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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*{symmetry of a Lagrangian density} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory} [[!include AQFT and operator algebra contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Throughout, let \begin{enumerate}% \item $\Sigma$ be a [[smooth manifold]] of [[dimension]] $p+1$, thought of as [[spacetime]]; \item $E \overset{fb}{\longrightarrow} \Sigma$ a [[fiber bundle]] thought of as a \emph{[[field bundle]]} \item $\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E)$ a [[Lagrangian density]]. \end{enumerate} \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 $[d, \iota_{\hat v_i}]$, 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 a variation $v$ (def. \ref{Variation}) which arises as the prolongation $\hat v$ (prop. \ref{EvolutionaryVectorFieldProlongation}) of an [[evolutionary vector field]] $v$ (def. \ref{EvolutionaryVectorField}) such that the [[Lie derivative]] $\mathcal{L}_v$ of the Lagrangian density along $\hat v$ is a [[total derivative|total spacetime derivative]] \begin{displaymath} \mathcal{L}_v \mathbf{L} = d \tilde J_v \end{displaymath} \item an \emph{[[on-shell]] [[conserved current]]} is a horizontal $p$-form $J \in \Omega^{p,0}_\Sigma(E)$ 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 $\mathcal{L}_v \mathbf{L} = d \tilde J_v$, then \begin{displaymath} J_v \coloneqq \tilde J_v - \iota_v \Theta_{BFV} \end{displaymath} is an [[on-shell]] [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}), for $\Theta_{BFV}$ a presymplectic potential \eqref{PresymplecticPotential} from def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}. \end{prop} \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_v & = \mathcal{L}_v \mathbf{L} \\ & = \iota_v \underset{= \delta_{EL}\mathbf{L} - d \Theta_{BFV}}{\underbrace{\mathbf{d} \mathbf{L}}} + \mathbf{d} \underset{= 0}{\underbrace{\iota_v \mathbf{L}}} \\ & = \iota_v \delta_{EL} \mathbf{L} + d \iota_v \Theta_{BFV} \end{aligned} \end{displaymath} which is equivalent to \begin{displaymath} d(\underset{= J_v}{\underbrace{\tilde J_v - \iota_v \Theta_{BFV}}}) = \iota_v \delta_{EL}\mathbf{L} \end{displaymath} Since, by definition of the [[shell]] $\mathcal{E}$, the form $\frac{\delta_{EL} \mathbf{L}}{\delta v}$ vanishes on $\mathcal{E}$ this yields the claim. \end{proof} \begin{prop} \label{FlowAlongImplicitInfinitesimalGaugeSymmetryPreservesOnShellSpaceofFieldHistories}\hypertarget{FlowAlongImplicitInfinitesimalGaugeSymmetryPreservesOnShellSpaceofFieldHistories}{} \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]] $\delta_{EL} \mathbf{L}$. For this it is sufficient to show that the [[derivative]] of the components of the [[Euler-Lagrange form]] along $\hat v$ vanish on the [[prolonged shell]] \begin{displaymath} \hat v \left( \frac{\delta_{EL} L}{\delta \phi^a} \right) \;=\; 0 \phantom{AA} on \mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E) \,. \end{displaymath} This is the statement of \hyperlink{Olver95}{Olver 95, theorem 5.53}. \end{proof} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Peter Olver]], \emph{Applications of Lie groups to differential equations}, Springer; \emph{Equivalence, invariants, and symmetry}, Cambridge Univ. Press 1995. \end{itemize} [[!redirects symmetries of a Lagrangian density]] [[!redirects symmetry of the Lagrangian density]] [[!redirects symmetries of the Lagrangian density]] [[!redirects symmetry of a Lagrangian]] [[!redirects symmetries of a Lagrangian]] [[!redirects symmetry of the Lagrangian]] [[!redirects symmetries of the Lagrangian]] [[!redirects infinitesimal symmetry of a Lagrangian density]] [[!redirects infinitesimal symmetries of a Lagrangian density]] [[!redirects infinitesimal symmetry of the Lagrangian density]] [[!redirects infinitesimal symmetries of the Lagrangian density]] [[!redirects infinitesimal symmetry of a Lagrangian]] [[!redirects infinitesimal symmetries of a Lagrangian]] [[!redirects infinitesimal symmetry of the Lagrangian]] [[!redirects infinitesimal symmetries of the Lagrangian]] [[!redirects infinitesimal symmetries of Lagrangians]] [[!redirects rigid infinitesimal symmetry of the Lagrangian]] [[!redirects rigid infinitesimal symmetries of the Lagrangian]] \end{document}