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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 -- Gauge fixing} \hypertarget{GaugeFixing}{}\subsection*{{Gauge fixing}}\label{GaugeFixing} In this chapter we discuss the following topics: \begin{enumerate}% \item \emph{\hyperlink{QuasiIsomorphismsBetweenBVBRSTComplexes}{Quasi-isomorphisms between local BV-BRST complexes}} \begin{enumerate}% \item \hyperlink{GaugeFixingChainMaps}{gauge fixing chain maps}; \item \hyperlink{AdjoiningAuiliaryFields}{adjoining contractible complexes of auxiliary fields} \end{enumerate} \item \emph{\hyperlink{GaugeFixingExamples}{Example: gauge fixed electromagnetic field}} \end{enumerate} $\,$ While in the \hyperlink{ReducedPhaseSpace}{previous chapter} we had constructed the [[reduced phase space]] of a [[Lagrangian field theory]], embodied by the [[local BV-BRST complex]] (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}), as the [[homotopy quotient]] by the [[infinitesimal gauge symmetries]] of the [[homotopy intersection]] with the [[shell]], this in general still does not yield a [[covariant phase space]] of [[on-shell]] [[field histories]] (prop. \ref{CovariantPhaseSpace}), since [[Cauchy surfaces]] for the [[equations of motion]] may still not exist (def. \ref{CauchySurface}). However, with the [[homological resolution]] constituted by the [[BV-BRST complex]] in hand, we now have the freedom to adjust the [[field (physics)|field]]-content of the theory without changing its would-be [[reduced phase space]], namely without changing its [[BV-BRST cohomology]]. In particular we may adjoin further ``[[auxiliary fields]]'' in various degrees, as long as they contribute only a [[contractible chain complex|contractible cochain complex]] to the [[BV-BRST complex]]. If such a \emph{[[quasi-isomorphism]]} of [[BV-BRST complexes]] brings the [[Lagrangian field theory]] into a form such that the [[equations of motion]] of the combined [[field (physics)|fields]], [[ghost fields]] and potential further [[auxiliary fields]] are [[Green hyperbolic differential equations]] after all, and thus admit a [[covariant phase space]], then this is called a \emph{[[gauge fixing]]} (def. \ref{GaugeFixingLagrangianDensity} below), since it is the [[infinitesimal gauge symmetries]] which [[obstruction|obstruct]] the existence of [[Cauchy surfaces]] (by prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces} and remark \ref{GaugeParametrizedInfinitesimalGaugeTransformation}). The archetypical example is the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixing]] of the [[electromagnetic field]] (example \ref{NLGaugeFixingOfElectromagnetism} below) which reveals that the gauge-invariant content of [[electromagnetic waves]] is only in their transversal [[wave polarization]] (prop. \ref{GaugeInvariantPolynomialOnShellObservablesOfFreeElectromagneticField} below). The tool of [[gauge fixing]] via [[quasi-isomorphisms]] of [[BV-BRST complexes]] finally brings us in position to consider, in the following chapters, the [[quantization]] also of [[gauge theories]]: We use [[gauge fixing]] [[quasi-isomorphisms]] to bring the [[BV-BRST complexes]] of the given [[Lagrangian field theories]] into a form that admits degreewise [[quantization]] of a [[graded manifold|graded]] [[covariant phase space]] of [[fields (physics)|fields]], [[ghost fields]] and possibly further [[auxiliary fields]], compatible with the gauge-fixed [[BV-BRST differential]]: $\,$ \begin{displaymath} \itexarray{ \underline{\mathbf{\text{pre-quantum geometry}}} && \underline{\mathbf{\text{higher pre-quantum geometry}}} \\ \, \\ \left\{ \itexarray{ \text{Lagrangian field theory with} \\ \text{infinitesimal gauge transformations} } \right\} &\overset{ \text{homotopy quotient by} \atop \text{gauge transformations} }{\longrightarrow}& \left\{ \itexarray{ \text{dg-Lagrangian field theory with} \\ \text{quotiented by gauge transformations} \\ \text{embodied by BRST complex } } \right\} \\ && \Big\downarrow{}^{\mathrlap{ \text{pass to} \atop \text{derived critical locus} }} \\ && \left\{ \itexarray{ \text{dg-reduced phase space} \\ \text{ embodied by BV-BRST complex } } \right\} \\ && {}^{\mathllap{\simeq}}\Big\downarrow{}^{\mathrlap{\text{fix gauge} }} \\ \left\{ \itexarray{ \text{ decategorified } \\ \text{ covariant } \\ \text{ reduced phase space } } \right\} &\underset{\text{pass to cohomology}}{\longleftarrow}& \left\{ \itexarray{ \text{ dg-covariant} \\ \text{reduced phase space } } \right\} \\ && \Big\downarrow{}^{\mathrlap{ \itexarray{ \text{ quantize } \\ \text{degreewise} } }} \\ \left\{ \itexarray{ \text{gauge invariant} \\ \text{quantum observables} } \right\} &\underset{\text{pass to cohomology}}{\longleftarrow}& \left\{ \itexarray{ \text{quantum} \\ \text{BV-BRST complex} } \right\} } \end{displaymath} Here: \begin{tabular}{l|l} term&meaning\\ \hline ``phase space''&[[derived critical locus]] of [[Lagrangian density\\ ``reduced''&[[gauge transformations]] have been [[homotopy quotient\\ ``covariant''&[[Cauchy surfaces]] exist degreewise\\ \end{tabular} $\,$ \textbf{[[quasi-isomorphisms]] between [[local BV-BRST complexes]]} Recall (prop. \ref{CochainCohomologyOfBVBRSTComplexInDegreeZero}) that given a [[local BV-BRST complex]] (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}) with [[BV-BRST differential]] $s$, then the space of [[local observables]] which are [[on-shell]] and [[gauge invariance|gauge invariant]] is the [[cochain cohomology]] of $s$ in degree zero: \begin{displaymath} H^0(s \vert d) \;=\; \left\{ \itexarray{ \text{gauge invariant on-shell} \\ \text{local observables} } \right\} \end{displaymath} The key point of having [[resolution|resolved]] (in chapter \emph{\hyperlink{ReducedPhaseSpace}{Reduced phase space}}) the naive [[quotient]] by [[infinitesimal gauge symmetries]] of the naive [[intersection]] with the [[shell]] by the [[L-infinity algebroid]] whose [[Chevalley-Eilenberg algebra]] is called the \emph{[[local BV-BRST complex]]}, is that placing the [[reduced phase space]] into the [[(infinity,1)-category|context]] of [[homotopy theory]]/[[homological algebra]] this way provides the freedom of changing the choice of [[field bundle]] and of [[Lagrangian density]] without actually changing the [[Lagrangian field theory]] \emph{up to [[equivalence]]}, namely without changing the [[cochain cohomology]] of the [[BV-BRST complex]]. A [[homomorphism]] of [[differential graded-commutative superalgebras]] (such as [[BV-BRST complexes]]) which induces an [[isomorphism]] in [[cochain cohomology]] is called a \emph{[[quasi-isomorphism]]}. We now discuss two classes of [[quasi-isomorphisms]] between [[BV-BRST complexes]]: \begin{enumerate}% \item \emph{[[gauge fixing]]} (def. \ref{GaugeFixingLagrangianDensity} below) \item \emph{adjoining [[auxiliary fields]]} (def. \ref{AuxiliaryFields} below). \end{enumerate} $\,$ \textbf{[[gauge fixing]] [[chain maps]]} \begin{prop} \label{ExponentialOfLocalAntibracket}\hypertarget{ExponentialOfLocalAntibracket}{} \textbf{([[local antibracket|local anti-]][[Hamiltonian flow]] is [[automorphism]] of [[local antibracket]])} Let \begin{displaymath} CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)_{\delta_{EL} L \simeq 0} \right) \;=\; \left( \Omega^{0,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1] \right) \;,\; d_{CE} = \underset{s}{ \underbrace{ \left\{ -\mathbf{L} + \mathbf{L}_{BRST} \,,\, - \right\} } } \;+\; d \right) \end{displaymath} be a [[local BV-BRST complex]] of a [[Lagrangian field theory]] $(E,\mathbf{L})$ (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}). Then for \begin{displaymath} \mathbf{L}_{gf} \;\in\; \Omega^{p+1,0} \left( T^\ast_{\Sigma,inf}\left(E \times_\Sigma \mathcal{G}\right) \times_\Sigma T \Sigma[1] \right) \end{displaymath} a [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) on the [[graded manifold|graded]] [[field bundle]] \begin{displaymath} \mathbf{L}_{gf} \;=\; L_{gf} \ dvol_\Sigma \end{displaymath} of degree \begin{displaymath} deg(L) = (-1, even) \end{displaymath} then the [[exponential]] of forming the [[local antibracket]] (def. \ref{LocalAntibracket}) with $\mathbf{L}_{gf}$ \begin{displaymath} \itexarray{ \Omega^{p+1,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma \mathcal{G}[1]\right) \right) & \overset{ e^{\left\{ \mathbf{L}_{gf} \,,\, -\right\}}(-) }{\longrightarrow} & \Omega^{p+1,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma \mathcal{G}[1]\right) \right) \\ \mathbf{K} &\mapsto& \left\{ \mathbf{L}_{gf} , \mathbf{K} \right\} + \tfrac{1}{2} \left\{ \mathbf{L}_{gf} \,,\, \left\{ \mathbf{L}_{gf} \,,\, \mathbf{K} \right\} \right\} + \tfrac{1}{6} \left\{ \mathbf{L}_{gf} \,,\,\left\{ \mathbf{L}_{gf} \,,\, \left\{ \mathbf{L}_{gf} \,,\,\mathbf{K} \right\} \right\} \right\} + \cdots } \end{displaymath} is an [[endomorphism]] of the [[local antibracket]] (def. \ref{LocalAntibracket}) in that \begin{displaymath} e^{ \left\{ \mathbf{\psi} \,,\, - \right\} } \left( \left\{ \mathbf{A} \,,\, \mathbf{B} \right\} \right) \;=\; \left\{ e^{ \left\{ \mathbf{\psi} \,,\, - \right\} } \left(\mathbf{A}\right) \,,\, e^{ \left\{ \psi \,,\, - \right\} } \left(\mathbf{B}\right) \right\} \end{displaymath} and in fact an [[automorphism]], with [[inverse morphism]] given by \begin{displaymath} \left(e^{\left\{ \psi \,,\, -\right\}}(-)\right)^{-1} \;=\; e^{\left\{ -\psi \,,\, -\right\}}(-) \,. \end{displaymath} We may think of this as the \emph{[[Hamiltonian flow]]} of $\mathbf{L}_{gf}$ under the [[local antibracket]]. In particular when applied to the [[BV-Lagrangian density]] \begin{displaymath} s_{gf} \;\coloneqq\; \left\{ e^{\left\{ \mathbf{L}_{gf},-\right\}}\left(- \mathbf{L} + \mathbf{L}_{BRST}\right) \,,\, - \right\} \end{displaymath} this yields another [[differential]] \begin{displaymath} \left( s_{gf}\right)^2 \;=\; 0 \end{displaymath} and hence another [[differential graded-commutative superalgebra]] (def. \ref{differentialgradedcommutativeSuperalgebra}) \begin{displaymath} CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)^{gf}_{\delta_{EL} L \simeq 0} \right) \;=\; \left( \Omega^{0,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1] \right) \;,\; d_{CE} = \underset{s_{gf}}{ \underbrace{ \left\{ e^{\left\{ \mathbf{L}_{gf}, - \right\}}\left( - \mathbf{L} + \mathbf{L}_{BRST} \right) \,,\, - \right\} } } \;+\; d \right) \end{displaymath} Finally, $e^{\left\{\mathbf{L}_{gf},-\right\}}$ constitutes a [[chain map]] from the [[local BV-BRST complex]] to this deformed version, in fact a [[homomorphism]] of [[differential graded-commutative superalgebras]], in that \begin{displaymath} s_{gf} \circ e^{ \left\{ \mathbf{L}_{gf}\,,\, - \right\} } \;=\; e^{ \left\{ \mathbf{L}_{gf}\,,\, - \right\} } \circ s \,. \end{displaymath} \end{prop} \begin{proof} By prop. \ref{BasicPropertiesOfTheLocalAntibracket} the [[local antibracket]] $\left\{ -,-\right\}$ is a graded [[derivation]] in its second argument, of degree one more than the degree of its first argument \eqref{LocalAntibracketGradedDerivationInSecondArgument}. Hence for the first argument of degree -1 this implies that $e^{\{\mathbf{L}_{gf}, - \}}$ is an automorphism of the local antibracket. Moreover, it is clear from the definition that $\left\{ \mathbf{L}_{gf},-\right\}$ is a [[derivation]] with respect to the pointwise product of smooth functions, so that $e^{\{\mathbf{L}_{gf},-\}}$ is also a homomorpism of graded algebras. Since $e^{\{\mathbf{L}_{gf}, -\}}$ is an automorphism of the local antibracket, and since $s$ and $s_{gf}$ are themselves given by applying the local antibracket in the second argument, this implies that $e^{\{\mathbf{L}_{gf},-\}}$ respects the differentials: \begin{displaymath} \itexarray{ \mathbf{A} &\overset{e^{\{\mathbf{L}_{gf},-\}}}{\longrightarrow}& e^{\{\mathbf{L}_{gf},-\}}\left( \mathbf{A} \right) \\ {}^{\mathllap{s}}\downarrow && \downarrow^{\mathrlap{s_{gf}}} \\ \left\{ \left(-\mathbf{L} + \mathbf{L}_{BRST}\right)\,,\, \mathbf{A}\right\} &\underset{ e^{\{\mathbf{L}_{gf}\,,\,-\}} }{\longrightarrow}& \left\{ e^{\{\mathbf{L}_{gf},-\}}\left(-\mathbf{L} + \mathbf{L}_{BRST}\right) \,,\, e^{\{\mathbf{L}_{gf},-\}}(\mathbf{A}) \right\} } \end{displaymath} \end{proof} \begin{defn} \label{GaugeFixingLagrangianDensity}\hypertarget{GaugeFixingLagrangianDensity}{} \textbf{([[gauge fixing Lagrangian density]])} Let \begin{displaymath} CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)_{\delta_{EL} L \simeq 0} \right) \;=\; \left( \Omega^{0,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1] \right) \;,\; d_{CE} = \underset{s}{ \underbrace{ \left\{ -\mathbf{L} + \mathbf{L}_{BRST} \,,\, - \right\} } } \;+\; d \right) \end{displaymath} be a [[local BV-BRST complex]] of a [[Lagrangian field theory]] $(E,\mathbf{L})$ (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}) and let \begin{displaymath} \mathbf{L}_{gf} \;\in\; \Omega^{p+1,0} \left( T^\ast_{\Sigma,inf}\left(E \times_\Sigma \mathcal{G}\right) \times_\Sigma T \Sigma[1] \right) \end{displaymath} be a [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) on the [[graded manifold|graded]] [[field bundle]] such that \begin{displaymath} deg(L_{gf}) = -1 \,. \end{displaymath} If the [[quasi-isomorphism]] of [[BV-BRST complexes]] given by the [[local antibracket|local anti-]][[Hamiltonian flow]] $\mathbf{L}_{gf}$ via prop. \ref{ExponentialOfLocalAntibracket} \begin{displaymath} e^{\left\{ \mathbf{L}_{gf},-\right\}} \;\colon\; CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)^{gf}_{\delta_{EL} L \simeq 0} \right) \overset{\phantom{A}\simeq_{qi}\phantom{A}}{\longrightarrow} CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)^{gf}_{\delta_{EL} L \simeq 0} \right) \end{displaymath} is such that for the transformed graded [[Lagrangian field theory]] \begin{equation} -\underset{deg_{af} = 0}{\underbrace{\mathbf{L}' }} + \mathbf{L}'_{BRST} \;\coloneqq\; e^{\{\mathbf{L}_{gf},-\}}(-\mathbf{L} + \mathbf{L}_{BRST}) \label{GaugeFixedLagrangianDensity}\end{equation} (with [[Lagrangian density]] $\mathbf{L}'$ the part independent of [[antifields]]) the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) admit [[Cauchy surfaces]] (def. \ref{CauchySurface}), then we call $\mathbf{L}_{gf}$ a \emph{[[gauge fixing Lagrangian density]]} for the original Lagrangian field theory, and $\mathbf{L}'$ the corresponding \emph{gauge fixed} form of the original [[Lagrangian density]] $\mathbf{L}$. \end{defn} \begin{remark} \label{}\hypertarget{}{} \textbf{(warning on terminology)} What we call a \emph{[[gauge fixing Lagrangian density]]} $\mathbf{L}_{gf}$ in def. \ref{GaugeFixingLagrangianDensity} is traditionally called a \emph{[[gauge fixing fermion]]} and denoted by ``$\psi$'' (\href{BRST+complex#Henneaux90}{Henneaux 90, section 8.3, 8.4}). Here ``fermion'' is meant as a reference to the fact that the cohomological degree $deg(L_{gf}) = -1$, which is reminiscent of the odd [[supergeometry|super-degree]] of [[fermion]] fields such as the [[Dirac field]] (example \ref{DiracFieldBundle}); see at \emph{[[signs in supergeometry]]} the section \emph{\href{signs+in+supergeometry#SuperOddConvention}{The super odd sign rule}}. \end{remark} \begin{example} \label{GaugeFixingViaAntiLagrangianSubspaces}\hypertarget{GaugeFixingViaAntiLagrangianSubspaces}{} \textbf{([[gauge fixing]] via [[local antibracket|anti-]][[Lagrangian subspaces]])} Let $\mathbf{L}_{gf}$ be a [[gauge fixing Lagrangian density]] as in def. \ref{GaugeFixingLagrangianDensity} such that \begin{enumerate}% \item its [[local antibracket]]-square vanishes \begin{displaymath} \left\{ \mathbf{L}_{gf},\, \left\{ \mathbf{L}_{gf}, \, -\right\} \right\} = 0 \end{displaymath} hence its [[local antibracket|anti-]][[Hamiltonian flow]] has at most a linear component in its argument $\mathbf{A}$: \begin{displaymath} e^{\left\{ \mathbf{L}_{gf} \,,\, \mathbf{A} \right\}} \;=\; \mathbf{A} + \left\{ \mathbf{L}_{gf} \,,\, \mathbf{A} \right\} \end{displaymath} \item it is independent of the [[antifields]] \begin{displaymath} deg_{af}\left( L_{gf} \right) \;=\; 0 \,. \end{displaymath} \end{enumerate} Then with \begin{itemize}% \item $(\phi^A)$ collectively denoting all the [[field (physics)|field]] coordinates (including the actual fields $\phi^a$, the [[ghost fields]] $c^\alpha$ as well as possibly further [[auxiliary fields]]) \item $(\phi^\ddagger_A)$ collectively denoting all the [[antifield]] coordinates (includion the antifields $\phi^\ddagger_a$ of the actual fields, the antifields $c^\ddagger_\alpha$ of the [[ghost fields]] as well as those of possibly further [[auxiliary fields]] ) \end{itemize} we have \begin{displaymath} \begin{aligned} (\phi')^A & \coloneqq e^{\left\{ \mathbf{L}_{gf}\,,\, - \right\}}(\phi^A) \\ & = \phi^A \\ \phantom{A} \\ (\phi')^\ddagger_A & \coloneqq e^{\left\{ \mathbf{L}_{gf}\,,\, - \right\}} \left( \phi^\ddagger_A \right) \\ & = \phi^\ddagger_A - \frac{\overset{\leftarrow}{\delta}_{EL} \mathbf{L}_{gf}}{\delta \phi^a} \end{aligned} \end{displaymath} (and similarly for the higher jets); and the corresponding transformed [[Lagrangian density]] \eqref{GaugeFixedLagrangianDensity} may be written as \begin{displaymath} \begin{aligned} -\mathbf{L}' + \mathbf{L}'_{BRST} & \coloneqq e^{\left\{ \mathbf{L}_{gf}\,,\, - \right\}}\left( -\mathbf{L} + \mathbf{L}_{BRST} \right) \\ & = \left( -\mathbf{L} + \mathbf{L}_{BRST} \right) \left( \phi', (\phi')^\ddagger \right) \end{aligned} \,, \end{displaymath} where the notation on the right denotes that $\phi'$ is [[substitution|substituted]] for $\phi$ and $\phi'_\ddagger$ for $\phi_\ddagger$. This means that the defining condition that $\mathbf{L}'$ be the antifield-independent summand \eqref{GaugeFixedLagrangianDensity}, which we may write as \begin{displaymath} \mathbf{L}' \coloneqq \left( -\mathbf{L} + \mathbf{L}_{BRST} \right) \left( \phi'(\phi), \phi_\ddagger = 0 \right) \end{displaymath} translates into \begin{displaymath} \mathbf{L}' \coloneqq \left( -\mathbf{L} + \mathbf{L}_{BRST} \right) \left( \phi', (\phi')^\ddagger_A = -\frac{\overset{\leftarrow}{\delta}_{EL} L_{gf}}{\delta \phi^A} \right) \,. \end{displaymath} In this form BV-gauge fixing is considered traditionally (e.g. \href{gauge+fixing#Henneaux90}{Hennaux 90, section 8.3, page 83, equation (76b) and item (iii)}). \end{example} $\,$ \textbf{adjoining [[contractible chain complexes|contractible cochain complexes]] of [[auxiliary fields]]} Typically a [[Lagrangian field theory]] $(E,\mathbf{L})$ for given choice of [[field bundle]], even after finding appropriate [[gauge parameter bundles]] $\mathcal{G}$, does not yet admit a [[gauge fixing Lagrangian density]] (def. \ref{GaugeFixingLagrangianDensity}). But if the [[gauge parameter bundle]] has been chosen suitably, then the remaining [[obstruction]] vanishes ``up to [[homotopy]]'' in that a [[gauge fixing Lagrangian density]] does exist if only one adjoins sufficiently many [[auxiliary fields]] forming a [[contractible chain complex|contractible complex]], hence without changing the [[cochain cohomology]] of the [[BV-BRST complex]]: \begin{defn} \label{AuxiliaryFields}\hypertarget{AuxiliaryFields}{} \textbf{([[auxiliary fields]] and [[antighost fields]])} Over [[Minkowski spacetime]] $\Sigma$, let \begin{displaymath} A \overset{aux}{\longrightarrow} \Sigma \end{displaymath} be any [[graded manifold|graded]] [[vector bundle]] (remark \ref{dgManifolds}), to be regarded as a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) for \emph{[[auxiliary fields]]}. If this is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) we denote its field [[coordinates]] by $(b^i)$. On the corresponding graded bundle with degrees shifted down by one \begin{displaymath} A[-1] \overset{aux[-1]}{\longrightarrow} \Sigma \end{displaymath} we write $(\overline{c}^i)$ for the induced field coordinates. Accordingly, the shifted infinitesimal [[vertical cotangent bundle]] (def. \ref{InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle}) of the [[fiber product]] of these bundles \begin{displaymath} T^\ast_{\Sigma,inf}[-1]\left( A \times_\Sigma A^\ast[-1] \right) \end{displaymath} has the following coordinates: \begin{displaymath} \itexarray{ \text{name:} & \itexarray{ \text{antifield of} \\ \text{antighost field} } & \itexarray{ \text{antifield of} \\ \text{auxiliary field} } & \text{antighost field} & \text{auxiliary field} \\ \text{symbol:} & \overline{c}^\ddagger_i & b^\ddagger_i & \overline c^i & b^i \\ deg = & -(deg(b^i)-1)-1 & -deg(b^i)-1 & deg(b^i)-1 & deg(b^i) \\ & = -deg(b^i) } \end{displaymath} On this [[fiber bundle]] consider the [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) \begin{equation} \mathbf{L}_{aux} \;\in\; \Omega^{p+1,0}_\Sigma( T^\ast_{\Sigma,inf}[-1]\left( A \times_\Sigma A[-1] \right) ) \label{LagrangianDensityForAuxiliaryFields}\end{equation} given in [[local coordinates]] by \begin{displaymath} \mathbf{L}_{aux} \;\coloneqq\; \overline{c}^\ddagger_i b^i \, dvol_\Sigma \,. \end{displaymath} This is such that the [[local antibracket]] (def. \ref{LocalAntibracket}) with this Lagrangian acts on generators as follows: \begin{equation} \itexarray{ && \left\{ \mathbf{L}_{aux},- \right\} \\ \text{auxiliary field} & b^i &\mapsto& 0 \\ \text{antighost field} & \overline{c}^i &\mapsto& b^i \\ \text{antifield of auxiliary field} & b^\ddagger_i &\mapsto& - \overline{c}^\ddagger_i \\ \text{antifield of antighost field} & \overline{c}^\ddagger_i &\mapsto& 0 } \label{BVDifferentialOnauxiliaryFields}\end{equation} \end{defn} \begin{remark} \label{}\hypertarget{}{} \textbf{(warning on terminology)} Beware that when adjoining [[antifields]] as in def. \ref{AuxiliaryFields} to a [[Lagrangian field theory]] which also has [[ghost fields]] $(c^\alpha)$ adjoined (example \ref{LocalOffShellBRSTComplex}) then there is \emph{no} relation, a priori, between \begin{itemize}% \item the ``antighost field'' $\overline{c}^i$ \end{itemize} and \begin{itemize}% \item the ``antifield of the ghost field'' $c^\ddagger_\alpha$ \end{itemize} In particular there is also the \begin{itemize}% \item ``antifield of the antighost field'' $\overline{c}^\ddagger_i$ \end{itemize} The terminology and notation is maybe unfortunate but entirely established. \end{remark} The following is immediate from def. \ref{AuxiliaryFields}, in fact this is the purpose of the definition: \begin{prop} \label{QuasiIsomorphismAdjoiningAuxiliaryFields}\hypertarget{QuasiIsomorphismAdjoiningAuxiliaryFields}{} \textbf{(adjoining [[auxiliary fields]] is [[quasi-isomorphism]] of [[BV-BRST complexes]])} Let \begin{displaymath} CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)_{\delta_{EL} L \simeq 0} \right) \;=\; \left( \Omega^{0,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1] \right) \;,\; d_{CE} = \underset{s}{ \underbrace{ \left\{ -\mathbf{L} + \mathbf{L}_{BRST} \,,\, - \right\} } } \;+\; d \right) \end{displaymath} be a [[local BV-BRST complex]] of a [[Lagrangian field theory]] $(E,\mathbf{L})$ (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}). Let moreover $A \overset{aux}{\longrightarrow} \Sigma$ be any [[auxiliary field bundle]] (def. \ref{AuxiliaryFields}). Then on the [[fiber product]] of the original [[field bundle]] $E$ and the shifted [[gauge parameter bundle]] $\mathcal{G}[1]$ with the [[auxiliary field bundle]] $A$ the sum of the original [[BV-Lagrangian density]] $-\mathbf{L} + \mathbf{L}_{BRST}$ with the auxiliary Lagrangian density $\mathbf{L}_{aux}$ \eqref{LagrangianDensityForAuxiliaryFields} induce a new [[differential graded-commutative superalgebra]]: \begin{displaymath} \begin{aligned} & CE\left( E/(\mathcal{G} \times_\Sigma (A \times_\Sigma A[-1]) \times_\Sigma T \Sigma)^{aux}_{\delta_{EL} L \simeq 0} \right) \\ & \coloneqq\; \left( \Omega^{0,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1] \left( E \times_\Sigma \mathcal{G}[1] \times_\Sigma \left( A \times_\Sigma A[-1]\right) \right) \times_\Sigma T \Sigma[1] \right) \;,\; d_{CE} = \underset{s}{ \underbrace{ \left\{ \left( - L + L_{BRST} + \mathbf{L}_{aux} \right) dvol_\Sigma \,,\, - \right\} } } \;+\; d \right) \end{aligned} \end{displaymath} with generators \begin{displaymath} \itexarray{ \text{fields} & \phi^a & E & \phi^\ddagger_a & \text{antifields} \\ \\ \text{ghost fields} & c^\alpha & \mathcal{G}[1] & c^\ddagger_\alpha & \itexarray{ \text{antifields of} \\ \text{ghost fields} } \\ \\ \text{ auxiliary fields } & b^i & A & b^\ddagger_i & \itexarray{ \text{antifields of} \\ \text{auxiliary fields} } \\ \\ \text{ antighost fields } & \overline{c}^i & A[-1] & \overline{c}^{\ddagger}_i & \itexarray{ \text{antifields of} \\ \text{antighost fields} } } \end{displaymath} Moreover, the [[differential graded-commutative superalgebra]] of [[auxiliary fields]] and their [[antighost fields]] is a [[contractible chain complex]] \begin{displaymath} \left( \Omega^{0,0}_\Sigma( A \times_{\Sigma} A[-1] ) \,,\, d_{CE} = \left\{ \overline{c}^\ddagger_i b^i \, dvol_\Sigma \,,\, - \right\} \right) \overset{\simeq_{qi}}{\longrightarrow} 0 \end{displaymath} and thus the canonical inclusion map \begin{displaymath} CE\left( E/(\mathcal{G} \times_\Sigma \times_\Sigma T \Sigma)_{\delta_{EL} L \simeq 0} \right) \overset{\phantom{AA} \simeq_{qi} \phantom{aa}}{\hookrightarrow} CE\left( E/(\mathcal{G} \times_\Sigma (A \times_\Sigma A[-1]) \times_\Sigma T \Sigma)^{aux}_{\delta_{EL} L \simeq 0} \right) \end{displaymath} (of the original [[BV-BRST complex]] into its [[tensor product]] with that for the [[auxiliary fields]] and their [[antighost fields]]) is a [[quasi-isomorphism]]. \end{prop} \begin{proof} From \eqref{BVDifferentialOnauxiliaryFields} we read off that \begin{enumerate}% \item the map $s_{aux} \coloneqq \left\{ \mathbf{L}_{aux},- \right\}$ is a [[differential]] (squares to zero), and the auxiliary [[Lagrangian density]] satisfies its [[classical master equation]] (remark \ref{ClassicalMasterEquationLocal}) strictly \begin{displaymath} \{\mathbf{L}_{aux}, \mathbf{L}_{aux}\} = 0 \end{displaymath} \item the [[cochain cohomology]] of this differential is trivial: \begin{displaymath} H^\bullet( s_{aux} )\;=\;0 \end{displaymath} \item The [[local antibracket]] of the [[BV-Lagrangian density]] with the auxiliary Lagrangian density vanishes: \begin{displaymath} \left\{ - \mathbf{L} + \mathbf{L}_{BRST} \,,\, \mathbf{L}_{aux} \right\} \;=\; 0 \end{displaymath} \end{enumerate} Together this implies that the sum $-\mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}$ satisfies the [[classical master equation]] (remark \ref{ClassicalMasterEquationLocal}) \begin{displaymath} \left\{ \left( - \mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} \right) \,,\, \left( - \mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} \right) \right\} \;=\; 0 \end{displaymath} and hence that \begin{displaymath} s + s_{aux} \;\coloneqq\; \left\{ - \mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} \,,\, - \right\} \end{displaymath} is indeed a [[differential]]; such that its [[cochain cohomology]] is identified with that of $s = \left\{-\mathbf{L} + \mathbf{L}_{BRST},-\right\}$ under the canonical inclusion map. \end{proof} \begin{remark} \label{FieldBundleBVBRST}\hypertarget{FieldBundleBVBRST}{} \textbf{([[gauge fixing|gauge fixed]] [[BV-BRST formalism|BV-BRST]] [[field bundle]])} In conclusion, we have that, given \begin{enumerate}% \item $(E,\mathbf{L})$ a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), with [[field bundle]] $E$ (def. \ref{FieldsAndFieldBundles}); \item $\mathcal{G}$ a choice of [[gauge parameters]] (def. \ref{GaugeParameters}), hence $\mathcal{G}[1]$ a choice of [[ghost fields]] (example \ref{LocalOffShellBRSTComplex}); \item $A$ a choice of [[auxiliary fields]] (def. \ref{AuxiliaryFields}), hence $A[-1]$ a choice of [[antighost fields]] (def. \ref{AuxiliaryFields}) \item $T^\ast_{\Sigma,inf}[-1](\cdots)$ the corresponding [[antifields]] (def. \ref{InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle}) \item a [[gauge fixing Lagrangian density]] $\mathbf{L}_{gf}$ (def. \ref{GaugeFixingLagrangianDensity}) \end{enumerate} then the result is a new [[Lagrangian field theory]] \begin{displaymath} \left( E_{\text{BV-BRST}}, \mathbf{L}' \right) \end{displaymath} now with [[graded manifold|graded]] [[field bundle]] (remark \ref{dgManifolds}) the [[fiber product]] \begin{displaymath} E_{\text{BV-BRST}} \;\coloneqq\; \underset{ \itexarray{ \text{anti-} \\ \text{fields} } }{ \underbrace{ T^\ast_{\Sigma,inf} } } \left( \underset{\text{fields}}{\underbrace{E}} \times_\Sigma \underset{ \itexarray{ \text{ghost} \\ \text{fields} }}{\underbrace{\mathcal{G}[1]}} \times_\Sigma \underset{\itexarray{ \text{auxiliary} \\ \text{fields} }}{\underbrace{A}} \times_{\Sigma} \underset{ \itexarray{ \text{antighost} \\ \text{fields} } }{\underbrace{A[-1]}} \right) \end{displaymath} and with [[Lagrangian density]] $\mathbf{L'}$ independent of the [[antifields]], but complemented by an auxiliary Lagrangian density $\mathbf{L}'_{BRST}$. The key point being that $\mathbf{L}'$ admits a [[covariant phase space]] (while $\mathbf{L}$ may not), while in [[BV-BRST cohomology]] both theories still have the same gauge-invariant on-shell observables. \end{remark} $\,$ \textbf{Gauge fixed electromagnetic field} As an example of the general theory of BV-BRST [[gauge fixing]] above we now discuss the gauge fixing of the [[electromagnetic field]]. \begin{example} \label{NLGaugeFixingOfElectromagnetism}\hypertarget{NLGaugeFixingOfElectromagnetism}{} \textbf{([[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixing]] of [[vacuum]] [[electromagnetism]])} Consider the [[local BV-BRST complex]] for the [[free field theory|free]] [[electromagnetic field]] on [[Minkowski spacetime]] from example \ref{LocalBVComplexOfVacuumElectromagnetismOnMinkowskiSpacetime}: The [[field bundle]] is $E \coloneqq T^\ast \Sigma$ and the [[gauge parameter bundle]] is $\mathcal{G} \coloneqq \Sigma \times \mathbb{R}$. The 0-jet field coordinates are \begin{displaymath} \itexarray{ & c^\ddagger & (a^\ddagger)^\mu & a_\mu & c \\ deg = & -2 & -1 & 0 & 1 } \end{displaymath} the [[Lagrangian density]] is \eqref{ElectromagnetismLagrangian} \begin{equation} \mathbf{L}_{EM} \coloneqq \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu} \label{VacuumEMLagrangianDensityRecalledForNLFields}\end{equation} and the [[BV-BRST differential]] acts as: \begin{displaymath} \itexarray{ & &\itexarray{ \text{BV-BRST} \\ \text{differential} }& \\ \itexarray{ \text{ electromagnetic field } \\ \text{ ("vector potential") } } & a_\mu &\mapsto& c_{,\mu} & \text{gauge transformation} \\ \phantom{A} \\ \text{ ghost field } & c &\mapsto& 0 & \text{abelian Lie algebra} \\ \phantom{A} \\ \itexarray{ \text{antifield of} \\ \text{electromagnetic field} } & (a^\ddagger)^\mu &\mapsto& f^{\nu \mu}_{,\nu} & \text{equations of motion} \\ \phantom{A} \\ \itexarray{ \text{antifield of} \\ \text{ghostfield} } & c^\ddagger &\mapsto& (a^\ddagger)^\mu_{,\mu} & \text{Noether identity} \\ \phantom{A} \\ \text{Nakanishi-Lautrup field} & b &\mapsto& 0 & \text{vanishing of auxiliary fields...} \\ \phantom{A} \\ \text{antighost field} & \overline{c} &\mapsto& b & \text{... in cohomology} \\ \phantom{A} \\ \itexarray{ \text{antifield of} \\ \text{ Nakanishi-Lautrup field } } & b^\ddagger &\mapsto& -\overline{c}^\ddagger & \text{vanishing of antifields of auxiliary fields...} \\ \phantom{A} \\ \itexarray{ \text{antifield of} \\ \text{antighost field} } & \overline{c}^\ddagger &\mapsto& 0 & \text{... in cohomology} } \end{displaymath} Introduce a [[trivial vector bundle|trivial]] [[real line bundle]] for [[auxiliary fields]] $b$ in degree 0 and their [[antighost fields]] $\overline{c}$ (def. \ref{AuxiliaryFields}) in degree -1: \begin{displaymath} \itexarray{ & \Sigma \times \langle \overline{c}\rangle &\overset{ \overline{c} \mapsto b}{\longrightarrow}& \Sigma \times\langle b\rangle \\ deg = & -1 && 0 } \,. \end{displaymath} In the present context the [[auxiliary field]] $b$ is called the \emph{[[abelian Lie algebra|abelian]] [[Nakanishi-Lautrup field]]}. The corresponding [[BV-BRST complex]] with [[auxiliary fields]] adjoined, which, by prop. \ref{QuasiIsomorphismAdjoiningAuxiliaryFields}, is [[quasi-isomorphism|quasi-isomorphic]] to the original one above, has coordinate generators \begin{displaymath} \itexarray{ & c^\ddagger & (a^\ddagger)^\mu & a_\mu & c \\ & & \overline{c} & b \\ & & b^{\ddagger} & \overline{c}^\ddagger \\ deg = & -2 & -1 & 0 & 1 } \,. \end{displaymath} and [[BV-BRST differential]] given by the [[local antibracket]] (def. \ref{LocalAntibracket}) with $-\mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}$: \begin{displaymath} s \;=\; \left\{ \left( - \underset{ = L_{EM}}{\underbrace{\tfrac{1}{2}f_{\mu \nu} f^{\mu \nu}}} + \underset{ = L_{BRST} }{\underbrace{ c_{,\mu} (a^\ddagger)^\mu }} + \underset{ = L_{aux} }{\underbrace{ b \overline{c}^{\ddagger} }} \right) dvol_\Sigma \,,\, (-) \right\} \end{displaymath} We say that the [[gauge fixing Lagrangian]] (def. \ref{GaugeFixingLagrangianDensity}) for [[Gaussian-averaged Lorenz gauge]]\_ for the [[electromagnetic field]] \begin{displaymath} \mathbf{L}_{gf} \;\in\; \Omega^{p+1}_\Sigma\left( E \times_\Sigma \mathcal{G}[1] \times_\Sigma A \times_\Sigma A[-1] \right) \,. \end{displaymath} is given by (\href{Nakanishi-Lautrup+field#Henneaux90}{Henneaux 90 (103a)}) \begin{equation} \mathbf{L}_{gf} \;\coloneqq\; \underset{deg = -1}{ \underbrace{ \phantom{A}\overline{c}\phantom{A} }} \underset{deg = 0}{\underbrace{( b - a^{\mu}_{,\mu} )}} \, dvol_\Sigma \,. \label{GaugeFixingLagrangianForGaussianAveragedLorentzGauge}\end{equation} We check that this really is a [[gauge fixing Lagrangian density]] according to def. \ref{GaugeFixingLagrangianDensity}: From \eqref{VacuumEMLagrangianDensityRecalledForNLFields} and \eqref{GaugeFixingLagrangianForGaussianAveragedLorentzGauge} we find the [[local antibracket|local antibrackets]] (def. \ref{LocalAntibracket}) with this [[gauge fixing Lagrangian density]] to be \begin{displaymath} \begin{aligned} \left\{\mathbf{L}_{gf}\,,\,\left( - \mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} \right) \right\} & = \left\{ \overline{c}\left( b - a^\mu_{,\mu}\right) \, dvol_\Sigma \,,\, \left( -\tfrac{1}{2}f_{\mu \nu}f^{\mu \nu} + c_{,\mu} (a^\ddagger)^\mu + b \overline{c}^\ddagger \right) dvol_\Sigma \right\} \\ & = \left\{ \overline{c}\left( b - a^\mu_{,\mu}\right) \, dvol_\Sigma \,,\, b \overline{c}^{\ddagger} \, dvol_\Sigma \right\} + \left\{ \overline{c}\left( b - a^\mu_{,\mu}\right) \, dvol_\Sigma \,,\, c_{,\mu} (a^{\ddagger})^\mu \, dvol_\Sigma \right\} \\ & = - \left( b ( b - a^{\mu}_{,\mu} ) + \overline{c}_{,\mu} c^{,\mu} \right) \, dvol_\Sigma \\ \phantom{A} \\ \{ \mathbf{L}_{gf}, \{ \mathbf{L}_{gf} , (-\mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} )\}\} & = 0 \end{aligned} \end{displaymath} (So we are in the traditional situation of example \ref{GaugeFixingViaAntiLagrangianSubspaces}.) Therefore the corresponding [[gauge fixing|gauge fixed]] [[Lagrangian density]] \eqref{GaugeFixedLagrangianDensity} is (see also \href{Nakanishi-Lautrup+field#Henneaux90}{Henneaux 90 (103b)}): \begin{equation} \begin{aligned} -\mathbf{L}' + \mathbf{L}'_{BRST} & \coloneqq e^{\left\{ \mathbf{L}_{gf} ,-\right\}}\left( -\mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} \right) \\ & = - \underset{ = \mathbf{L}' }{ \underbrace{ \left( \underset{ = L_{EM} }{ \underbrace{ \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu} } } + \underset{ = -\left\{ L_{gf}, L_{BRST} + L_{aux} \right\} }{ \underbrace{ b ( b - a^{\mu}_{,\mu} ) + \overline{c}_{,\mu} c^{,\mu} } } \right) dvol_\Sigma } } \;+\; \underset{ = \mathbf{L}'_{BRST} }{ \underbrace{ \left( \underset{ = L_{BRST} }{ \underbrace{ c_{,\mu} (a^\ddagger)^\mu } } + \underset{ = L_{aux} }{ \underbrace{ b \overline{c}^\ddagger } } \right) dvol_\Sigma } } \end{aligned} \,. \label{GaussianAveragedLorentzianGaugeFixOfElectromagneticFieldOnMinkowskiSpacetime}\end{equation} The [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] (def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}) induced by the gauge fixed Lagrangian density $\mathbf{L}'$ at antifield degree 0 are (using \eqref{ElectromagneticFieldEulerLagrangeForm}): \begin{equation} \delta_{EL} \mathbf{L}' \;=\; 0 \phantom{AAA} \Leftrightarrow \phantom{AAA} \left\{ \begin{aligned} -\frac{d}{d x^\mu} f^{\mu \nu} & = b^{,\nu} \\ b & = \tfrac{1}{2} a^\mu_{,\mu} \\ c_{,\mu}{}^{,\mu} & = 0 \\ \overline{c}_{,\mu}{}^{,\mu} & = 0 \end{aligned} \right. \phantom{AAA} \Leftrightarrow \phantom{AAA} \left\{ \begin{aligned} \Box a^\mu & = 0 \\ b & = \tfrac{1}{2} div a \\ \Box c & = 0 \\ \Box \overline{c} & = 0 \end{aligned} \right. \label{LorenzGaugeFixedEOMForVacuumElectromagnetism}\end{equation} (e.g. \href{Nakanishi-Lautrup+field#Rejzner16}{Rejzner 16 (7.15) and (7.16)}). (Here in the middle we show the equations as the appear directly from the [[Euler-Lagrange variational derivative]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}). The [[differential operator]] $\Box = \eta^{\mu \nu} \frac{d}{d x^\mu} \frac{d}{d x^\nu}$ on the right is the [[wave operator]] (example \ref{EquationOfMotionOfFreeRealScalarField}) and $div$ denotes the [[divergence]]. The equivalence to the equations on the right follows from using in the first equation the derivative of the second equation on the left, which is $b^{,\nu} = \tfrac{1}{2} a^{\mu,\nu}{}_{,\mu}$ and recalling the definition of the universal [[Faraday tensor]] \eqref{FaradayTensorJet}: $\frac{d}{d x^\mu} f^{\mu \nu} = \tfrac{1}{2} \left( a^{\nu,\mu}{}_{,\mu} - a^{\mu,\nu}{}_{,\mu} \right)$.) Now the [[differential equations]] for [[gauge fixing|gauge-fixed]] [[electromagnetism]] on the right in \eqref{LorenzGaugeFixedEOMForVacuumElectromagnetism} are nothing but the [[wave equations]] [[equations of motion|of motion]] of $(p+1) + 1 + 1$ [[free field theory|free]] [[mass|massless]] [[scalar fields]] (example \ref{EquationOfMotionOfFreeRealScalarField}). As such, by example \ref{GreenHyperbolicKleinGordonEquation} they are a system of [[Green hyperbolic differential equations]] (def. \ref{GreenHyperbolicDifferentialOperator}), hence admit [[Cauchy surfaces]] (def. \ref{CauchySurface}). Therefore \eqref{GaussianAveragedLorentzianGaugeFixOfElectromagneticFieldOnMinkowskiSpacetime} indeed is a [[gauge fixing]] of the [[Lagrangian density]] of the [[electromagnetic field]] on [[Minkowski spacetime]] according to def. \ref{GaugeFixingLagrangianDensity}. The gauge-fixed [[BRST operator]] induced from the gauge fixed Lagrangian density \eqref{GaussianAveragedLorentzianGaugeFixOfElectromagneticFieldOnMinkowskiSpacetime} acts as \begin{equation} \itexarray{ & \itexarray{ s'_{BRST} = \\ \left\{ \left( c_{,\mu} (a^\ddagger)^\mu + b \overline{c}^{\ddagger}\right) dvol_\Sigma, (-) \right\} } \\ a_\mu &\mapsto& c_{,\mu} \\ b &\mapsto& 0 \\ \overline{c} &\mapsto & b } \label{GaussianAveragedLorentzGaugeFixedBRSTOperator}\end{equation} \end{example} From this we immediately obtain the [[propagators]] for the gauge-fixed [[electromagnetic field]]: \begin{prop} \label{PhotonPropagatorInGaussianAveragedLorenzGauge}\hypertarget{PhotonPropagatorInGaussianAveragedLorenzGauge}{} \textbf{([[photon propagator]] in [[Gaussian-averaged Lorenz gauge]])} After [[gauge fixing|fixing]] [[Gaussian-averaged Lorenz gauge]] (example \ref{NLGaugeFixingOfElectromagnetism}) of the [[electromagnetic field]] on [[Minkowski spacetime]], the [[causal propagator]] (prop. \ref{GreenFunctionsAreContinuous}) of the combined [[electromagnetic field]] and [[Nakanishi-Lautrup field]] is of the form \begin{displaymath} \Delta^{EM, EL} \;=\; \left( \itexarray{ \Delta^{photon} & \ast \\ \ast & \ast } \right) \end{displaymath} with \begin{displaymath} \Delta^{photon}_{\mu \nu}(x,y) \;=\; \eta_{\mu \nu} \Delta(x,y) \,, \end{displaymath} where \begin{enumerate}% \item $\eta_{\mu \nu}$ is the [[Minkowski metric]] [[tensor]] (def. \ref{MinkowskiSpacetime}); \item $\Delta(x,y)$ is the [[causal propagator]] of the [[free field theory]] [[mass|massless]] [[real scalar field]] (prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}). \end{enumerate} Accordingly the [[Feynman propagator]] of the [[electromagnetic field]] in [[Gaussian-averaged Lorenz gauge]] is \begin{displaymath} (\Delta^{photon}_F)_{\mu \nu}(x,y) \;=\; \eta_{\mu \nu} \Delta_F(x,y) \,, \end{displaymath} where on the right $\Delta_F(x,y)$ is the [[Feynman propagator]] of the [[free field theory|free]] [[mass|massless]] [[real scalar field]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}). This is also called the \emph{[[photon propagator]]}. Hence by prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} the [[Fourier transform of distributions|distributional Fourier transform]] of the photon propagator is \begin{displaymath} \widehat{\Delta^{photon}_F}_{\mu \nu}(k) \;=\; \frac{1}{- k^\mu k_\mu + i 0^+} \,. \end{displaymath} \end{prop} (this is a special case of \href{gauge+fixing#Khavkine14}{Khavkine 14 (99)}, see also \href{perturbative+algebraic+quantum+field+theory#Rejzner16}{Rejzner 16, (7.20)}) \begin{proof} The Gaussian-averaged Lorenz gauge-fixed equations of motion \eqref{LorenzGaugeFixedEOMForVacuumElectromagnetism} of the electromagnetic field are just $(p+1)$ uncoupled [[mass|massless]] [[Klein-Gordon equations]], hence [[wave equations]] (example \ref{EquationOfMotionOfFreeRealScalarField}) for the $(p+1)$ real components of the [[electromagnetic field]] (``[[vector potential]]'') \begin{displaymath} \Box A_\mu = 0\phantom{AAAA} \mu \in \{0,1,\cdots, p\} \,. \end{displaymath} This shows that the propoagator is proportional to that of the [[real scalar field]]. To see that the index structure is as claimed, recall that the [[domain]] and [[codomain]] of the [[advanced and retarded propagators]] in def. \ref{AdvancedAndRetardedGreenFunctions} is \begin{displaymath} \itexarray{ \Gamma_\Sigma(T\Sigma) &\overset{\left( (\mathrm{G}_{\pm})_{\mu \nu} \right)}{\longrightarrow}& \Gamma_\Sigma(T^\ast \Sigma) } \end{displaymath} corresponding to a [[differential operator]] for the [[equations of motion]] which by \eqref{ElectromagneticFieldEulerLagrangeForm} and \eqref{LorenzGaugeFixedEOMForVacuumElectromagnetism} is given by \begin{displaymath} \itexarray{ \Gamma_\Sigma(T^\ast \Sigma) &\overset{ \eta^{-1} \circ \Box }{\longrightarrow}& \Gamma_\Sigma(T \Sigma) \\ A_\mu &\mapsto& \eta^{\mu \nu} \Box A_\nu } \end{displaymath} Then the defining equation \eqref{AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator} for the [[advanced and retarded Green functions]] is, in terms of their [[integral kernels]], the [[advanced and retarded propagators]] $\Delta_{\pm}$ \begin{displaymath} \eta^{\mu' \mu} \Box \underset{y \in X}{\int} (\Delta_{\pm})_{\mu \nu}((-),y) A^{\nu}(y) \, dvol_\Sigma(x) = A^\nu(x) \,. \end{displaymath} This shows that \begin{displaymath} (\Delta_{\pm})_{\mu \nu} \;=\; \eta_{\mu\nu} \Delta_{\pm} \end{displaymath} with $\Delta_{\pm}$ the [[advanced and retarded propagator]] of the [[free field theory|free]] [[real scalar field]] on [[Minkowski spacetime]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}), and hence \begin{displaymath} \begin{aligned} \Delta_{\mu \nu} &= (\Delta_+)_{\mu \nu} - (\Delta_-)_{\mu \nu} \\ & = \eta_{\mu \nu} (\Delta_+ - \Delta_-) \\ & = \eta_{\mu \nu} \Delta \end{aligned} \end{displaymath} \end{proof} Next we compute the gauge-invariant on-shell polynomial observables of the electromagnetic field. The result will involve the following concept: \begin{defn} \label{LinearObservablesOfElectromagenticFieldWavePolarization}\hypertarget{LinearObservablesOfElectromagenticFieldWavePolarization}{} \textbf{([[wave polarization]] of linear [[observables]] of the [[electromagnetic field]])} Consider the [[electromagnetic field]] on [[Minkowski spacetime]] $\Sigma$, with [[field bundle]] the [[cotangent bundle]] The space of off-shell linear observables is spanned by the point evaluation observables \begin{displaymath} e^\mu \mathbf{A}_\mu(x) \;\in\; LinObs(T^\ast \Sigma) \end{displaymath} where \begin{enumerate}% \item $e = (e^\mu) \in \mathbb{R}^{p,1}$ is some vector; \item $x \in \mathbb{R}^{p,1}$ is some point in Minkowski spacetime \item $\mathbf{A}_\mu(x) \;\colon\; A \mapsto A_\mu(x)$ is the functional which sends a section $A \in \Gamma_\Sigma(E) = \Omega^1(\Sigma)$ to its $\mu$-component at $x$. \end{enumerate} After [[Fourier transform of distributions]] this is \begin{displaymath} e^\mu \widehat{\mathbf{A}}_\mu(k) \;\in\; LinObs(T^\ast \Sigma) \end{displaymath} for $k = (k_\mu) \in (\mathbb{R}^{p,1})^\ast$ the \emph{[[wave vector]]} for $e = (e^\mu) \in \mathbb{R}^{p,1}$ the \emph{[[wave polarization]]} The linear [[on-shell]] observables are spanned by the same expressions, but subject to the condition that \begin{displaymath} {\vert k\vert}_\eta^2 = k^\mu k_\mu = 0 \end{displaymath} hence \begin{displaymath} LinObs(T^\ast \Sigma,\mathbf{L}_{EM}) \;=\; \left\langle e^\mu \widehat{\mathbf{A}}_\mu(k) \;\vert\; k^\mu k_\mu = 0 \right\rangle \end{displaymath} We say that the space of \emph{[[transversal polarization|transversally polarized]]} linear on-shell observables is the [[quotient vector space]] \begin{equation} LinObs(T^\ast \Sigma,\mathbf{L}_{EM})_{trans} \;\coloneqq\; \frac{ \langle e^\mu \widehat{\mathbf{A}}_\mu(k) \;\vert\; k^\mu k_\mu = 0 \,\, \text{and} \,\, e^\mu k_\mu = 0 \rangle }{ \langle e^\mu \widehat{\mathbf{A}}_\mu(k) \;\vert\; k^\mu k_\mu = 0 \,\, \text{and} \,\, e_\mu \propto k_\mu \rangle } \label{ElectromagneticFieldLinearObservablesTransversallyPolarized}\end{equation} of those observables whose [[Fourier transform|Fourier modes]] involve [[wave polarization]] vectors $e$ that vanish when contracted with the [[wave vector]] $k$, modulo those whose [[wave polarization]] vector $e$ is proportional to the [[wave vector]]. For example if $k = (\kappa, 0, \cdots, \kappa)$, then the corresponding space of transversal polarization vectors may be identified with $\left\{e \,\vert\, e = (0,e_1, e_2, \cdots, e_{p-1}, 0) \right\}$. \end{defn} \begin{prop} \label{GausianAveragedLorenzGaugeFixedLinearObservablesOfTheElectromagneticField}\hypertarget{GausianAveragedLorenzGaugeFixedLinearObservablesOfTheElectromagneticField}{} \textbf{([[BRST cohomology]] on linear [[on-shell]] [[observables]] of the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixed]] [[electromagnetic field]])} After [[gauge fixing|fixing]] [[Gaussian-averaged Lorenz gauge]] (example \ref{NLGaugeFixingOfElectromagnetism}) of the [[electromagnetic field]] on [[Minkowski spacetime]], the global [[BRST cohomology]] (def. \ref{ComplexBVBRSTGlobal}) on the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixed]] (def. \ref{NLGaugeFixingOfElectromagnetism}) [[on-shell]] [[linear observables]] (def. \ref{LinearObservables}) at $deg_{gh} = 0$ (prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}) is [[isomorphism|isomorphic]] to the space of transversally polarized linear observables, def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}: \begin{displaymath} H^0( LinObs( T^\ast \Sigma \times_\Sigma A \times_\Sigma A[-1] \times_\Sigma \mathcal{G}[1], \mathbf{L}' ), s'_{BRST} ) \;\simeq\; LinObs( T^\ast \Sigma, \mathbf{L}_{EM})_{trans} \,. \end{displaymath} \end{prop} (e.g. \href{wave+polarization#Dermisek09}{Dermisek 09 II-5, p. 325}) \begin{proof} The gauge fixed BRST differential \eqref{GaussianAveragedLorentzGaugeFixedBRSTOperator} acts on the [[Fourier transform|Fourier modes]] of the linear observables (def. \ref{LinearObservables}) as follows \begin{displaymath} \itexarray{ & & s'_{BRST} \\ \itexarray{ \text{antighost} \\ \text{field} } & \widehat{\overline{\mathbf{C}}}(k) &\mapsto& \widehat{\mathbf{B}}(k) & \itexarray{ \text{Nakanishi-Lautrup} \\ \text{field} } \\ \phantom{a} \\ &&& \underset{\text{on-shell}}{=} \tfrac{i}{2} k^\mu \widehat{\mathbf{A}}_\mu(k) & \itexarray{ \text{Lorenz gauge} \\ \text{condition} } \\ \phantom{A} \\ \itexarray{ \text{electromagnetic} \\ \text{field} } & e^\mu \widehat{\mathbf{A}}_\mu(k) &\mapsto& i \left(e^\mu k_\mu\right) \widehat{\mathbf{C}}(k) & \itexarray{ \text{polarization contracted} \\ \text{with wave vector} \\ \text{times ghost field} } \\ \phantom{A} \\ \itexarray{ \text{Nakanishi-Lautrup} \\ \text{field} } & \widehat{\mathbf{B}} &\mapsto& 0 } \end{displaymath} This impies that the gauge fixed [[BRST cohomology]] on linear on-shell observables at $deg_{gh} = 0$ is the space of transversally polarized linear observables (def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}): \begin{equation} \begin{aligned} H^0(LinObs(E,\mathbf{L}_{EM}), s'_{BRST}) & = \left\langle \frac{ \left\{ e^\mu \widehat{\mathbf{A}}_{\mu}(k) \,\vert\, k^\mu k_\mu = 0 \,\,\text{and}\,\,0 = d_{BRST}\left( e^\mu \widehat{\mathbf{A}}_\mu(k) \right) = i (e^\mu k_\mu) \widehat{\mathbf{C}}(k) \right\} }{ \left\{ e^\mu \widehat{\mathbf{A}}_\mu(k) \,\vert\, k^\mu k_\mu = 0 \,\,\text{and}\,\, e^\mu \widehat{\mathbf{A}}_\mu(k) \propto s'_{BRST}( \widehat{\overline{\mathbf{C}}}(k) ) = \tfrac{i}{2} k^\mu \widehat{ \mathbf{A} }_\mu(k) \right\} } \right\rangle \\ & = \left\langle \frac{ \left\{ e^\mu \widehat{\mathbf{A}}_\mu(k) \,\vert \, k^\mu k_\mu = 0 \,\, \text{and} \,\, e^\mu k_\mu = 0 \right\} } { \left\{ e^\mu \widehat{\mathbf{A}}_\mu(k) \,\vert \, k^\mu k_\mu = 0 \,\, \text{and} \,\, e^\mu \propto k^\mu \right\} } \right\rangle \\ & = LinObs(T^\ast \Sigma,\mathbf{L}_{EM})_{trans} \end{aligned} \label{LinearOnShellObservablesGaugeFixedBRSTCohomologyForEMField}\end{equation} Here the first line is the definition of [[cochain cohomology]] (using that both $\widehat{\mathbf{B}}$ and $\widehat{\overline{\mathbf{C}}}$ are immediately seen to vanish in cohomology), the second line is spelling out the action of the BRST operator and using the on-shell relations \eqref{LorenzGaugeFixedEOMForVacuumElectromagnetism} for $\widehat{\mathbf{B}}$ and the last line is by def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}. \end{proof} As a corollary we obtain: \begin{prop} \label{BRSTCohomologyOnPolynomialOnShellObservabledOfTheGaussianAveragedLorenzGaugeFixedElectromagneticField}\hypertarget{BRSTCohomologyOnPolynomialOnShellObservabledOfTheGaussianAveragedLorenzGaugeFixedElectromagneticField}{} \textbf{([[BRST cohomology]] on polynomial [[on-shell]] [[observables]] of the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixed]] [[electromagnetic field]])} After [[gauge fixing|fixing]] [[Gaussian-averaged Lorenz gauge]] (example \ref{NLGaugeFixingOfElectromagnetism}) of the [[electromagnetic field]] on [[Minkowski spacetime]], the global [[BRST cohomology]] (def. \ref{ComplexBVBRSTGlobal}) on the [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixed]] (def. \ref{NLGaugeFixingOfElectromagnetism}) polynomial on-shell observables (def. \ref{PolynomialObservables}) at $deg_{gh} = 0$ (prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}) is [[isomorphism|isomorphic]] to the distributional polynomial algebra on transversally polarized linear observables, def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}: \begin{equation} H^0(PolyObs( T^\ast \Sigma \times_\Sigma \mathcal{G}[1] \times_\Sigma A \times_\Sigma A[-1] ,\mathbf{L}), s'_{BRST}) \;\simeq\; Sym\left( LinObs(T^\ast \Sigma,\mathbf{L}_{EM})_{trans} \right) \label{EMBRSTCohomologyOnPolynomialOnShellObservables}\end{equation} \end{prop} \begin{proof} Generally, if $(V^\bullet,d)$ is a cochain complex over a [[ground field]] of [[characteristic zero]] (such as the [[real numbers]] in the present case) and $Sym(V^\bullet,d)$ the differential graded-[[symmetric algebra]] that it induces (\href{symmetric+algebra#SymmetricAlgebraInCoChainComplexes}{this example}), then \begin{displaymath} H^\bullet(Sym(V,d)) = Sym(H^\bullet(V,d)) \,. \end{displaymath} (by \href{CochainCohomologyOfSymmetricAlgebraOnCochainComplex}{this prop.}). \end{proof} In conclusion we finally obtain: \begin{prop} \label{GaugeInvariantPolynomialOnShellObservablesOfFreeElectromagneticField}\hypertarget{GaugeInvariantPolynomialOnShellObservablesOfFreeElectromagneticField}{} \textbf{(gauge-invariant polynomial [[on-shell]] [[observables]] of the [[free field theory]] [[electromagnetic field]])} The [[BV-BRST cohomology]] on infinitesimal observables (def. \ref{LocalObservablesOnInfinitesimalNeighbourhood}) of the [[free field theory|free]] [[electromagnetic field]] on [[Minkowski spacetime]] (example \ref{LocalBVComplexOfVacuumElectromagnetismOnMinkowskiSpacetime}) at $deg_{gh} = 0$ is the distributional polynomial algebra in the transversally polarized linear on-shell observables, def. \ref{LinearObservablesOfElectromagenticFieldWavePolarization}, as in prop. \ref{BRSTCohomologyOnPolynomialOnShellObservabledOfTheGaussianAveragedLorenzGaugeFixedElectromagneticField}. \end{prop} \begin{proof} By the classes of [[quasi-isomorphisms]] of prop. \ref{ExponentialOfLocalAntibracket} and prop. \ref{QuasiIsomorphismAdjoiningAuxiliaryFields} we may equivalently compute the cohomology if the [[BV-BRST complex]] with differential $s'$, obtained after [[Gaussian-averaged Lorenz gauge]] [[gauge fixing|fixing]] from example \ref{NLGaugeFixingOfElectromagnetism}. Since the [[equations of motion]] \eqref{LorenzGaugeFixedEOMForVacuumElectromagnetism} are manifestly [[Green hyperbolic differential equations]] after this gauge fixing [[Cauchy surfaces]] for the [[equations of motion]] exist and hence prop. \ref{NonTrivialImplicitInfinitesimalGaugeSymmetriesPbstructExistenceOfCauchySurfaces} together with prop. \ref{BVComplexIsHomologicalResolutionPreciselyIfNoNonTrivialImplicitGaugeSymmetres} implies that the gauge fixed BV-complex $s'_{BV}$ has its cohomology concentrated in degree zero on the [[on-shell]] observables. Therefore prop. \ref{CochainCohomologyOfBVBRSTComplexInDegreeZero} (i.e. the collapsing of the [[spectral sequence]] for the BV/BRST [[bicomplex]]) implies that the gauge fixed BV-BRST cohomology at ghost number zero is given by the on-shell BRST-cohomology. This is characterized by prop. \ref{BRSTCohomologyOnPolynomialOnShellObservabledOfTheGaussianAveragedLorenzGaugeFixedElectromagneticField}. \end{proof} $\,$ This concludes our discussion of [[gauge fixing]]. With the [[covariant phase space]] for [[gauge theories]] obtained thereby, we may finally pass to the [[quantization]] of field theory to [[quantum field theory]] proper, in the \hyperlink{Quantization}{next chapter}. \end{document}