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\section*{quantum master equation}

\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{in_causal_perturbation_theory}{In causal perturbation theory}\dotfill \pageref*{in_causal_perturbation_theory} \linebreak
\noindent\hyperlink{background}{Background}\dotfill \pageref*{background} \linebreak
\noindent\hyperlink{interacting_quantum_bvdifferential}{Interacting quantum BV-differential}\dotfill \pageref*{interacting_quantum_bvdifferential} \linebreak
\noindent\hyperlink{RenormalizationAndMasterWardIdentity}{Renormalization and Master ward identity}\dotfill \pageref*{RenormalizationAndMasterWardIdentity} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[perturbative quantum field theory]] formulated in terms of [[BV-BRST formalism]], the \emph{classical master equation} expresses the nilpotency of the [[BV-differential]] before [[quantization]], with the latter regarded as a [[Hamiltonian vector field]] with respect to the \emph{[[antibracket]]} $\{-,-\}$, for ``Hamiltonian'' the BV-BRST-extended [[action functional]] ``S + S\_\{BRST\}'':

\begin{displaymath}
\left( 
    s_{BV}
  \right)^2 = 0
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  \left( \{S + S_{BRST},(-)\}\right)^2 = 0
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  \{S + S_{BRST}, S + BRST\} = 0
  \,.
\end{displaymath}
The \emph{quantum master equation} (prop. \ref{QuantumMasterEquation} below) is the version of this equation after [[quantization]], in which case the the [[BV-differential]] picks up a quantum correction of order $\hbar$ ([[Planck's constant]]) by the [[BV-operator]] $\Delta_{BV}$:

\begin{displaymath}
\left(\, \{S + S_{BRST},(-)\} + i \hbar \Delta_{BV} \, \right)^2 = 0
 \,.
\end{displaymath}
\hypertarget{in_causal_perturbation_theory}{}\subsection*{{In causal perturbation theory}}\label{in_causal_perturbation_theory}

We discuss the quantum master equation in the rigorous formulation of [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] via [[causal perturbation theory]]/[[perturbative AQFT]] (\hyperlink{FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b}, \hyperlink{Rejzner11}{Rejzner 11}).

First we consider all structure just on [[regular polynomial observables]], hence excluding non-linear [[local observables]] such as the usual point-[[interaction]] [[action functionals]].

Then the [[extension]] of all structures from regular to [[local observables]] is the [[renormalization]] step, discussed \hyperlink{RenormalizationAndMasterWardIdentity}{furthter below}.

\hypertarget{background}{}\subsubsection*{{Background}}\label{background}

Throughout, let $(E_{\text{BV-BST}},\mathbf{L}')$ be a [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theory]] with global [[BV-differential]] (\href{A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal}{this def.})

\begin{displaymath}
\{-S', -\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\end{displaymath}
Hence $S'$ denotes the gauge fixed free action functional.

Moreover, let $\Delta_H$ be a compatible choice of [[Wightman propagator]] with associated [[Feynman propagator]] $\Delta_F$.

\begin{lemma}
\label{DerivationBVDifferentialForWickAlgebra}\hypertarget{DerivationBVDifferentialForWickAlgebra}{}
\textbf{(global [[BV-differential]] is [[derivation]] on [[Wick algebra]])}

The global [[BV-differential]]

\begin{displaymath}
\{-S',-\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
   \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\end{displaymath}
is a [[derivation]] also with respect to the [[Wick algebra]] [[star product]] $\star_H$:

\begin{displaymath}
\left\{
    -S', A_1 \star_H A_2
  \right\}
  \;=\;
  \left\{
    -S', A_1
  \right\}
    \star_H
  A_2
    \;+\;
  A_1
    \star_H
  \left\{
   -S', A_2
  \right\}
  \,.
\end{displaymath}
\end{lemma}
(\hyperlink{FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b, below (37)}, \hyperlink{Rejzner11}{Rejzner 11, below (5.28)}) For \textbf{proof} see \href{A+first+idea+of+quantum+field+theory#OnMicrocausalObservablesGlobalBVDifferential}{this prop}

\begin{defn}
\label{OnRegularObservablesPerturbativeSMatrix}\hypertarget{OnRegularObservablesPerturbativeSMatrix}{}
\textbf{([[perturbative S-matrix]] on [[regular polynomial observables]])}

The \emph{[[perturbative S-matrix]]} on [[regular polynomial observables]] is the [[exponential]] with respect to the [[time-ordered product]]

\begin{displaymath}
\mathcal{S} 
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))
\end{displaymath}
given by

\begin{displaymath}
\mathcal{S}(S_{int})
  =
  \exp_{\star_F}
  \left(
    \tfrac{1}{i \hbar} S_{int})
  \right)
  \coloneqq
  1 
    + 
  \tfrac{1}{\i \hbar} S_{int} 
    +
  \tfrac{1}{2}
  \tfrac{1}{(i \hbar)^2} 
  S_{int} \star_F S_{int}
  + 
  \cdots
  \,.
\end{displaymath}
We think of $S_{int}$ here as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]].

We write $\mathcal{S}(S_{int})^{-1}$ for the [[inverse]] with respect to the [[Wick algebra|Wick product]] (which exists by \href{S-matrix#PerturbativeSMatrixInverse}{this remark})

\begin{displaymath}
\mathcal{S}(S_{int})^{-1} \star_H \mathcal{S}(S_{int})
  = 
  1
  \,.
\end{displaymath}
Notice that this is in general different form the inverse with respect to the [[time-ordered product]] $\star_F$, which is $\mathcal{S}(-S_{int})$:

\begin{displaymath}
\mathcal{S}(-S_{int}) 
    \star_F
  \mathcal{S}(S_{int})
  =
  1
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{MollerOperatorOnRegularPolynomialObservables}\hypertarget{MollerOperatorOnRegularPolynomialObservables}{}
\textbf{([[quantum Møller operator]] on [[regular polynomial observables]])}

Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] of degree 0

\begin{displaymath}
S_{int} 
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ]
\end{displaymath}
then the corresponding \emph{[[quantum Møller operator]]} on [[regular polynomial observables]]

\begin{displaymath}
\mathcal{R}^{-1}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\end{displaymath}
is given by the [[derivative]] of [[Bogoliubov's formula]]

\begin{displaymath}
\mathcal{R}^{-1}
    \;\coloneqq\;
  \mathcal{S}(S_{int})^{-1} 
    \star_H 
  (\mathcal{S}(S_{int}) \star_F (-))
  \,,
\end{displaymath}
where $\mathcal{S}(S_{int}) = \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)$ is the [[perturbative S-matrix]] from def. \ref{OnRegularObservablesPerturbativeSMatrix}.

This indeed lands in [[formal power series]] in [[Planck's constant]] $\hbar$ (by \href{Bogoliubov's+formula#PowersInPlancksConstant}{this remark}), instead of in more general [[Laurent series]] as the [[perturbative S-matrix]] does (def. \ref{OnRegularObservablesPerturbativeSMatrix}).

Hence the inverse map is

\begin{displaymath}
\mathcal{R}
   \;=\;
  \mathcal{S}(-S_{int}) \star_F ( \mathcal{S}(S_{int}) \star(-) )
  \,.
\end{displaymath}
\end{defn}
(\href{Bogoliubov's+formula#BogoliubovShirkov59}{Bogoliubov-Shirkov 59}; the above terminology follows \href{Møller+operator#HawkinsRejzner16}{Hawkins-Rejzner 16, below def. 5.1})

Notice that compared to Fredenhagen-Rejzner et. al. we have changed notation conventions $\mathcal{R} \leftrightarrow \mathcal{R}^{-1}$ in order to bring out the analogy to (the conventions for the) [[time-ordered product]] $A_1 \star_F A_2 = \mathcal{T}(\mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2))$ on regular polynomial observables.

notice the implicit dependencies

\newline | $\phantom{AA}\mathcal{S}$ | [[S-matrix]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] | | $\phantom{AA}\mathcal{R}$ | [[quantum Møller operator]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] and [[interaction]] |

\begin{defn}
\label{FieldAlgebraObservablesInteracting}\hypertarget{FieldAlgebraObservablesInteracting}{}
\textbf{([[interacting field algebra]])}

Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] in degree 0

\begin{displaymath}
S_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ]
  \,,
\end{displaymath}
then the \emph{[[interacting field algebra]]} [[structure]] on [[regular polynomial observables]]

\begin{displaymath}
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] 
    \otimes 
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] 
    \overset{ \star_{int} }{\longrightarrow}
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\end{displaymath}
is the [[conjugation]] of the [[Wick algebra]]-[[structure]] by the [[quantum Møller operator]] (def. \ref{MollerOperatorOnRegularPolynomialObservables}):

\begin{displaymath}
A_1 \star_{int} A_2
  \;\coloneqq\;
  \mathcal{R}
  \left(
    \mathcal{R}^{-1}(A_1) 
     \star_H 
    \mathcal{R}^{-1}(A_2)
  \right)
\end{displaymath}
\end{defn}
(e.g. \hyperlink{FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b, (19)})

\hypertarget{interacting_quantum_bvdifferential}{}\subsubsection*{{Interacting quantum BV-differential}}\label{interacting_quantum_bvdifferential}

Recall how the global BV-differential

\begin{displaymath}
\{S',-\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\end{displaymath}
on [[regular polynomial observables]] (\href{A+first+idea+of+quantum+field+theory#BVDifferentialGlobal}{this def.}) is conjugated into the [[time-ordered product]] via the time ordering operator $\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-}$ (\href{BV-operator#GaugeFixedActionFunctionalTimeOrderedAntibracket}{this prop.}).

In the same way we may use the [[quantum Møller operators]] to conjugate the BV-differential into the regular part of the [[interacting field algebra of observables]]:

\begin{defn}
\label{BVDifferentialInteractingQuantum}\hypertarget{BVDifferentialInteractingQuantum}{}
\textbf{(interacting quantum [[BV-differential]])}

Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] $S_{int}$, then the \emph{interacting quantum BV-differential} on the [[interacting field algebra]] (def. \ref{FieldAlgebraObservablesInteracting}) on [[regular polynomial observables]] is the [[conjugation]] of the plain [[BV-differential]] $\{-S',-\}$ by the [[quantum Møller operator]] induced by $S_{int}$ (def. \ref{MollerOperatorOnRegularPolynomialObservables}):

\begin{displaymath}
\mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Rejzner11}{Rejzner 11, (5.38)})

\begin{prop}
\label{QuantumMasterEquation}\hypertarget{QuantumMasterEquation}{}
\textbf{([[quantum master equation]] and [[quantum master Ward identity]] on [[regular polynomial observables]])}

Consider an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] in degree 0

\begin{displaymath}
S_{int} 
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ]
  \,,
\end{displaymath}
Then the following are equivalent:

\begin{enumerate}%
\item The \emph{[[quantum master equation]]} (QME)

\begin{equation}
\tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}}
   +
   i \hbar  \Delta_{BV}( S' + S_{int} )
  \;=\;
  0
  \,.
\label{OnRegularObservablesQuantumMasterEquation}\end{equation}

\item The [[perturbative S-matrix]] (def. \ref{OnRegularObservablesPerturbativeSMatrix}) is $BV$-closed

\begin{displaymath}
\{-S', \mathcal{S}(S_{int})\} = 0
  \,.
\end{displaymath}

\item The quantum \emph{[[master Ward identity]]} (MWI) on [[regular polynomial observables]] \emph{in terms of [[retarded products]]}:

\begin{equation}
\mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1}   
  \;=\;
  \left\{ -(S' + S_{int}) \,,\, (-) \right\}_{\mathcal{T}} 
  - 
  i \hbar   \Delta_{BV}
\label{OnRegularObservablesQuantumMasterWardIdentity}\end{equation}
(\href{Ward+identity#Duetsch18}{Dütsch 18, (4.2)})

expressing the interacting quantum [[BV-differential]] (def. \ref{BVDifferentialInteractingQuantum}) as the sum of the [[time-ordered product|time-ordered]] [[antibracket]] (\href{BV-operator#AntibracketTimeOrdered}{this def.}) with the \emph{total} [[action functional]] $S' + S_{int}$ and $i \hbar$ times the [[BV-operator]] (\href{BV-operator#ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}{BV-operator}).


\item The quantum \emph{[[master Ward identity]]} (MWI) on [[regular polynomial observables]] \emph{in terms of [[time-ordered products]]}:

\begin{equation}
\mathcal{S}(-S_{int}) 
    \star_F 
  \{-S', \mathcal{S}(S_{int}) \star_F  (-)\}
  \;=\;
  -
  \left(
    \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}} 
    + i \hbar   \Delta_{BV}
  \right)
\label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}\end{equation}
(\href{Ward+identity#Duetsch18}{Dütsch 18, (4.8)})



\end{enumerate}
\end{prop}
(\hyperlink{Rejzner11}{Rejzner 11, (5.35) - (5.38)}, following \href{Ward+identity#Hollands07}{Hollands 07, (342)-(345)})

\begin{proof}
To see that the first two conditions are equivalent, we compute as follows

\begin{equation}
\begin{aligned}
    \left\{
      -S',  \mathcal{S}(S_{int})
    \right\}
    & =
    \left\{
      -S'
      , 
      \exp_{\mathcal{T}}
      \left(
        \tfrac{1}{i \hbar} S_{int}
      \right)
    \right\}
    \\
    & =  
    \underset{
      {
        \tfrac{-1}{i \hbar} \{S',S\}_{\mathcal{T}}
      }
      \atop
      {
        \star_F 
        \exp_{\mathcal{T}}
        \left(
          \tfrac{1}{i \hbar} S_{int}
        \right)
      }
    }{
    \underbrace{
    \left\{
      -S'
      , 
      \exp_{\mathcal{T}}
      \left(
        \tfrac{1}{i \hbar} S_{int}
      \right)
    \right\}_{\mathcal{T}}
    }
    }
    - 
    i \hbar 
    \underset{
      {
        \left( 
          \tfrac{1}{i \hbar}
          \Delta_{BV}(S_{int})
          +
          \tfrac{1}{2 (i \hbar)^2}
          \left\{
            S_{int}, S_{int}
          \right\}_{\mathcal{T}}
        \right)
      }
      \atop
      {
        \star_{F} 
        \exp_{\mathcal{T}}
        \left(
          \tfrac{1}{i \hbar}
          S_{int}
        \right)
      }
    }{
    \underbrace{
    \Delta_{BV}
    \left(
      \exp_{\mathcal{T}}
      \left(
        \tfrac{1}{i \hbar} S_{int}
      \right)
    \right)    
    }
    }
    \\
    & = 
    \tfrac{-1}{i \hbar}
    \underset{ \text{QME} }{
    \underbrace{
    \left(
      \{S',S_{int}\} 
      + 
      \tfrac{1}{2}\{S_{int}, S_{int}\}
      +
      i \hbar \Delta_{BV}(S_{int})
    \right)
    }
    }
      \star_F
    \exp_{\mathcal{T}}
    \left(
      \tfrac{1}{i \hbar}
      S_{int}
    \right)
  \end{aligned}
\label{QuantumMasterOnRegularObservablesBVDifferentialOfSMatrixInTerms}\end{equation}
Here in the first step we used the definition of the [[BV-operator]] (\href{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}{this def.}) to rewrite the plain antibracket in terms of the time-ordered antibracket (\href{BV-operator#AntibracketTimeOrdered}{this def.}), then under the second brace we used that the time-ordered antibracket is the failure of the BV-operator to be a derivation (\href{BV-operator#AntibracketBVOperatorRelation}{this prop}) and under the first brace the consequence of this statement for application to exponentials (\href{BV-operator#TimeOrderedExponentialBVOperator}{this example}). Finally we collected terms, and to ``complete the square'' we added the terms on the left of

\begin{displaymath}
\frac{1}{2} \underset{= 0}{\underbrace{\{S', S'\}_{\mathcal{T}}}} 
  - 
  i \hbar \underset{ = 0}{\underbrace{ \Delta_{BV}(S')}} = 0
\end{displaymath}
which vanish because, by definition of [[gauge fixing]] (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}), the free gauge-fixed action functional $S'$ is independent of [[antifields]].

But since the operation $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)$ has the [[inverse]] $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar}  S_{int} \right)$, this implies the claim.

Next we show that the [[quantum master equation]] implies the [[quantum master Ward identities]].

We use that the BV-differential $\{-S',-\}$ is a [[derivation]] of the [[Wick algebra]] product $\star_H$ (lemma \ref{DerivationBVDifferentialForWickAlgebra}).

First of all this implies that with $\{-S', \mathcal{S}(S_{int})\} = 0$ also $\{-S', \mathcal{S}(S_{int})^{-1}\} = 0$.

Thus we compute as follows:

\begin{displaymath}
\begin{aligned}
    \{-S', -\} \circ \mathcal{R}^{-1}(A)
    & =
    \{-S', \mathcal{R}^{-1}(A)\}
    \\
    & = 
    \left\{
      { \, \atop \, }
      -S', 
      \mathcal{S}(S_{int})^{-1} 
        \star_H 
      \left( \mathcal{S}(S_{int}) \star_F a \right)
      {\, \atop \,}
    \right\}
    \\
    & = \phantom{+}
    \underset{
      = 0
    }{ 
    \underbrace{
    \left\{
      -S', \mathcal{S}(S_{int})^{-1}
    \right\}
    }
    }
      \star_H
    \left(
      \mathcal{S}(S_{int}) \star_F A
    \right)    
    \\
    & \phantom{=}
    +
    \mathcal{S}(S_{int})^{-1}
      \star_H
    \left\{
      -S',
      \mathcal{S}(S_{int}) \star_F A
    \right\}
    \\
    & = 
    \mathcal{S}(S_{int})^{-1}
      \star_H
    \left(
      \underset{
        = 1
      }{
      \underbrace{
        \mathcal{S}(+ S_{int}) 
          \star_F 
        \mathcal{S}(- S_{int})
      }
      }
        \star_F
      \left\{
        -S',
        \mathcal{S}(S_{int}) \star_F A
      \right\}
   \right)
   \\
   & =
    \mathcal{S}(S_{int})^{-1}
      \star_H
    \left(
      \mathcal{S}(+ S_{int}) 
        \star_F 
      \underset{ (\ast) }{
      \underbrace{
        \mathcal{S}(- S_{int})
          \star_F
        \left\{
          -S',
          \mathcal{S}(S_{int}) \star_F A
        \right\}
     }
     }
   \right)
   \\
   & =
   \mathcal{R}^{-1}
   \left(
     \underset{ (\ast) }{
     \underbrace{
       \phantom{\, \atop \,}
       \mathcal{S}(-S_{int})
         \star_F
       \left\{
         -S',
         \mathcal{S}(S_{int}) \star_F A
       \right\}
       }
       }
   \right)
  \end{aligned}
\end{displaymath}
By applying $\mathcal{R}$ to both sides of this equation, this means first of all that the interacting quantum BV-differential is equivalently given by

\begin{displaymath}
\mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}
  \;=\;
  \mathcal{S}(-S_{int}) \star_F \{-S', \mathcal{S}(S_{int}) \star_F  (-)\}
  \,,
\end{displaymath}
hence that if either version \eqref{OnRegularObservablesQuantumMasterWardIdentity} or \eqref{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered} of the [[master Ward identity]] holds, it implies the other.

Now expanding out the definition of $\mathcal{S}$ (def. \ref{OnRegularObservablesPerturbativeSMatrix}) and expressing $\{-S',-\}$ via the [[time-ordered product|time-ordered]] [[antibracket]] (\href{BV-operator#AntibracketTimeOrdered}{this def.}) and the [[BV-operator]] $\Delta_{BV}$ (\href{BV-operator#ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}{this prop.}) as

\begin{displaymath}
\{-S',-\}  
  \;=\;
  \{-S',-\}_{\mathcal{T}} - i \hbar \Delta_{BV}
\end{displaymath}
(on [[regular polynomial observables]]), we continue computing as follows:

\begin{equation}
\begin{aligned}
    & \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}( A )
    \\
    & =
   \exp_{\mathcal{T}}
    \left( 
      \tfrac{-1}{i \hbar} S_{int} 
    \right)
      \star_F 
    \left\{
      -S', 
      \exp_{\mathcal{T}}
      \left(
         \tfrac{1}{i \hbar} S_{int}
      \right) 
        \star_F  
      A 
    \right\}
    \\
    & =
    \exp_{\mathcal{T}}
    \left( 
      \tfrac{-1}{i \hbar} 
        S_{int}
    \right)
      \star_F
    \left(
      \left\{
        -S', 
        \exp_{\mathcal{T}}
        \left( 
          \tfrac{ 1 }{i \hbar} S_{int}
        \right) 
          \star_F  
        A 
      \right\}_{\mathcal{T}}
      - 
      i \hbar 
      \Delta_{BV}
      \left(
        \exp_{\mathcal{T}}
        \left( 
           \tfrac{ 1 }{i \hbar} S_{int}
        \right) 
          \star_F 
        A
      \right)
    \right)
    \\
    & 
    \phantom{+}
    = 
    \tfrac{1}{i \hbar} \{ -S', S_{int}  \}_{\mathcal{T}} \star_F   A
    + 
    \{-S', A\}_{\mathcal{T}}
    \\
    &
    \phantom{=}
    - 
    i \hbar
    \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int}\right)
    \star_F
    \left(
      \underset{
        {
          \left(
          \tfrac{1}{i \hbar}\Delta_{BV}(S_{int})
          +
          \tfrac{1}{2 (i \hbar)^2} \left\{ S_{int}, S_{int} \right\}
          \right)
        }
        \atop
        {
          \star_F \exp_{\mathcal{T}}\left( \tfrac{ 1 }{i \hbar} S_{int} \right)
        }
      }{
      \underbrace{
        \Delta_{BV}
        \left(
          \exp_{\mathcal{T}}
          \left(
            \tfrac{ 1}{i \hbar} S_{int}
          \right)
        \right)
      }
      }
      \star_F A
      \,+\,
       \exp_{\mathcal{T}}
       \left(
         \tfrac{ 1}{i \hbar} S_{int}
       \right)
       \star_F
       \Delta_{BV}(A)
       \,+\,
       \underset{
         {
           \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)
         }
         \atop
         {
           \star_F
           \tfrac{ 1}{i \hbar}
           \{S_{int}, A\}
         }
       }{ 
       \underbrace{
         \left\{
           \exp_{\mathcal{T}}
           \left(
             \tfrac{ 1}{i \hbar} S_{int}
           \right)
           \,,\,
           A
         \right\}_{\mathcal{T}}
      }
      }
    \right)
    \\
    & =
    -
    \left(
      \{ S' + S_{int}\,,\, A\}_{\mathcal{T}} 
        +
      i \hbar \Delta_{BV}(A)
    \right)
    \\
    & \phantom{=}
    -
    \tfrac{1}{i \hbar}
    \underset{ \text{QME} }{
    \underbrace{
    \left(
      \tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}}
      +
      i \hbar  \Delta_{BV}( S' + S_{int} )
    \right)
    }}
      \star_F A
    \\
    & =
    -
    \left(
      \{ S' + S_{int}\,,\, A\}_{\mathcal{T}} 
        +
      i \hbar \Delta_{BV}(A)
    \right)
  \end{aligned}
\label{QMESecondStep}\end{equation}
Here in the line with the braces we used that the [[BV-operator]] is a [[derivation]] of the [[time-ordered product]] up to correction by the time-ordered [[antibracket]] (\href{BV-operator#AntibracketBVOperatorRelation}{this prop.}), and under the first brace we used the effect of that property on time-ordered exponentials (\href{BV-operator#TimeOrderedExponentialBVOperator}{this example}), while under the second brace we used that $\{(-),A\}_{\mathcal{T}}$ is a derivation of the time-ordered product. Finally we have collected terms, added $0 = \{S',S'\} + i \hbar \Delta_{BV}(S')$ as before, and then used the QME.

This shows that the quantum [[master Ward identities]] follow from the [[quantum master equation]]. To conclude, it is now sufficient to show that, conversely, the MWI in terms of, say, retarded products implies the QME.

To see this, observe that with the BV-differential being nilpotent, also its conjugation by $\mathcal{R}$ is, so that with the above we have:

\begin{displaymath}
\begin{aligned}
  & \left( \{-S',-\}\right)^2 = 0
  \\
  \Leftrightarrow
  \;
  &
  \left( \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \right)^2 = 0
  \\
  \Leftrightarrow  
  \;
  &
  \underset{
    \left\{
      {\, \atop \,}
      \tfrac{1}{2}\{S' + S_{int}, S' + S_{int}\}_{\mathcal{T}} 
      + 
      i \hbar \Delta_{BV}(S' + S_{int})
      \,,\,
      (-)
    \right\}
  }{
  \underbrace{
    \left(
      \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV}
    \right)^2 
  }
  } = 0
  \end{aligned}
\end{displaymath}
Here under the brace we computed as follows:

\begin{displaymath}
\begin{aligned}
  \left(
    \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV}
  \right)^2 
  & =
  \phantom{+}
  \underset{
    \tfrac{1}{2} \{ \{S' + S, S'+ S\}_{\mathcal{T}}, (-) \}_{\mathcal{T}}
  }{
  \underbrace{
    \{S' + S_{int}, \{S' + S_{int}\}_{\mathcal{T}}, (-) \}_{\mathcal{T}}
  }}
  \\
  &
  \phantom{=}
  +
  i \hbar
  \underset{
    \{  \Delta_{BV}(S'+ S)\,,\, (-)  \}_{\mathcal{T}}
  }{
  \underbrace{
  \left(
    \{S' + S_{int}, (-)\}_{\mathcal{T}} \circ \Delta_{BV}
    + 
    \Delta_{BV} \circ \{S' + S_{int}, (-)\}_{\mathcal{T}}
  \right)
  }}
  \\
  &
  \phantom{=}
  +   
  (i \hbar)^2 
   \underset{= 0}
   {
     \underbrace{
       \Delta_{BV} \circ \Delta_{BV}
      }
    }
  \end{aligned}
  \,.
\end{displaymath}
where, in turn, the term under the first brace follows by the graded [[Jacobi identity]], the one under the second brace by Henneaux-Teitelboim (15.105c) and the one under the third brace by Henneaux-Teitelboim (15.105b).

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

The [[classical limit]] $\hbar \to 0$ of the [[quantum master Ward identity]] \eqref{OnRegularObservablesQuantumMasterWardIdentity} is

\begin{displaymath}
\{-S',(-)\} \circ \mathcal{R}^{-1}   
  \;=\;
  -
  \mathcal{R}^{-1}
  \left(
    \left\{ S' + S_{int} \,,\, (-) \right\}
  \right)
  \,.
\end{displaymath}
Applied to an observable which is linear in the [[antifields]]

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

\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}
In this form the \emph{classical Master Ward identity} was originally identified in (\href{master+Ward+identity#DuetschFredenhagen02}{Dütsch-Fredenhagen 02, (90)}, \href{master+Ward+identity#BrennecketDuetsch07}{Brennecke-Dütsch 07, (5.5)}, following \href{master+Ward+identity#DuetschBoas02}{Dütsch-Boas 02}).

\end{example}
\hypertarget{RenormalizationAndMasterWardIdentity}{}\subsubsection*{{Renormalization and Master ward identity}}\label{RenormalizationAndMasterWardIdentity}

The quantum master equation in the form of prop. \ref{QuantumMasterEquation} is derived on [[regular polynomial observables]], in particular hence for non-point-[[interaction]] [[action functionals]] $S_{int}$. But the [[interaction]] terms of interest are point-interactions, hence are [[local observables]]. The [[extension]] of the [[time-ordered product]] and hence of the [[perturbative S-matrix]] from regular to local onservables exsists but involves choices, these are the \emph{[[renormalization]]} choices in the formulation of [[causal perturbation theory]].

Since for [[gauge fixing|gauged fixed]] [[gauge theories]] this physically relevant [[observables]] are not the plain ([[microcausal polynomial observables|mcirocausal]]) [[polynomial observables]], but the [[cochain cohomology]] of the [[BV-BRST differential]] on them, one needs to require for gauge theories that the [[quantum master equation]] still holds after [[renormalization]]. This is closely related to the [[renormalization condition]] called the \emph{[[master Ward identity]]} (\hyperlink{Rejzner11}{Rejzner 11 (prop. 5.3.1) and following paragraphs}). If the quantum master equation cannot be retained in [[renormalization]] one says that the field theory suffers from a \emph{[[quantum anomaly]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[master Ward identity]]

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

The concept originates with

\begin{itemize}%
\item [[Igor Batalin]], [[Grigori Vilkovisky]], \emph{Gauge Algebra and Quantization}, Phys. Lett. B 102 (1): 27--31, 1981 ()

\end{itemize}
Traditional review includes

\begin{itemize}%
\item [[Marc Henneaux]], [[Claudio Teitelboim]], section 15.5.3 of \emph{[[Quantization of Gauge Systems]]}, Princeton University Press, 1992

\end{itemize}
Discussion in the rigorous context of [[relativistic field theory|relativistic]] [[perturbative QFT]] formulated in [[causal perturbation theory]]/[[perturbative AQFT]] is in:

\begin{itemize}%
\item [[Klaus Fredenhagen]], [[Kasia Rejzner]], \emph{Batalin-Vilkovisky formalism in the functional approach to classical field theory}, Commun. Math. Phys. 314(1), 93--127 (2012) (\href{https://arxiv.org/abs/1101.5112}{arXiv:1101.5112})


\item [[Klaus Fredenhagen]], [[Kasia Rejzner]], \emph{Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory}, Commun. Math. Phys. 317(3), 697--725 (2012) (\href{https://arxiv.org/abs/1110.5232}{arXiv:1110.5232})


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


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



\end{itemize}
and surveyed in

\begin{itemize}%
\item [[Kasia Rejzner]], section 7 of \emph{Perturbative algebraic quantum field theory} Springer 2016 (\href{https://link.springer.com/book/10.1007%2F978-3-319-25901-7}{web})

\end{itemize}
[[!redirects classical master equation]]



\end{document}