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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 -- Symmetries} \hypertarget{Symmetries}{}\subsection*{{Symmetries}}\label{Symmetries} In this chapter we discuss these topics: \begin{itemize}% \item \emph{\hyperlink{InfinitesimalSymmetriesOfTheLagrangianDensity}{Infinitesimal symmetries of the Lagrangian density}} \item \emph{\hyperlink{InfinitesimalSymmetriesOfThePresymplecticPotentialCurrent}{Infinitesimal symmetries of the presymplectic potential current}} \end{itemize} $\,$ We have introduced the concept of \emph{[[Lagrangian field theories]]} $(E,\mathbf{L})$ in terms of a [[field bundle]] $E$ equipped with a [[Lagrangian density]] $\mathbf{L}$ on its [[jet bundle]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). Generally, given any [[object]] equipped with some [[structure]], it is of paramount interest to determine the [[symmetries]], hence the [[isomorphisms]]/[[equivalences]] of the object that preserve the given [[structure]] (this is the ``[[Erlanger program]]'', \href{Erlangen+program#Klein1872}{Klein 1872}). The [[infinitesimal symmetries of the Lagrangian density]] (def. \ref{SymmetriesAndConservedCurrents} below) send one [[field history]] to an [[infinitesimal|infinitesimally]] nearby one which is ``[[equivalence|equivalent]]'' for all purposes of [[field theory]]. Among these are the \emph{[[infinitesimal gauge symmetries]]} which will be of concern \hyperlink{GaugeSymmetries}{below}. A central theorem of [[variational calculus]] says that [[infinitesimal symmetries of the Lagrangian]] correspond to \emph{[[conserved currents]]}, this is [[Noether theorem|Noether's theorem I]], prop. \ref{NoethersFirstTheorem} below. These conserved currents constitute an [[Lie algebra extension|extension]] of the [[Lie algebra]] of symmetries, called the \emph{[[Dickey bracket]]}. But in \eqref{DerivativeOfLepageForm} we have seen that the [[Lagrangian density]] of a [[Lagrangian field theory]] is just one component, in [[codimension]] 0, of an inhomogeneous ``[[Lepage form]]'' which in [[codimension]] 1 is given by the [[presymplectic potential current]] $\Theta_{BFV}$ \eqref{PresymplecticPotential}. (This will be conceptually elucidated, after we have introduced the [[local BV-complex]], in example \ref{DerivedPresymplecticCurrentOfRealScalarField} below.) This means that in [[codimension]] 1 we are to consider infinitesimal [[on-shell]] symmetries of the [[Lepage form]] $\mathbf{L} + \Theta_{BFV}$. These are known as \emph{[[Hamiltonian vector fields]]} (def. \ref{HamiltonianForms} below) and the analog of [[Noether's theorem|Noether's theorem I]] now says that these correspond to \emph{[[Hamiltonian differential forms]]}. The [[Lie algebra]] of these infinitesimal symmetries is called the \emph{[[Poisson bracket Lie n-algebra|local Poisson bracket]]} (prop. \ref{LocalPoissonBracket} below). \textbf{[[Noether theorem]] and [[Hamiltonian Noether theorem]]} \newline | [[Lagrangian density]] $\mathbf{L}$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) | $\mathcal{L}_v \mathbf{L} = d \tilde J$ | $d(\underset{= J_v}{\underbrace{\tilde J - \iota_v \Theta_{BFV}}}) = \iota_v \, \delta_{EL}\mathbf{L}$ | [[conserved current]] $J_v$ (def. \ref{SymmetriesAndConservedCurrents}) | [[Dickey bracket]] | | [[presymplectic current]] $\Omega_{BFV}$ (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) | $\mathcal{L}^{var}_v \Theta_{BFV} = \delta \tilde H$ | $\delta(\underset{= H_v}{\underbrace{\tilde H_v - \iota_v \Theta_{BFV}}}) = \iota_v \Omega_{BFV}$ | [[Hamiltonian differential form|Hamiltonian form]] $H_v$ (def. \ref{HamiltonianForms}) | [[Poisson bracket Lie n-algebra|local Poisson bracket]] (prop. \ref{LocalPoissonBracket}) | $\,$ In the chapter \emph{\hyperlink{PhaseSpace}{Phase space}} below we [[transgression|transgress]] this [[Poisson bracket Lie n-algebra|local Poisson bracket]] of [[infinitesimal symmetries]] of the [[presymplectic potential current]] to the ``global'' [[Poisson bracket]] on the \emph{[[covariant phase space]]} (def. \ref{PoissonBracketOnHamiltonianLocalObservables} below). This is the structure which then \hyperlink{Quantization}{further below} leads over to the [[quantization]] ([[deformation quantization]]) of the [[prequantum field theory]] to a genuine [[perturbative quantum field theory]]. However, it will turn out that there may be an [[obstruction]] to this construction, namely the existence of special infinitesimal symmetries of the Lagrangian densities, called \emph{implicit [[gauge symmetries]]} (discussed \hyperlink{GaugeSymmetries}{further below}). $\,$ \textbf{[[infinitesimal symmetries]] of the [[Lagrangian density]]} \begin{defn} \label{Variation}\hypertarget{Variation}{} \textbf{(variation)} Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}). A \emph{variation} is a [[vertical vector field]] $v$ on the [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) hence a vector field which vanishes when evaluated in the [[horizontal differential forms]]. In the special case that the [[field bundle]] is [[trivial vector bundle]] over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle}, a variation is of the form \begin{displaymath} v = v^a \partial_{\phi^a} + v^a_{,\mu} \partial_{\phi^a_{,\mu}} + v^a_{\mu_1 \mu_2} \partial_{\phi^a_{\mu_1 \mu_2}} + \cdots \end{displaymath} \end{defn} The concept of variation in def. \ref{Variation} is very general, in that it allows to vary the field coordinates independently from the corresponding jets. This generality is necessary for discussion of symmetries of [[presymplectic currents]] in def. \ref{HamiltonianForms} below. But for discussion of symmetries of [[Lagrangian densities]] we are interested in explicitly varying just the [[field (physics)|field]] coordinates (def. \ref{EvolutionaryVectorField} below) and inducing from this the corresponding variations of the field derivatives (prop. \ref{EvolutionaryVectorFieldProlongation}) below. In order to motivate the following definition \ref{EvolutionaryVectorField} of \emph{[[evolutionary vector fields]]} we follow remark \ref{ReplacingBundleMorphismsByDifferentialOperators} saying that concepts in [[variational calculus]] are obtained from their analogous concepts in plain [[differential calculus]] by replacing plain [[bundle morphisms]] by morphisms out of the [[jet bundle]]: Given a [[fiber bundle]] $E \overset{fb}{\to} \Sigma$, then a \emph{[[vertical vector field]]} on $E$ is a [[section]] of its [[vertical tangent bundle]] $T_\Sigma E$ (def. \ref{VerticalTangentBundle}), hence is a [[bundle morphism]] of this form \begin{displaymath} \itexarray{ E && \overset{\text{vertical vector field}}{\longrightarrow} && T_\Sigma E \\ & {}_{\mathllap{id}}\searrow && \swarrow \\ && E } \end{displaymath} The variational version replaces the vector bundle on the left with its jet bundle: \begin{defn} \label{EvolutionaryVectorField}\hypertarget{EvolutionaryVectorField}{} \textbf{([[evolutionary vector fields]])} Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}). Then an \emph{[[evolutionary vector field]]} $v$ on $E$ is ``variational vertical vector field'' on $E$, hence a smooth [[bundle]] [[homomorphism]] out of the [[jet bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) \begin{displaymath} \itexarray{ J^\infty_\Sigma E && \overset{v}{\longrightarrow} && T_\Sigma E \\ & {}_{\mathllap{jb_{\infty,0}}}\searrow && \swarrow_{\mathllap{}} \\ && E } \end{displaymath} to the [[vertical tangent bundle]] $T_\Sigma E \overset{}{\to} \Sigma$ (def. \ref{VerticalTangentBundle}) of $E \overset{fb}{\to} \Sigma$. In the special case that the [[field bundle]] is a [[trivial vector bundle]] over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle}, this means that an evolutionary vector field is a [[tangent vector field]] (example \ref{TangentVectorFields}) on $J^\infty_\Sigma(E)$ of the special form \begin{displaymath} \begin{aligned} v & = v^a \partial_{\phi^a} \\ & = v^a\left( (x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots \right) \partial_{\phi^a} \end{aligned} \,, \end{displaymath} where the [[coefficients]] $v^a \in C^\infty(J^\infty_\Sigma(E))$ are general [[smooth functions]] on the [[jet bundle]] (while the cmponents are [[tangent vectors]] along the field coordinates $(\phi^a)$, but not along the spacetime coordinates $(x^\mu)$ and not along the jet coordinates $\phi^a_{,\mu_1 \cdots \mu_k}$). We write \begin{displaymath} \Gamma_E^{ev}\left( T_\Sigma E \right) \;\in\; \Omega^{0,0}_\Sigma(E) Mod \end{displaymath} for the space of evolutionary vector fields, regarded as a [[module]] over the $\mathbb{R}$-[[associative algebra|algebra]] \begin{displaymath} \Omega^{0,0}_\Sigma(E) \;=\; C^\infty\left( J^\infty_\Sigma(E) \right) \end{displaymath} of [[smooth functions]] on the [[jet bundle]]. \end{defn} An [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField}) describes an infinitesimal change of field values \emph{depending} on, possibly, the point in spacetime and the values of the field and all its derivatives (locally to finite order, by prop. \ref{JetBundleIsLocallyProManifold}). This induces a corresponding infinitesimal change of the derivatives of the fields, called the \emph{prolongation} of the evolutionary vector field: \begin{prop} \label{EvolutionaryVectorFieldProlongation}\hypertarget{EvolutionaryVectorFieldProlongation}{} \textbf{(prolongation of [[evolutionary vector field]])} Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]]. Given an [[evolutionary vector field]] $v$ on $E$ (def. \ref{EvolutionaryVectorField}) there is a unique [[tangent vector field]] $\hat v$ (example \ref{TangentVectorFields}) on the [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) such that \begin{enumerate}% \item $\hat v$ agrees on field coordinates (as opposed to jet coordinates) with $v$: \begin{displaymath} (jb_{\infty,0})_\ast(\hat v) = v \,, \end{displaymath} which means in the special case that $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) that $\hat v$ is of the form \begin{equation} \hat v \;=\; \underset{ = v }{ \underbrace{ v^a \partial_{\phi^a} }} \,+\, \hat v^a_{\mu} \partial_{\phi^a_{,\mu}} + \hat v^a_{\mu_1 \mu_2} \partial_{\phi^a_{,\mu_1 \mu_2}} + \cdots \label{GenericComponentsForProlongationOfEvolutionaryVectorField}\end{equation} \item contraction with $\hat v$ (def. \ref{ContractionOfFormsWithVectorFields}) anti-commutes with the [[total derivative|total spacetime derivative]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}): \begin{equation} \iota_{\hat v} \circ d + d \circ \iota_{\hat v} = 0 \,. \label{ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative}\end{equation} \end{enumerate} In particular [[Cartan's homotopy formula]] (prop. \ref{CartanHomotopyFormula}) for the [[Lie derivative]] $\mathcal{L}_{\hat v}$ holds with respect to the [[variational derivative]] $\delta$: \begin{equation} \mathcal{L}_{\hat v} = \delta \circ \iota_{\hat v} + \iota_{\hat v} \circ \delta \label{HomotopyFormulaForLieDerivativeAlongProlongationOfEvolutionaryVectorField}\end{equation} Explicitly, in the special case that the [[field bundle]] is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) $\hat v$ is given by \begin{equation} \hat v = \underoverset{n = 0}{\infty}{\sum} \frac{d^n v^a}{ d x^{\mu_1} \cdots d x^{\mu_n} } \partial_{\phi^a_{\mu_1 \cdots \mu_n}} \,. \label{ProlongationOfEvolutionaryVectorFieldExplicit}\end{equation} \end{prop} \begin{proof} It is sufficient to prove the coordinate version of the statement. We prove this by [[induction]] over the maximal jet order $k$. Notice that the coefficient of $\partial_{\phi^a_{\mu_1 \cdots \mu_k}}$ in $\hat v$ is given by the contraction $\iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k}$ (def. \ref{ContractionOfFormsWithVectorFields}). Similarly (at ``$k = -1$'') the component of $\partial_{\mu_1}$ is given by $\iota_{\hat v} d x^{\mu}$. But by the second condition above this vanishes: \begin{displaymath} \begin{aligned} \iota_{\hat v} d x^\mu & = d \iota_{\hat v} x^\mu \\ & = 0 \end{aligned} \end{displaymath} Moreover, the coefficient of $\partial_{\phi^a}$ in $\hat v$ is fixed by the first condition above to be \begin{displaymath} \iota_{\hat v} \delta \phi^a = v^a \,. \end{displaymath} This shows the statement for $k = 0$. Now assume that the statement is true up to some $k \in \mathbb{N}$. Observe that the coefficients of all $\partial_{\phi^a_{\mu_1 \cdots \mu_{k+1}}}$ are fixed by the contractions with $\delta \phi^a_{\mu_1 \cdots \mu_{k} \mu_{k+1}} \wedge d x^{\mu_{k+1}}$. For this we find again from the second condition and using $\delta \circ d + d \circ \delta = 0$ as well as the induction assumption that \begin{displaymath} \begin{aligned} \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_{k+1}} \wedge d x^{\mu_{k+1}} & = \iota_{\hat v} \delta d \phi^a_{\mu_1 \cdots \mu_k} \\ & = d \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k} \\ & = d \frac{d^k v^a}{d x^{\mu_1} \cdots d x^{\mu_k}} \\ & = \frac{d^{k+1}v^a }{d x^{\mu_1} \cdots d x^{\mu_{k+1}}} d x^{\mu_{k+1}} \,. \end{aligned} \end{displaymath} This shows that $\hat v$ satisfying the two conditions given exists uniquely. Finally formula \eqref{HomotopyFormulaForLieDerivativeAlongProlongationOfEvolutionaryVectorField} for the [[Lie derivative]] follows from the second of the two conditions with [[Cartan's homotopy formula]] $\mathcal{L}_{\hat v} = \mathbf{d} \circ \iota_{\hat v} + \iota_{\hat v} \circ \mathbf{d}$ (prop. \ref{CartanHomotopyFormula}) together with $\mathbf{d} = \delta + d$ \eqref{VariationalDerivative}. \end{proof} \begin{prop} \label{EvolutionaryVectorFieldLieAlgebra}\hypertarget{EvolutionaryVectorFieldLieAlgebra}{} \textbf{([[evolutionary vector fields]] form a [[Lie algebra]])} Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]]. For any two [[evolutionary vector fields]] $v_1$, $v_2$ on $E$ (def. \ref{EvolutionaryVectorField}) the [[Lie bracket]] of [[tangent vector fields]] of their prolongations $\hat v_1$, $\hat v_2$ (def. \ref{EvolutionaryVectorFieldProlongation}) is itself the prolongation $\widehat{[v_1, v_2]}$ of a unique evolutionary vector field $[v_1,v_2]$. This defines the structure of a [[Lie algebra]] on evolutionary vector fields. \end{prop} \begin{proof} It is clear that $[\hat v_1, \hat v_2]$ is still [[vertical vector field|vertical]], therefore, by prop. \ref{EvolutionaryVectorFieldProlongation}, it is sufficient to show that contraction $\iota_{[v_1, v_2]}$ with this vector field (def. \ref{ContractionOfFormsWithVectorFields}) anti-commutes with the [[horizontal derivative]] $d$, hence that $[d, \iota_{[\hat v_1, \hat v_2]}] = 0$. Now $[d, \iota_{[\hat v_1, \hat v_2]}]$ is an operator that sends vertical 1-forms to horizontal 1-forms and vanishes on horizontal 1-forms. Therefore it is sufficient to see that this operator in fact also vanishes on all vertical 1-forms. But for this it is sufficient that it commutes with the vertical derivative. This we check by [[Cartan calculus]], using $[d,\delta] = 0$ and $[d, \iota_{\hat v_i}]=0$, by assumption: \begin{displaymath} \begin{aligned} {[ \delta, [ d,\iota_{[\hat v_1, \hat v_2]}] ]} & = - [d, [\delta, \iota_{[\hat v_1, \hat v_2]}]] \\ & = - [d, \mathcal{L}_{[\hat v_1, \hat v_2]}] \\ & = -[d, [\mathcal{L}_{\hat v_1}, \iota_{\hat v_2}] ] \\ & = - [d, [ [\delta, \iota_{\hat v_1}], \iota_{\hat v_2} ]] \\ & = 0 \,. \end{aligned} \end{displaymath} \end{proof} Now given an evolutionary vector field, we want to consider the [[flow]] that it induces on the [[space of field histories]]: \begin{defn} \label{FlowOfFieldHistoriesAlongEvolutionaryVectorField}\hypertarget{FlowOfFieldHistoriesAlongEvolutionaryVectorField}{} \textbf{([[flow]] of [[field histories]] along [[evolutionary vector field]])} Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and let $v$ be an [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField}) such that the ordinary [[flow]] of its prolongation $\hat v$ (prop. \ref{EvolutionaryVectorFieldProlongation}) \begin{displaymath} \exp(t \hat v) \;\colon\; J^\infty_\Sigma(E) \longrightarrow J^\infty_\Sigma(E) \end{displaymath} exists on the [[jet bundle]] (e.g. if the order of derivatives of field coordinates that it depends on is bounded). For $\Phi_{(-)} \colon U_1 \to \Gamma_\Sigma(E)$ a collection of [[field histories]] (hence a plot of the [[space of field histories]] (def. \ref{SupergeometricSpaceOfFieldHistories}) ) the \emph{[[flow]]} of $v$ through $\Phi_{(-)}$ is the [[smooth function]] \begin{displaymath} U_1 \times \mathbb{R}^1 \overset{\exp(v)(\Phi_{(-)})}{\longrightarrow} \Gamma_\Sigma(E) \end{displaymath} whose unique factorization $\widehat{\exp(v)}(\Phi_{(-)})$ through the space of jets of field histories (i.e. the [[image]] $im(j^\infty_\Sigma)$ of [[jet prolongation]], def. \ref{JetProlongation}) \begin{displaymath} \itexarray{ && im(j^\infty_\Sigma) &\hookrightarrow& \Gamma_\Sigma(J^\infty_\Sigma(E)) \\ & {}^{\mathllap{\widehat{\exp(v)}(\Phi_{(-)})}} \nearrow& \downarrow^{\mathrlap{\simeq}} \\ U_1 \times \mathbb{R}^1 &\underset{ \exp(v)(\Phi) }{\longrightarrow}& \Gamma_{\Sigma}(E)_{} } \end{displaymath} takes a plot $t_{(-)} \;\colon\; U_2 \to \mathbb{R}^1$ of the [[real line]] (regarded as a [[super formal smooth set|super smooth set]] via example \ref{SuperSmoothSetSuperCartesianSpaces}), to the plot \begin{equation} (\exp(t(-) \hat v) \circ j^\infty_\Sigma(\Phi_{(-)}) \;\colon\: U_1 \times U_2 \longrightarrow \Gamma_\Sigma\left( J^\infty_\Sigma(E) \right) \label{LocalDataForFlowOfImplicitInfinitesimalGaugeSymmetry}\end{equation} of the [[smooth space|smooth]] [[space of sections]] of the [[jet bundle]]. (That $\exp(t(-) \hat v)$ indeed flows jet prolongations $j^\infty_\Sigma(\Phi(-))$ again to jet prolongations is due to its defining relation to the [[evolutionary vector field]] $v$ from prop. \ref{EvolutionaryVectorFieldProlongation}.) \end{defn} \begin{defn} \label{SymmetriesAndConservedCurrents}\hypertarget{SymmetriesAndConservedCurrents}{} \textbf{([[infinitesimal symmetries of the Lagrangian]] and [[conserved currents]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). Then \begin{enumerate}% \item an \emph{[[infinitesimal symmetry of the Lagrangian]]} is an [[evolutionary vector field]] $v$ (def. \ref{EvolutionaryVectorField}) such that the [[Lie derivative]] of the [[Lagrangian density]] along its prolongation $\hat v$ (prop. \ref{EvolutionaryVectorFieldProlongation}) is a [[total spacetime derivative]]: \begin{displaymath} \mathcal{L}_{\hat v} \mathbf{L} \;=\; d \tilde J_{\hat v} \end{displaymath} \item an \emph{[[on-shell]] [[conserved current]]} is a horizontal $p$-form $J \in \Omega^{p,0}_\Sigma(E)$ (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) whose [[total derivative|total spacetime derivative]] vanishes on the [[prolonged shell]] \eqref{ShellInJetBundle} \begin{displaymath} d J\vert_{\mathcal{E}^\infty} \;=\; 0 \,. \end{displaymath} \end{enumerate} \end{defn} \begin{prop} \label{NoethersFirstTheorem}\hypertarget{NoethersFirstTheorem}{} \textbf{([[Noether's theorem]] I)} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). If $v$ is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}) with $\mathcal{L}_{\hat v} \mathbf{L} = d \tilde J_{\hat v}$, then \begin{equation} J_{\hat v} \coloneqq \tilde J_{\hat v} - \iota_{\hat v} \Theta_{BFV} \label{NoetherCurrent}\end{equation} is an [[on-shell]] [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}), for $\Theta_{BFV}$ a presymplectic potential \eqref{PresymplecticPotential} from def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}. \end{prop} ([[Noether's theorem|Noether's theorem II]] is prop. \ref{NoetherIdentities} below.) \begin{proof} By [[Cartan's homotopy formula]] for the [[Lie derivative]] (prop. \ref{CartanHomotopyFormula}) and the decomposition of the variational derivative $\delta \mathbf{L}$ \eqref{dLDecomposition} and the fact that contraction $\iota_{\hat v}$ with the prolongtion of an evolutionary vector field vanishes on horizontal differential forms \eqref{GenericComponentsForProlongationOfEvolutionaryVectorField} and anti-commutes with the horizontal differential \eqref{ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative}, by def. \ref{EvolutionaryVectorField}, we may re-express the defining equation for the symmetry as follows: \begin{displaymath} \begin{aligned} d \tilde J_{\hat v} & = \mathcal{L}_{\hat v} \mathbf{L} \\ & = \iota_{\hat v} \underset{= \delta_{EL}\mathbf{L} - d \Theta_{BFV}}{\underbrace{\mathbf{d} \mathbf{L}}} + \mathbf{d} \underset{= 0}{\underbrace{\iota_v \mathbf{L}}} \\ & = \iota_{\hat v} \delta_{EL} \mathbf{L} + d \iota_{\hat v} \Theta_{BFV} \end{aligned} \end{displaymath} which is equivalent to \begin{equation} d(\underset{= J_{\hat v}}{\underbrace{\tilde J_{\hat v} - \iota_{\hat v} \Theta_{BFV}}}) \;=\; \iota_{\hat v} \delta_{EL}\mathbf{L} \label{CurrentNoetherConservation}\end{equation} Since, by definition of the [[shell]] $\mathcal{E}$, the differential form on the right vanishes on $\mathcal{E}$ this yields the claim. \end{proof} \begin{example} \label{ScalarFieldEnergyMomentum}\hypertarget{ScalarFieldEnergyMomentum}{} \textbf{([[energy-momentum]] of the [[scalar field]])} Consider the [[Lagrangian field theory]] of the [[free field|free]] [[scalar field]] from def. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}: \begin{displaymath} \mathbf{L} \;=\; \tfrac{1}{2} \left( \eta^{\mu \nu}\phi_{,\mu} \phi_{,\nu} - m^2 \phi^2 \right) dvol_\Sigma \,. \end{displaymath} For $\nu \in \{0, 1, \cdots, p\}$ consider the vector field on the jet bundle given by \begin{displaymath} v_\nu \;\coloneqq\; \phi_{,\nu} \partial_{\phi} + \phi_{,\mu \nu} \partial_{\phi_{,\mu}} + \cdots \,. \end{displaymath} This describes infinitesimal translations of the fields in the direction of $\partial_\nu$. And this is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}), since \begin{displaymath} \iota_{v_\nu} \mathbf{d}\mathbf{L} = d L \wedge \iota_{\partial_\nu} dvol_\Sigma \,. \end{displaymath} With the formula \eqref{PresymplecticPotentialOfFreeScalarField} for the presymplectic potential \begin{displaymath} \Theta_{BFV} = \eta^{\mu \nu} \phi_{,\mu} \delta \phi \iota_{\partial_{\nu}} dvol_\Sigma \end{displaymath} it hence follows from [[Noether's theorem]] (prop. \ref{NoethersFirstTheorem}) that the corresponding [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}) is \begin{displaymath} \begin{aligned} T_\nu & = L \, \iota_{\partial_\nu} dvol_\Sigma - \iota_{v_\nu}\Theta_{BFV} \\ & = L \, \iota_{\partial_\nu} dvol_\Sigma - \eta^{\rho \mu} \phi_{,\rho} \phi_{,\nu} \, \iota_{\partial_\mu} dvol_\Sigma \\ & = ( \underset{=: T^\mu_\nu}{ \underbrace{ \delta^\mu_\nu L - \eta^{\rho \mu} \phi_{,\rho} \phi_{,\nu} } } ) \, \iota_{\partial_\mu} dvol_\Sigma \end{aligned} \,. \end{displaymath} This [[conserved current]] is called the \emph{[[energy-momentum tensor]]}. \end{example} \begin{example} \label{DiracCurrent}\hypertarget{DiracCurrent}{} \textbf{([[Dirac current]])} Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] in spacetime dimension $p + 1 = 3+1$ (example \ref{LagrangianDensityForDiracField}) \begin{displaymath} \mathbf{L} = i \overline{\psi} \gamma^\mu \psi_{,\mu} \, dvol_\Sigma \,. \end{displaymath} Then the prolongation (prop. \ref{EvolutionaryVectorFieldProlongation}) of the [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField}) \begin{displaymath} v \;\coloneqq\; i \psi_\alpha \partial_{\psi_\alpha} \end{displaymath} is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}). The [[conserved current]] that corresponds to this under [[Noether's theorem|Noether's theorem I]] (prop. \ref{NoethersFirstTheorem}) is \begin{displaymath} i \overline{\psi} \gamma^\mu \psi \, \iota_{\partial_\mu} dvol_\Sigma \;\in\; \Omega^{p,0}_{\Sigma}(E) \,. \end{displaymath} This is called the \emph{[[Dirac current]]}. \end{example} \begin{proof} By equation \eqref{ProlongationOfEvolutionaryVectorFieldExplicit} the prolongation of $v$ is \begin{displaymath} \hat v = i \psi_\alpha \partial_{\psi_\alpha} + i \psi_{\alpha,\mu} \partial_{\psi_{\alpha,\mu}} + \cdots \,. \end{displaymath} Therefore the [[Lagrangian density]] is strictly invariant under the [[Lie derivative]] along $\hat v$ \begin{displaymath} \begin{aligned} \mathcal{L}_{\hat v} \left( i \overline{\psi} \gamma^\mu \psi_{,\mu} \right) dvol_\Sigma & = \underset{ = i \cdot (-i) \overline{\psi} \gamma^\mu \psi_{,\mu} }{ \underbrace{ i \overline{i \psi} \gamma^\mu \psi_{,\mu} } } dvol_\Sigma + \underset{ i \cdot i \overline{\psi} \gamma^\mu \psi_{,\mu} }{ \underbrace{ i \overline{\psi} \gamma^\mu (i \psi_{,\mu}) } } dvol_\Sigma \\ & = 0 \,. \end{aligned} \end{displaymath} and so the formula for the corresponding conserved current \eqref{NoetherCurrent} is \begin{displaymath} \begin{aligned} J_v & = - \iota_{\hat v} \left( \underset{ - \overline{\psi} \gamma^\mu \delta \psi \, \iota_{\partial_\mu} dvol_\Sigma }{ \underbrace{ \Theta_{BFV} } } \right) \\ & = + i \overline{\psi}\gamma^\mu \psi \, \iota_{\partial_\mu} dvol_\Sigma \end{aligned} \,, \end{displaymath} where under the brace we used example \ref{PresymplecticCurrentDiracField} to identify the [[presymplectic potential]] for the [[free field theory|free]] [[Dirac field]]. \end{proof} $\,$ Since an [[infinitesimal symmetry of a Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}) by definition changes the Lagrangian only up to a [[total spacetime derivative]], and since the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] by construction depend on the [[Lagrangian density]] only up to a [[total spacetime derivative]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}), it is plausible that and [[infinitesimal symmetry of the Lagrangian]] preserves the [[equations of motion]] \eqref{EulerLagrangeEquationGeneral}, hence the [[shell]] \eqref{ProlongedShellInJetBundle}. That this is indeed the case is the statement of prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion} below. To make the proof transparent, we now first introduce the concept of the \emph{[[evolutionary derivative]]} (def. \ref{FieldDependentDifferentialOperatorDerivative}) below and then observe that in terms of these the [[Euler-Lagrange derivative]] is in fact a [[derivation]] (prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}). \begin{defn} \label{FieldDependentSections}\hypertarget{FieldDependentSections}{} \textbf{([[field (physics)|field]]-dependent [[sections]])} For \begin{displaymath} E \overset{fb}{\longrightarrow} \Sigma \end{displaymath} a [[fiber bundle]] (def. \ref{FiberBundle}), regarded as a [[field bundle]] (def. \ref{FieldsAndFieldBundles}), and for \begin{displaymath} E' \overset{fb'}{\longrightarrow} \Sigma \end{displaymath} any other [[fiber bundle]] over the same base space ([[spacetime]]), we write \begin{displaymath} \Gamma_{J^\infty_\Sigma(E)}(E') \;\coloneqq\; \Gamma_{J^\infty_\Sigma(E)}( jb^\ast E' ) \;=\; Hom_\Sigma(J^\infty_\Sigma(E), E') \;\simeq\; DiffOp(E,E') \end{displaymath} for the [[space of sections]] of the [[pullback of bundles]] of $E'$ to the [[jet bundle]] $J^\infty_\Sigma(E) \overset{jb}{\longrightarrow} \Sigma$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) along $jb$. \begin{displaymath} \Gamma_{J^\infty_\Sigma(E)}(E') \;=\; \left\{ \itexarray{ && E' \\ & {}^{\mathllap{}}\nearrow & \downarrow \mathrlap{fb'} \\ J^\infty_\Sigma(E) &\underset{jb}{\longrightarrow}& \Sigma } \phantom{A}\,\, \right\} \,. \end{displaymath} (Equivalently this is the space of [[differential operators]] from sections of $E$ to sections of $E'$, according to prop. \ref{DifferentialOperator}. ) \end{defn} In (\href{evolutionary+derivative#Olver93}{Olver 93, section 5.1, p. 288}) the field dependent sections of def. \ref{FieldDependentSections}, considered in [[local coordinates]], are referred to as [[tuples]] of \emph{differential functions}. \begin{example} \label{EvolutionaryVectorFieldsAsFieldDependentSections}\hypertarget{EvolutionaryVectorFieldsAsFieldDependentSections}{} \textbf{([[source forms]] and [[evolutionary vector fields]] are field-dependent sections)} For $E \overset{fb}{\to} \Sigma$ a [[field bundle]], write $T_\Sigma E$ for its [[vertical tangent bundle]] (example \ref{VerticalTangentBundle}) and $T_\Sigma^\ast E$ for its [[dual vector bundle]] (def. \ref{DualVectorBundle}), the [[vertical cotangent bundle]]. Then the field-dependent sections of these bundles according to def. \ref{FieldDependentSections} are identified as follows: \begin{itemize}% \item the space $\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)$ contains the space of [[evolutionary vector fields]] $v$ (def. \ref{EvolutionaryVectorField}) as those bundle morphism which respect not just the projection to $\Sigma$ but also its factorization through $E$: \begin{displaymath} \left( \itexarray{ && T_\Sigma E \\ & {}^{\mathllap{v}}\nearrow & \downarrow^{\mathrlap{tb_\Sigma}} \\ J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \underset{fb}{\longrightarrow}& \Sigma } \right) \;\in\; \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E) \end{displaymath} \item $\Gamma_{J^\infty_\Sigma(E)}( T^\ast_\Sigma E) \otimes \wedge^{p+1}_\Sigma(T^\ast \Sigma)$ contains the space of [[source forms]] $E$ (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) as those bundle morphisms which respect not just the projection to $\Sigma$ but also its factorization through $E$: \begin{displaymath} \left( \itexarray{ && T^\ast_\Sigma E \\ & {}^{E}\nearrow & \downarrow^{\mathrlap{ctb_\Sigma}} \\ J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \underset{fb}{\longrightarrow}& \Sigma } \right) \;\in\; \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E) \end{displaymath} \end{itemize} This makes manifest the duality pairing between [[source forms]] and [[evolutionary vector fields]] \begin{displaymath} \itexarray{ \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E) \otimes \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E) &\longrightarrow& C^\infty(J^\infty_\Sigma(E)) } \end{displaymath} which in local coordinates is given by \begin{displaymath} (v^a \partial_{\phi^a} \,,\, \omega_a \delta \phi^a) \mapsto v^a \omega_a \end{displaymath} for $v^a, \omega_a \in C^\infty(J^\infty_\Sigma(E))$ [[smooth functions]] on the [[jet bundle]] (as in prop. \ref{JetBundleIsLocallyProManifold}). \end{example} \begin{defn} \label{FieldDependentDifferentialOperatorDerivative}\hypertarget{FieldDependentDifferentialOperatorDerivative}{} \textbf{([[evolutionary derivative of field-dependent section]])} Let \begin{displaymath} E \overset{fb}{\to} \Sigma \end{displaymath} be a [[fiber bundle]] regarded as a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and let \begin{displaymath} V \overset{vb}{\to} \Sigma \end{displaymath} be a [[vector bundle]] (def. \ref{VectorBundle}). Then for \begin{displaymath} P \in \Gamma_{J^\infty_\Sigma(E)}(V) \end{displaymath} a field-dependent section of $E$ according to def. \ref{FieldDependentSections}, its \emph{evolutionary derivative} is the morphism \begin{displaymath} \itexarray{ \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E) & \overset{ \mathrm{D}P }{\longrightarrow} & \Gamma_{J^\infty_\Sigma(E)}(V) \\ v &\mapsto& \hat v(P) } \end{displaymath} which, under the identification of example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}, sense an [[evolutionary vector field]] $v$ to the [[derivative]] of $P$ (example \ref{TangentVectorFields}) along the prolongation [[tangent vector field]] $\hat v$ of $v$ (prop. \ref{EvolutionaryVectorFieldProlongation}). In the case that $E$ and $V$ are [[trivial vector bundles]] over [[Minkowski spacetime]] with coordinates $((x^\mu), (\phi^a))$ and $((x^\mu), (\rho^b))$, respectively (example \ref{TrivialVectorBundleAsAFieldBundle}), then by \eqref{ProlongationOfEvolutionaryVectorFieldExplicit} this is given by \begin{displaymath} ((\mathrm{D}P)(v))^b \;=\; \left( v^a \frac{\partial P^b}{\partial \phi^a} + \frac{d v^a}{d x^\mu} \frac{\partial P^b}{\partial \phi^a_{,\mu}} + \frac{d^2 v^a}{d x^\mu d x^\nu} \frac{\partial P^b}{\partial \phi^a_{,\mu \nu}} + \cdots \right) \end{displaymath} This makes manifest that $\mathrm{D}P$ may equivalently be regarded as a $J^\infty_\Sigma(E)$-dependent [[differential operator]] (def. \ref{DifferentialOperator}) from the [[vertical tangent bundle]] $T_\Sigma E$ (def. \ref{VerticalTangentBundle}) to $V$, namely a [[bundle homomorphism]] over $\Sigma$ of the form \begin{displaymath} \mathrm{D}_P \;\colon\; J^\infty_\Sigma(E) \times_\Sigma J^\infty_\Sigma T_\Sigma E \longrightarrow V \end{displaymath} in that \begin{equation} \mathrm{D}_P(-,v) = \mathrm{D}P(v) = \hat v (P) \,. \label{FrechetDerivativeAsDifferentialOperatorEquality}\end{equation} \end{defn} (\href{evolutionary+derivative#Olver93}{Olver 93, def. 5.24}) \begin{example} \label{DifferentialOperatorDerivativeOfLagrangianFunction}\hypertarget{DifferentialOperatorDerivativeOfLagrangianFunction}{} \textbf{([[evolutionary derivative]] of [[Lagrangian function]])} Over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), let $\mathbf{L} = L dvol \in \Omega^{p+1,0}_\Sigma(E)$ be a [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), with coefficient function regarded as a field-dependent section (def. \ref{FieldDependentSections}) of the [[trivial bundle|trivial]] [[real line bundle]]: \begin{displaymath} L \;\in \; \Gamma_{J^\infty_\Sigma}(\Sigma \times \mathbb{R}) \,, \end{displaymath} Then the [[formally adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators}) \begin{displaymath} (\mathrm{D}_L)^\ast \;\colon\; J^\infty_\Sigma(E)\times_\Sigma (\Sigma \times \mathbb{R})^\ast \longrightarrow T_\Sigma^\ast E \end{displaymath} of its [[evolutionary derivative]], def. \ref{FieldDependentDifferentialOperatorDerivative}, regarded as a $J^\infty_\Sigma(E)$-dependent differential operator $\mathrm{D}_P$ from $T_\Sigma$ to $V$ and applied to the constant section \begin{displaymath} 1 \in \Gamma_\Sigma(\Sigma \times \mathbb{R}^\ast) \end{displaymath} is the [[Euler-Lagrange derivative]] \eqref{EulerLagrangeEquationGeneral} \begin{displaymath} \delta_{EL}\mathbf{L} \;=\; \left(\mathrm{D}_{L}\right)^\ast(1) \;\in\; \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma^\ast) \simeq \Omega^{p+1,1}_\Sigma(E)_{source} \end{displaymath} via the identification from example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}. \end{example} \begin{prop} \label{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}\hypertarget{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}{} \textbf{([[Euler-Lagrange derivative]] is [[derivation]] via [[evolutionary derivatives]])} Let $V \overset{vb}{\to} \Sigma$ be a [[vector bundle]] (def. \ref{VectorBundle}) and write $V^\ast \overset{}{\to} \Sigma$ for its [[dual vector bundle]] (def. \ref{DualVectorBundle}). For field-dependent sections (def. \ref{FieldDependentSections}) \begin{displaymath} \alpha \in \Gamma_{J^\infty_\Sigma(E)}(V) \end{displaymath} and \begin{displaymath} \beta^\ast \in \Gamma_{J^\infty_\Sigma(E)}(V^\ast) \end{displaymath} we have that the [[Euler-Lagrange derivative]] \eqref{EulerLagrangeEquationGeneral} of their canonical pairing to a [[smooth function]] on the [[jet bundle]] (as in prop. \ref{JetBundleIsLocallyProManifold}) is the sum of the derivative of either one via the [[formally adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators}) of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}) of the other: \begin{displaymath} \delta_{EL}( \alpha \cdot \beta^\ast ) \;=\; (\mathrm{D}_\alpha)^\ast(\beta^\ast) + (\mathrm{D}_{\beta^\ast})^\ast(\alpha) \end{displaymath} \end{prop} \begin{proof} It is sufficient to check this in [[local coordinates]]. By the [[product law]] for [[differentiation]] we have \begin{displaymath} \begin{aligned} \frac{ \delta_{EL} \left(\alpha \cdot \beta^\ast \right) } { \delta \phi^a } & = \frac{\partial \left(\alpha \cdot \beta^\ast \right)}{\partial \phi^a} - \frac{d}{d x^\mu} \left( \frac{\partial \left( \alpha \cdot \beta^\ast \right)}{\partial \phi^a_{,\mu}} \right) + \frac{d}{d x^\mu d x^\nu} \left( \frac{\partial \left( \alpha \cdot \beta^\ast \right) }{\partial \phi^a_{,\mu \nu}} \right) - \cdots \\ & = \phantom{+} \frac{\partial \alpha }{\partial \phi^a} \cdot \beta^\ast - \frac{d}{d x^\mu} \left( \frac{\partial \alpha }{\partial \phi^a_{,\mu}} \cdot \beta^\ast \right) + \frac{d}{d x^\mu d x^\nu} \left( \frac{\partial \alpha }{\partial \phi^a_{,\mu \nu}} \cdot \beta^\ast \right) - \cdots \\ & \phantom{=} + \frac{\partial \beta^\ast }{\partial \phi^a} \cdot \alpha - \frac{d}{d x^\mu} \left( \frac{\partial \beta^\ast }{\partial \phi^a_{,\mu}} \cdot \alpha \right) + \frac{d}{d x^\mu d x^\nu} \left( \frac{\partial \beta^\ast }{\partial \phi^a_{,\mu \nu}} \cdot \alpha \right) - \cdots \\ & = (\mathrm{D}_\alpha)_a^\ast(\beta^\ast) + (\mathrm{D}_{\beta^\ast})_a^\ast(\alpha) \end{aligned} \end{displaymath} \end{proof} \begin{prop} \label{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}\hypertarget{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}{} \textbf{([[evolutionary derivative]] of [[Euler-Lagrange forms]] is [[formally self-adjoint differential operator|formally self-adjoint]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) and regard the [[Euler-Lagrange derivative]] \begin{displaymath} \delta_{EL}\mathbf{L} \;=\; \delta_{EL}L \wedge dvol_\Sigma \end{displaymath} (from prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) as a field-dependent section of the [[vertical cotangent bundle]] \begin{displaymath} \delta_{EL}L \;\in\; \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E) \end{displaymath} as in example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}. Then the corresponding [[evolutionary derivative]] field-dependent [[differential operator]] $D_{\delta_{EL}L}$ (def. \ref{FieldDependentDifferentialOperatorDerivative}) is [[formally self-adjoint differential operator|formally self-adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}): \begin{displaymath} (D_{\delta_{EL}L})^\ast \;=\; D_{\delta_{EL}L} \,. \end{displaymath} (In terms of the [[Euler-Lagrange complex]], remark \ref{EulerLagrangeComplex}, this says that the [[Helmholtz operator]] vanishes on the image of the [[Euler-Lagrange operator]].) \end{prop} (\href{evolutionary+derivative#Olver93}{Olver 93, theorem 5.92}) The following proof is due to [[Igor Khavkine]]. \begin{proof} By definition of the [[Euler-Lagrange form]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) we have \begin{displaymath} \frac{\delta_{EL} L }{\delta \phi^a} \delta \phi^a \, \wedge dvol_\Sigma \;=\; \delta L \,\wedge dvol_\Sigma \;+\; d(...) \,. \end{displaymath} Applying the [[variational derivative]] $\delta$ (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) to both sides of this equation yields \begin{displaymath} \left(\delta \frac{\delta_{EL} L }{\delta \phi^a}\right) \wedge \delta \phi^a \, \wedge dvol_\Sigma \;=\; \underset{= 0}{\underbrace{\delta \delta L}} \wedge dvol_\Sigma \;+\; d(...) \,. \end{displaymath} It follows that for $v,w$ any two [[evolutionary vector fields]] the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of their prolongations $\hat v$ and $\hat w$ (def. \ref{EvolutionaryVectorFieldProlongation}) into the [[variational differential form|differential 2-form]] on the left is \begin{displaymath} \left( \delta \frac{\delta_{EL} L }{\delta \phi^a} \wedge \delta \phi^a \right)(v,w) = w^a (\mathrm{D}_{\delta_{EL}})_a(v) - v^b(\mathrm{D}_{\delta_{EL}})_b(w) \,, \end{displaymath} by inspection of the definition of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}). Moreover, their contraction into the differential form on the right is \begin{displaymath} \iota_{\hat v} \iota_{\hat w} d(...) \;=\; d(...) \end{displaymath} by the fact (prop. \ref{EvolutionaryVectorFieldProlongation}) that contraction with prolongations of evolutionary vector fields anti-commutes with the [[total spacetime derivative]] \eqref{ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative}. Hence the last two equations combined give \begin{displaymath} w^a (\mathrm{D}_{\delta_{EL}})_a(v) - v^b(\mathrm{D}_{\delta_{EL}})_b(w) \;=\; d(...) \,. \end{displaymath} This is the defining condition for $\mathrm{D}_{\delta_{EL}}$ to be [[formally self-adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators}). \end{proof} $\,$ Now we may finally prove that an [[infinitesimal symmetry of the Lagrangian]] is also an infinitesimal symmetry of the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]]: \begin{prop} \label{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}\hypertarget{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}{} \textbf{([[infinitesimal symmetries of the Lagrangian]] are also [[infinitesimal symmetries]] of the [[equations of motion]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]]. If an [[evolutionary vector field]] $v$ is an [[infinitesimal symmetry of the Lagrangian]] then the [[flow]] along its prolongation $\hat v$ preserves the [[prolonged shell]] $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ \eqref{ProlongedShellInJetBundle} in that the [[Lie derivative]] of the [[Euler-Lagrange form]] $\delta_{EL}\mathbf{L}$ along $\hat v$ vanishes on $\mathcal{E}^\infty$: \begin{displaymath} \mathcal{L}_{\hat v}\mathbf{L} = d(...) \phantom{AAA} \Rightarrow \phantom{AAA} \mathcal{L}_{\hat v} \, \delta_{EL}\mathbf{L}\vert_{\mathcal{E}^\infty} = 0 \,. \end{displaymath} \end{prop} \begin{proof} Notice that for any vector field $\hat v$ the [[Lie derivative]] (prop. \ref{CartanHomotopyFormula})$\mathcal{L}_{\hat v}$ of the [[Euler-Lagrange form]] $\delta_{EL}\mathbf{L} = \frac{\delta_{EL}L}{\delta \phi^a} \delta \phi^a \wedge dvol_\Sigma$ differs from that of its component functions $\frac{\delta_{EL}L}{\delta \phi^a} dvol_\Sigma$ by a term proportional to these component functions, which by definition vanishes on-shell: \begin{displaymath} \mathcal{L}_{\hat v} \left( \frac{\delta_{EL} L}{\delta \phi^a} \delta \phi^a \wedge dvol_\Sigma \right) \;=\; \underset{ = \hat v\left( \frac{\delta_{EL}L}{\delta \phi^a} \right) }{ \underbrace{ \left( \mathcal{L}_{\hat v} \frac{\delta_{EL}L}{\delta \phi^a} \right) } } \delta \phi^a \wedge dvol_\Sigma + \underset{ = 0 \, \text{on} \, \mathcal{E}^\infty }{ \underbrace{ \frac{\delta_{EL}L}{\delta \phi^a} } } \left( \mathcal{L}_{\hat v} \delta \phi^a \right) \wedge dvol_\Sigma \end{displaymath} But the Lie derivative of the component functions is just their plain derivative. Therefore it is sufficient to show that \begin{displaymath} \hat v \left( \frac{\delta_{EL} L}{\delta \phi^a} \right) \vert_{\mathcal{E}^\infty} \;=\; 0 \,. \end{displaymath} Now by [[Noether's theorem|Noether's theorem I]] (prop. \ref{NoethersFirstTheorem}) the condition $\mathcal{L}_{\hat v} = d \tilde J_{\hat v}$ for an [[infinitesimal symmetry of the Lagrangian]] implies that the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of the [[Euler-Lagrange form]] with the corresponding [[evolutionary vector field]] is a [[total spacetime derivative]]: \begin{displaymath} \iota_{\hat v} \, \delta_{EL}\mathbf{L} \;=\; d J_{\hat v} \,. \end{displaymath} Since the [[Euler-Lagrange derivative]] vanishes on [[total spacetime derivative]] (example \ref{TrivialLagrangianDensities}) also its application on the contraction on the left vanishes. But via example \ref{EvolutionaryVectorFieldsAsFieldDependentSections} that contraction is a pairing of field-dependent sections as in prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}. Hence we use this proposition to compute: \begin{equation} \begin{aligned} 0 & = \frac{\delta_{EL} \left( v \cdot \delta_{EL} L\right) }{ \delta \phi^a } \\ & = (\mathrm{D}_{v})^\ast_a( \delta_{EL}L ) + (\mathrm{D}_{\delta_{EL}L})^\ast_a(v) \\ & = (\mathrm{D}_{v})^\ast_a( \delta_{EL}L ) + (\mathrm{D}_{\delta_{EL}L})_a(v) \\ & = (\mathrm{D}_{v})^\ast_a( \delta_{EL}L ) + \hat v\left( \frac{\delta_{EL}L}{\delta \phi^a} \right) \,. \end{aligned} \label{TowardsProofThatSymmetriesPreserveTheShell}\end{equation} Here the first step is by prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}, the second step is by prop. \ref{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint} and the third step is \eqref{FrechetDerivativeAsDifferentialOperatorEquality}. Hence \begin{displaymath} \begin{aligned} \hat v(\delta_{EL}L) \vert_{\mathcal{E}^\infty} & = - (\mathrm{D}_{v})^\ast( \delta_{EL}L ) \vert_{\mathcal{E}^\infty} \\ & = 0 \end{aligned} \,, \end{displaymath} where in the last line we used that on the [[prolonged shell]] $\delta_{EL}L$ and all its horizontal derivatives vanish, by definition. \end{proof} As a corollary we obtain: \begin{prop} \label{FlowAlongInfinitesimalSymmetryOfLagrangianPreservesOnShellSpaceOfFieldHistories}\hypertarget{FlowAlongInfinitesimalSymmetryOfLagrangianPreservesOnShellSpaceOfFieldHistories}{} \textbf{([[flow]] along [[infinitesimal symmetry of the Lagrangian]] preserves [[on-shell]] [[space of field histories]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). For $v$ an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}) the [[flow]] on the [[space of field histories]] (example \ref{DiffeologicalSpaceOfFieldHistories}) that it induces by def. \ref{FlowOfFieldHistoriesAlongEvolutionaryVectorField} preserves the space of [[on-shell]] field histories (from prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}): \begin{displaymath} \itexarray{ \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} &\hookrightarrow& \Gamma_\Sigma(E) \\ {\mathllap{\exp(\hat v)\vert_{\delta_{EL}\mathbf{L} = 0} }} \uparrow && \uparrow {\mathrlap{\exp(\hat v)}} \\ \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} &\hookrightarrow& \Gamma_\Sigma(E) } \end{displaymath} \end{prop} \begin{proof} By def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} a field history $\Phi \in \Gamma_\Sigma(E)$ is [[on-shell]] precisely if its [[jet prolongation]] $j^\infty_\Sigma(E)$ (def. \ref{JetProlongation}) factors through the [[shell]] $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ \eqref{ShellInJetBundle}. Hence by def. \ref{FlowOfFieldHistoriesAlongEvolutionaryVectorField} the statement is equivalently that the ordinary flow (prop. \ref{CartanHomotopyFormula}) of $\hat v$ (def. \ref{EvolutionaryVectorFieldProlongation}) on the [[jet bundle]] $J^\infty_\Sigma(E)$ preserves the [[shell]]. This in turn means that it preserves the vanishing locus of the [[Euler-Lagrange form]] $\delta_{EL} \mathbf{L}$, which is the case by prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}. \end{proof} $\,$ \textbf{[[infinitesimal symmetries]] of the [[presymplectic potential current]]} Evidently [[Noether's theorem|Noether's theorem I]] in [[variational calculus]] (prop. \ref{NoethersFirstTheorem}) is the special case for horizontal $p+1$-forms of a more general phenomenon relating symmetries of variational forms to forms that are closed up to a contraction. The same phenomenon applied instead to the [[presymplectic current]] yields the following: \begin{defn} \label{LieDerivativeVariational}\hypertarget{LieDerivativeVariational}{} \textbf{(variational Lie derivative)} Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) with [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). For $v$ a vertical [[tangent vector field]] on the [[jet bundle]] (a variation def. \ref{Variation}) write \begin{equation} \mathcal{L}^{var}_{v} \;\coloneqq\; \delta \circ \iota_v + \iota_v \circ \delta \label{LieDerivativeVariational}\end{equation} for the \emph{variational Lie derivative} along $v$, analogous to [[Cartan's homotopy formula]] (prop. \ref{CartanHomotopyFormula}) but defined in terms of the variational derivative $\delta$ \eqref{VariationalDerivative} as opposed to the full [[de Rham differential]]. Then for $v_1$ and $v_2$ two vertical vector fields, write \begin{displaymath} [v_1, v_2]^{var} \;\in \; \Gamma( T_{vert} J^\infty_\Sigma(E) ) \end{displaymath} for the vector field whose contraction operator (def. \ref{ContractionOfFormsWithVectorFields}) is given by \begin{displaymath} \begin{aligned} \iota_{[v_1,v_2]^{var}} & = \left[ \mathcal{L}^{var}_{v_1}, \iota_{v_2} \right] \\ & \coloneqq \mathcal{L}^{var}_{v_1} \circ \iota_{v_2} - \iota_{v_2} \circ \mathcal{L}^{var}_{v_1} \end{aligned} \,, \end{displaymath} \end{defn} \begin{defn} \label{HamiltonianForms}\hypertarget{HamiltonianForms}{} \textbf{([[Hamiltonian vector fields|infinitesimal symmetry of the presymplectic potential]] and [[Hamiltonian differential forms]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) with [[presymplectic potential current]] $\Theta_{BFV}$ \eqref{PresymplecticPotential}. Write $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ for the [[shell]] \eqref{ShellInJetBundle}. Then: \begin{enumerate}% \item An [[on-shell]] variation $v$ (def. \ref{Variation}) is an \emph{[[infinitesimal symmetry]] of the [[presymplectic current]]} or \emph{[[Hamiltonian vector field]]} if [[on-shell]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) its variational Lie derivative along $v$ (def. \ref{LieDerivativeVariational}) is a [[variational derivative]]: \begin{displaymath} (\delta \circ \iota_v + \iota_v \circ \delta) \Theta_{BFV} = \delta \tilde H_v \phantom{AAA} \text{on}\, \mathcal{E} \end{displaymath} for some variational form $\tilde H_v$. \item A \emph{[[Hamiltonian differential form]]} $H$ (or \emph{local Hamiltonian current}) is a variational form on the shell such that there exists a variation $v$ with \begin{displaymath} \delta H = \iota_v \Omega_{BFV} \phantom{AA} \, \text{on}\, \mathcal{E} \,. \end{displaymath} \end{enumerate} We write \begin{displaymath} \Omega^{p,0}_{\Sigma, Ham}(E) \;\coloneqq\; \left\{ (H,v) \;\vert\; v \, \text{is a variation and}\, \iota_v \Omega_{BFV} = \delta H \right\} \end{displaymath} for the space of pairs consisting of a Hamiltonian differential forms [[on-shell]] and a corresponding variation. \end{defn} \begin{prop} \label{HamiltonianDifferentialForms}\hypertarget{HamiltonianDifferentialForms}{} \textbf{([[Hamiltonian Noether's theorem]])} A variation $v$ is an infinitesimal symmetry of the presymplectic potential (def. \ref{HamiltonianForms}) with $\mathcal{L}^{var}_v ( \Theta_{BFV} ) = \delta \tilde H_v$ precisely if \begin{displaymath} H_v \coloneqq \tilde H_v - \iota_v \Theta_{BFV} \end{displaymath} is a [[Hamiltonian differential form]] for $v$. \end{prop} \begin{proof} From the definition \eqref{LieDerivativeVariational} of $\mathcal{L}^{var}_v$ we have \begin{displaymath} \begin{aligned} & \mathcal{L}^{var}_v \Theta_{BFV} = \delta \tilde H_v \\ \Leftrightarrow\;\; & \delta \iota_v \Theta_{BFV} + \iota_v \underset{= \Omega_{BFV}}{\underbrace{\delta \Theta_{BFV}}} = \delta \tilde H_v \\ \Leftrightarrow\;\; & \delta \left( \tilde H_v - \iota_v \Theta_{BFV} \right) = \iota_v \Omega_{BFV} \,, \end{aligned} \end{displaymath} where we used the definition \eqref{PresymplecticCurrent} of $\Omega_{BFV}$ . \end{proof} $\,$ Since therefore both the [[conserved currents]] from [[Noether's theorem]] as well as the [[Hamiltonian differential forms]] are generators of infinitesimal [[symmetries]] of certain variational forms (namely of the [[Lagrangian density]] and of the [[presymplectic current]], respectively) they form a [[Lie algebra]]. For the conserved currents this is sometimes known as the \emph{[[Dickey bracket]] Lie algebra}. For the Hamiltonian forms it is the \emph{[[Poisson bracket Lie n-algebra|Poisson bracket Lie p+1-algebra]]}. Since here for simplicity we are considering just vertical variations, we have just a plain [[Lie algebra]]. The [[transgression of variational differential forms|transgression]] of this Lie algebra of Hamiltonian forms on the jet bundle to [[Cauchy surfaces]] yields a [[presymplectic structure]] on [[phase space]], this we discuss \hyperlink{PhaseSpace}{below}. \begin{prop} \label{LocalPoissonBracket}\hypertarget{LocalPoissonBracket}{} \textbf{([[Poisson bracket Lie n-algebra|local Poisson bracket]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). On the space $\Omega^{p,0}_{\Sigma,Ham}(E)$ pairs $(H,v)$ of [[Hamiltonian differential forms]] $H$ with compatible variation $v$ (def. \ref{HamiltonianForms}) the following operation constitutes a [[Lie bracket]]: \begin{equation} \left\{(H_1, v_1),\, (H_2, v_2)\right\} \;\coloneqq\; (\iota_{v_1} \iota_{v_2} \Omega_{BFV},\, [v_1,v_2]^{var}) \,, \label{LocalPoissonLieBracket}\end{equation} where $[v_1, v_2]^{var}$ is the variational Lie bracket from def. \ref{LieDerivativeVariational}. We call this the \emph{local Poisson Lie bracket}. \end{prop} \begin{proof} First we need to check that the bracket is well defined in itself. It is clear that it is linear and skew-symmetric, but what needs proof is that it does indeed land in $\Omega^{p,0}_{\Sigma,Ham}(E)$, hence that the following equation holds: \begin{displaymath} \delta \iota_{v_2} \iota_{v_1} \Omega_{BFV} \;=\; \iota_{[v_1, v_2]^{var}} \Omega_{BFV} \,. \end{displaymath} With def. \ref{LieDerivativeVariational} for $\mathcal{L}^{var}$ and $[-,-]^{var}$ we compute this as follows: \begin{displaymath} \begin{aligned} \delta \iota_{v_1} \iota_{v_2} \Omega_{BFV} & = \tfrac{1}{2} \delta \iota_{v_1} \iota_{v_2} \Omega_{BFV} - \tfrac{1}{2} (v_1 \leftrightarrow v_2) \\ & = \tfrac{1}{2} \left( \mathcal{L}^{var}_{v_1} \iota_{v_2} \Omega_{BFV} - \iota_{v_1} \delta \iota_{v_2} \Omega_{BFV} \right) - \tfrac{1}{2} (v_1 \leftrightarrow v_2) \\ & = \tfrac{1}{2} \left( \mathcal{L}^{var}_{v_1} \iota_{v_2} \Omega_{BFV} - \iota_{v_1} \mathcal{L}^{var}_{v_2} \Omega_{BFV} + \iota_{v_1} \iota_{v_2} \underset{= 0}{\underbrace{\delta \Omega_{BFV}}} \right) - \tfrac{1}{2} (v_1 \leftrightarrow v_2) \\ & = [\mathcal{L}^{var}_{v_2}, \iota_{v_1}] \Omega_{BFV} \\ & = \iota_{[v_1, v_2]^{var}} \Omega_{BFV} \,. \end{aligned} \end{displaymath} This shows that the bracket is well defined. It remains to see that the bracket satifies the [[Jacobi identity]]: \begin{displaymath} \left\{ (H_1, v_1), \left\{ (H_2, v_2), (H_3,v_3) \right\} \right\} \;+\; (cyclic) \;=\; 0 \end{displaymath} hence that \begin{displaymath} \left( \iota_{v_1} \iota_{[v_2,v_3]^{var}} \Omega_{BFV} ,\, [v_1, [v_2, v_2]^{var}]^{var} \right) \;+\; (cyclic) \;=\; 0 \,. \end{displaymath} Here $[v_1, [v_2, v_3]^{var}]^{var} + (cyclic) = 0$ holds because by def. \ref{LieDerivativeVariational} $[v_1,-]^{var}$ acts as a derivation, and hence what remains to be shown is that \begin{displaymath} \iota_{v_1} \iota_{\left([v_2, v_3]^{var}\right)} \Omega_{BFV} + (cyclic) = 0 \end{displaymath} We check this by repeated uses of def. \ref{LieDerivativeVariational}, using in addition that \begin{enumerate}% \item $\delta \iota_{v_i} \Omega_{BFV} = 0$ (since $\iota_{v_i} \Omega_{BFV} = \delta H_i$ by $v_i$ being Hamiltonian) \item $\mathcal{L}^{var}_{v_i} \Omega_{BFV} = 0$ (since in addition $\delta \Omega_{BFV} = 0$) \item $\iota_{v_1} \iota_{v_2} \iota_{v_3} \Omega_{BFV} = 0$ (since $\Omega_{BFV} \in \Omega^{p,2}_\Sigma(E)$ is of vertical degree 2, and since all variations $v_i$ are vertical by assumption). \end{enumerate} So we compute as follows (a special case of \href{Poisson+bracket+Lie+n-algebra#FRS13b}{FRS 13b, lemma 3.1.1}): \begin{displaymath} \begin{aligned} 0 & = \delta \iota_{v_1} \iota_{v_2} \iota_{v_3} \Omega_{BFV} \\ & = \mathcal{L}^{var}_{v_1} \iota_{v_2} \iota_{v_3} \Omega_{BFV} - \iota_{v_1} \delta \iota_{v_2} \iota_{v_3} \Omega_{BFV} \\ & = \iota_{[v_1, v_2]^{var}} \iota_{v_3} \Omega_{BFV} + \iota_{v_2} \mathcal{L}^{var}_{v_1} \iota_{v_3} \Omega_{BFV} - \iota_{v_1} \mathcal{L}^{var}_{v_2} \iota_{v_3} \Omega_{BFV} + \iota_{v_1} \iota_{v_2} \delta \iota_{v_3} \Omega_{BFV} \\ & = \iota_{[v_1, v_2]^{var}} \iota_{v_3} \Omega_{BFV} + \iota_{v_2} \iota_{[v_1,v_3]^{var}} \Omega_{BFV} - \iota_{v_1} \iota_{[v_2, v_3]^{var}} \Omega_{BFV} \\ & = - \iota_{v_1} \iota_{[v_2, v_3]^{var}} \Omega_{BFV} - \iota_{v_2} \iota_{[v_3, v_1]^{var}} \Omega_{BFV} - \iota_{v_3} \iota_{[v_1, v_2]^{var}} \Omega_{BFV} \,. \end{aligned} \end{displaymath} \end{proof} $\,$ The [[Poisson bracket Lie n-algebra|local Poisson bracket]] [[Lie algebra]] $(\Omega^{p,0}_{\Sigma,Ham}(E), [-,-]^{var})$ from prop. \ref{LocalPoissonBracket} is but the lowest stage of a [[higher Lie theory|higher Lie theoretic]] structure called the \emph{[[Poisson bracket Lie n-algebra|Poisson bracket Lie p-algebra]]}. Here we will not go deeper into this [[schreiber:Higher Structures|higher structure]] (see at \emph{[[schreiber:Higher Prequantum Geometry]]} for more), but below we will need the following simple shadow of it: \begin{lemma} \label{HorizontallyExactFormsDropOutOfLocalLieBracket}\hypertarget{HorizontallyExactFormsDropOutOfLocalLieBracket}{} The horizontally exact Hamiltonian forms constitute a [[Lie ideal]] for the local Poisson Lie bracket \eqref{LocalPoissonLieBracket}. \end{lemma} \begin{proof} Let $E$ be a horizontally exact Hamiltonian form, hence \begin{displaymath} E = d K \end{displaymath} for some $K$. Write $e$ for a [[Hamiltonian vector field]] for $E$. Then for $(H,v)$ any other pair consisting of a Hamiltonian form and a corresponding Hamiltonian vector field, we have \begin{displaymath} \begin{aligned} \iota_v \, \iota_e \, \Omega_{BFV} & = \phantom{-}\iota_v \, \delta E \\ & = \phantom{-}\iota_v \, \delta \, d \, K \\ & = - \iota_v \, d \, \delta K \\ & = \phantom{-}d \, \iota_v \, \delta \, K \,. \end{aligned} \end{displaymath} Here we used that the horizontal derivative anti-commutes with the vertical one by construction of the [[variational bicomplex]], and that $\iota_e$ anti-commutes with the horizontal derivative $d$ since the variation $e$ (def. \ref{Variation}) is by definition vertical. \end{proof} \begin{example} \label{LocalPoissonBracketForRealScalarField}\hypertarget{LocalPoissonBracketForRealScalarField}{} \textbf{([[Poisson bracket Lie n-algebra|local Poisson bracket]] for [[real scalar field]])} Consider the [[Lagrangian field theory]] for the [[free field|free]] [[real scalar field]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}. By example \ref{FreeScalarFieldEOM} its [[presymplectic current]] is \begin{displaymath} \Omega_{BFV} = \eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \wedge \iota_{\partial_\mu} dvol_\Sigma \, \end{displaymath} The corresponding [[Poisson bracket Lie n-algebra|local Poisson bracket algebra]] (prop. \ref{LocalPoissonBracket}) has in degree 0 [[Hamiltonian forms]] (def. \ref{HamiltonianDifferentialForms}) such as \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} The corresponding [[Hamiltonian vector fields]] are \begin{displaymath} v_Q = -\partial_{\phi_{,0}} \end{displaymath} and \begin{displaymath} v_P = - \partial_{\phi} \,. \end{displaymath} Hence the corresponding [[Poisson bracket Lie n-algebra|local Poisson bracket]] is \begin{displaymath} \{P,Q\} = \iota_{v_P} \iota_{v_Q} \omega = \iota_{\partial_0} dvol_\Sigma \,. \end{displaymath} More generally for $b_1, b_2 \in C^\infty_{cp}(\Sigma)$ two [[bump functions]] then \begin{displaymath} \{ b_1 P, b_2 Q \} = b_1 b_2 \iota_{\partial_0} dvol_\Sigma \,. \end{displaymath} \end{example} \begin{example} \label{LocalPoissonBracketForDiracField}\hypertarget{LocalPoissonBracketForDiracField}{} \textbf{([[Poisson bracket Lie n-algebra|local Poisson bracket]] for [[free field theory|free]] [[Dirac field]])} Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] (example \ref{LagrangianDensityForDiracField}), whose [[presymplectic current]] is, according to example \ref{PresymplecticCurrentDiracField}, given by \begin{equation} \Omega_{BFV} \;=\; (\overline{\delta \psi}) \, \gamma^\mu \, (\delta \psi) \, \iota_{\partial_\mu} dvol_\Sigma \,. \label{RecallPresymplecticCurrentOfDiracField}\end{equation} Consider this specifically in [[spacetime]] [[dimension]] $p + 1 = 4$ in which case the components $\psi_\alpha$ are [[complex number]]-valued (by prop./def. \ref{SpacetimeAsMatrices}), so that the [[tuple]] $(\psi_\alpha)$ amounts to 8 real-valued coordinate functions. By changing complex coordinates, we may equivalently consider $(\psi_\alpha)$ as four coordinate functions, and $(\overline{\psi}^\alpha)$ as another four independent coordinate functions. Using this coordinate transformation, it is immediate to find the following [[pairs]] of [[Hamiltonian vector fields]] and their [[Hamiltonian differential forms]] from def. \ref{HamiltonianForms} applied to \eqref{RecallPresymplecticCurrentOfDiracField} \begin{tabular}{l|l} [[Hamiltonian vector field]]&[[Hamiltonian differential form]]\\ \hline $\phantom{AA} \partial_{\psi_\alpha}$&$\phantom{AA}\left(\overline{\delta \psi}\gamma^\mu\right)^\alpha\, \iota_{\partial_\mu} dvol_\Sigma$\\ $\phantom{AA} \partial_{\overline{\psi}_\alpha}$&$\phantom{AA}\left( \gamma^\mu \psi \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma$\\ \end{tabular} and to obtain the following non-trivial [[Poisson bracket Lie n-algebra|local Poisson brackets]] (prop. \ref{LocalPoissonBracket}) (the other possible brackets vanish): \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} Notice the signs: Due to the odd-grading of the field coordinate function $\psi$, its variational derivative $\delta \psi$ has bi-degree $(1,odd)$ and the contraction operation $\iota_{\psi}$ has bi-degree $(-1,odd)$, so that commuting it past $\overline{\psi}$ picks up \emph{two} minus signs, a ``cohomological'' sign due to the differential form degrees, and a ``supergeometric'' one (def. \ref{DifferentialFormOnSuperCartesianSpaces}): \begin{displaymath} \iota_{\partial_\psi} \overline{\delta \psi} \cdots = (-1) (-1) \overline{\delta \psi} \,\iota_{\partial_\psi} \cdots \,. \end{displaymath} For the same reason, the [[Poisson bracket Lie n-algebra|local Poisson bracket]] is a \emph{[[super Lie algebra]]} with \emph{symmetric} 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\{ \left(\overline{\psi}\gamma^\mu\right)^\beta\, \iota_{\partial_\mu} dvol_\Sigma \,,\, \left( \gamma^\mu \psi \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma \right\} \,. \end{displaymath} \end{example} $\,$ This concludes our discussion of general [[infinitesimal symmetries of a Lagrangian]]. We pick this up again in the discussion of \emph{\hyperlink{GaugeSymmetries}{Gauge symmetries}} below. First, in the \hyperlink{Observables}{next chapter} we discuss the concept of [[observables]] in [[field theory]]. \end{document}