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] (example \ref{LagrangianDensityForDiracField}) with [[field bundle]] the odd-shifted [[spinor bundle]] $E = \Sigma \times S_{odd}$ (example \ref{DiracFieldBundle}) and with \begin{displaymath} \theta \Psi_\alpha(x) \;\colon\; \mathbb{R}^{0\vert 1} \longrightarrow \left[ \Gamma_\Sigma(\Sigma \times S_{odd})_{\delta_{EL}\mathbf{L} = 0}, \mathbb{C} \right] \end{displaymath} the corresponding odd-graded point-evaluation observable (example \ref{PointEvaluationObservables}). Then consider the [[Cauchy surfaces]] in [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) given by $x^0 = t$ for $t \in \mathbb{R}$. Under [[transgression of variational differential forms|transgression]] to this Cauchy surface via def. \ref{PoissonBracketOnHamiltonianLocalObservables}, the [[Poisson bracket Lie n-algebra|local Poisson bracket]], which by example \ref{LocalPoissonBracketForDiracField} is given by the [[super Lie algebra|super Lie bracket]] \begin{displaymath} \left\{ \left( \gamma^\mu \psi \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma \,,\, \left(\overline{\psi}\gamma^\mu\right)^\beta\, \iota_{\partial_\mu} dvol_\Sigma \right\} \;=\; \left(\gamma^\mu\right)_\alpha{}^{\beta} \, \iota_{\partial_\mu} dvol_\Sigma \,, \end{displaymath} has [[integral kernel]] \begin{displaymath} \left\{ \psi_\alpha(t,\vec x) , \overline{\psi}^\beta(t,\vec y) \right\} \;=\; (\gamma^0)_{\alpha}{}^\beta \delta(\vec y - \vec x) \,. \end{displaymath} \end{prop} $\,$ This concludes our discussion of the [[phase space]] and the [[Poisson-Peierls bracket]] for well behaved [[Lagrangian field theories]]. In the \hyperlink{Propagators}{next chapter} we discuss in detail the [[integral kernels]] corresponding to the [[Poisson-Peierls bracket]] for key classes of examples. These are the \emph{[[propagators]]} of the theory. \end{document}