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\begin{document}

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\section*{Ward identity}

\begin{quote}%
under construction


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{BeforeRenormalization}{Before renormalization}\dotfill \pageref*{BeforeRenormalization} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[perturbative quantum field theory]] the \emph{quantum master Ward identity} (def. \ref{OnRegularPolynomialObservablesMasterWardIdentity} below) expresses the relation between the [[quantum field theory|quantum]] (measured by [[Planck's constant]] $\hbar$) [[interacting field theory|interacting]] (measured by the [[coupling constant]] $g$) [[equations of motion]] to the [[classical field theory|classical]] [[free field]] [[equations of motion]] at $\hbar, g\to 0$ (remark \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below). As such it generalizes the [[Schwinger-Dyson equation]], to which it reduces for $g = 0$ (example \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below) as well as the \emph{classical master Ward identity}, which is the case for $\hbar = 0$ (example \ref{MasterWardIdentityClassical} below).

Applied to products of the [[equations of motion]] with any given [[observable]], the master Ward identity becomes a particular \emph{Ward identity}.

This is of interest notably in view of [[Noether's theorem]], which says that every [[infinitesimal symmetry of the Lagrangian]] of, in particular, the given [[free field theory]], corresponds to a [[conserved current]], hence a [[horizontal differential form]] whose [[total spacetime derivative]] vanishes up to a term proportional to the [[equations of motion]]. Under [[transgression of variational differential forms|transgression]] to [[local observables]] this is a relation of the form

\begin{displaymath}
div \mathbf{J} = 0 \phantom{AAA} \text{on-shell}
  \,,
\end{displaymath}
where ``on shell'' means up to the ideal generated by the [[classical field theory|classical]] [[free field theory|free]] [[equations of motion]]. Hence for the case of [[local observables]] of the form $div \mathbf{J}$, the quantum Ward identity expresses the possible failure of the original [[conserved current]] to actually be conserved, due to both quantum effects ($\hbar$) and interactions ($g$). This is the form in which Ward identities are usually understood (example \ref{NoetherCurrentConservationQuantumCorrection} below).

In terms of [[BV-BRST formalism]], the master Ward identity is equivalent to the \emph{[[quantum master equation]]} on [[regular polynomial observables]] (\href{quantum+master+equation#QuantumMasterEquation}{this prop.}).

Neither of these equations is guaranteed to hold for any choice of [[renormalization|(``re''-)normalization]]. If a Ward identity is violated by the [[renormalization|(``re''-)normalized]] [[perturbative QFT]], specifically if there is no possible choice of [[renormalization|(``re''-)normalization]] that preserves it, the one speaks of a \emph{[[quantum anomaly]]}. Specifically if the [[conserved current]] corresponding to a [[gauge symmetry]] is \emph{anomalous} in this way, one speaks of a \emph{[[gauge anomaly]]}.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{BeforeRenormalization}{}\subsubsection*{{Before renormalization}}\label{BeforeRenormalization}

\begin{defn}
\label{OnRegularPolynomialObservablesMasterWardIdentity}\hypertarget{OnRegularPolynomialObservablesMasterWardIdentity}{}
Consider a [[free field theory|free]] [[gauge fixing|gauge fixed]] [[Lagrangian field theory]] $(E_{\text{BV-BRST}}, \mathbf{L}')$ (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}) with global [[BV-differential]] on [[regular polynomial observables]]

\begin{displaymath}
\{-S',(-)\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\end{displaymath}
(\href{A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal}{this def.}).

Let moreover

\begin{displaymath}
g S_{int} 
   \;\in\; 
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar , g  ] ]
\end{displaymath}
be a [[regular polynomial observable]] (regarded as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]]) such that the total action $S' + g S_{int}$ satisfies the [[quantum master equation]] (\href{quantum+master+equation#QuantumMasterEquation}{this prop.}); and write

\begin{displaymath}
\mathcal{R}^{-1}(-)
  \;\coloneqq\;
  \mathcal{S}(g S_{int})^{-1} 
    \star_H 
  (\mathcal{S}(g S_{int}) \star_F (-))
\end{displaymath}
for the corresponding [[quantum Møller operator]] (\href{quantum+master+equation#MollerOperatorOnRegularPolynomialObservables}{this def.}).

Then by \href{quantum+master+equation#QuantumMasterEquation}{this prop.} we have

\begin{equation}
\{-S',(-)\} \circ \mathcal{R}^{-1}   
  \;=\;
  \mathcal{R}^{-1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, (-) \right\}_{\mathcal{T}} 
    - 
    i \hbar   \Delta_{BV}
  \right)
\label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}\end{equation}
This is the \emph{quantum master Ward identity} on [[regular polynomial observables]], i.e. before [[renormalization]].

\end{defn}
(\hyperlink{Rejzner13}{Rejzner 13, (37)})

\begin{remark}
\label{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations}\hypertarget{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations}{}
\textbf{([[quantum master Ward identity]] relates [[quantum field theory|quantum]] [[interacting field theory|interacting field]] [[equation of motion|EOMs]] to [[classical field theory|classical]] [[free field]] [[equation of motion|EOMs]])}

For $A \in PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar, g] ]$ the [[quantum master Ward identity]] on [[regular polynomial observables]] \eqref{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered} reads

\begin{equation}
\mathcal{R}^{-1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}} 
    - 
    i \hbar   \Delta_{BV}(A)
  \right)
  \;=\;
  \{-S', \mathcal{R}^{-1}(A) \}
\label{RearrangedMasterQuantumWard}\end{equation}
The term on the right is manifestly in the [[image]] of the global [[BV-differential]] $\{-S',-\}$ of the [[free field theory]] (\href{A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal}{this def.}) and hence vanishes when passing to [[on-shell]] observables along the [[isomorphism]] (\href{A+first+idea+of+quantum+field+theory#eq:OnShellPolynomialObservablesAsBVCohomology}{this equation})

\begin{displaymath}
\underset{
    \text{on-shell}
  }{
  \underbrace{
    PolyObs(E_{\text{BV-BRST}}, \mathbf{L}')
  }}
  \;\simeq\;
  \underset{
    \text{off-shell}
  }{
  \underbrace{
    PolyObs(E_{\text{BV-BRST}})_{def(af = 0)}
  }}/im(\{-S',-\})
\end{displaymath}
(by \href{A+first+idea+of+quantum+field+theory#BVDifferentialGlobal}{this example}).

Hence

\begin{displaymath}
\mathcal{R}^{-1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}} 
    - 
    i \hbar   \Delta_{BV}(A)
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
\end{displaymath}
In contrast, the left hand side is the [[interacting field observable]] (via \href{S-matrix#MollerOperatorOnRegularPolynomialObservables}{this def.}) of the sum of the [[time-ordered product|time-ordered]] [[antibracket]] with the [[action functional]] of the [[interacting field theory]] and a quantum correction given by the [[BV-operator]]. If we use the definition of the [[BV-operator]] $\Delta_{BV}$ (\href{BV-operator#RearrangedMasterWardWithOnShell}{this def.}) we may equivalently re-write this as

\begin{equation}
\mathcal{R}^{-1}
  \left(
    \left\{ -S'  \,,\, A \right\}
    +
    \left\{ -g S_{int} \,,\, A  \right\}_{\mathcal{T}}
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
\label{RearrangedMasterWardWithOnShell}\end{equation}
Hence the [[quantum master Ward identity]] expresses a relation between the ideal spanned by the [[classical field theory|classical]] [[free field theory|free field]] [[equations of motion]] and the [[quantum field theory|quantum]] [[interacting field theory|interacting field]] equations of motion.

\end{remark}
\begin{example}
\label{SchwingerDysonReductionOfQuantumMasterWardIdentity}\hypertarget{SchwingerDysonReductionOfQuantumMasterWardIdentity}{}
\textbf{([[free field]]-limit of [[master Ward identity]] is [[Schwinger-Dyson equation]])}

In the [[free field]]-limit $g \to 0$ (noticing that in this limit $\mathcal{R}^{-1} = id$) the [[quantum master Ward identity]] \eqref{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered} reduces to

\begin{displaymath}
\left\{ -S'  \,,\, A \right\}_{\mathcal{T}} 
  - 
  i \hbar   \Delta_{BV}(A)
  \;=\;
  \{-S', A  \}
\end{displaymath}
which is the defining equation for the [[BV-operator]] (\href{BV-operator#eq:BVOperatorDefiningRelation}{this equation}), hence is isomorphic (under $\mathcal{T}$) to the [[Schwinger-Dyson equation]] (\href{BV-operator#DysonSchwinger}{this prop.})

\end{example}
\begin{example}
\label{MasterWardIdentityClassical}\hypertarget{MasterWardIdentityClassical}{}
\textbf{([[classical limit]] of [[quantum master Ward identity]])}

In the [[classical limit]] $\hbar \to 0$ (noticing that the classical limit of $\{-,-\}_{\mathcal{T}}$ is $\{-,-\}$) the [[quantum master Ward identity]] \eqref{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered} reduces to

\begin{displaymath}
\mathcal{R}^{1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, A \right\} 
  \right)
  \;=\;
  \{-S', \mathcal{R}^{-1}(A) \}
\end{displaymath}
This says that the [[interacting field observable]] corresponding to the global [[antibracket]] with the action functional of the [[interacting field theory]] vanishes on-shell, classically.

Applied to an observable which is [[linear map|linear]] in the [[antifields]]

\begin{displaymath}
A 
    \;=\; 
  \underset{\Sigma}{\int}
    A^a(x) 
    \mathbf{\Phi}^\ddagger_a(x)
  \, 
  dvol_\Sigma(x)
\end{displaymath}
this yields

\begin{displaymath}
\begin{aligned}
    0
    & =
    \{-S', \mathcal{R}^{-1}(A)\} 
    +
    \mathcal{R}^{-1}
    \left(
      \left\{ -(S' + S_{int}) \,,\, A \right\}_{\mathcal{T}}
    \right)
    \\
    & = 
    \underset{\Sigma}{\int} 
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \mathcal{R}^{-1}(A^a(x))
    \, dvol_\Sigma(x)
    +   
    \mathcal{R}^{-1}
    \left(
      \underset{\Sigma}{\int}
          A^a(x) \frac{\delta (S' + S_{int})}{\delta \mathbf{\Phi}^a(x)}
        \, dvol_\Sigma(x)
    \right)
  \end{aligned}
\end{displaymath}
This is the \emph{classical master Ward identity} according to (\hyperlink{DuetschFredenhagen02}{Dütsch-Fredenhagen 02}, \hyperlink{BrennecketDuetsch07}{Brennecke-Dütsch 07, (5.5)}), following (\hyperlink{DuetschBoas02}{Dütsch-Boas 02}).

\end{example}
\begin{example}
\label{NoetherCurrentConservationQuantumCorrection}\hypertarget{NoetherCurrentConservationQuantumCorrection}{}
\textbf{(quantum correction to [[Noether's theorem|Noether current]] [[conserved current|conservation]])}

Let $v \in \Gamma^{ev}_\Sigma(T_\Sigma(E_{\text{BRST}}))$ be an [[evolutionary vector field]], which is an [[infinitesimal symmetry of the Lagrangian]] $\mathbf{L}'$, and let $J_{\hat v} \in \Omega^{p,0}_\Sigma(E_{\text{BV-BRST}})$ the corresponding [[conserved current]], by [[Noether's theorem|Noether's theorem I]] (\href{A+first+idea+of+quantum+field+theory#NoethersFirstTheorem}{this prop.}), so that

\begin{displaymath}
\begin{aligned}
    d J_{\hat v} 
    & =  
    \iota_{\hat v}  \delta \mathbf{L}'
    \\
    & = 
    (v^a dvol_\Sigma) \frac{\delta_{EL} L'}{\delta \phi^a}
    \phantom{AAA}
    \in 
    \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})
  \end{aligned}
\end{displaymath}
(by \href{A+first+idea+of+quantum+field+theory#eq:CurrentNoetherConservation}{this equation}), where in the second line we just rewrote the expression in components (using \href{A+first+idea+of+quantum+field+theory#eq:EulerLagrangeEquationGeneral}{this equation})

\begin{displaymath}
v^a
  \,,
  \frac{\delta_{EL} L'}{\delta \phi^a} 
  \;\in \Omega^{0,0}_\Sigma(E_{\text{BV-BRST}})
\end{displaymath}
and re-arranged suggestively.

Then for $a_{sw} \in C^\infty_{cp}(\Sigma)$ any choice of [[bump function]], we obtain the [[local observables]]

\begin{displaymath}
\begin{aligned}
    A_{sw}
    & \coloneqq
    \underset{\Sigma}{\int}
      \underset{
        A^a(x)
      }{
      \underbrace{
        a_{sw}(x)
        v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
      }
      }
      \mathbf{\Phi}^\ddagger_a(x)
      \,
      dvol_\Sigma(x)
    \\ 
    & \coloneqq
    \tau_\Sigma( a_{sw}  v^a \phi^{\ddagger}_a \, dvol_\Sigma)
  \end{aligned}
\end{displaymath}
and

\begin{displaymath}
\begin{aligned}
    (div \mathbf{J})_{sw}
    & \coloneqq
    \underset{\Sigma}{\int}
      \underset{
        A^a(x)
      }{
      \underbrace{
        a_{sw}(x)
        v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
      }
      }
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
      \,
      dvol_\Sigma(x)
    \\ 
    & \coloneqq
    \tau_\Sigma
    \left(
      a_{sw}  v^a \frac{\delta_{EL} \mathbf{L}'}{\delta \phi^a} 
      \, dvol_\Sigma  
    \right)
  \end{aligned}
\end{displaymath}
by [[transgression of variational differential forms]].

This is such that

\begin{displaymath}
\left\{
    -S' , A_{sw}
  \right\}
  = 
  (div \mathbf{J})_{sw}
  \,.
\end{displaymath}
Hence applied to this choice of local observable $A$, the quantum master Ward identity \eqref{RearrangedMasterWardWithOnShell} now says that

\begin{displaymath}
\mathcal{R}^{-1}
  \left(
    {\, \atop \,}
    (div \mathbf{J})_{sw}
  \right)
  \;=\;
  \mathcal{R}^{-1}
  \left(
      \{g S_{int}, A_{sw} \}_{\mathcal{T}}

    {\, \atop \,}
  \right)
    \phantom{AAA}
  \text{on-shell}
\end{displaymath}
Hence the [[interacting field observable]]-version $\mathcal{R}^{-1}(div\mathbf{J})$ of $div \mathbf{J}$ need not vanish itself on-shell, instead there may be a correction as shown on the right.

\end{example}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[chiral anomaly]]


\item [[conformal blocks]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after \emph{[[John Clive Ward]]}.

Discussion of the [[master Ward identity]] in the rigorous context of [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] formulated via [[causal perturbation theory]]/[[perturbative AQFT]] is in

\begin{itemize}%
\item [[Michael Dütsch]], F.-M. Boas, \emph{The Master Ward Identity}, Rev. Math. Phys 14, (2002) 977-1049 (\href{http://cds.cern.ch/record/526377/files/0111101.pdf}{pdf})


\item [[Michael Dütsch]], [[Klaus Fredenhagen]], equation (90) in \emph{The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory}, Commun.Math.Phys. 243 (2003) 275-314 (\href{https://arxiv.org/abs/hep-th/0211242}{arXiv:hep-th/0211242})


\item Ferdinand Brennecke, [[Michael Dütsch]], equation (5.5) in \emph{Removal of violations of the Master Ward Identity}, in perturbative QFT, Rev.Math.Phys. 20 (2008) 119-172 (\href{https://arxiv.org/abs/0705.3160}{arXiv:https://arxiv.org/abs/0705.3160})


\item [[Stefan Hollands]], around (322) and (333) and (345) of \emph{Renormalized Quantum Yang-Mills Fields in Curved Spacetime}, Rev. Math. Phys.20:1033-1172, 2008 (\href{https://arxiv.org/abs/0705.3340}{arXiv:0705.3340})


\item [[Katarzyna Rejzner]], section 5.3 of \emph{Batalin-Vilkovisky formalism in locally covariant field theory} (\href{https://arxiv.org/abs/1111.5130}{arXiv:1111.5130})


\item [[Katarzyna Rejzner]], equation (37) of \emph{Remarks on local symmetry invariance in perturbative algebraic quantum field theory} (\href{https://arxiv.org/abs/1301.7037}{arXiv:1301.7037})


\item [[Michael Dütsch]], equation (4.2) of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Ward%E2%80%93Takahashi_identity}{Ward-Takahashi identity}}

\end{itemize}
[[!redirects Ward identities]]

[[!redirects Ward–Takahashi identity]] [[!redirects Ward–Takahashi identities]]

[[!redirects classical master Ward identity]] [[!redirects classical master Ward identities]]

[[!redirects quantum master Ward identity]] [[!redirects quantum master Ward identities]]

[[!redirects master Ward identity]] [[!redirects master Ward identities]]

[[!redirects quantum Ward identity]] [[!redirects quantum Ward identities]]



\end{document}