\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\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*{BV-operator}

\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{InPerturbativeQuantumFieldTheory}{In perturbative quantum field theory}\dotfill \pageref*{InPerturbativeQuantumFieldTheory} \linebreak
\noindent\hyperlink{ForFiniteDimensionalToyPathIntegrals}{For finite-dimensional toy path integrals}\dotfill \pageref*{ForFiniteDimensionalToyPathIntegrals} \linebreak
\noindent\hyperlink{InCausalPerturbationTheory}{In causal perturbation theory}\dotfill \pageref*{InCausalPerturbationTheory} \linebreak
\noindent\hyperlink{background}{Background}\dotfill \pageref*{background} \linebreak
\noindent\hyperlink{BVOperatorInCausalPerturbationTheory}{The BV-Operator}\dotfill \pageref*{BVOperatorInCausalPerturbationTheory} \linebreak
\noindent\hyperlink{InCausalPerturbationTheoryRelationToTimeOrderedAntibracket}{Relation to time-ordered antibracket}\dotfill \pageref*{InCausalPerturbationTheoryRelationToTimeOrderedAntibracket} \linebreak
\noindent\hyperlink{SchwingerDysonEquation}{Schwinger-Dyson equation}\dotfill \pageref*{SchwingerDysonEquation} \linebreak
\noindent\hyperlink{quantum_master_equation}{Quantum master equation}\dotfill \pageref*{quantum_master_equation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The concept of \emph{BV-operator} has two different but (somewhat subtly) related meanings:

\begin{enumerate}%
\item In [[perturbative quantum field theory]] (discussed \hyperlink{InPerturbativeQuantumFieldTheory}{below}) the \emph{BV-operator} or \emph{BV-Laplacian} (\hyperlink{BatalinVilkovisky81}{Batalin-Vilkovisky 81}) is essentially the difference between the action of the [[BV differential]] on the [[algebra of observables]] before and after [[quantization]] of the [[free field theory]] around which the [[perturbative quantum field theory|perturbative quantization]] is considered; it is a quantum correction $i \hbar \Delta_{BV}$ (of order of [[Planck's constant]] $\hbar$) to the [[BV differential]] $s_{BV} = \{-S',-.\}$. Where the condition $\left(\{-S',-\}\right)^2 = 0$ for the [[BV-differential]] to be a [[differential]] is called the ``[[master equation]]'' in [[BV-BRST theory]], the quantum corrected version $\left( \{-S',-\} + i \hbar \Delta\right)^2 = 0$ is called the \emph{[[quantum master equation]]}.


\item In [[higher algebra]], under the identification of a [[BV-algebra]] with the [[chain homology]] of a [[little k-cubes operad|E2-algebra]], the \emph{BV-operator} corresponds to the operation of rotating a little disk around.



\end{enumerate}
For the relation between the two see \emph{[[relation between BV and BD]]}.

\hypertarget{InPerturbativeQuantumFieldTheory}{}\subsection*{{In perturbative quantum field theory}}\label{InPerturbativeQuantumFieldTheory}

In [[perturbative quantum field theory]] the BV-operator $i \hbar \Delta$ may be understood intuitively as reflecting the contribution of the [[Gaussian measure]] in the [[path integral]] of the [[free field theory]] around which the perturbative quantization takes place. This intuition may be made precise for finite-dimensional toy path integrals. This we discuss in:

\begin{itemize}%
\item \emph{\hyperlink{ForFiniteDimensionalToyPathIntegrals}{In finite-dimensional toy path integrals}}

\end{itemize}
In the rigorous construction of [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] via [[causal perturbation theory]]/[[perturbative AQFT]] there is a rigorous incarnation of the BV-operator (\hyperlink{FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b, section 2}, \hyperlink{Rejzner11}{Rejzner 11, section 5.1.2}): The would-be path integral is reflected in the [[perturbative S-matrix]], hence in the [[time ordered products]], and the BV-operator on [[regular polynomial observables]] is the difference between the classical [[BV-differential]] and its [[conjugation]] into the [[time-ordered products]] (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator} below). This we discuss in

\begin{itemize}%
\item \emph{\hyperlink{InCausalPerturbationTheory}{In causal perturbation theory}}

\end{itemize}
\hypertarget{ForFiniteDimensionalToyPathIntegrals}{}\subsubsection*{{For finite-dimensional toy path integrals}}\label{ForFiniteDimensionalToyPathIntegrals}

If $X$ is a [[finite number|finite]] [[dimension|dimensional]] [[closed manifold|closed]] [[orientation|oriented]] [[smooth manifold]], then [[integration of differential forms]] of top degree over $X$ may be identified with sending such differential forms to their [[image]] in the [[de Rham cohomology]] of $X$.

This may be slightly reformulated: Fixing a [[volume form]] $\mu$ on $X$ it induces by contraction with [[vector fields]] degreewise a [[linear isomorphism]]

\begin{displaymath}
\mu \;\colon\;
  \Gamma_X(\wedge^n T^\ast X)
   \overset{\simeq}{\longrightarrow}
  \Gamma_X(\wedge^{dim(X)-n} T X)
\end{displaymath}
between the spaces of [[differential n-forms]] on $X$ and the space of [[multivector fields]] on $X$ of degree $n-dim(X)$. Under this isomorphism the [[de Rham differential]] induces a differential on [[multivector fields]]:

\begin{displaymath}
\Delta_{BV}
  \;\coloneqq\;
  \mu \circ d_{dR} \circ \mu^{-1}
\end{displaymath}
This $\Delta_{BV}$ is the \emph{BV-operator} in this simple situation.

The above statement about [[integration]] now translates into saying that for $f$ any [[smooth function]] on $X$, then its [[integration of differential forms]] $\int f \mu$ may be identified with the [[image]] of $f$ in the [[chain homology]] of the BV-operator $\Delta_{BV}$.

If one thinks of $X$ as a space of [[field configurations]] and of $f = \exp(i \hbar S)$ as an exponentiated [[action functional]], the one may think of this integral $\int \exp(i \hbar S) \mu$ as the finite-dimensional toy version of a [[path integral]].

While this is in general not defined in the actual non-finite dimensional situations in [[field theory]], the above re-formulation in terms of the [[chain homology]] of a BV-operator does make sense whenever an appropriate kind of differential is given. One may then try to axiomatize which chain complexes qualify as BV-complex and try to interpret their chain homology as computing perturbative [[path integrals]].

For more on this perspective see at \emph{[[BV-BRST formalism]]} the section \emph{\href{BV-BRST+formalism#HomologicalIntegration}{Quantum BV as homological (path-)integration}}

\hypertarget{InCausalPerturbationTheory}{}\subsubsection*{{In causal perturbation theory}}\label{InCausalPerturbationTheory}

For more context for the following see at \emph{[[A first idea of quantum field theory]]} the chapter \emph{\href{A+first+idea+of+quantum+field+theory#FreeQuantumFields}{Free quantum fields}}.

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

Recall the following context

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] admitting a [[gauge fixing]], and let $\mathbf{L}' - \mathbf{L}'_{BRST}$ be its BV-BRST [[Lagrangian density]] after [[gauge fixing]] (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}), so that the gauge-fixed [[local BRST complex|local BV-BRST differential]] is given by the [[local antibracket]] as

\begin{displaymath}
s' 
    \;=\;
  \left\{ -\mathbf{L}' + \mathbf{L}'_{BRST}, - \right\}
\end{displaymath}
The corresponding global BV-BRST differential on [[regular polynomial observables]] is (\href{A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal}{this def.})

\begin{equation}
\left\{ -S' + S'_{BRST} \;,\; -\right\}
  \;\coloneqq\;
  \left\{ -\tau_\Sigma\mathbf{L}'(x) + \tau_\Sigma\mathbf{L}'_{BRST}(x), - \right\} 
  \;\colon\;
  PolyObs(E)_{reg} \longrightarrow PolyObs(E)_{reg}
  \,.
\label{GaugeFixedGlobalBVBRSTDifferentialRecalled}\end{equation}
By definition of [[gauge fixing]] (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}), the [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] for $\mathbf{L}'$ are [[Green hyperbolic differential equations|Green hyperbolic]] and hence have a [[causal propagator]] $\Deta = \Delta_+ - \Delta_-$ and admit a compatible \emph{[[Wightman propagator]]} $\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H$ and the corresponding [[Feynman propagator]] $\Delta_F \coloneqq \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$. The [[star products]] with respect to these (\href{star+product#PropagatorStarProduct}{this def.}) on [[regular polynomial observables]]

\begin{displaymath}
\star_H, \star_F
  \;\colon\;
  PolyObs(E)_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{reg}[ [\hbar] ]
\end{displaymath}
are, respectively, the [[Wick algebra]] product ([[operator product]], see \href{Wick+algebra#NormalOrderedProductNotation}{this def.})

\begin{displaymath}
A_1 A_2
  \;\coloneqq\;
  A_1 \star_H A_2 
  \;\coloneqq\;
  ((-) \cdot (-))
  \circ
  \exp\left(
    \hbar 
    \underset{\Sigma \times \Sigma}{\int}
      \Delta_{H}(x,y)^{a b} 
      \frac{\delta}{\delta \mathbf{\Phi}^a(x)} 
        \otimes 
      \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
      \, dvol_\Sigma(x) \, dvol_\Sigma(y)
  \right)
  (A_1 \otimes A_2)
\end{displaymath}
and the [[time-ordered product]] (see again \href{Wick+algebra#NormalOrderedProductNotation}{this def.})

\begin{displaymath}
T(A_1 A_2)
  \;\coloneqq\;
  A_1 \star_F A_2 
  \;\coloneqq\;
  ((-) \cdot (-))
  \circ
  \exp\left(
    \hbar 
    \underset{\Sigma \times \Sigma}{\int}
      \Delta_{F}(x,y)^{a b} 
      \frac{\delta}{\delta \mathbf{\Phi}^a(x)} 
        \otimes 
      \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
      \, dvol_\Sigma(x)\, dvol_\Sigma(y)
  \right)
  (A_1 \otimes A_2)
  \,,
\end{displaymath}
Since the [[Feynman propagator]] is symmetric (\href{A+first+idea+of+quantum+field+theory#SymmetricFeynmanPropagator}{this prop.}), the latter [[time ordered product]] on [[regular polynomial observables]] is [[isomorphism|isomorphic]] (via \href{star+product#SymmetricContribution}{this prop.}) to the pointwise product, via

\begin{equation}
\mathcal{T}A
  \;\coloneqq\;
  \exp\left(
    \tfrac{1}{2}
    \hbar 
    \underset{\Sigma}{\int} 
     \Delta_F(x,y)^{a b} 
     \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)}
  \right)
  A
\label{RecallIsomorphsimTimeOrdering}\end{equation}
as

\begin{displaymath}
A_1 \star_{F} A_2
  \;=\;
  \mathcal{T}( \mathcal{T}^{-1}A_1 \cdot \mathcal{T}^{-1}A_2 )
\end{displaymath}
(\href{time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}{this prop.}).

\hypertarget{BVOperatorInCausalPerturbationTheory}{}\paragraph*{{The BV-Operator}}\label{BVOperatorInCausalPerturbationTheory}

\begin{defn}
\label{AntibracketTimeOrdered}\hypertarget{AntibracketTimeOrdered}{}
\textbf{([[time-ordered product|time-ordered]] [[antibracket]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (\href{A+first+idea+of+quantum+field+theory#FreeFieldTheory}{this def.}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.

Then the \emph{time-ordered global [[antibracket]]} on [[regular polynomial observables]]

\begin{displaymath}
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
   \otimes
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \overset{\{-,-\}_{\mathcal{T}}}{\longrightarrow}
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\end{displaymath}
is the [[conjugation]] of the global [[antibracket]] (\href{A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal}{this def.}) by the time-ordering operator $\mathcal{T}$ (from \href{time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}{this prop.}):

\begin{displaymath}
\{-,-\}_{\mathcal{T}}
  \;\coloneqq\;
  \mathcal{T}\left(\left\{ \mathcal{T}^{-1}(-), \mathcal{T}^{-1}(-)\right\}\right)
\end{displaymath}
hence

\begin{displaymath}
\itexarray{
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      &\overset{\{-,-\}}{\longrightarrow}&
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mathcal{T}}}_\simeq
    \\
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      &\overset{ \{-,-\}_{\mathcal{T}} }{\longrightarrow}&
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  }
\end{displaymath}
\end{defn}
(\hyperlink{FredenhagenRejzner11}{Fredenhagen-Rejzner 11, (27)}, \hyperlink{Rejzner11}{Rejzner 11, (5.14)})

\begin{prop}
\label{GaugeFixedActionFunctionalTimeOrderedAntibracket}\hypertarget{GaugeFixedActionFunctionalTimeOrderedAntibracket}{}
\textbf{([[time-ordered product|time-ordered]] [[antibracket]] with [[gauge fixing|gauge fixed]] [[action functional]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (\href{A+first+idea+of+quantum+field+theory#FreeFieldTheory}{this def.}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.

Then the [[time-ordered product|time-ordered]] [[antibracket]] (def. \ref{AntibracketTimeOrdered}) with the gauge fixed BV-[[action functional]] $-S'$ (def. \ref{ComplexBVBRSTGlobal}) equals the [[conjugation]] of the global [[BV-differential]] with the [[isomorphism]] $\mathcal{T}$ from the pointwise to the [[time-ordered product]] of [[observables]] (from \href{time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}{this prop.})

\begin{displaymath}
\{-S',-\}_{\mathcal{T}}
  \;=\;
  \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
  \,,
\end{displaymath}
hence

\begin{displaymath}
\itexarray{
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
     &\overset{ \{-S',-\} }{\longrightarrow}&
    PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T}}}\downarrow 
      &&
    \downarrow^{\mathrlap{\mathcal{T}}}
    \\
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
     &\overset{ \{-S',-\}_{\mathcal{T}} }{\longrightarrow}&
    PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  }
\end{displaymath}
\end{prop}
\begin{proof}
By the assumption that $(E,\mathbf{L})$ is a [[free field theory]] its [[Euler-Lagrange equations]] are linear in the fields, and hence $S'$ is quadratic in the fields. This means that

\begin{displaymath}
\mathcal{T}^{-1}S' = S' + const
  \,,
\end{displaymath}
where the second term on the right is independent of the fields, and hence that

\begin{displaymath}
\{\mathcal{T}^{-1}(-S'),-\}
  = 
  \{-S', - \}
  \,.
\end{displaymath}
This implies the claim:

\begin{displaymath}
\begin{aligned}
    \{-S',-\}_{\mathcal{T}}
    & \coloneqq
    \mathcal{T}\left(\{ \mathcal{T}^{-1}(-S'), \mathcal{T}^{-1}(-) \}\right)
    \\
   & =
   \mathcal{T}\left(\{ -S', \mathcal{T}^{-1}(-) \}\right)
   \\
   & =
   \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
   \,.
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{defn}
\label{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}\hypertarget{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}{}
\textbf{([[BV-operator]] for [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theory]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (\href{A+first+idea+of+quantum+field+theory#FreeFieldTheory}{this def.}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ and with corresponding gauge-fixed global [[BV-BRST differential]] on graded [[regular polynomial observables]]

\begin{displaymath}
\{-S' + S'_{BRST}, -\}
  \;\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#GaugeFixingLagrangianDensity}{this def.}).

Then the corresponding \emph{[[BV-operator]]}

\begin{displaymath}
\Delta_{BV}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\end{displaymath}
on [[regular polynomial observables]] is, up to a factor of $i \hbar$, the difference between the free component $\{-S',-\}$ of the gauge fixed global BV differential and its time-ordered version (def. \ref{AntibracketTimeOrdered})

\begin{displaymath}
\Delta_{BV}
    \;\coloneqq\;
  \tfrac{1}{i \hbar}
  \left(
    \left\{
      -S',-
    \right\}_{\mathcal{T}}
    -
    \left\{ -S',(-) \right\}
  \right)
  \,,
\end{displaymath}
hence

\begin{equation}
\{-S',-\}_{\mathcal{T}}
  \;=\;
  \{-S',-\} + i \hbar \Delta_{BV}
  \,.
\label{BVOperatorDefiningRelation}\end{equation}
\end{defn}
\begin{prop}
\label{ComponentsBVOperator}\hypertarget{ComponentsBVOperator}{}
\textbf{([[BV-operator]] in components)}

If the [[field bundles]] of all [[field (physics)|fields]], [[ghost fields]] and [[auxiliary fields]] are [[trivial vector bundles]], with field/ghost-field/auxiliary-field coordinates collectively denoted $(\phi^A)$ then the [[BV-operator]] $\Delta_{BV}$ from prop. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator} is given explicitly by

\begin{displaymath}
\Delta_{BV}
  \;=\;
  \underset{a}{\sum} (-1)^{deg(\Phi^A)}
  \underset{\Sigma}{\int} 
    \frac{\delta}{\delta \Phi^A(x)} 
    \frac{\delta}{\delta \Phi^{\ddagger}_A(y)}
  dvol_\Sigma
\end{displaymath}
Since this formula exhibits a graded [[Laplace operator]], the BV-operator is also called the \emph{BV-Laplace operator} or \emph{BV-Laplacian}, for short.

\end{prop}
(\hyperlink{FredenhagenRejzner11}{Fredenhagen-Rejzner 11, (29)}, \hyperlink{Rejzner11}{Rejzner 11, (5.20)})

\begin{proof}
By prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket} we have equivalently

\begin{displaymath}
i \hbar \Delta_{BV}
  \;=\;
  \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
  \,-\,
  \{-S',-\}
\end{displaymath}
and by \href{A+first+idea+of+quantum+field+theory#BVDifferentialGlobal}{this example} the second term on the right is

\begin{displaymath}
\begin{aligned}
    \left\{ -S', -\right\}
    & =
    \underset{\Sigma}{\int}
    j^{\infty}\left(\mathbf{\Phi}\right)^\ast
    \left(
      \frac{\overset{\leftarrow}{\delta}_{EL} L}{\delta \phi^A}
    \right)(x) 
    \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} 
    \,
    dvol_\Sigma(x)
    \\
    & =
    \underset{a}{\sum}
      (-1)^{deg(\phi^A)}
    \underset{}{\int}
      (P\mathbf{\Phi})_A(x)
      \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
    \,
    dvol_\Sigma(x)
  \end{aligned}
\end{displaymath}
With this we compute as follows:

\begin{equation}
\begin{aligned}
    \{-S',-\}_{\mathcal{T}}
    & =
    \mathcal{T} \circ \left\{ -S,-\right\} \circ \mathcal{T}^{-1}
    \\
    & = 
    \exp\left( 
     \left[
       \hbar \tfrac{1}{2} \Delta_F
       \left( 
         \frac{\delta}{\delta \mathbf{\Phi}}, 
         \frac{\delta}{\delta \mathbf{\Phi}}      
        \right)
        \,,\,
        -
      \right]
    \right)
    \left(
      \{-S',-\}
    \right)
    \\
    & = 
    \{-S',-\}
    +
    \left[
      \hbar \tfrac{1}{2}
      \Delta_F
      \left( 
        \frac{\delta}{\delta \mathbf{\Phi}}, 
        \frac{\delta}{\delta \mathbf{\Phi}}      
      \right)
      \,,
      \{-S',-\}
    \right]
    + \underset{ = 0 }{\underbrace{\hbar^2(...)}}
    \\
    & =
    \phantom{+}
    \left\{ -S' , -\right\}
    \\
    &
    \phantom{=}
    +
    \left[
      \tfrac{1}{2}\hbar 
      \underset{\Sigma \times \Sigma}{\int} 
      \Delta_F^{A B}(x,y)
      \frac{\delta^2}{\delta \mathbf{\Phi}^A(x) \delta \mathbf{\Phi}^B(y)}
      \, dvol_\Sigma(x)
      \, dvol_\Sigma(y)
      \;,\;
      \underset{a}{\sum}
      (-1)^{deg(\phi^A)}
      \underset{\Sigma}{\int}
        (P\mathbf{\Phi})_A(x)
        \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
      \,
      dvol_\Sigma(x)
    \right]
   \\
   & =
   \left\{ -S', -\right\}
   \\
   & \phantom{=}
   +
   \underset{A}{\sum} 
    (-1)^{deg(\phi^A)}
   \underset{\Sigma \times \Sigma}{\int} 
     \underset{ =  i \delta(x-y) }{\underbrace{P_x \Delta_F(x,y)}}
     \frac{\delta}{\delta \mathbf{\Phi}^A(x)}
     \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(y)}
    \, dvol_\Sigma(x)
    \, dvol_\Sigma(y)
    \\
   & = \left\{ -S', -\right\} 
   + i \hbar 
   \underset{A}{\sum}
   (-1)^{deg(\phi^A)}
   \underset{\Sigma}{\int}
    \frac{\delta}{\delta \mathbf{\Phi}^A(x)}
    \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
    \,
    dvol_\Sigma(x)
  \end{aligned}
\label{AAA}\end{equation}
Here we used

\begin{enumerate}%
\item under the first brace that by assumption of a [[free field theory]], $\{-S',-\}$ is linear in the fields, so that the first [[commutator]] with the [[Feynman propagator]] is independent of the fields, and hence all the higher commutators vanish;


\item under the second brace that the [[Feynman propagator]] is $+i$ times the [[Green function]] for the [[Green hyperbolic differential equation|Green hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] (\href{A+first+idea+of+quantum+field+theory#GreenFunctionFeynmanPropagator}{this cor.}).



\end{enumerate}
\end{proof}
\hypertarget{InCausalPerturbationTheoryRelationToTimeOrderedAntibracket}{}\paragraph*{{Relation to time-ordered antibracket}}\label{InCausalPerturbationTheoryRelationToTimeOrderedAntibracket}

\begin{prop}
\label{AntibracketBVOperatorRelation}\hypertarget{AntibracketBVOperatorRelation}{}
\textbf{(global [[antibracket]] exhibits failure of [[BV-operator]] to be a [[derivation]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (\href{A+first+idea+of+quantum+field+theory#FreeFieldTheory}{this def.}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$

The [[BV-operator]] $\Delta_{BV}$ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) and the global [[antibracket]] $\{-,-\}$ (\href{A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal}{this def.}) satisfy for all [[polynomial observables]] (\href{A+first+idea+of+quantum+field+theory#PolynomialObservables}{this def.}) $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})[ [\hbar] ]$ the relation

\begin{equation}
\{A_1, A_2\}
  \;=\;
  (-1)^{deg(A_2)}
  \,
  \Delta_{BV}(A_1 \cdot A_2)
  - 
  (-1)^{deg(A_2)}
  \,
  \Delta_{BV}(A_1) \cdot A_2
  -
  A_1 \cdot \Delta_{BV}(A_2)
\label{GlobalAntibracketInteractingWithBVOperator}\end{equation}
for $(-) \cdot (-)$ the pointwise product of observables (def. \ref{Observable}).

Moreover, it commutes on [[regular polynomial observables]] with the [[time-ordered product|time-ordering operator]] $\mathcal{T}$ (\href{time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}{this prop.})

\begin{displaymath}
\Delta_{BV} \circ \mathcal{T}
  =
  \mathcal{T} \circ \Delta_{BV}
  \phantom{AAA}
  \text{on}
  \,\,
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\end{displaymath}
and hence satisfies the analogue of relation \eqref{GlobalAntibracketInteractingWithBVOperator} also for the time-ordered antibracket $\{-,-\}_{\mathcal{T}}$ (def. \ref{AntibracketTimeOrdered}) and the [[time-ordered product]] $\star_F$ on regular polynomial observables

\begin{displaymath}
\{A_1, A_2\}_{\mathcal{T}}
  \;=\;
  (-1)^{deg(A_2)}
  \,
  \Delta_{BV}(A_1 \star_F A_2)
  - 
  (-1)^{deg(A_2)}
  \Delta_{BV}(A_1) \star_F A_2
  -
  A_1 \star_F \Delta_{BV}(A_2)  
  \,.
\end{displaymath}
\end{prop}
(e.g. \href{antibracket#HenneauxTeitelboim92}{Henneaux-Teitelboim 92, (15.105d)})

\begin{proof}
With prop. \ref{ComponentsBVOperator} the first statement is a graded version of the analogous relation for an ordinary [[Laplace operator]] $\Delta \coloneqq g^{a b} \partial_a \partial_b$ acting on [[smooth functions]] on [[Cartesian space]], which on [[smooth functions]] $f,g$ satisfies

\begin{displaymath}
\Delta(f \cdot g)
  \;=\;
  (\nabla f, \nabla g) - \Delta(f) g - f \Delta(g)
  \,,
\end{displaymath}
by the [[product law]] for [[differentiation]], where now $\nabla f \coloneqq (g^{a b} \partial_b f)$ is the [[gradient]] and $(v,w) \coloneqq g_{a b} v^a w b$ the [[inner product]]. Here one just needs to carefully record the relative signs that appear.

That the BV-operator commutes with the time-ordering operator is clear from the fact that both of these are given by [[partial derivative|partial]] [[functional derivatives]] with \emph{[[constant function|constant]]} [[coefficients]]. This immediately implies the last statement from the first.

\end{proof}
\begin{example}
\label{TimeOrderedExponentialBVOperator}\hypertarget{TimeOrderedExponentialBVOperator}{}
\textbf{([[BV-operator]] on [[time-ordered product|time-ordered]] [[exponentials]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (\href{A+first+idea+of+quantum+field+theory#FreeFieldTheory}{this def.}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.

Let moreover $V \in PolyObs(E_{\text{BV-BRST}})_{reg, deg = 0}[ [\hbar] ]$ be a [[regular polynomial observable]] (def. \ref{PolynomialObservables}) of degree zero. Then the application of the [[BV-operator]] $\Delta_{BV}$ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) to the [[time-ordered product|time-ordered]] [[exponential]] $\exp_{\mathcal{T}}(V)$ (\href{time-ordered+product#RegularObservablesExponentialTimeOrdered}{this example}) is the [[time-ordered product]] of the time-ordered exponential with the sum of $\Delta_{BV}(V)$ and the global [[time-ordered product|time-ordered]] [[antibracket]] $\tfrac{1}{2}\{V,V\}_{\mathcal{T}}$ (def. \ref{AntibracketTimeOrdered}) of $V$ with itself:

\begin{displaymath}
\Delta_{BV}
  \left(
    \exp_{\mathcal{T}}(V)
  \right)
  \;=\;
  \left(
    \Delta_{BV}(V)
    +
    \tfrac{1}{2}\{V,V\}_{\mathcal{T}}
  \right)
   \star_F
  \exp_{\mathcal{T}}(V)
\end{displaymath}
\end{example}
\begin{proof}
By prop. \ref{AntibracketBVOperatorRelation} $\Delta_{BV}$ acts as a [[derivation]] on the [[time-ordered product]] up to a correction given by the antibracket of the two factors. This yields the result by the usual combinatorics of [[exponentials]].

\begin{displaymath}
\begin{aligned}
    \Delta_{BV}
    \left(
      1 + V + \tfrac{1}{2}V \star_F V + \cdots
    \right)
    & =
    \Delta_{BV}(V) 
      + 
    \tfrac{1}{2}\left( \Delta_{BV}(V) \star_F V + V \star_F \Delta_{BV}(V) \right)
      +
    \tfrac{1}{2}\{V,V\}_{\mathcal{T}}
      +
    \cdots
    \\
    & = 
    \Delta_{BV}(V) +  \Delta_{BV}(V) \star_F V + \tfrac{1}{2}\{V,V\}_{\mathcal{T}}
    + \cdots
  \end{aligned}
\end{displaymath}
\end{proof}
\hypertarget{SchwingerDysonEquation}{}\paragraph*{{Schwinger-Dyson equation}}\label{SchwingerDysonEquation}

A special case of the general occurence of the [[BV-operator]] is the following important property of [[on-shell]] [[time-ordered products]]:

\begin{prop}
\label{DysonSchwinger}\hypertarget{DysonSchwinger}{}
\textbf{([[Schwinger-Dyson equation]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (\href{A+first+idea+of+quantum+field+theory#FreeFieldTheory}{this def.}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (\href{A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity}{this def.}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.

Let

\begin{equation}
A
  \;\coloneqq\;
  \underset{\Sigma}{\int}
    A^a(x) \cdot \mathbf{\Phi}^\ddagger_a(x)
  \, dvol_\Sigma(x)
  \;\in\;
  PolyObs_{reg}(E_{\text{BV-BRST}})
\label{SchwingerDysonTestObservable}\end{equation}
be an [[off-shell]] [[regular polynomial observable]] which is [[linear map|linear]] in the [[antifield]] [[field observables]] $\mathbf{\Phi}^\ddagger$. Then

\begin{equation}
\mathcal{T}^{\pm 1}
  \left(
    \underset{\Sigma}{\int}
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
      \cdot
      A^a(x)
      \, dvol_\Sigma(x)
  \right)
  \;=\;
  \pm i \hbar
  \,
  \mathcal{T}^{\pm}
  \left(
    \underset{\Sigma}{\int}
    \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)}
    \,
    dvol_\Sigma(x)     
  \right)  
  \phantom{A}
  \in
  \underset{
    \text{on-shell}
  }{
  \underbrace{
    PolyObs_{reg}(E_{\text{BV-BRST}}, \mathbf{L'})
  }}
  \,.
\label{EquationSchwingerDyson}\end{equation}
This is called the \emph{[[Schwinger-Dyson equation]]}.

\end{prop}
The proof below is due to (\hyperlink{Rejzner16}{Rejzner 16, remark 7.7}), following the informal traditional argument (\hyperlink{HenneauxTeitelboim92}{Henneaux-Teitelboim 92, (15.108b)}).

\begin{proof}
Applying the inverse time-ordering map $\mathcal{T}^{-1}$ (\href{time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}{this prop.}) to equation \eqref{BVOperatorDefiningRelation} applied to $A$ yields

\begin{equation}
\underset{
    \mathcal{T}^{-1}
    \underset{\Sigma}{\int}
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
      \cdot
      A^a(x)
    dvol_\Sigma(x)
  }{
  \underbrace{
    \mathcal{T}^{-1}\left\{ -S', A\right\}
  }
  }
  \;=\;
  -
  \underset{
    i \hbar
    \mathcal{T}^{-1}
    \underset{\Sigma}{\int} 
      \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)} 
    \, dvol_\Sigma
  }{
  \underbrace{
    i \hbar \mathcal{T}^{-1}\Delta_{BV}(A)
  }
  }
  +
  \underset{
    \{-S',\mathcal{T}^{-1}(A)\}
  }{
  \underbrace{
    \mathcal{T}^{-1}\left\{ -S',A\right\}_{\mathcal{T}}
  }
  }
\label{BVDerivationOfSchwingerDyson}\end{equation}
where we have identified the terms under the braces by 1) the component expression for the BV-differential $\{-S',-\}$ from \href{A+first+idea+of+quantum+field+theory#BVDifferentialGlobal}{this prop}, 2) prop. \ref{ComponentsBVOperator} and 3) prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket}.

The last term is manifestly in the [[image]] of the [[BV-differential]] $\{-S',-\}$ 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{equation}
\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',-\})
\label{OnShellObservablesAsQuotientOfOffShellObservablesRecalled}\end{equation}
(by \href{A+first+idea+of+quantum+field+theory#BVDifferentialGlobal}{this example.}).

The same argument with the replacement $\mathcal{T} \leftrightarrow \mathcal{T}^{-1}$ throughout yields the other version of the equation (with time-ordering instead of reverse time ordering and the sign of the $\hbar$-term reversed).

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{(``Schwinger-Dyson operator'')}

The proof of the [[Schwinger-Dyson equation]] in prop. \ref{DysonSchwinger} shows that, up to [[time-ordered product|time-ordering]], the [[Schwinger-Dyson equation]] is the on-shell vanishing of the ``quantized'' [[BV-differential]] \eqref{BVOperatorDefiningRelation}

\begin{displaymath}
\{-S',-\}_{\mathcal{T}}
  \;=\;
  \{-S', -\}
  \,+\,
  i \hbar \, \Delta_{BV}
  \,,
\end{displaymath}
where the [[BV-operator]] is the quantum correction of order $\hbar$. Therefore this is also called the \emph{Schwinger-Dyson operator} (\hyperlink{HenneauxTeitelboim92}{Henneaux-Teitelboim 92, (15.111)}).

\end{remark}
\begin{remark}
\label{SchwingerDysonDistributional}\hypertarget{SchwingerDysonDistributional}{}
\textbf{([[distribution|distributional]] [[Schwinger-Dyson equation]])}

Often the [[Schwinger-Dyson equation]] (prop. \ref{DysonSchwinger}) is displayed before spacetime-smearing of [[field observables]] in terms of [[operator products]] of [[operator-valued distributions]], taking the observable $A$ in \eqref{SchwingerDysonTestObservable} to be

\begin{displaymath}
A^a(x) 
    \;\coloneqq\;
  \delta(x-x_0) \delta^a_{a_0} 
  \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \,.
\end{displaymath}
This choice makes \eqref{EquationSchwingerDyson} become the [[distribution|distributional]] [[Schwinger-Dyson equation]]

\begin{displaymath}
\begin{aligned}
  & T
  \left(
    \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
      \cdot 
    \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \right)
  \\
  & 
  \underset{\text{on-shell}}{=}
  - i \hbar
  \underset{k}{\sum}
  T
  \left(
    \mathbf{\Phi}^{a_1}(x_1)
      \cdots
    \mathbf{\Phi}^{a_{k-1}}(x_{k-1})
      \cdot
    \delta(x_0 - x_k) \delta^{a_0}_{a_k}
      \cdot
    \mathbf{\Phi}^{a_{k+1}}(x_{k+1})
      \cdots
    \mathbf{\Phi}^{a_n}(x_m)
  \right)
  \end{aligned}
\end{displaymath}
(e.q. \href{Schwinger-Dyson+equation#Dermisek09}{Dermisek 09}).

In particular this means that if $(x_0,a_0) \neq (x_k, a_k)$ for all $k \in \{1,\cdots ,n\}$ then

\begin{displaymath}
T
  \left(
    \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
      \cdot 
    \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
\end{displaymath}
Since by the [[principle of extremal action]] (\href{A+first+idea+of+quantum+field+theory#PrincipleOfExtremalAction}{this prop.}) the equation

\begin{displaymath}
\frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
  \;=\;
  0
\end{displaymath}
is the [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] (for the [[classical field theory]]) ``at $x_0$'', this may be interpreted as saying that the classical equations of motion for fields at $x_0$ still hold for [[time-ordered product|time-ordered]] [[quantum theory|quantum]] [[expectation values]], as long as all other observables are evaluated away from $x_0$; while if observables do coincide at $x_0$ then there is a correction measured by the [[BV-operator]].

\end{remark}
\begin{remark}
\label{FreeFieldQuantumShell}\hypertarget{FreeFieldQuantumShell}{}
\textbf{(the ``quantum shell'')}

Beware that, superficially, it might seem that in equation \eqref{BVDerivationOfSchwingerDyson} not only the term $\{-S',\mathcal{T}^{-1}(A)\}$ on the right vanishes [[on-shell]], but also the term $\mathcal{T}^{-1}\left\{ -S', A\right\}$ on the left, since the latter is the image under the linear map $\mathcal{T}^{-1}$ of an observable that vanishes on-shell.

To sort this out, notice that the isomorphism \eqref{OnShellObservablesAsQuotientOfOffShellObservablesRecalled} tells us that the observables that vanish when passing from off-shell to on-shell observables are precisely those in the ideal generated by the image of $\{-S',(-)\}$. But while $\mathcal{T}^{-1}$ is an isomorphism on (regular off-shell observables), it need not (and in general does not) preserve this ideal! Hence $\mathcal{T}^{-1}(\{-S',A\})$ need not (and in general is not) an element of that ideal, and this is why it remains when passing to the algebra of on-shell observables and thus makes its crucial appearance in the [[Schwinger-Dyson equation]].

\end{remark}
\hypertarget{quantum_master_equation}{}\paragraph*{{Quantum master equation}}\label{quantum_master_equation}

Passing from [[free field theory]] to [[perturbative quantum field theory|perturbative]] [[quantization]] of [[interacting field theory]], the above BV-operator of the underying free field theory appears as a quantum correction to the ``[[classical master equation]]'' which expresses the nilpotency of the [[BV-differential]]. This ``master equation'' corrected by the BV-operator is called the \emph{[[quantum master equation]]}. See \href{quantum+master+equation#QuantumMasterEquation}{this prop.}.

\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}
The understanding of the BV-operator in the rigorous formulation of [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] via [[causal perturbation theory]]/[[perturbative AQFT]] is due to

\begin{itemize}%
\item [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], section 2 of \emph{Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory}, Commun. Math. Phys. (2013) 317: 697 (\href{https://arxiv.org/abs/1110.5232}{arXiv:1110.5232}, \href{https://doi.org/10.1007/s00220-012-1601-1}{doi:10.1007/s00220-012-1601-1})


\item [[Katarzyna Rejzner]], section 5.1.2 of \emph{Batalin-Vilkovisky formalism in locally covariant field theory} (\href{https://arxiv.org/abs/1111.5130}{arXiv:1111.5130})



\end{itemize}
surveyed in

\begin{itemize}%
\item [[Katarzyna Rejzner]], section 7.3 of \emph{[[Perturbative Algebraic Quantum Field Theory]]}, Mathematical Physics Studies, Springer 2016 (\href{https://link.springer.com/book/10.1007%2F978-3-319-25901-7}{web})

\end{itemize}
See at \emph{[[BV-formalism]]} for more references.

[[!redirects BV-operators]]

[[!redirects BV-Laplacian]] [[!redirects BV-Laplacians]]

[[!redirects BV-Laplace operator]] [[!redirects BV-Laplace operators]]

[[!redirects Schwinger-Dyson operator]] [[!redirects Schwinger-Dyson operators]]



\end{document}