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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 -- Lagrangians} \hypertarget{Lagrangians}{}\subsection*{{Lagrangians}}\label{Lagrangians} In this chapter we discuss the following topics: \begin{itemize}% \item \emph{\hyperlink{LagrangianDensities}{Lagrangian densities}} \item \emph{\hyperlink{ELFormsAndPresymplecticCurrents}{Euler-Lagrange forms and Presymplectic currents}} \item \emph{\hyperlink{ELEquationsOfMotion}{Euler-Lagrange equations of motion}} \end{itemize} $\,$ Given any [[type]] of [[field (physics)|fields]] (def. \ref{FieldsAndFieldBundles}), those [[field histories]] that are to be regarded as ``physically realizable'' (if we think of the field theory as a description of the [[observable universe]]) should satisfy some [[differential equation]] -- the \emph{[[equation of motion]]} -- meaning that realizability of any field histories may be checked upon restricting the configuration to the [[infinitesimal neighbourhoods]] (example \ref{InfinitesimalNeighbourhood}) of each spacetime point. This expresses the physical absence of ``action at a distance'' and is one aspect of what it means to have a \emph{[[local field theory]]}. By remark \ref{JetBundleInTermsOfSyntheticDifferentialGeometry} this means that [[equations of motion]] of a field theory are [[equations]] among the [[coordinates]] of the [[jet bundle]] of the [[field bundle]]. For many field theories of interest, their [[differential equation|differential]] [[equation of motion]] is not a random [[partial differential equations]], but is of the special kind that exhibits the ``[[principle of extremal action]]'' (prop. \ref{PrincipleOfExtremalAction} below) determined by a \emph{[[local Lagrangian density]]} (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} below). These are called \emph{[[Lagrangian field theories]]}, and this is what we consider here. Namely among all the [[variational differential forms]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) two kinds stand out, namley the 0-forms in $\Omega^{0,0}_\Sigma(E)$ -- the smooth functions -- and the horizontal $p+1$-forms $\Omega^{p+1,0}_\Sigma(E)$ -- to be called the \emph{[[Lagrangian densities]] $\mathbf{L}$} (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} below) -- since these occupy the two ``corners'' of the [[variational bicomplex]] \eqref{VariationalBicomplexDiagram}. There is not much to say about the 0-forms, but the [[Lagrangian densities]] $\mathbf{L}$ do inherit special structure from their special position in the [[variational bicomplex]]: Their [[variational derivative]] $\delta \mathbf{L}$ uniquely decomposes as \begin{enumerate}% \item the \emph{[[Euler-Lagrange derivative]]} $\delta_{EL} \mathbf{L}$ which is proportional to the variation of the fields (instead of their derivatives) \item the [[total derivative|total spacetime derivative]] $d \Theta_{BFV}$ of a potential $\Theta_{BFV}$ for a \emph{[[presymplectic current]]} $\Omega_{BFV} \coloneqq \delta \Theta_{BFV}$. \end{enumerate} This is prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime} below: \begin{displaymath} \delta \mathbf{L} \;=\; \underset{ \text{Euler-Lagrange variation} }{\underbrace{\delta_{EL}\mathbf{L}}} - d \underset{\text{presymplectic current}}{\underbrace{\Theta_{BFV}}} \,. \end{displaymath} These two terms play a pivotal role in the theory: The condition that the first term vanishes on [[field histories]] is a [[differential equation]] on field histories, called the \emph{[[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]]} (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} below). The space of solutions to this [[differential equation]], called the \emph{[[on-shell]] [[space of field histories]]} \begin{equation} \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \overset{\phantom{AAA}}{\hookrightarrow} \Gamma_\Sigma(E) \label{InclusionOfOnShellSpaceOfFieldHistories}\end{equation} has the interpretation of the space of ``physically realizable field histories''. This is the key object of study in the following chapters. Often this is referred to as the space of \emph{[[classical field theory|classical]] field histories}, indicating that this does not yet reflect the full [[quantum field theory]]. Indeed, there is also the second term in the variational derivative of the Lagrangian density, the [[presymplectic current]] $\Theta_{BFV}$, and this implies a [[presymplectic structure]] on the on-shell space of field histories (def. \ref{PhaseSpaceAssociatedWithCauchySurface} below) which encodes [[deformations]] of the algebra of smooth functions on $\Gamma_\Sigma(E)$. This deformation is the \emph{[[quantization]]} of the field theory to an actual [[quantum field theory]], which we discuss \hyperlink{Quantization}{below}. \begin{displaymath} \itexarray{ &&& \delta \mathbf{L} \\ &&& = \\ & & \delta_{EL}\mathbf{L} &- & d \Theta_{BFV} & \\ & \swarrow && && \searrow \\ \itexarray{ \text{classical} \\ \text{field theory} } && && && \itexarray{ \text{deformation to} \\ \text{quantum} \\ \text{field theory} } } \end{displaymath} $\,$ \textbf{[[Lagrangian densities]]} \begin{defn} \label{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}\hypertarget{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}{} \textbf{([[local Lagrangian density]])} Given a [[field bundle]] $E$ over a $(p+1)$-dimensional [[Minkowski spacetime]] $\Sigma$ as in example \ref{TrivialVectorBundleAsAFieldBundle}, then a \emph{[[local Lagrangian density]]} $\mathbf{L}$ (for the type of field thus defined) is a [[horizontal differential form]] of degree $(p+1)$ (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) on the corresponding [[jet bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}): \begin{displaymath} \mathbf{L} \;\in \; \Omega^{p+1,0}_{\Sigma}(E) \,. \end{displaymath} By example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle} in terms of the given [[volume form]] on spacetimes, any such Lagrangian density may uniquely be written as \begin{equation} \mathbf{L} = L \, dvol_\Sigma \label{LagrangianFunctionViaVolumeForm}\end{equation} where the [[coefficient]] function (the \emph{Lagrangian function}) is a smooth function on the spacetime and field coordinates: \begin{displaymath} L = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots ) \,. \end{displaymath} where by prop. \ref{JetBundleIsLocallyProManifold} $L((x^\mu), \cdots)$ depends locally on an arbitrary but finite order of derivatives $\phi^a_{,\mu_1 \cdots \mu_k}$. We say that a [[field bundle]] $E \overset{fb}{\to} \Sigma$ (def. \ref{FieldsAndFieldBundles}) equipped with a [[local Lagrangian density]] $\mathbf{L}$ is (or defines) a \emph{[[prequantum field theory|prequantum]] [[Lagrangian field theory]]} on the [[spacetime]] $\Sigma$. \end{defn} \begin{remark} \label{ParameterizedLagrangianDensities}\hypertarget{ParameterizedLagrangianDensities}{} \textbf{(parameterized and [[physical unit]]-less [[Lagrangian densities]])} More generally we may consider parameterized collections of [[Lagrangian densities]], i.e. functions \begin{displaymath} \mathbf{L}_{(-)} \;\colon\; U \longrightarrow \Omega^{p+1,0}_\Sigma(E) \end{displaymath} for $U$ some [[Cartesian space]] or generally some [[super Cartesian space]]. For example all [[Lagrangian densities]] considered in [[relativistic field theory]] are naturally [[smooth functions]] of the scale of the [[metric]] $\eta$ (def. \ref{SpacetimeAsMatrices}) \begin{displaymath} \itexarray{ \mathbb{R}_{\gt 0} &\overset{}{\longrightarrow}& \Omega^{p+1,0}_\Sigma(E) \\ r &\mapsto& \mathbf{L}_{r^2\eta} } \end{displaymath} But by the discussion in remark \ref{MinkowskiMetricAndPhysicalUnitOfLength}, in [[physics]] a rescaling of the [[metric]] is interpreted as reflecting but a change of [[physical units]] of [[length]]/[[distance]]. Hence if a [[Lagrangian density]] is supposed to express intrinsic content of a [[theory (physics)|physical theory]], it should remain unchanged under such a change of [[physical units]]. This is achieved by having the Lagrangian be parameterized by \emph{further} parameters, whose corresponding [[physical units]] compensate that of the metric such as to make the Lagrangian density ``[[physical unit]]-less''. This means to consider parameter spaces $U$ equipped with an [[action]] of the multiplicative [[group]] $\mathbb{R}_{\gt 0}$ of [[positive real numbers]], and parameterized Lagrangians \begin{displaymath} \mathbf{L}_{(-)} \;\colon\; U \longrightarrow \Omega^{p+1,0}_\Sigma(E) \end{displaymath} which are [[invariant]] under this [[action]]. \end{remark} \begin{remark} \label{LocallyVariationalFieldTheory}\hypertarget{LocallyVariationalFieldTheory}{} \textbf{([[locally variational field theory]] and Lagrangian [[circle n-bundle with connection|p-gerbe connection]])} If the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) is not just a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) then a Lagrangian density for a given [[equation of motion]] may not exist as a globally defined differential $(p+1)$-form, but only as a [[circle n-bundle with connection|p-gerbe connection]]. This is the case for \emph{[[locally variational field theories]]} such as the \emph{[[charged particle]]}, the \emph{[[WZW model]]} and generally theories involving \emph{[[higher WZW terms]]}. For more on this see the exposition at \emph{[[schreiber:Higher Structures|Higher Structures in Physics]]}. \end{remark} \begin{example} \label{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}\hypertarget{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}{} \textbf{([[local Lagrangian density]] for [[free field|free]] [[real scalar field]] on [[Minkowski spacetime]])} Consider the [[field bundle]] for the [[real scalar field]] from example \ref{RealScalarFieldBundle}, i.e. the [[trivial line bundle]] over [[Minkowski spacetime]]. According to def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} its [[jet bundle]] $J^\infty_\Sigma(E)$ has canonical coordinates \begin{displaymath} \left\{ \{x^\mu\}, \phi, \{\phi_{,\mu}\}, \{\phi_{,\mu_1 \mu_2}\}, \cdots \right\} \,. \end{displaymath} In these coordinates, the [[local Lagrangian density]] $L \in \Omega^{p+1,0}(\Sigma)$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) defining the [[free field|free]] [[real scalar field]] of [[mass]] $m \in \mathbb{R}$ on $\Sigma$ is \begin{displaymath} L \coloneqq \tfrac{1}{2} \left( \eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu} - m^2 \phi^2 \right) \mathrm{dvol}_\Sigma \,. \end{displaymath} This is naturally thought of as a collection of Lagrangians smoothly parameterized by the [[metric]] $\eta$ and the [[mass]] $m$. For this to be [[physical unit]]-free in the sense of remark \ref{ParameterizedLagrangianDensities} the [[physical unit]] of the parameter $m$ must be that of the inverse metric, hence must be an inverse [[length]] according to remark \ref{MinkowskiMetricAndPhysicalUnitOfLength} This is the \emph{inverse [[Compton wavelength]]} $\ell_m = \hbar / m c$ \eqref{ComptonWavelength} and hence the [[physical unit]]-free version of the Lagrangian density for the free scalar particle is \begin{displaymath} \mathbf{L}_{\eta,\ell_m} \:\coloneqq\; \tfrac{\ell_m^2}{2} \left( \eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu} - \left( \tfrac{m c}{\hbar} \right)^2 \phi^2 \right) \mathrm{dvol}_\Sigma \,. \end{displaymath} \end{example} \begin{example} \label{phintheoryLagrangian}\hypertarget{phintheoryLagrangian}{} \textbf{([[phi{\tt \symbol{94}}n theory]])} Consider the [[field bundle]] for the [[real scalar field]] from example \ref{RealScalarFieldBundle}, i.e. the [[trivial line bundle]] over [[Minkowski spacetime]]. More generally we may consider adding to the [[free field]] [[Lagrangian density]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime} some power of the field coordinate \begin{displaymath} \mathbf{L}_{int} \;\coloneqq\; g \phi^n \, dvol_\Sigma \,, \end{displaymath} for $g \in \mathbb{R}$ some number, here called the \emph{[[coupling constant]]}. The [[interacting field theory|interacting]] [[Lagrangian field theory]] defined by the resulting [[Lagrangian density]] \begin{displaymath} \mathbf{L} + \mathbf{L}_{int} \;=\; \tfrac{1}{2} \left( \eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu} - m^2 \phi^2 + g \phi^n \right) \mathrm{dvol}_\Sigma \end{displaymath} is usually called just \emph{[[phi{\tt \symbol{94}}n theory]]}. \end{example} \begin{example} \label{ElectromagnetismLagrangianDensity}\hypertarget{ElectromagnetismLagrangianDensity}{} \textbf{([[local Lagrangian density]] for [[free field|free]] [[electromagnetic field]])} Consider the [[field bundle]] $T^\ast \Sigma \to \Sigma$ for the [[electromagnetic field]] on [[Minkowski spacetime]] from example \ref{Electromagnetism}, i.e. the [[cotangent bundle]], which over Minkowski spacetime happens to be a [[trivial vector bundle]] of [[rank of a vector bundle|rank]] $p+1$. With [[fiber]] coordinates taken to be $(a_\mu)_{\mu = 0}^p$, the induced fiber coordinates on the corresponding [[jet bundle]] $J^\infty_\Sigma(T^\ast \Sigma)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) are $( (x^\mu), (a_\mu), (a_{\mu,\nu}), (a_{\mu,\nu_1 \nu_2}), \cdots )$. Consider then the [[local Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) given by \begin{equation} \mathbf{L} \;\coloneqq\; \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu} dvol_\Sigma \;\in\; \Omega^{p+1,0}_\Sigma(T^\ast \Sigma) \,, \label{ElectromagnetismLagrangian}\end{equation} where $f_{\mu \nu} \coloneqq \tfrac{1}{2}(a_{\nu,\mu} - a_{\mu,\nu})$ are the components of the universal [[Faraday tensor]] on the [[jet bundle]] from example \ref{JetFaraday}. This is the [[Lagrangian density]] that defines the Lagrangian field theory of \emph{[[free field|free]] [[electromagnetism]]}. Here for $A \in \Gamma_\Sigma(T^\ast \Sigma)$ an [[electromagnetic field]] history ([[vector potential]]), then the [[pullback of differential forms|pullback]] of $f_{\mu \nu}$ along its [[jet prolongation]] (def. \ref{JetProlongation}) is the corresponding component of the [[Faraday tensor]] \eqref{TensorFaraday}: \begin{displaymath} \begin{aligned} \left( j^\infty_\Sigma(A) \right)^\ast(f_{\mu \nu}) & = (d A)_{\mu \nu} \\ & = F_{\mu \nu} \end{aligned} \end{displaymath} It follows that the pullback of the Lagrangian \eqref{ElectromagnetismLagrangian} along the jet prologation of the electromagnetic field is \begin{displaymath} \begin{aligned} \left( j^\infty_\Sigma(A) \right)^\ast \mathbf{L} & = \tfrac{1}{2} F_{\mu \nu} F^{\mu \nu} dvol_\Sigma \\ & = \tfrac{1}{2} F \wedge \star_\eta F \end{aligned} \end{displaymath} Here $\star_\eta$ denotes the [[Hodge star operator]] of [[Minkowski spacetime]]. \end{example} More generally: \begin{example} \label{YangMillsLagrangian}\hypertarget{YangMillsLagrangian}{} \textbf{([[Lagrangian density]] for [[Yang-Mills theory]] on [[Minkowski spacetime]])} Let $\mathfrak{g}$ be a [[finite number|finite]] [[dimension|dimensional]] [[Lie algebra]] which is [[semisimple Lie algebra|semisimple]]. This means that the [[Killing form]] [[invariant polynomial]] \begin{displaymath} k \colon \mathfrak{g} \otimes \mathfrak{g} \longrightarrow \mathbb{R} \end{displaymath} is a non-degenerate [[bilinear form]]. Examples include the [[special unitary Lie algebras]] $\mathfrak{so}(n)$. Then for $E = T^\ast \Sigma \otimes \mathfrak{g}$ the [[field bundle]] for [[Yang-Mills theory]] as in example \ref{YangMillsFieldOverMinkowski}, the [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) $\mathfrak{g}$-[[Yang-Mills theory]] on [[Minkowski spacetime]] is \begin{displaymath} \mathbf{L} \;\coloneqq\; \tfrac{1}{2} k_{\alpha \beta} f^\alpha_{\mu \nu} f^{\beta \mu \nu} \, dvol_\Sigma \;\in\; \Omega^{p+1,0}_\Sigma(T^\ast \Sigma) \,, \end{displaymath} where \begin{displaymath} f^\alpha_{\mu \nu} \;\coloneqq\; \tfrac{1}{2} \left( a^\alpha_{\nu,\mu} - a^\alpha_{\mu,\nu} + \gamma^{\alpha}{}_{\beta \gamma} a^\beta_{\mu} a^\gamma_{\nu} \right) \;\in\; \Omega^{0,0}_\Sigma(E) \end{displaymath} is the universal [[Yang-Mills theory|Yang-Mills]] [[field strength]] \eqref{YangMillsJetFieldStrengthMinkowski}. For the purposes of [[perturbative quantum field theory]] (to be discussed below in chapter \emph{\hyperlink{InteractingQuantumFields}{15. Interacting quantum fields}}) we may allow for a rescaling of the structure constants by (at this point) a [[real number]] $g$, to be called the \emph{[[coupling constant]]}, and decompose the Lagrangian into a sum of a [[free field theory|free field theory Lagrangian]] (def. \ref{FreeFieldTheory}) and an [[interaction]] term: \begin{displaymath} \begin{aligned} \mathbf{L} & = \tfrac{1}{2} k_{\alpha \beta} \tfrac{1}{2} \left( a^\alpha_{\nu,\mu} - a^\alpha_{\mu,\nu} + g \gamma^{\alpha}{}_{\beta' \gamma'} a^{\beta'}_{\mu} a^{\gamma'}_{\nu} \right) \tfrac{1}{2} \left( a^{\beta\nu,\mu} - a^{\beta \mu,\nu} + g \gamma^{\beta}{}_{\beta'' \gamma''} a^{\beta''}_{\mu} a^{\gamma''}_{\nu} \right) \,dvol_\Sigma \\ & = \underset{ \mathbf{L}_{\mathrm{free}} }{ \underbrace{ \tfrac{1}{2} k_{\alpha \beta} \tfrac{1}{2} \left( a^\alpha_{\nu,\mu} - a^\alpha_{\mu,\nu} \right) \tfrac{1}{2} \left( a^{\beta\nu,\mu} - a^{\beta \mu,\nu} \right) \,dvol_\Sigma } } \\ & \phantom{=} + \underset{ \mathbf{L}_{int} }{ \underbrace{ g \, k_{\alpha \beta} \tfrac{1}{2} \left( a^\alpha_{\nu,\mu} - a^\alpha_{\mu,\nu} \right) \tfrac{1}{2} \left( \gamma^{\beta}{}_{\beta'' \gamma''} a^{\beta''}_{\mu} a^{\gamma''}_{\nu} \right) \,dvol_\Sigma \; + \; g^2 \, \tfrac{1}{2} k_{\alpha \beta} \tfrac{1}{2} \left( \gamma^{\alpha}{}_{\beta' \gamma'} a^{\beta'}_{\mu} a^{\gamma'}_{\nu} \right) \tfrac{1}{2} \left( \gamma^{\beta}{}_{\beta'' \gamma''} a^{\beta''}_{\mu} a^{\gamma''}_{\nu} \right) \,dvol_\Sigma } } \\ \end{aligned} \,, \end{displaymath} Notice that $\mathbf{L}_{free}$ is equivalently a sum of $dim(\mathfrak{g})$-copies of the Lagrangian for the [[electromagnetic field]] (example \ref{ElectromagnetismLagrangianDensity}). On the other hand, for the purpose of exhibiting ``[[non-perturbative effects]] due to [[instantons]]'' in [[Yang-Mills theory]], one consider the rescaled Yang-Mills field coordinate \begin{displaymath} \tilde a^\alpha_\mu \;\coloneqq\; \frac{1}{g} a^\alpha_\mu \end{displaymath} with corresponding [[field strength]] \begin{displaymath} \tilde f^\alpha_{\mu \nu} \;\coloneqq\; \tfrac{1}{2} \left( \tilde a^\alpha_{\nu,\mu} - \tilde a^\alpha_{\mu,\nu} + \gamma^{\alpha}{}_{\beta \gamma} \tilde a^\beta_{\mu} \tilde a^\gamma_{\nu} \right) \;\in\; \Omega^{0,0}_\Sigma(E) \,. \end{displaymath} In terms of this the expression for the Lagrangian is brought back to the abstract form it had before rescaling the structure constants by the [[coupling constant]], up to a \emph{global} rescaling of all terms by the \emph{inverse square} of the coupling constant: \begin{equation} \mathbf{L} \;=\; \frac{1}{g^2} \tfrac{1}{2} k_{\alpha \beta} \tilde f^\alpha_{\mu \nu} \tilde f^{\beta \mu \nu} \, dvol_\Sigma \,. \label{MinkowskiYangMillsLagrangianWithCouplingConstantPulledOut}\end{equation} \end{example} \begin{example} \label{BFieldLagrangianDensity}\hypertarget{BFieldLagrangianDensity}{} \textbf{([[local Lagrangian density]] for [[free field|free]] [[B-field]])} Consider the [[field bundle]] $\wedge^2_\Sigma T^\ast \Sigma \to \Sigma$ for the [[B-field]] on [[Minkowski spacetime]] from example \ref{BField}. With [[fiber]] coordinates taken to be $(b_{\mu \nu})$ with \begin{displaymath} b_{\mu \nu} = - b_{\nu \mu} \,, \end{displaymath} the induced fiber coordinates on the corresponding [[jet bundle]] $J^\infty_\Sigma(T^\ast \Sigma)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) are $( (x^\mu), (b_{\mu \nu}), (b_{\mu \nu, \mu_1}), (b_{\mu \nu, \mu_1 \mu_2}), \cdots )$. Consider then the [[local Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) given by \begin{equation} \mathbf{L} \;\coloneqq\; \tfrac{1}{2} h_{\mu_1 \mu_2 \mu_3} h^{\mu_1 \mu_2 \mu_3} \, dvol_\Sigma \;\in\; \Omega^{p+1,0}_\Sigma(\wedge^2_\Sigma T^\ast \Sigma) \,, \label{LagrangianForBField}\end{equation} where $h_{\mu_1 \mu_2 \mu_3}$ are the components of the universal [[B-field|B-]][[field strength]] on the [[jet bundle]] from example \ref{BFieldJetFaraday}. \end{example} \begin{example} \label{LagrangianDensityForDiracField}\hypertarget{LagrangianDensityForDiracField}{} \textbf{([[Lagrangian density]] for [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]])} For $\Sigma$ [[Minkowski spacetime]] of [[dimension]] $p + 1 \in \{3,4,6,10\}$ (def. \ref{MinkowskiSpacetime}), consider the [[field bundle]] $\Sigma \times S_{odd} \to \Sigma$ for the [[Dirac field]] from example \ref{DiracFieldBundle}. With the two-component [[spinor]] [[field fiber]] coordinates from remark \ref{TwoComponentSpinorNotation}, the [[jet bundle]] has induced fiber coordinates as follows: \begin{displaymath} \left( \left(\psi^\alpha\right) , \left( \psi^\alpha_{,\mu} \right) , \cdots \right) \;=\; \left( \left( (\chi_a), (\chi_{a,\mu}), \cdots \right), \left( ( \xi^{\dagger \dot a}), (\xi^{\dagger \dot a}_{,\mu}), \cdots \right) \right) \end{displaymath} All of these are odd-graded elements (def. \ref{SupercommutativeSuperalgebra}) in a [[Grassmann algebra]] (example \ref{GrassmannAlgebra}), hence anti-commute with each other, in generalization of \eqref{DiracFieldCoordinatesAnticommute}: \begin{equation} \psi^\alpha_{,\mu_1 \cdots \mu_r} \psi^\beta_{,\mu_1 \cdots \mu_s} \;=\; - \psi^\beta_{,\mu_1 \cdots \mu_s} \psi^\alpha_{,\mu_1 \cdots \mu_r} \,. \label{DiracFieldJetCoordinatesAnticommute}\end{equation} The [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) of the \emph{massless [[free field theory|free]] [[Dirac field]]} on [[Minkowski spacetime]] is \begin{equation} \mathbf{L} \;\coloneqq\; \overline{\psi} \, \gamma^\mu \psi_{,\mu}\, dvol_\Sigma \,, \label{DiracFieldLagrangianMassless}\end{equation} given by the bilinear pairing $\overline{(-)}\Gamma(-)$ from prop. \ref{RealSpinorPairingsViaDivisionAlg} of the field coordinate with its first spacetime derivative and expressed here in two-component spinor field coordinates as in \eqref{TwoComponentNotationForSpinorToVectorPairing}, hence with the [[Dirac conjugate]] $\overline{\psi}$ \eqref{DiracConjugate} on the left. Specifically in [[spacetime]] [[dimension]] $p + 1 = 4$, the [[Lagrangian function]] for the \emph{massive [[Dirac field]]} of [[mass]] $m \in \mathbb{R}$ is \begin{displaymath} \begin{aligned} L & \coloneqq \underset{ \text{kinetic term} }{ \underbrace{ i \, \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} } } + \underset{ \text{mass term} }{ \underbrace{ m \overline{\psi} \psi }} \end{aligned} \end{displaymath} This is naturally thought of as a collection of Lagrangians smoothly parameterized by the [[metric]] $\eta$ and the [[mass]] $m$. For this to be [[physical unit]]-free in the sense of remark \ref{ParameterizedLagrangianDensities} the [[physical unit]] of the parameter $m$ must be that of the inverse metric, hence must be an inverse [[length]] according to remark \ref{MinkowskiMetricAndPhysicalUnitOfLength} This is the \emph{inverse [[Compton wavelength]]} $\ell_m = \hbar / m c$ \eqref{ComptonWavelength} and hence the [[physical unit]]-free version of the Lagrangian density for the free Dirac field is \begin{displaymath} \mathbf{L}_{\eta,\ell_m} \;\coloneqq\; \ell_m \left( i \overline{\psi} \gamma^\mu \psi_{,\mu} + \left( \tfrac{m c}{\hbar} \right) \overline{\psi} \psi \right) dvol_\Sigma \,. \end{displaymath} \end{example} \begin{remark} \label{RealityOfLagrangianDensityOfTheDiracField}\hypertarget{RealityOfLagrangianDensityOfTheDiracField}{} \textbf{([[real part|reality]] of the [[Lagrangian density]] of the [[Dirac field]])} The kinetic term of the [[Lagrangian density]] for the [[Dirac field]] form def. \ref{LagrangianDensityForDiracField} is a sum of two contributions, one for each [[chiral spinor]] component in the full [[Dirac spinor]] (remark \ref{TwoComponentSpinorNotation}): \begin{displaymath} \begin{aligned} i \overline{\psi} \gamma^\mu \psi_{,\mu} & = i \underset{ -(\partial_\mu \xi^a ) \sigma^\mu_{a \dot c} \xi^{\dagger \dot c} + \partial_\mu(\chi^a \sigma^\mu_{a \dot c} \chi^{\dagger \dot c}) }{ \underbrace{ \xi^a \sigma^\mu_{a \dot c} \partial_\mu \xi^{\dagger \dot c} } } + \xi^\dagger_{\dot a} \tilde \sigma^{\mu \dot a c} \partial_\mu \xi_c \\ & = \xi^\dagger \tilde \sigma^\mu \partial_\mu \xi + \chi^\dagger \tilde \sigma^\mu \partial_\mu \chi + \partial_\mu(\xi \sigma^\mu \xi^\dagger) \end{aligned} \end{displaymath} Here the computation shown under the brace crucially uses that all these jet coordinates for the Dirac field are anti-commuting, due to their [[supergeometry|supergeometric]] nature \eqref{DiracFieldJetCoordinatesAnticommute}. Notice that a priori this is a function on the jet bundle with values in $\mathbb{K}$. But in fact for $\mathbb{K} = \mathbb{C}$ it is real up to a [[total spacetime derivative]]:, because \begin{displaymath} \begin{aligned} \left( i \chi^\dagger \tilde \sigma^\mu \partial_\mu\chi \right)^\dagger & = -i \left( \partial_\mu \chi\right)^\dagger \sigma^\mu \chi \\ & = i \chi^\dagger \sigma^\mu \partial_\mu \chi + i \partial_\mu\left( \chi^\dagger \sigma^\mu \chi \right) \end{aligned} \end{displaymath} and similarly for $i \xi^\dagger \tilde \sigma^\mu \partial_\mu\xi$ \end{remark} (e.g. \href{Dirac+field#DermisekI9}{Dermisek I-9}) \begin{example} \label{LagrangianQED}\hypertarget{LagrangianQED}{} \textbf{([[Lagrangian density]] for [[quantum electrodynamics]])} Consider the [[fiber product]] of the [[field bundles]] for the [[electromagnetic field]] (example \ref{Electromagnetism}) and the [[Dirac field]] (example \ref{DiracFieldBundle}) over 4-dimensional [[Minkowski spacetime]] $\Sigma \coloneqq \mathbb{R}^{3,1}$ (def. \ref{MinkowskiSpacetime}): \begin{displaymath} E \;\coloneqq\; \underset{ \itexarray{ \text{electromagnetic} \\ \text{field} } }{\underbrace{T^\ast \Sigma}} \times \underset{ \itexarray{ \text{Dirac} \\ \text{field} } }{ \underbrace{ S_{odd} } } \,. \end{displaymath} This means that now a [[field history]] is a [[pair]] $(A,\Psi)$, with $A$ a field history of the [[electromagnetic field]] and $\Psi$ a field history of the [[Dirac field]]. On the resulting [[jet bundle]] consider the [[Lagrangian density]] \begin{equation} L_{int} \;\coloneqq\; i g \, \overline{\psi} \gamma^\mu \psi a_\mu \label{ElectronPhotonInteractionLocalLagrangian}\end{equation} for $g \in \mathbb{R}$ some number, called the \emph{[[coupling constant]]}. This is called the \emph{[[electron-photon interaction]]}. Then the sum of the [[Lagrangian densities]] for \begin{enumerate}% \item the [[free field|free]] [[electromagnetic field]] (example \ref{ElectromagnetismLagrangianDensity}); \item the [[free field|free]] [[Dirac field]] (example \ref{LagrangianDensityForDiracField}) \item the above [[electron-photon interaction]] \end{enumerate} \begin{displaymath} \mathbf{L}_{EM} + \mathbf{L}_{Dir} + \mathbf{L}_{int} \;=\; \left( \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu} \;+\; i \, \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} + m \overline{\psi} \psi \;+\; i g \, \overline{\psi} \gamma^\mu \psi a_\mu \right) \, dvol_\Sigma \end{displaymath} defines the [[interacting field theory]] [[Lagrangian field theory]] whose [[perturbative quantum field theory|perturbative quantization]] is called \emph{[[quantum electrodynamics]]}. In this context the square of the [[coupling constant]] \begin{displaymath} \alpha \coloneqq \frac{g^2}{4 \pi} \end{displaymath} is called the \emph{[[fine structure constant]]}. \end{example} $\,$ \textbf{[[Euler-Lagrange forms]] and [[presymplectic currents]]} The beauty of [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) is that a choice of [[Lagrangian density]] determines both the [[equations of motion]] of the fields as well as a [[presymplectic manifold|presymplectic structure]] on the space of solutions to this equation (the ``[[shell]]''), making it the ``[[covariant phase space]]'' of the theory. All this we discuss \hyperlink{PhaseSpace}{below}. But in fact all this key structure of the field theory is nothing but the shadow (under ``[[transgression of variational differential forms]]'', def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces} below) of the following simple relation in the [[variational bicomplex]]: \begin{prop} \label{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}\hypertarget{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}{} \textbf{([[Euler-Lagrange form]] and [[presymplectic current]])} Given a [[Lagrangian density]] $\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E)$ as in def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}, then its de Rham differential $\mathbf{d}\mathbf{L}$, which by degree reasons equals $\delta \mathbf{L}$, has a \emph{unique} decomposition as a sum of two terms \begin{equation} \mathbf{d} \mathbf{L} = \delta_{EL} \mathbf{L} - d \Theta_{BFV} \label{dLDecomposition}\end{equation} such that $\delta_{EL}\mathbf{L}$ is proportional to the [[variational derivative]] of the fields (but not their derivatives, called a ``[[source form]]''): \begin{displaymath} \delta_{EL} \mathbf{L} \;\in\; \Omega^{p+1,0}_{\Sigma}(E) \wedge \delta C^\infty(E) \;\subset\; \Omega^{p+1,1}_{\Sigma}(E) \,. \end{displaymath} The map \begin{displaymath} \delta_{EL} \;\colon\; \Omega^{p+1,0}_{\Sigma}(E) \longrightarrow \Omega^{p+1,0}_{\Sigma}(E) \wedge \delta \Omega^{0,0}_{\Sigma}(E) \end{displaymath} thus defined is called the \emph{[[Euler-Lagrange operator]]} and is explicitly given by the \emph{[[Euler-Lagrange derivative]]}: \begin{equation} \begin{aligned} \delta_{EL} L \, dvol_\Sigma & \coloneqq \frac{\delta_{EL} L}{\delta \phi^a} \delta \phi^a \wedge dvol_\Sigma \\ & \coloneqq \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} + \frac{d^2}{d x^{\mu_1} d x^{\mu_2}} \frac{\partial L}{\partial \phi^a_{\mu_1, \mu_2}} - \cdots \right) \delta \phi^a \wedge dvol_\Sigma \,. \end{aligned} \label{EulerLagrangeEquationGeneral}\end{equation} The [[smooth space|smooth subspace]] of the [[jet bundle]] on which the [[Euler-Lagrange form]] vanishes \begin{equation} \mathcal{E} \;\coloneqq\; \left\{ x \in J^\infty_\Sigma(E) \;\vert\; \delta_{EL}\mathbf{L}(x) = 0 \right\} \;\overset{i_{\mathcal{E}}}{\hookrightarrow}\; J^\infty_\Sigma(E) \,. \label{ShellInJetBundle}\end{equation} is called the \emph{[[shell]]}. The smaller subspace on which also all [[total spacetime derivatives]] vanish (the ``[[formally integrable PDE|formally integrable prolongation]]'') is the \emph{prolonged [[shell]]} \begin{equation} \mathcal{E}^\infty \;\coloneqq\; \left\{ x \in J^\infty_\Sigma(E) \;\vert\; \left( \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \delta_{EL}\mathbf{L} \right)(x) = 0 \right\} \overset{i_{\mathcal{E}^\infty}}{\hookrightarrow} J^\infty_\Sigma(E) \,. \label{ProlongedShellInJetBundle}\end{equation} Saying something holds ``[[on-shell]]'' is to mean that it holds after restriction to this subspace. For example a [[variational differential form]] $\alpha \in \Omega^{\bullet,\bullet}_\Sigma(E)$ is said to \emph{vanish on shell} if $\alpha\vert_{\mathcal{E}^\infty} = 0$. The remaining term $d \Theta_{BFV}$ in \eqref{dLDecomposition} is unique, while the \emph{presymplectic potential} \begin{equation} \Theta_{BFV} \in \Omega^{p,1}_{\Sigma}(E) \label{PresymplecticPotential}\end{equation} is not unique. (For a [[field bundle]] which is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle} over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}), prop. \ref{HorizontalVariationalComplexOfTrivialFieldBundleIsExact} says that $\Theta_{BFV}$ is unique up to addition of total spacetime derivatives $d \kappa$, for $\kappa \in \Omega^{p-1,1}_\Sigma(E)$.) One possible choice for the presymplectic current $\Theta_{BFV}$ is \begin{equation} \begin{aligned} \Theta_{BFV} & \coloneqq \phantom{+} \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a \; \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & \phantom{=} + \left( \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} - \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu} \right) \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & \phantom{=} + \cdots \,, \end{aligned} \label{StandardThetaForTrivialVectorFieldBundleOnMinkowskiSpacetime}\end{equation} where \begin{displaymath} \iota_{\partial_{\mu}} dvol_\Sigma \;\coloneqq\; (-1)^{\mu} d x^0 \wedge \cdots d x^{\mu-1} \wedge d x^{\mu+1} \wedge \cdots \wedge d x^p \end{displaymath} denotes the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of the [[volume form]] with the [[vector field]] $\partial_\mu$. The [[vertical derivative]] of a chosen presymplectic potential $\Theta_{BFV}$ is called a \emph{[[pre-symplectic current]]} for $\mathbf{L}$: \begin{equation} \Omega_{BFV} \;\coloneqq\; \delta \Theta_{BFV} \;\;\; \in \Omega^{p,2}_{\Sigma}(E) \,. \label{PresymplecticCurrent}\end{equation} Given a choice of $\Theta_{BFV}$ then the sum \begin{equation} \mathbf{L} + \Theta_{BFV} \;\in\; \Omega^{p+1,0}_\Sigma(E) \oplus \Omega^{p,1}_\Sigma(E) \label{TheLepage}\end{equation} is called the corresponding \emph{[[Lepage form]]}. Its de Rham derivative is the sum of the Euler-Lagrange variation and the presymplectic current: \begin{equation} \mathbf{d}( \mathbf{L} + \Theta_{BFV} ) \;=\; \delta_{EL} \mathbf{L} + \Omega_{BFV} \,. \label{DerivativeOfLepageForm}\end{equation} (Its conceptual nature will be elucidated after the introduction of the [[local BV-complex]] in example \ref{DerivedPresymplecticCurrentOfRealScalarField} below.) \end{prop} \begin{proof} Using $\mathbf{L} = L dvol_\Sigma$ and that $d \mathbf{L} = 0$ by degree reasons (example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}), we find \begin{displaymath} \begin{aligned} \mathbf{d}\mathbf{L} & = \left( \frac{\partial L}{\partial \phi^a} \delta \phi^a + \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} + \frac{\partial L}{\partial \phi^a_{,\mu_1 \mu_2}} \delta \phi^a_{,\mu_1 \mu_2} + \cdots \right) \wedge dvol_{\Sigma} \end{aligned} \,. \end{displaymath} The idea now is to have $d \Theta_{BFV}$ pick up those terms that would appear as [[boundary]] terms under the [[integral]] $\int_\Sigma j^\infty_\Sigma(\Phi)^\ast \mathbf{d}L$ if we were to consider [[integration by parts]] to remove spacetime derivatives of $\delta \phi^a$. We compute, using example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}, the total horizontal derivative of $\Theta_{BFV}$ from \eqref{StandardThetaForTrivialVectorFieldBundleOnMinkowskiSpacetime} as follows: \begin{displaymath} \begin{aligned} d \Theta_{BFV} & = \left( d \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a \right) + d \left( \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} - \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{\mu \nu}} \delta \phi^a \right) + \cdots \right) \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & = \left( \left( \left( d \frac{\partial L}{\partial \phi^a_{,\mu}} \right) \wedge \delta \phi^a - \frac{\partial L}{\partial \phi^a_{,\mu}} \delta d \phi^a \right) + \left( \left(d \frac{\partial L}{\partial \phi^a_{,\nu \mu}}\right) \wedge \delta \phi^a_{,\nu} - \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta d \phi^a_{,\nu} - \left( d \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \right) \wedge \delta \phi^a + \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta d \phi^a \right) + \cdots \right) \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & = - \left( \left( \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a + \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} \right) + \left( \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} + \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu \mu} - \frac{d^2}{ d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a - \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu} \right) + \cdots \right) \wedge dvol_\Sigma \,, \end{aligned} \end{displaymath} where in the last line we used that \begin{displaymath} d x^{\mu_1} \wedge \iota_{\partial_{\mu_2}} dvol_\Sigma = \left\{ \itexarray{ dvol_\Sigma &\vert& \text{if}\, \mu_1 = \mu_2 \\ 0 &\vert& \text{otherwise} } \right. \end{displaymath} Here the two terms proportional to $\frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu}$ cancel out, and we are left with \begin{displaymath} d \Theta_{BFV} \;=\; - \left( \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} - \frac{d^2}{ d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} + \cdots \right) \delta \phi^a \wedge dvol_\Sigma - \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} + \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu \mu} + \cdots \right) \wedge dvol_\Sigma \end{displaymath} Hence $-d \Theta_{BFV}$ shares with $\mathbf{d} \mathbf{L}$ the terms that are proportional to $\delta \phi^a_{,\mu_1 \cdots \mu_k}$ for $k \geq 1$, and so the remaining terms are proportional to $\delta \phi^a$, as claimed: \begin{displaymath} \mathbf{d}\mathbf{L} + d \Theta_{BFV} = \underset{ = \delta_{EL}\mathbf{L} }{ \underbrace{ \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu}\frac{\partial L}{\partial \phi^a_{,\mu}} + \frac{d^2}{d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu\nu}} + \cdots \right) \delta \phi^a \wedge dvol_\Sigma }} \,. \end{displaymath} \end{proof} The following fact is immediate from prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}, but of central importance, we futher amplify this in remark \ref{PresymplecticCurrentInterpretation} below: \begin{prop} \label{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell}\hypertarget{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell}{} \textbf{([[total derivative|total spacetime derivative]] of [[presymplectic current]] vanishes [[on-shell]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}). Then the [[Euler-Lagrange form]] $\delta_{EL} \mathbf{L}$ and the [[presymplectic current]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) are related by \begin{displaymath} d \Omega_{BFV} = - \delta(\delta_{EL}\mathbf{L}) \,. \end{displaymath} In particular this means that restricted to the prolonged shell $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ \eqref{ProlongedShellInJetBundle} the total spacetime derivative of the [[presymplectic current]] vanishes: \begin{equation} d \Omega_{BFV} \vert_{\mathcal{E}^\infty} \;=\; 0 \,. \label{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell}\end{equation} \end{prop} \begin{proof} By prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime} we have \begin{displaymath} \delta \mathbf{L} = \delta_{EL} \mathbf{L} - d \Theta_{BFV} \,. \end{displaymath} The claim follows from applying the [[variational derivative]] $\delta$ to both sides, using \eqref{HorizontalAndVerticalDerivativeAnticommute}: $\delta^2 = 0$ and $\delta \circ d = - d \circ \delta$. \end{proof} Many examples of interest fall into the following two special cases of prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}: \begin{prop} \label{ShellForSpacetimeIndependentLagrangians}\hypertarget{ShellForSpacetimeIndependentLagrangians}{} \textbf{([[Euler-Lagrange form]] for [[spacetime]]-independent [[Lagrangian densities]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) whose [[field bundle]] $E$ is a [[trivial vector bundle]] $E \simeq \Sigma \times F$ over [[Minkowski spacetime]] $\Sigma$ (example \ref{TrivialVectorBundleAsAFieldBundle}). In general the [[Lagrangian density]] $\mathbf{L}$ is a function of all the spacetime and field coordinates \begin{displaymath} \mathbf{L} = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots) dvol_\Sigma \,. \end{displaymath} Consider the special case that $\mathbf{L}$ is \emph{[[spacetime]]-independent} in that the Lagrangian function $L$ is independent of the spacetime coordinate $(x^\mu)$. Then the same evidently holds for the [[Euler-Lagrange form]] $\delta_{EL}\mathbf{L}$ (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}). Therefore in this case the [[shell]] \eqref{ProlongedShellInJetBundle} is itself a [[trivial bundle]] over spacetime. In this situation every point $\varphi$ in the jet fiber defines a constant section of the shell: \begin{equation} \Sigma \times \{\varphi\} \subset \mathcal{E}^\infty \,. \label{ConstantSectionOfTrivialShellBundle}\end{equation} \end{prop} \begin{example} \label{CanonicalMomentum}\hypertarget{CanonicalMomentum}{} \textbf{([[canonical momentum]])} Consider a [[Lagrangian field theory]] $(E, \mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) whose [[Lagrangian density]] $\mathbf{L}$ \begin{enumerate}% \item does not depend on the [[spacetime]]-[[coordinates]] (example \ref{ShellForSpacetimeIndependentLagrangians}); \item depends on spacetime derivatives of [[field (physics)|field]] coordinates (hence on [[jet bundle]] coordinates) at most to first order. \end{enumerate} Hence if the [[field bundle]] $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) this means to consider the case that \begin{displaymath} \mathbf{L} \;=\; L\left( (\phi^a), (\phi^a_{,\mu}) \right) \wedge dvol_\Sigma \,. \end{displaymath} Then the [[presymplectic current]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) is (up to possibly a horizontally exact part) of the form \begin{equation} \Omega_{BFV} \;=\; \delta p_a^\mu \wedge \delta \phi^a \wedge \iota_{\partial_\mu} dvol_\Sigma \label{CanonicalMomentumPresymplecticCurrent}\end{equation} where \begin{equation} p_a^\mu \;\coloneqq\; \frac{\partial L}{ \partial \phi^a_{,\mu}} \label{CanonicalMomentumInCoordinates}\end{equation} denotes the [[partial derivative]] of the [[Lagrangian density|Lagrangian function]] with respect to the spacetime-[[derivatives]] of the [[field (physics)|field]] [[coordinates]]. Here \begin{displaymath} \begin{aligned} p_a & \coloneqq p_a^0 \\ & = \frac{\partial L}{\partial \phi^a_{,0}} \end{aligned} \end{displaymath} is called the \emph{[[canonical momentum]]} corresponding to the ``[[canonical coordinate|canonical field coordinate]]'' $\phi^a$. In the language of [[multisymplectic geometry]] the full expression \begin{displaymath} p_a^\mu \wedge \iota_{\partial_\mu} dvol_\Sigma \;\in\; \Omega^{p,1}_\Sigma(E) \end{displaymath} is also called the ``canonical multi-momentum'', or similar. \end{example} \begin{proof} We compute: \begin{displaymath} \begin{aligned} \mathbf{d} \mathbf{L} & = \left( \frac{\partial L}{\partial \phi^a} \delta \phi^a + \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} \right) \delta \phi^a \wedge dvol_\Sigma \\ & = \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} \right) \wedge dvol_\Sigma - d \underset{ \Theta_{BFV} }{ \underbrace{ \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a \right) \wedge \iota_{\partial_\mu} dvol_\Sigma } } \end{aligned} \,. \end{displaymath} Hence \begin{displaymath} \begin{aligned} \Omega_{BFV} & \coloneqq \delta \Theta_{BFV} \\ & = \delta \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} \wedge \iota_{\partial_\mu} dvol_\Sigma \right) \\ & = \delta \frac{\partial L}{\partial \phi^a_{,\mu}} \wedge \delta \phi^a_{,\mu} \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & = \delta p_a^\mu \wedge \delta \phi^a \wedge \iota_{\partial_\mu} dvol_\Sigma \end{aligned} \end{displaymath} \end{proof} \begin{remark} \label{PresymplecticCurrentInterpretation}\hypertarget{PresymplecticCurrentInterpretation}{} \textbf{([[presymplectic current]] is local version of ([[presymplectic form|pre-]])[[symplectic form]] of [[Hamiltonian mechanics]])} In the simple but very common situation of example \ref{CanonicalMomentum} the [[presymplectic current]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) takes the form \eqref{CanonicalMomentumInCoordinates} \begin{displaymath} \Omega_{BFV} \;=\; \delta p_a^\mu \wedge \delta \phi^a \wedge \iota_{\partial_\mu} dvol_\Sigma \end{displaymath} with $\phi^a$ the [[field (physics)|field]] [[coordinates]] (``[[canonical coordinates]]'') and $p_a^\mu$ the ``[[canonical momentum]]'' \eqref{CanonicalMomentumInCoordinates}. Notice that this is of the schematic form ``$(\delta p_a \wedge \delta q^a) \wedge dvol_{\Sigma_p}$'', which is reminiscent of the wedge product of a [[symplectic form]] expressed in [[Darboux coordinates]] with a [[volume form]] for a $p$-dimensional [[manifold]]. Indeed, below in \emph{\hyperlink{PhaseSpace}{Phase space}} we discuss that this [[presymplectic current]] ``[[transgression of variational differential forms|transgresses]]'' (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces} below) to a [[presymplectic form]] of the schematic form ``$d P_a \wedge d Q^a$'' on the [[on-shell]] [[space of field histories]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) by [[integration of differential forms|integrating]] it over a [[Cauchy surface]] of [[dimension]] $p$. In good situations this [[presymplectic form]] is in fact a [[symplectic form]] on the [[on-shell]] [[space of field histories]] (theorem \ref{PPeierlsBracket} below). This shows that the [[presymplectic current]] $\Omega_{BFV}$ is the [[local field theory|local]] (i.e. [[jet bundle|jet level]]) avatar of the [[symplectic form]] that governs the formulation of [[Hamiltonian mechanics]] in terms of [[symplectic geometry]]. In fact prop. \ref{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell} may be read as saying that the [[presymplectic current]] is a \emph{[[conserved current]]} (def. \ref{SymmetriesAndConservedCurrents} below), only that it takes values not in [[smooth functions]] of the field coordinates and jets, but in [[variational differential form|variational 2-forms]] on fields. There is a [[conserved charge]] associated with every [[conserved current]] (prop. \ref{ConservedCharge} below) and the conserved charge associated with the [[presymplectic current]] is the ([[presymplectic form|pre-]])[[symplectic form]] on the [[phase space]] of the field theory (def. \ref{PhaseSpaceAssociatedWithCauchySurface} below). \end{remark} \begin{example} \label{FreeScalarFieldEOM}\hypertarget{FreeScalarFieldEOM}{} \textbf{([[Euler-Lagrange form]] and [[presymplectic current]] for [[free field|free]] [[real scalar field]])} Consider the [[Lagrangian field theory]] of the [[free field|free]] [[real scalar field]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}. Then the [[Euler-Lagrange operator|Euler-Lagrange form]] and [[presymplectic current]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) are \begin{equation} \delta_{EL}\mathbf{L} \;=\; \left(\eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \right) \delta \phi \wedge dvol_\sigma \;\in\; \Omega^{p+1,1}_{\Sigma}(E) \,. \label{RealScalarFieldLEForm}\end{equation} and \begin{displaymath} \Omega_{BFV} \;=\; \left(\eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma} \;\in\; \Omega^{p,2}_{\Sigma}(E) \,, \end{displaymath} respectively. \end{example} \begin{proof} This is a special case of example \ref{CanonicalMomentum}, but we spell it out in detail again: We need to show that [[Euler-Lagrange operator]] $\delta_{EL} \colon \Omega^{p+1,0}(\Sigma) \to \Omega^{p+1,1}_S(\Sigma)$ takes the [[local Lagrangian density]] for the [[free field|free]] [[scalar field]] to \begin{displaymath} \delta_EL L \;=\; \left( \eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \phi \right) \delta \phi \wedge \mathrm{dvol}_\Sigma \,. \end{displaymath} First of all, using just the [[variational derivative]] ([[vertical derivative]]) $\delta$ is a graded [[derivation]], the result of applying it to the local Lagrangian density is \begin{displaymath} \delta L \;=\; \left( \eta^{\mu \nu} \phi_{,\mu} \delta \phi_{,\nu} - m^2 \phi \delta \phi \right) \wedge \mathrm{dvol}_\Sigma \,. \end{displaymath} By definition of the [[Euler-Lagrange operator]], in order to find $\delta_{EL}\mathbf{L}$ and $\Theta_{BFV}$, we need to exhibit this as the sum of the form $(-) \wedge \delta \phi - d \Theta_{BFV}$. The key to find $\Theta_{BFV}$ is to realize $\delta \phi_{,\nu}\wedge \mathrm{dvol}_\Sigma$ as a [[total derivative|total spacetime derivative]] ([[horizontal derivative]]). Since $d \phi = \phi_{,\mu} d x^\mu$ this is accomplished by \begin{displaymath} \delta \phi_{,\nu} \wedge \mathrm{dvol}_\Sigma = \delta d \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \,, \end{displaymath} where on the right we have the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of the [[tangent vector field]] along $x^\nu$ into the [[volume form]]. Hence we may take the presymplectic potential \eqref{PresymplecticPotential} of the free scalar field to be \begin{equation} \Theta_{BFV} \coloneqq \eta^{\mu \nu} \phi_{,\mu} \delta \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \,, \label{PresymplecticPotentialOfFreeScalarField}\end{equation} because with this we have \begin{displaymath} d \Theta_{BFV} = \eta^{\mu \nu} \left( \phi_{,\mu \nu} \delta \phi - \eta^{\mu \nu} \phi_{,\mu} \delta \phi_{,\nu} \right) \wedge \mathrm{dvol}_\Sigma \,. \end{displaymath} In conclusion this yields the decomposition of the vertical differential of the Lagrangian density \begin{displaymath} \delta L = \underset{ = \delta_{EL} \mathcal{L} }{ \underbrace{ \left( \eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \phi \right) \delta \phi \wedge \mathrm{dvol}_\Sigma } } - d \Theta_{BFV} \,, \end{displaymath} which shows that $\delta_{EL} L$ is as claimed, and that $\Theta_{BFV}$ is a presymplectic potential current \eqref{PresymplecticPotential}. Hence the presymplectic current itself is \begin{displaymath} \begin{aligned} \Omega_{BFV} &\coloneqq \delta \Theta_{BFV} \\ & = \delta \left( \eta^{\mu \nu} \phi_{,\mu} \delta \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \right) \\ & = \left(\eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma} \end{aligned} \,. \end{displaymath} \end{proof} \begin{example} \label{ElectromagnetismEl}\hypertarget{ElectromagnetismEl}{} \textbf{([[Euler-Lagrange form]] for [[free field|free]] [[electromagnetic field]])} Consider the [[Lagrangian field theory]] of [[free field|free]] [[electromagnetism]] from example \ref{ElectromagnetismLagrangianDensity}. The [[Euler-Lagrange variational derivative]] is \begin{equation} \delta_{EL} \mathbf{L} \;=\; - \frac{d}{d x^\mu} f^{\mu \nu} \delta a_\nu \,. \label{ElectromagneticFieldEulerLagrangeForm}\end{equation} Hence the [[shell]] \eqref{ShellInJetBundle} in this case is \begin{displaymath} \mathcal{E} = \Sigma \times \left\{ \left( (a_\mu) , (a_{\mu,\mu_1}), (a_{\mu,\mu_1 \mu_2}), \cdots \right) \;\vert\; f^{\mu \nu}{}_{,\mu} = 0 \right\} \;\subset\; J^\infty_\Sigma(T^\ast \Sigma) \,. \end{displaymath} \end{example} \begin{proof} By \eqref{EulerLagrangeEquationGeneral} we have \begin{displaymath} \begin{aligned} \frac{\delta_{EL} L}{\delta a_\mu} \delta a_\mu & = \left( \underset{ = 0 }{ \underbrace{ \frac{\partial}{\partial a_\mu} \tfrac{1}{2} a_{[\mu,\nu]} a^{[\mu,\nu]} } } - \frac{d}{d x^\rho} \frac{\partial}{\partial a_{\alpha,\rho}} \tfrac{1}{2} a_{[\mu,\nu]} a^{[\mu,\nu]} \right) \delta a_\alpha \\ & = - \tfrac{1}{2} \left( \frac{d}{d x^\rho} \frac{\partial}{\partial a_{\alpha,\rho}} a_{\mu,\nu} a^{[\mu,\nu]} \right) \delta a_\alpha \\ & = - \left( \frac{d}{d x^\rho} a^{[\alpha,\rho]} \right) \delta a_{\alpha} \\ & = - f^{\mu \nu}{}_{,\mu} \delta a_{\nu} \,. \end{aligned} \end{displaymath} \end{proof} More generally: \begin{example} \label{YangMillsOnMinkowskiEl}\hypertarget{YangMillsOnMinkowskiEl}{} \textbf{([[Euler-Lagrange form]] for [[Yang-Mills theory]] on [[Minkowski spacetime]])} Let $\mathfrak{g}$ be a [[semisimple Lie algebra]] and consider the [[Lagrangian field theory]] $(E,\mathbf{L})$ of $\mathfrak{g}$-[[Yang-Mills theory]] from example \ref{YangMillsLagrangian}. Its [[Euler-Lagrange form]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) is \begin{displaymath} \begin{aligned} \delta_{EL}\mathbf{L} & = - \left( f^{\mu \nu \alpha}_{,\mu} + \gamma^\alpha{}_{\beta' \gamma} a_\mu^{\beta'} f^{\mu \nu \gamma} \right) k_{\alpha \beta} \,\delta a_\mu^\beta \, dvol_\Sigma \,, \end{aligned} \end{displaymath} where \begin{displaymath} f^\alpha_{\mu \nu} \;\in\; \Omega^{0,0}_\Sigma(E) \end{displaymath} is the universal [[Yang-Mills theory|Yang-Mills]] [[field strength]] \eqref{YangMillsJetFieldStrengthMinkowski}. \end{example} \begin{proof} With the explicit form \eqref{EulerLagrangeEquationGeneral} for the [[Euler-Lagrange derivative]] we compute as follows: \begin{displaymath} \begin{aligned} \delta_{EL} \left( \tfrac{1}{2} k_{\alpha \beta} f^\alpha_{\mu\nu} f^{\beta \mu \nu} \right) & = \left( \left( \frac{\partial}{\partial a_{\mu'}^{\alpha'}} \left( a_{\nu,\mu}^\alpha + \tfrac{1}{2} \gamma^{\alpha}{}_{\alpha_2 \alpha_3} a_{\mu}^{\alpha_2} a_\nu^{\alpha_3} \right) \right) k_{\alpha \beta} f^{\beta \mu \nu} - \left( \frac{d}{d x^{\nu'}} \frac{\partial}{\partial a_{\mu',\nu'}^{\alpha'}} \left( a_{\nu,\mu}^\alpha + \tfrac{1}{2} \gamma^{\alpha}{}_{\alpha_2 \alpha_3} a_{\mu}^{\alpha_2} a_\nu^{\alpha_3} \right) \right) k_{\alpha \beta} f^{\beta \mu \nu} \right) \delta a_{\mu'}^{\alpha'} \\ & = \gamma^{\alpha}{}_{\alpha' \alpha_3} a_\nu^{\alpha_3} f^{\beta \mu \nu} k_{\alpha \beta} \delta a_{\mu}^{\alpha'} - \left( \frac{d}{d x^{\mu}} f^{\beta \mu \nu} \right) k_{\alpha \beta} \delta a_{\nu}^{\alpha} \\ &= - \left( f^{\alpha \mu \nu}_{,\mu} + \gamma^\alpha{}_{\beta \gamma} a_\mu^\beta f^{\gamma \mu \nu} \right) k_{\alpha \beta} \delta a_\nu^\beta \end{aligned} \end{displaymath} In the last step we used that for a [[semisimple Lie algebra]] $\gamma_{\alpha \beta \gamma} \coloneqq k_{\alpha \alpha'} \gamma^{\alpha'}{}_{\beta \gamma}$ is totally skew-symmetric in its indices (this being the coefficients of the [[Lie algebra cocycle]]) which is in transgression with the [[Killing form]] [[invariant polynomial]] $k$. \end{proof} \begin{example} \label{EulerLagrangeFormBField}\hypertarget{EulerLagrangeFormBField}{} \textbf{([[Euler-Lagrange form]] of [[free field|free]] [[B-field]])} Consider the [[Lagrangian field theory]] of the [[free field|free]] [[B-field]] from example \ref{BField}. The [[Euler-Lagrange variational derivative]] is \begin{displaymath} \delta_{EL} \mathbf{L} \;=\; h^{\mu \nu \rho}{}_{,\rho} \delta b_{\mu \nu} \,, \end{displaymath} where $h_{\mu_1 \mu_2 \mu_3}$ is the universal [[B-field|B-]][[field strength]] from example \ref{BFieldJetFaraday}. \end{example} \begin{proof} By \eqref{EulerLagrangeEquationGeneral} we have \begin{displaymath} \begin{aligned} \frac{\delta_{EL} L}{\delta b_{\mu \nu}} \delta b_{\mu \nu} & = \left( \underset{ = 0 }{ \underbrace{ \frac{\partial}{\partial b_{\mu \nu}} \tfrac{1}{2} b_{[\mu_1 \mu_2, \mu_3]} b^{[\mu_1 \mu_2, \mu_3]} } } - \frac{d}{d x^\rho} \frac{\partial}{\partial b_{\mu \nu, \rho}} \tfrac{1}{2} b_{[\mu_1 \mu_2, \mu_3]} b^{[\mu_1 \mu_2, \mu_3]} \right) \delta b_{\mu \nu} \\ & = - \left( \frac{d}{d x^\rho} \frac{\partial}{\partial b_{\mu \nu, \rho}} \tfrac{1}{2} b_{\mu_1 \mu_2, \mu_3} b^{[\mu_1 \mu_2, \mu_3]} \right) \delta b_{\mu \nu} \\ & = - \left( \frac{d}{d x^\rho} b^{[\mu \nu, \rho]} \right) \delta b_{\mu \nu} \\ & = - h^{\mu \nu \rho}{}_{,\rho} \delta b_{\mu \nu} \,. \end{aligned} \end{displaymath} \end{proof} \begin{example} \label{PresymplecticCurrentDiracField}\hypertarget{PresymplecticCurrentDiracField}{} \textbf{([[Euler-Lagrange form]] and [[presymplectic current]] of [[Dirac field]])} Consider the [[Lagrangian field theory]] of the [[Dirac field]] on [[Minkowski spacetime]] of [[dimension]] $p + 1 \in \{3,4,6,10\}$ (example \ref{LagrangianDensityForDiracField}). Then \begin{itemize}% \item the [[Euler-Lagrange variational derivative]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) in the case of vanishing [[mass]] $m$ is \begin{displaymath} \delta_{EL} \mathbf{L} \;=\; 2 i\, \overline{\delta \psi} \,\gamma^\mu\, \psi_{,\mu} \, \wedge dvol_\Sigma \end{displaymath} and in the case that [[spacetime]] [[dimension]] is $p +1 = 4$ and arbitrary [[mass]] $m\in \mathbb{R}$, it is \begin{displaymath} \delta_{EL} \mathbf{L} \;=\; \left( \overline{\delta \psi} \left( i \gamma^\mu \psi_{,\mu} + m \psi \right) + \left( - i \gamma^\mu\overline{\psi_{,\mu}} + m \overline{\psi} \right) (\delta \psi) \right) \, dvol_\Sigma \end{displaymath} \item its [[presymplectic current]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) is \begin{displaymath} \Omega_{BFV} \;=\; \overline{\delta \psi}\,\gamma^\mu \,\delta \psi \, \iota_{\partial_\mu} dvol_\Sigma \end{displaymath} \end{itemize} \end{example} \begin{proof} In any case the [[canonical momentum]] of the [[Dirac field]] according to example \ref{CanonicalMomentum} is \begin{displaymath} \begin{aligned} p^\alpha_\mu & \coloneqq \frac{\partial }{\partial \psi^\alpha_{,\mu}} \left( i \overline {\psi} \, \gamma^\nu \, \psi_{,\nu} + m \overline{\psi} \psi \right) \\ & = \overline{\psi}^\beta (\gamma^\mu)_\beta{}^\alpha \end{aligned} \end{displaymath} This yields the [[presymplectic current]] as claimed, by example \ref{CanonicalMomentum}. Now regarding the [[Euler-Lagrange form]], first consider the massless case in spacetime dimension $p+1 \in \{3,4,6,10\}$, where \begin{displaymath} L \;=\; i \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} \,. \end{displaymath} Then we compute as follows: \begin{displaymath} \begin{aligned} \delta_{EL} L & = i \,\overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu} \underset{ = + i \,\overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu} }{ \underbrace{ - i \overline{\psi_{,\mu}} \, \gamma^\mu \, \delta \psi } } \\ & = 2 i \, \overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu} \end{aligned} \end{displaymath} Here the first equation is the general formula \eqref{EulerLagrangeEquationGeneral} for the Euler-Lagrange variation, while the identity under the braces combines two facts (as in remark \ref{LagrangianDensityOfDiracFieldSupergeometricNature} above): \begin{enumerate}% \item the symmetry \eqref{SpinorToVectorPairingIsSymmetric} of the spinor pairing $\overline{(-)}\gamma^\mu(-)$ (prop. \ref{RealSpinorPairingsViaDivisionAlg}); \item the anti-commutativity \eqref{DiracFieldJetCoordinatesAnticommute} of the Dirac field and jet coordinates, due to their [[supergeometry|supergeometric]] nature (remark \ref{DiracFieldSupergeometric}). \end{enumerate} Finally in the special case of the massive Dirac field in spacetime dimension $p+1 = 4$ the Lagrangian function is \begin{displaymath} L \;=\; i \, \overline{\psi} \gamma^\mu \psi_{,\mu} + m \overline{\psi}\psi \end{displaymath} where now $\psi_\alpha$ takes values in the [[complex numbers]] $\mathbb{C}$ (as opposed to in $\mathbb{R}$, $\mathbb{H}$ or $\mathbb{O}$). Therefore we may now form the [[derivative]] equivalently by treeating $\psi$ and $\overline{\psi}$ as independent components of the field. This immediately yields the claim. \end{proof} \begin{example} \label{TrivialLagrangianDensities}\hypertarget{TrivialLagrangianDensities}{} \textbf{(trivial [[Lagrangian densities]] and the [[Euler-Lagrange complex]])} If a [[Lagrangian density]] $\mathbf{L}$ (def. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}) is in the image of the [[total spacetime derivative]], hence horizontally exact (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) \begin{displaymath} \mathbf{L} \;=\; d \mathbf{\ell} \end{displaymath} for any $\mathbf{\ell} \in \Omega^{p,0}_\Sigma(E)$, then both its [[Euler-Lagrange form]] as well as its [[presymplectic current]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) vanish: \begin{displaymath} \delta_{EL}\mathbf{L} = 0 \phantom{AA} \,, \phantom{AA} \Omega_{BFV} = 0 \,. \end{displaymath} This is because with $\delta \circ d = - d \circ \delta$ \eqref{HorizontalAndVerticalDerivativeAnticommute} the defining unique decomposition \eqref{dLDecomposition} of $\delta \mathbf{L}$ is given by \begin{displaymath} \begin{aligned} \delta \mathbf{L} & = \delta d \mathbf{\ell} \\ & = \underset{= \delta_{EL}\mathbf{L}}{\underbrace{0}} - d \underset{\Theta_{BFV}}{\underbrace{\delta \mathbf{l}}} \end{aligned} \end{displaymath} which then implies with \eqref{PresymplecticCurrent} that \begin{displaymath} \begin{aligned} \Omega_{BFV} & \coloneqq \delta \Theta_{BFV} \\ & = \delta \delta \mathbf{\ell} \\ & = 0 \end{aligned} \end{displaymath} Therefore the [[Lagrangian densities]] which are [[total spacetime derivatives]] are also called \emph{trivial Lagrangian densities}. If the [[field bundle]] $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) then also the converse is true: Every Lagrangian density whose [[Euler-Lagrange form]] vanishes is a total spacetime derivative. Stated more [[category theory|abstractly]], this means that the [[exact sequence]] of the total spacetime from prop. \ref{HorizontalVariationalComplexOfTrivialFieldBundleIsExact} extends to the right via the [[Euler-Lagrange variational derivative]] $\delta_{EL}$ to an [[exact sequence]] of the form \begin{displaymath} \mathbb{R} \overset{}{\hookrightarrow} \Omega^{0,0}_\Sigma(E) \overset{d}{\longrightarrow} \Omega^{1,0}_\Sigma(E) \overset{d}{\longrightarrow} \Omega^{2,0}_\Sigma(E) \overset{d}{\longrightarrow} \cdots \overset{d}{\longrightarrow} \Omega^{p,0}_\Sigma(E) \overset{d}{\longrightarrow} \Omega^{p+1,0}_\Sigma(E) \overset{\delta_{EL}}{\longrightarrow} \Omega^{p+1,0}_\Sigma(E) \wedge \delta(C^\infty(E)) \overset{\delta_{H}}{\longrightarrow} \cdots \,. \end{displaymath} In fact, as shown, this [[exact sequence]] keeps going to the right; this is also called the \emph{[[Euler-Lagrange complex]]}. (\href{Euler-Lagrange+complex#Anderson89}{Anderson 89, theorem 5.1}) The next [[differential]] $\delta_{H}$ after the [[Euler-Lagrange variational derivative]] $\delta_{EL}$ is known as the \emph{[[Helmholtz operator]]}. By definition of [[exact sequence]], the [[Helmholtz operator]] detects whether a [[partial differential equation]] on [[field histories]], induced by a [[variational differential form]] $P \in \Omega^{p+1,0}_\Sigma(E) \wedge \delta(C^\infty(E))$ as in \eqref{EquationOfMotionEL} comes from varying a [[Lagrangian density]], hence whether it is the [[equation of motion]] of a [[Lagrangian field theory]] via def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}. This way [[homological algebra]] is brought to bear on core questions of [[field theory]]. For more on this see the exposition at \emph{[[schreiber:Higher Structures|Higher Structures in Physics]]}. \end{example} \begin{remark} \label{LagrangianDensityOfDiracFieldSupergeometricNature}\hypertarget{LagrangianDensityOfDiracFieldSupergeometricNature}{} \textbf{([[supergeometry|supergeometric]] nature of [[Lagrangian density]] of the [[Dirac field]])} Observe that the [[Lagrangian density]] for the [[Dirac field]] (def. \ref{LagrangianDensityForDiracField}) makes sense (only) due to the [[supergeometry|supergeometric]] nature of the [[Dirac field]] (remark \ref{DiracFieldSupergeometric}): If the field jet coordinates $\psi_{,\mu_1 \cdots \mu_k}$ were not anti-commuting \eqref{DiracFieldJetCoordinatesAnticommute} then the Dirac's field Lagrangian density (def. \ref{LagrangianDensityForDiracField}) would be a [[total spacetime derivative]] and hence be trivial according to example \ref{TrivialLagrangianDensities}. This is because \begin{displaymath} d \left( \tfrac{1}{2} \overline{\psi} \,\gamma^\mu\, \psi \, \iota_{\partial_\mu} dvol_\Sigma \right) = \tfrac{1}{2} \overline{\psi_{,\mu}} \,\gamma^\mu\, \psi \, dvol_\Sigma + \underset{ = (-1) \tfrac{1}{2} \overline{\psi_{,\mu}} \,\gamma^\mu\, \psi \, dvol_\Sigma }{ \underbrace{ \tfrac{1}{2}\overline{\psi} \,\gamma^\mu\, \psi_{,\mu} \, dvol_\Sigma }} \,. \end{displaymath} Here the identification under the brace uses two facts: \begin{enumerate}% \item the symmetry \eqref{SpinorToVectorPairingIsSymmetric} of the spinor bilinear pairing $\overline{(-)}\Gamma (-)$; \item the anti-commutativity \eqref{DiracFieldJetCoordinatesAnticommute} of the Dirac field and jet coordinates, due to their [[supergeometry|supergeometric]] nature (remark \ref{DiracFieldSupergeometric}). \end{enumerate} The second fact gives the minus sign under the brace, which makes the total expression vanish, if the Dirac field and jet coordinates indeed are anti-commuting (which, incidentally, means that we found an ``[[off-shell]] [[conserved current]]'' for the Dirac field, see example \ref{DiracCurrent} below). If however the Dirac field and jet coordinates did commute with each other, we would instead have a plus sign under the brace, in which case the total horizontal derivative expression above would equal the massless Dirac field Lagrangian \eqref{DiracFieldLagrangianMassless}, thus rendering it trivial in the sense of example \ref{TrivialLagrangianDensities}. The same [[supergeometry|supergeometric]] nature of the [[Dirac field]] will be necessary for its intended [[equation of motion]], the \emph{[[Dirac equation]]} (example \ref{EquationOfMotionOfDiracFieldIsDiracEquation}) to derive from a [[Lagrangian density]]; see the proof of example \ref{PresymplecticCurrentDiracField} below, and see remark \ref{SupergeometricNatureOfDiracEquation} below. \end{remark} $\,$ \textbf{[[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]} The key implication of the [[Euler-Lagrange form]] on the [[jet bundle]] is that it induces the \emph{[[equation of motion]]} on the [[space of field histories]]: \begin{defn} \label{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}\hypertarget{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}{} \textbf{([[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]])} Given a [[Lagrangian field theory]] $(E,\mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} then the corresponding \emph{[[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]} is the condition on [[field histories]] (def. \ref{SupergeometricSpaceOfFieldHistories}) \begin{displaymath} \Phi_{(-)} \;\colon\; U \longrightarrow \Gamma_\Sigma(E) \end{displaymath} to have a [[jet prolongation]] (def. \ref{JetProlongation}) \begin{displaymath} j^\infty_\Sigma(\Phi_{(-)}(-) ) \;\colon\; U \times \Sigma \longrightarrow J^\infty_\Sigma(E) \end{displaymath} that factors through the [[shell]] inclusion $\mathcal{E} \overset{i_{\mathcal{E}}}{\hookrightarrow} J^\infty_\Sigma(E)$ \eqref{ShellInJetBundle} defined by vanishing of the [[Euler-Lagrange form]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) \begin{equation} j^\infty_\Sigma(\Phi_{(-)}(-)) \;\colon\; U \times \Sigma \longrightarrow \mathcal{E} \overset{i_{\mathcal{E}}}{\hookrightarrow} J^\infty_\Sigma(E) \,. \label{EquationOfMotionEL}\end{equation} (This implies that $j^\infty_\Sigma(\Phi_{(-)})$ factors even through the prolonged shell $\mathcal{E}^\infty \overset{i_{\mathcal{E}^\infty}}{\hookrightarrow} J^\infty_\Sigma(E)$ \eqref{ProlongedShellInJetBundle}.) In the case that the field bundle is a [[trivial vector bundle]] over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle} this is the condition that $\Phi_{(-)}$ satisfies the following [[differential equation]] (again using prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}): \begin{displaymath} \frac{\delta_{EL} L}{\delta \phi^a} \;\coloneqq\; \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} + \frac{d^2}{d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu\nu}} - \cdots \right) \left( (x^\mu), (\Phi^a), \left( \frac{\partial \Phi^a_{(-)}}{\partial x^\mu}\right), \left( \frac{\partial^2 \Phi^a_{(-)}}{\partial x^\mu \partial x^\nu} \right), \cdots \right) \;=\; 0 \,, \end{displaymath} where the [[differential operator]] (def. \ref{DifferentialOperator}) \begin{equation} j^\infty_\Sigma(-)^\ast \left( \frac{\delta_{EL}L}{\delta \phi^{(-)}} \right) \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(T^\ast_\Sigma E) \label{DifferentialOperatorEulerLagrangeDerivative}\end{equation} from the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) to its [[vertical cotangent bundle]] (def. \ref{VerticalTangentBundle}) is given by the \emph{[[Euler-Lagrange derivative]]} \eqref{EulerLagrangeEquationGeneral}. The \emph{[[on-shell]] [[space of field histories]]} is the space of solutions to this condition, namely the the sub-[[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet}) of the full [[space of field histories]] \eqref{SpaceOfFieldHistories} (def. \ref{SupergeometricSpaceOfFieldHistories}) \begin{equation} \Gamma_\Sigma(E)_{\delta_{EL} L = 0} \overset{\phantom{AAA}}{\hookrightarrow} \Gamma_\Sigma(E) \label{OnShellFieldHistories}\end{equation} whose plots are those $\Phi_{(-)} \colon U \to \Gamma_\Sigma(E)$ that factor through the shell \eqref{EquationOfMotionEL}. More generally for $\Sigma_r \hookrightarrow \Sigma$ a [[submanifold]] of [[spacetime]], we write \begin{equation} \Gamma_{\Sigma_r}(E)_{\delta_{EL} L = 0} \overset{\phantom{AAA}}{\hookrightarrow} \Gamma_{\Sigma_r}(E) \label{OnShellFieldHistoriesInHigherCodimension}\end{equation} for the sub-[[super formal smooth set|super smooth ste]] of on-shell field histories restricted to the [[infinitesimal neighbourhood]] of $\Sigma_r$ in $\Sigma$ \eqref{SpaceOfFieldHistoriesInHigherCodimension}. \end{defn} \begin{defn} \label{FreeFieldTheory}\hypertarget{FreeFieldTheory}{} \textbf{([[free field theory]])} A [[Lagrangian field theory]] $(E, \mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) with [[field bundle]] $E \overset{fb}{\to} \Sigma$ a [[vector bundle]] (e.g. a [[trivial vector bundle]] as in example \ref{TrivialVectorBundleAsAFieldBundle}) is called a \emph{[[free field theory]]} if its [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) is a [[differential equation]] that is \emph{[[linear differential equation]]}, in that with \begin{displaymath} \Phi_1, \Phi_2 \;\in\; \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \end{displaymath} any two [[on-shell]] [[field histories]] \eqref{OnShellFieldHistories} and $c_1, c_2 \in \mathbb{R}$ any two [[real numbers]], also the [[linear combination]] \begin{displaymath} c_1 \Phi_1 + c_2 \Phi_2 \;\in\; \Gamma_\Sigma(E) \,, \end{displaymath} which a priori exists only as an element in the off-shell [[space of field histories]], is again a solution to the [[equations of motion]] and hence an element of $\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}$. A [[Lagrangian field theory]] which is not a [[free field theory]] is called an \emph{[[interaction|interacting]]} [[field theory]]. \end{defn} \begin{remark} \label{FreeFieldTheoryRelevance}\hypertarget{FreeFieldTheoryRelevance}{} \textbf{(relevance of [[free field theory]])} In [[perturbative quantum field theory]] one considers [[interaction|interacting]] [[field theories]] in the [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) of [[free field theories]] (def. \ref{FreeFieldTheory}) inside some [[super formal smooth set|super smooth set]] of general [[Lagrangian field theories]]. While [[free field theories]] are typically of limited interest in themselves, this [[perturbative quantum field theory|perturbation theory]] around them exhausts much of what is known about [[quantum field theory]] in general, and therefore [[free field theories]] are of paramount importance for the general theory. We discuss the [[covariant phase space]] of [[free field theories]] below in \emph{\hyperlink{Propagators}{Propagators}} and their [[quantization]] below in \emph{\hyperlink{FreeQuantumFields}{Free quantum fields}}. \end{remark} \begin{prop} \label{EquationOfMotionOfFreeRealScalarField}\hypertarget{EquationOfMotionOfFreeRealScalarField}{} \textbf{([[equation of motion]] of [[free field|free]] [[real scalar field]] is [[Klein-Gordon equation]])} Consider the [[Lagrangian field theory]] of the [[free field|free]] [[real scalar field]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}. By example \ref{FreeScalarFieldEOM} its [[Euler-Lagrange form]] is \begin{displaymath} \delta_{EL}\mathbf{L} \;=\; \left(\eta^{\mu \nu} \phi_{,\mu \nu} - m^2 \right) \delta \phi \wedge dvol_\sigma \end{displaymath} Hence for $\Phi \in \Gamma_\Sigma(E) = C^\infty(X)$ a [[field history]], its [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] according to def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} is \begin{displaymath} \eta^{\mu \nu} \frac{\partial^2 }{\partial x^\mu \partial x^\nu} \Phi - m^2 \Phi \;=\; 0 \end{displaymath} often abbreviated as \begin{equation} (\Box - m^2) \Phi \;=\; 0 \,. \label{KleinGordonEquation}\end{equation} This [[PDE]] is called the \emph{[[Klein-Gordon equation]]} on Minowski spacetime. If the [[mass]] $m$ vanishes, $m = 0$, then this is the \emph{relativistic [[wave equation]]}. Hence this is indeed a [[free field theory]] according to def. \ref{FreeFieldTheory}. The corresponding [[linear differential operator]] (def. \ref{DifferentialOperator}) \begin{equation} (\Box - m^2) \;\colon\; \Gamma_\Sigma(\Sigma \times \mathbb{R}) \longrightarrow \Gamma_\Sigma(\Sigma \times \mathbb{R}) \label{KleinGordonOperator}\end{equation} is called the \emph{[[Klein-Gordon operator]]}. \end{prop} For later use we record the following basic fact about the [[Klein-Gordon equation]]: \begin{example} \label{FormallySelfAdjointKleinGordonOperator}\hypertarget{FormallySelfAdjointKleinGordonOperator}{} \textbf{([[Klein-Gordon operator]] is [[formally adjoint differential operator|formally self-adjoint]] )} The [[Klein-Gordon operator]] \eqref{KleinGordonOperator} is its own [[formal adjoint differential operator|formal adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}) witnessed by the bilinear differential operator \eqref{FormallyAdjointDifferentialOperatorWitness} given by \begin{equation} K(\Phi_1, \Phi_2) \;\coloneqq\; \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \,. \label{WitnessForFormalSelfadjointnessOfKleinGordonEquation}\end{equation} \end{example} \begin{proof} \begin{displaymath} \begin{aligned} d K(\Phi_1, \Phi_2) & = d \left( \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 - \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \right) \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma \\ &= \left( \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 + \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu} \right) - \left( \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu} + \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) \right) dvol_\Sigma \\ & = \left( \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 - \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \right) dvol_\Sigma \\ & = \Box(\Phi_1) \Phi_2 - \Phi_1 \Box (\Phi_2) \end{aligned} \end{displaymath} \end{proof} \begin{prop} \label{MaxwellVacuumEquation}\hypertarget{MaxwellVacuumEquation}{} \textbf{([[equations of motion]] of [[vacuum]] [[electromagnetism]] are [[vacuum]] [[Maxwell's equations]])} Consider the [[Lagrangian field theory]] of [[free field|free]] [[electromagnetism]] on [[Minkowski spacetime]] from example \ref{ElectromagnetismLagrangianDensity}. By example \ref{ElectromagnetismEl} its [[Euler-Lagrange form]] is \begin{displaymath} \delta_{EL}\mathbf{L} \;=\; \frac{d}{d x^\mu}f^{\mu \nu} \delta a_\nu \,. \end{displaymath} Hence for $A \in \Gamma_{\Sigma}(T^\ast \Sigma) = \Omega^1(\Sigma)$ a [[field history]] (``[[vector potential]]''), its [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] according to def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} is \begin{displaymath} \begin{aligned} & \frac{\partial}{\partial x^\mu} F^{\mu \nu} = 0 \\ \Leftrightarrow\;\; & d \star_\eta F = 0 \end{aligned} \,, \end{displaymath} where $F = d A$ is the [[Faraday tensor]] \eqref{TensorFaraday}. (In the coordinate-free formulation in the second line ``$\star_\eta$'' denotes the [[Hodge star operator]] induced by the [[pseudo-Riemannian metric]] $\eta$ on [[Minkowski spacetime]].) These [[PDEs]] are called the \emph{[[vacuum]] [[Maxwell's equations]]}. This, too, is a [[free field theory]] according to def. \ref{FreeFieldTheory}. \end{prop} \begin{example} \label{EquationOfMotionOfDiracFieldIsDiracEquation}\hypertarget{EquationOfMotionOfDiracFieldIsDiracEquation}{} \textbf{([[equation of motion]] of [[Dirac field]] is [[Dirac equation]])} Consider the [[Lagrangian field theory]] of the [[Dirac field]] on [[Minkowski spacetime]] from example \ref{LagrangianDensityForDiracField}, with [[field fiber]] the [[spin representation]] $S$ regarded as a [[superpoint]] $S_{odd}$ and [[Lagrangian density]] given by the spinor bilinear pairing \begin{displaymath} L \;=\; i \overline{\psi} \gamma^\mu \partial_\mu \psi + m \overline{\psi}\psi \end{displaymath} (in spacetime dimension $p+1 \in \{3,4,6,10\}$ with $m = 0$ unless $p+1 = 4$). By example \ref{PresymplecticCurrentDiracField} the [[Euler-Lagrange derivative|Euler-Lagrange]] [[differential operator]] \eqref{DifferentialOperatorEulerLagrangeDerivative} for the [[Dirac field]] is of the form \begin{equation} \itexarray{ \Gamma_\Sigma(\Sigma \times S) &\overset{ }{\longrightarrow}& \Gamma_\Sigma(\Sigma \times S^\ast) \\ \Psi &\mapsto& \overline{(-)} D \psi } \label{DiracOperatorAsELOperator}\end{equation} so that the corresponding [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) is equivalently \begin{equation} \underset{D}{ \underbrace{ \left(-i \gamma^\mu \partial_\mu + m\right) }} \psi \;=\; 0 \,. \label{DiracEquation}\end{equation} This is the \emph{[[Dirac equation]]} and $D$ is called a \emph{[[Dirac operator]]}. In terms of the \emph{[[Feynman slash notation]]} from \eqref{FeynmanSlashNotationForMasslessDiracOperator} the corresponding [[differential operator]], the \emph{[[Dirac operator]]} reads \begin{displaymath} D \;=\; \left( - i \partial\!\!\!/\, + m \right) \,. \end{displaymath} Hence this is a [[free field theory]] according to def. \ref{FreeFieldTheory}. Observe that the ``square'' of the [[Dirac operator]] is the [[Klein-Gordon operator]] $\Box - m^2$ \eqref{KleinGordonEquation} \begin{displaymath} \begin{aligned} \left( +i \gamma^\mu \partial_\mu + m \right) \left(-i \gamma^\mu \partial_\mu + m\right)\psi & = \left(\partial_\mu \partial^\mu - m^2\right) \psi \\ & = \left(\Box - m^2\right) \psi \end{aligned} \,. \end{displaymath} This means that a [[Dirac field]] which solves the [[Dirac equations]] is in particular (on [[Minkowski spacetime]]) componentwise a [[solution]] to the [[Klein-Gordon equation]]. \end{example} \begin{prop} \label{SupergeometricNatureOfDiracEquation}\hypertarget{SupergeometricNatureOfDiracEquation}{} \textbf{([[supergeometry|supergeometric]] nature of the [[Dirac equation]] as an [[Euler-Lagrange equation]])} While the [[Dirac equation]] \eqref{DiracEquation} of example \ref{EquationOfMotionOfDiracFieldIsDiracEquation} would make sense in itself also if the field coordinates $\psi$ and jet coordinates $\psi_{,\mu}$ of the [[Dirac field]] were not anti-commuting \eqref{DiracFieldJetCoordinatesAnticommute}, due to their [[supergeometry|supergeometric]] nature (remark \ref{DiracFieldSupergeometric}), it would, by remark \ref{LagrangianDensityOfDiracFieldSupergeometricNature}, then no longer be the [[Euler-Lagrange equation]] of a [[Lagrangian density]], hence then Dirac field theory would not be a [[Lagrangian field theory]]. \end{prop} \begin{example} \label{DiracOperatorOnDiracSpinorsIsFormallySelfAdjointDifferentialOperator}\hypertarget{DiracOperatorOnDiracSpinorsIsFormallySelfAdjointDifferentialOperator}{} \textbf{([[Dirac operator]] on [[Dirac spinors]] is [[formally self-adjoint differential operator]])} The \emph{[[Dirac operator]], hence the [[differential operator]] corresponding to the [[Dirac equation]] of example \ref{EquationOfMotionOfDiracFieldIsDiracEquation} via def. \ref{DifferentialOperator} is a [[formally self-adjoint differential operator|formally anti-self adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}):} \begin{displaymath} D^\ast = - D \,. \end{displaymath} \end{example} \begin{proof} By \eqref{DiracOperatorAsELOperator} we are to regard the Dirac operator as taking values in the [[dual vector bundle|dual]] [[spin bundle]] by using the [[Dirac conjugate]] $\overline{(-)}$ \eqref{DiracConjugate}: \begin{displaymath} \itexarray{ \Gamma_\Sigma(\Sigma \times S) &\overset{}{\longrightarrow}& \Gamma_\Sigma(\Sigma \times S^\ast) \\ \Psi &\mapsto& \overline{(-)} D \Psi } \end{displaymath} Then we need to show that there is $K(-,-)$ such that for all [[pairs]] of [[spinor]] [[sections]] $\Psi_1, \Psi_2$ we have \begin{displaymath} \overline{\Psi_2}\gamma^\mu (\partial_\mu \Psi_1) - \overline{\Psi_1}\gamma^\mu (-\partial_\mu \Psi_2) \;=\; d K(\psi_1, \psi_2) \,. \end{displaymath} But the spinor-to-vector pairing is symmetric \eqref{SpinorToVectorPairingIsSymmetric}, hence this is equivalent to \begin{displaymath} \overline{\partial_\mu \Psi_1}\gamma^\mu \Psi_2 + \overline{\Psi_1}\gamma^\mu (\partial_\mu \Psi_2) \;=\; d K(\psi_1, \psi_2) \,. \end{displaymath} By the [[product law]] of [[differentiation]], this is solved, for all $\Psi_1, \Psi_2$, by \begin{displaymath} K(\Psi_1, \Psi_2) \;\coloneqq\; \left( \overline{\Psi_1} \gamma^\mu \Psi_2\right) \, \iota_{\partial_\mu} dvol \,. \end{displaymath} \end{proof} $\,$ This concludes our discussion of [[Lagrangian densities]] and their [[variational calculus]]. In the \hyperlink{Symmetries}{next chapter} we consider the [[infinitesimal symmetries of Lagrangians]] and the [[conserved currents]] that these induce via [[Noether's theorem]]. \end{document}