\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*{A first idea of quantum field theory -- Free quantum fields} \hypertarget{FreeQuantumFields}{}\subsection*{{Free quantum fields}}\label{FreeQuantumFields} In this chapter we discuss the following topics: \begin{itemize}% \item \emph{\hyperlink{WickAlgebraAbstract}{Wick algebra}} \item \emph{\hyperlink{AbstractTimeOrderedProduct}{Time-ordered product}} \item \emph{\hyperlink{OperatorProductAndNormalOrderedProduct}{Operator product notation}} \item \emph{\hyperlink{HadamardVacuumStatesOnWickAlgebras}{Hadamard vacuum state}} \item \emph{\hyperlink{FreeQuantumBVDifferential}{Free quantum BV-differential}} \item \emph{\hyperlink{SchwingerDysonEquation}{Schwinger-Dyson equation}} \end{itemize} $\,$ In the \hyperlink{Quantization}{previous chapter} we discussed \emph{[[quantization]]} of linear [[phase spaces]], which turns the [[algebra of observables]] into a [[noncommutative algebra]] of [[quantum observables]]. Here we apply this to the [[covariant phase spaces]] of [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theories]] (as discussed in the chapter \emph{\hyperlink{GaugeFixing}{Gauge fixing}}), obtaining genuine [[quantum field theory]] for [[free fields]]. For this purpose we first need to find a sub-algebra of all observables which is large enough to contain all [[local observables]] (such as the [[phi{\tt \symbol{94}}n interaction]], example \ref{InWickAlgebraphinInteraction} below, and the [[electron-photon interaction]], example \ref{InWickAlgebraElectronPhotonInteraction} below) but small enough for the [[star product]] [[deformation quantization]] to meet [[Hörmander's criterion]] for absence of [[UV-divergences]] (remark \ref{UltravioletDivergencesFromPaleyWiener}). This does exist (example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal} below): It is called the algebra of \emph{[[microcausal polynomial observables]]} (def. \ref{MicrocausalObservable} below). [[!include perturbative observables -- table]] While the [[star product]] of the [[causal propagator]] still violates [[Hörmander's criterion]] for absence of [[UV-divergences]] on [[microcausal polynomial observables]], we have seen in the \hyperlink{Quantization}{previous chapter} that qantization freedom allows to shift this [[Poisson tensor]] by a symmetric contribution. By prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime} such a shift is provided by passage from the [[causal propagator]] to the [[Wightman propagator]], and by prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski} this reduces the [[wave front set]] and hence the UV-singularities ``by half''. This way the [[deformation quantization]] of the [[Peierls-Poisson bracket]] exists on [[microcausal polynomial observables]] as the [[star product]] algebra induced by the [[Wightman propagator]]. The resulting [[non-commutative algebra|non-commutative]] [[algebra of observables]] is called the \emph{[[Wick algebra]]} (prop. \ref{MoyalStarProductOnMicrocausal} below). Its algebra structure may be expressed in terms of a commutative ``[[normal-ordered product]]'' (def. \ref{NormalOrderedProductNotation} below) and the [[vacuum expectation values]] of [[field observables]] in a canonically induced [[vacuum state]] (prop. \ref{WickAlgebraCanonicalState} below). The analogous [[star product]] induced by the [[Feynman propagator]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} below) acts by first [[causal ordering]] its arguments and then multiplying them with the [[Wick algebra|Wick algebra product]] (prop. \ref{CausalOrderingTimeOrderedProductOnRegular} below) and hence is called the \emph{[[time-ordered product]]} (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} below). This is the key structure in the discussion of [[interacting field theory]] discussed in the next chapter \emph{\hyperlink{InteractingQuantumFields}{Interacting quantum fields}}. Here we consider this on [[regular polynomial observables]] only, hence for averages of [[field observables]] that evaluate at distinct [[spacetime]] points. The [[extension]] of the [[time-ordered product]] to [[local observables]] is possible, but requires making choices: This is called \emph{[[renormalization]]}, which we turn to in the chapter \emph{\hyperlink{Renormalization}{Renormalization}} below. [[!include Wick algebra -- table]] While the [[Wick algebra]] with its [[vacuum state]] provides a [[quantization]] of the [[algebra of observables]] of [[free field theory|free]] [[gauge fixing|gauge fixed]] [[Lagrangian field theories]], the possible existence of [[infinitesimal gauge symmetries]] implies that the physically relevant observables are just the [[gauge invariance|gauge invariant]] [[on-shell]] ones, exhibited by the [[cochain cohomology]] of the [[BV-BRST differential]] $\{-S' + S'_{BRST}, (-)\}$. Hence to complete [[quantization]] of [[gauge theories]], the [[BV-BRST differential]] needs to be lifted to the [[noncommutative algebra]] of [[quantum observables]] -- this is called \emph{[[BV-BRST quantization]]}. To do so, we may regard the [[gauge fixing|gauge fixed]] [[BRST complex|BRST]]-[[action functional]] $S'_{BRST}$ as an [[interaction]] term, to be dealt with later via [[scattering theory]], and hence consider quantization of just the free BV-differential $\{-S',(-)\}$. One finds that this is equal to its [[time-ordered product|time-ordered]] version $\{-S',(-)\}_{\mathcal{T}}$ (prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket} below) plus a quantum correction, called the \emph{[[BV-operator]]} (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator} below) or \emph{[[BV-Laplacian]]} (prop. \ref{ComponentsBVOperator} below). Applied to [[observables]] this relation is the \emph{[[Schwinger-Dyson equation]]} (prop. \ref{DysonSchwinger} below), which expresses the [[quantum physics|quantum]]-correction to the [[equations of motion]] of the [[free field theory|free]] [[gauge fixing|gauge field]] [[Lagrangian field theory]] as seen by [[time-ordered products]] of [[observables]] (example \ref{SchwingerDysonDistributional} below.) After introducing [[field (physics)|field]]-[[interactions]] via [[scattering theory]] in the \hyperlink{InteractingQuantumFields}{next chapter} the quantum correction to the [[BV-differential]] by the [[BV-operator]] becomes the ``[[quantum master equation]]'' and the [[Schwinger-Dyson equation]] becomes the ``[[master Ward identity]]''. When choosing [[renormalization]] these identities become \emph{conditions} to be satisfied by [[renormalization]] choices in order for the interacting quantum BV-BRST differential, and hence for [[gauge invariance|gauge invariant]] quantum observables, to be well defined in [[perturbative quantum field theory]] of [[gauge theories]]. This we discuss below in \emph{\hyperlink{Renormalization}{Renormalization}}. $\,$ \textbf{[[Wick algebra]]} The abstract [[Wick algebra]] of a [[free field theory]] with [[Green hyperbolic differential equation]] is directly analogous to the [[star product]]-algebra induced by a [[finite dimensional vector space|finite dimensional]] [[Kähler vector space]] (def. \ref{WickAlgebraOfAlmostKaehlerVectorSpace}) under the following identification of the [[Wightman propagator]] with the [[Kähler space]]-[[structure]]: \begin{remark} \label{WightmanPropagatorAsKaehlerVectorSpaceStructure}\hypertarget{WightmanPropagatorAsKaehlerVectorSpaceStructure}{} \textbf{([[Wightman propagator]] as [[Kähler vector space]]-[[structure]])} Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] is a [[Green hyperbolic differential equation]]. Then the corresponding [[Wightman propagator]] is analogous to the rank-2 tensor on a [[Kähler vector space]] as follows: \newline | [[space of field histories]] $\Gamma_\Sigma(E)$ | $\mathbb{R}^{2n}$ | | [[symplectic form]] $\tau_{\Sigma_p} \Omega_{BFV}$ | [[Kähler form]] $\omega$ | | [[causal propagator]] $\Delta$ | $\omega^{-1}$ | | [[Peierls-Poisson bracket]] $\{A_1,A_2\} = \int \Delta^{a_1 a_2}(x_1,x_2) \frac{\delta A_1}{\delta \mathbf{\Phi}^{a_1}(x_1)} \frac{\delta A_2}{\delta \mathbf{\Phi}^{a_2}(x_2)} dvol_\Sigma(x)$ | [[Poisson bracket]] | | [[Wightman propagator]] $\Delta_H = \tfrac{i}{2} \Delta + H$ | [[Hermitian form]] $\pi = \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}$ | \end{remark} (\href{pAQFT#FredenhagenRejzner15}{Fredenhagen-Rejzner 15, section 3.6}, \href{pAQFT#Collini16}{Collini 16, table 2.1}) \begin{defn} \label{MicrocausalObservable}\hypertarget{MicrocausalObservable}{} \textbf{([[microcausal observable|microcausal]] [[polynomial observables]])} Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] which is a [[vector bundle]], over some [[spacetime]] $\Sigma$. A [[polynomial observable]] (def. \ref{PolynomialObservables}) \begin{displaymath} \begin{aligned} A & = \phantom{+} \alpha^{(0)} \\ & \phantom{=} + \int_{\Sigma} \mathbf{\Phi}^a(x) \alpha^{(1)}_a(x) \, dvol_\Sigma(x) \\ & \phantom{=} + \int_{\Sigma^2} \mathbf{\Phi}^{a_1}(x_1) \cdot \mathbf{\Phi}^{a_2}(x_2) \alpha^{(2)}_{a_1 a_2}(x_1, x_2) \, dvol_\Sigma(x_1) dvol_\Sigma(x_2) \\ & \phantom{=} + \int_{\Sigma^3} \mathbf{\Phi}^{a_1}(x_1) \cdot \mathbf{\Phi}^{a_2}(x_2) \cdot \mathbf{\Phi}^{a_3}(x_3) \alpha^{(3)}_{a_1 a_2 a_3}(x_1,x_2,x_3) \, dvol_\Sigma(x_1) dvol_\Sigma(x_2) dvol_\Sigma(x^3) \\ & \phantom{=} + \cdots \,. \end{aligned} \end{displaymath} is called \emph{[[microcausal polynomial observable|microcausal]]} if each [[distribution|distributional]] [[coefficient]] \begin{displaymath} \alpha^{(k)} \;\in\; \Gamma'_{\Sigma^k}(E^{\boxtimes^k}) \end{displaymath} as above has [[wave front set]] (def. \ref{WaveFrontSet}) \emph{not} containing those elements $(x_1, \cdots x_k, k_1, \cdots k_k)$ where the $k$ [[wave vectors]] are all in the [[closed future cone]] or all in the [[closed past cone]] (def. \ref{CausalPastAndFuture}). We write \begin{displaymath} \itexarray{ PolyObs(E)_{mc} &\hookrightarrow& PolyObs(E) \\ PolyObs(E,\mathbf{L})_{mc} \simeq PolyObs(E)_{mc}/im(P) &\hookrightarrow& PolyObs(E,\mathbf{L}) } \end{displaymath} for the [[subspace]] of [[off-shell]]/[[on-shell]] [[microcausal polynomial observables]] inside all [[off-shell]]/[[on-shell]] [[polynomial observables]]. \end{defn} The important point is that [[microcausal polynomial observables]] still contain all [[regular polynomial observables]] but also all polynomial [[local observables]]: \begin{example} \label{MicrocausalRegularObservables}\hypertarget{MicrocausalRegularObservables}{} \textbf{([[regular polynomial observables]] are [[microcausal observables|microcausal]])} Every [[regular polynomial observable]] (def. \ref{PolynomialObservables}) is [[microcausal polynomial observable|microcausal]] (def. \ref{MicrocausalObservable}). \end{example} \begin{proof} By definition of regular polynomial observables, their [[coefficients]] are [[non-singular distributions]] and because the [[wave front set]] of [[non-singular distributions]] is [[empty set|empty]] (example \ref{NonSingularDistributionTrivialWaveFrontSet}) \end{proof} \begin{example} \label{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}\hypertarget{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}{} \textbf{(polynomial [[local observables]] are [[microcausal polynomial observables|microcausal]])} Every polynomial [[local observable]] (def. \ref{LocalObservables}) is a [[microcausal polynomial observable]] (def. \ref{MicrocausalObservable}). \end{example} \begin{proof} For notational convenience, consider the case of the [[scalar field]] with $k = 2$; the general case is directly analogous. Then the [[local observable]] coming from $\phi^2$ (a [[phi{\tt \symbol{94}}n interaction]]-term), has, regarded as a [[polynomial observable]], the [[delta distribution]] $\delta(x_1-x_2)$ as [[coefficient]] in degree 2: \begin{displaymath} \begin{aligned} A(\Phi) & = \underset{\Sigma}{\int} g(x) (\Phi(x))^2 \,dvol_\Sigma(x) \\ & = \underset{\Sigma \times \Sigma}{\int} \underset{ = \alpha^{(2)}}{ \underbrace{ g(x_1) \delta(x_1 - x_2) }} \, \Phi(x_1) \Phi(x_2) \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \end{aligned} \,. \end{displaymath} Now for $(x_1, x_2) \in \Sigma \times \Sigma$ and $\mathbb{R}^{2n} \simeq U \subset X \times X$ a [[chart]] around this point, the [[Fourier transform of distributions]] of $g \cdot \delta(-,-)$ restricted to this chart is proportional to the Fourier transform $\hat g$ of $g$ evaluated at the sum of the two covectors: \begin{displaymath} \begin{aligned} (k_1, k_2) & \mapsto \underset{\mathbb{R}^{2n}}{\int} g(x_1) \delta(x_1, x_2) e^{i (k_1 \cdot x_1 + k_2 \cdot x_2 )} \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \\ & \propto \hat g(k_1 + k_2) \end{aligned} \,. \end{displaymath} Since $g$ is a plain [[bump function]], its [[Fourier transform]] $\hat g$ is quickly decaying (according to prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}) along $k_1 + k_2$, as long as $k_1 + k_2 \neq 0$. Only on the [[cone]] $k_1 + k_2 = 0$ the Fourier transform is [[constant function|constant]], and hence in particular not decaying. This means that the wave front set consists of the elements of the form $(x, (k, -k))$ with $k \neq 0$. Since $k$ and $-k$ are both in the [[closed future cone]] or both in the [[closed past cone]] precisely if $k = 0$, this situation is excluded in the wave front set and hence the distribution $g \cdot \delta(-,-)$ is [[microcausal observable|microcausal]]. \begin{quote}% (graphics grabbed from \href{microcausal+polynomial+observable#KhavkineMoretti14}{Khavkine-Moretti 14, p. 45}) \end{quote} \end{proof} \begin{prop} \label{MoyalStarProductOnMicrocausal}\hypertarget{MoyalStarProductOnMicrocausal}{} \textbf{([[Hadamard distribution|Hadamard]]-[[Moyal star product]] on [[microcausal observables]] -- [[abstract Wick algebra]])} Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$. Write $\Delta$ for the [[causal propagator]] and let \begin{displaymath} \Delta_H \;=\; \tfrac{i}{2}\Delta + H \end{displaymath} be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]). Then the [[star product]] induced by $\Delta_H$ \begin{displaymath} A \star_H A \;\coloneqq\; prod \circ \exp\left( \int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \mathbf{\Phi}^a(x_1)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(x_2)} dvol_g \right) (P_1 \otimes P_2) \end{displaymath} on [[off-shell]] [[microcausal observables]] $A_1, A_2 \in \mathcal{F}_{mc}$ (def. \ref{MicrocausalObservable}) is well defined in that the [[wave front sets]] involved in the [[products of distributions]] that appear in expanding out the [[exponential]] satisfy [[Hörmander's criterion]]. Hence by the general properties of [[star products]] (prop. \ref{AssociativeAndUnitalStarProduct}) this yields a [[unital algebra|unital]] [[associative algebra]] [[structure]] on the space of [[formal power series]] in $\hbar$ of [[off-shell]] [[microcausal observables]] \begin{displaymath} \left( PolyObs(E)_{mc}[ [\hbar] ] \,,\, \star_H \right) \,. \end{displaymath} This is the \emph{[[off-shell]] [[Wick algebra]]} corresponding to the choice of [[Wightman propagator]] $H$. Moreover the image of $P$ is an ideal with respect to this algebra structure, so that it descends to the [[on-shell]] [[microcausal observables]] to yield the \emph{[[on-shell]] [[Wick algebra]]} \begin{displaymath} \left( PolyObs(E,\mathbf{L})_{mc}[ [ \hbar ] ] \,,\, \star_H \right) \,. \end{displaymath} Finally, under [[complex conjugation]] $(-)^\ast$ these are [[star algebras]] in that \begin{displaymath} \left( A_1 \star_H A_2 \right)^\ast = A_2^\ast \star_H A_1^\ast \,. \end{displaymath} \end{prop} (e.g. \href{Wick+algebra#Collini16}{Collini 16, p. 25-26}) \begin{proof} By prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski} the [[wave front set]] of $\Delta_H$ has all cotangents on the first variables in the [[closed future cone]] (at the given base point, which itself is on the [[light cone]]) and hence all those on the second variables in the [[closed past cone]]. The first variables are integrated against those of $A_1$ and the second against $A_2$. By definition of [[microcausal observables]] (def. \ref{MicrocausalObservable}), the wave front sets of $A_1$ and $A_2$ are disjoint from the subsets where all components are in the [[closed future cone]] or all components are in the [[closed past cone]]. Therefore the relevant sum of of the wave front covectors never vanishes and hence [[Hörmander's criterion]] (prop. \ref{HoermanderCriterionForProductOfDistributions}) for partial [[products of distributions|products of]] [[distributions of several variables]] (prop. \ref{PartialProductOfDistributionsOfSeveralVariables}). It remains to see that the star product $A_1 \star_H A_2$ is itself again a [[microcausal observable]]. It is clear that it is again a [[polynomial observable]] and that it respects the ideal generated by the equations of motion. That it still satisfies the condition on the [[wave front set]] follows directly from the fact that the wave front set of a [[product of distributions]] is inside the fiberwise sum of elements of the factor wave front sets (prop. \ref{WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets}, prop. \ref{PartialProductOfDistributionsOfSeveralVariables}). Finally the [[star algebra]]-structure via [[complex conjugation]] follows via remark \ref{WightmanPropagatorAsKaehlerVectorSpaceStructure} as in prop. \ref{StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra}. \end{proof} \begin{remark} \label{WickAlgebraIsFormalDeformationQuantization}\hypertarget{WickAlgebraIsFormalDeformationQuantization}{} \textbf{([[Wick algebra]] is [[formal deformation quantization]] of [[Poisson-Peierls bracket|Poisson-Peierls algebra of observables]])} Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$ with [[causal propagator]] $\Delta$ and let $\Delta_H \;=\; \tfrac{i}{2}\Delta + H$ be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]). Then the [[Wick algebra]] $\left( PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ] \,,\, \star_H \right)$ from prop. \ref{MoyalStarProductOnMicrocausal} is a [[formal deformation quantization]] of the [[Poisson algebra]] on the [[covariant phase space]] given by the [[on-shell]] [[polynomial observables]] equipped with the [[Poisson-Peierls bracket]] $\{-,-\} \;\colon\; PolyObs(E,\mathbf{L})_{mc} \otimes PolyObs(E,\mathbf{L})_{mc} \to PolyObs(E,\mathbf{L})_{mc}$ in that for all $A_1, A_2 \in PolyObs(E,\mathbf{L})_{mc}$ we have \begin{displaymath} A_1 \star_H A_2 \;=\; A_1 \cdot A_2 \;mod\; \hbar \end{displaymath} and \begin{displaymath} A_1 \star_H A_2 - A_2 \star_H a_1 \;=\; i \hbar \{A_1, A_2\} \;mod\; \hbar^2 \,. \end{displaymath} \end{remark} (\href{Wick+algebra#Dito90}{Dito 90}, \href{Wick+algebra#DuetschFredenhagen00}{Dütsch-Fredenhagen 00}, \href{Wick+algebra#DuetschFredenhagen01}{Dütsch-Fredenhagen 01}, \href{Wick+algebra#HirschfeldHenselder02}{Hirshfeld-Henselder 02}) \begin{proof} By prop. \ref{MoyalStarProductOnMicrocausal} this is immediate from the general properties of the [[star product]] (example \ref{MoyalStarProductIsFormalDeformationQuantization}). Explicitly, consider, without restriction of generality, $A_1 = \int (\alpha_1)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ and $A_2 = \int (\alpha_2)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ be two linear observables. Then \begin{displaymath} \begin{aligned} & A_1 \star_H A_2 \\ & = \phantom{+} A_1 A_2 \\ & \phantom{=} + \hbar \int \left( \tfrac{i}{2} \Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1,x_2) \right) \frac{\partial A_1}{\partial \mathbf{\Phi}^{a_1}(x_1)} \frac{\partial A_2}{\partial \mathbf{\Phi}^{a_2}(x_2)} \;mod\; \hbar^2 \\ & = \phantom{+} A_1 A_2 \\ & \phantom{=} + \hbar \left( \int (\alpha_1)_{a_1}(x_1) \left( \tfrac{i}{2}\Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1, x_2) \right) (\alpha_2)_{a_2}(x_2) \right) \;mod\; \hbar^2 \end{aligned} \end{displaymath} Now since $\Delta$ is skew-symmetric while $H$ is symmetric (prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) it follows that \begin{displaymath} \begin{aligned} A_1 \star_H A_2 - A_2 \star_H A_1 & = i \hbar \left( \int (\alpha_1)_{a_1}(x_1) \Delta^{a_1 a_2}(x_1, x_2) (\alpha_2)_{a_2}(x_2) \right) \;mod\; \hbar^2 \\ & = i \hbar \, \left\{ A_1, A_2\right\} \end{aligned} \,. \end{displaymath} The right hand side is the [[integral kernel]]-expression for the [[Poisson-Peierls bracket]], as shown in the second line. \end{proof} $\,$ \textbf{[[time-ordered product]]} \begin{defn} \label{OnRegularPolynomialObservablesTimeOrderedProduct}\hypertarget{OnRegularPolynomialObservablesTimeOrderedProduct}{} \textbf{([[time-ordered product]] on [[regular polynomial observables]])} Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H$ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]]. Then the \emph{[[time-ordered product]]} on the space of [[off-shell]] [[regular polynomial observable]] $PolyObs(E)_{reg}$ is the [[star product]] induced by the [[Feynman propagator]] (via prop. \ref{PropagatorStarProduct}): \begin{displaymath} \itexarray{ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] \\ (A_1, A_2) &\mapsto& \phantom{\coloneqq} A_1 \star_F A_2 } \end{displaymath} hence \begin{displaymath} A_1 \star_F A_2 \; \coloneqq \; ((-)\cdot(-)) \circ \exp\left( \underset{\Sigma \times \Sigma}{\int} \Delta_F^{a b}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \, dvol_\Sigma(x) \, dvol_\Sigma(y) \right) \end{displaymath} (Notice that this does not descend to the [[on-shell]] observables, since the [[Feynman propagator]] is not a solution to the \emph{homogeneous} [[equations of motion]].) \end{defn} \begin{prop} \label{CausalOrderingTimeOrderedProductOnRegular}\hypertarget{CausalOrderingTimeOrderedProductOnRegular}{} \textbf{([[time-ordered product]] is indeed causally ordered [[Wick algebra]] product)} Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H$ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]]. Then the [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is indeed a time-ordering of the [[Wick algebra]] product $\star_H$ in that for all [[pairs]] of [[regular polynomial observables]] \begin{displaymath} A_1, A_2 \in PolyObs(E)_{reg}[ [\hbar] ] \end{displaymath} with [[disjoint subset|disjoint]] [[spacetime]] [[support]] we have \begin{displaymath} A_1 \star_F A_2 \;=\; \left\{ \itexarray{ A_1 \star_H A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \star_H A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \,. \end{displaymath} Here $S_1 {\vee\!\!\!\wedge} S_2$ is the [[causal order]] relation (``$S_1$ does not intersect the [[past cone]] of $S_2$''). Beware that for general [[pairs]] $(S_1, S_2)$ of subsets neither $S_1 {\vee\!\!\!\wedge} S_2$ nor $S_2 {\vee\!\!\!\wedge} S_1$. \end{prop} \begin{proof} Recall the following facts: \begin{enumerate}% \item the [[advanced and retarded propagators]] $\Delta_{\pm}$ by definition are [[support|supported]] in the [[future cone]]/[[past cone]], respectively \begin{displaymath} supp(\Delta_{\pm}) \subset \overline{V}^{\pm} \end{displaymath} \item they turn into each other under exchange of their arguments (cor. \ref{CausalPropagatorIsSkewSymmetric}): \begin{displaymath} \Delta_\pm(y,x) = \Delta_{\mp}(x,y) \,. \end{displaymath} \item the real part $H$ of the [[Feynman propagator]], which by definition is the real part of the [[Wightman propagator]] is symmetric (by definition or else by prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}): \begin{displaymath} H(x,y) = H(y,x) \end{displaymath} \end{enumerate} Using this we compute as follows: \begin{displaymath} \begin{aligned} A_1 \underset{\Delta_{F}}{\star} A_2 \; & = A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2 \\ & = \left\{ \itexarray{ A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \itexarray{ A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \itexarray{ A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \itexarray{ A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \end{aligned} \end{displaymath} \end{proof} \begin{prop} \label{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}\hypertarget{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}{} \textbf{([[time-ordered product]] on [[regular polynomial observables]] [[isomorphism|isomorphic]] to pointwise product)} The [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is [[isomorphism|isomorphic]] to the pointwise product of [[observables]] (def. \ref{Observable}) via the [[linear isomorphism]] \begin{displaymath} \mathcal{T} \;\colon\; PolyObs(E)_{reg}[ [\hbar] ] \longrightarrow PolyObs(E)_{reg}[ [\hbar] ] \end{displaymath} given by \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{OnregularPolynomialObservablesPointwiseTimeOrderedIsomorphism}\end{equation} in that \begin{displaymath} \begin{aligned} T(A_1 A_2) & \coloneqq A_1 \star_{F} A_2 \\ & = \mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2) ) \end{aligned} \end{displaymath} hence \begin{displaymath} \itexarray{ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{(-)\cdot (-)}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] \\ {}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow && \Big\downarrow{}^{\mathrlap{\mathcal{T}}}_\simeq \\ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{(-) \star_F (-)}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] } \end{displaymath} \end{prop} (\href{time-ordered+product#BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, (12)-(13)}, \href{time-ordered+product#FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b, (14)}) \begin{proof} Since the [[Feynman propagator]] is symmetric (prop. \ref{SymmetricFeynmanPropagator}), the statement is a special case of prop. \ref{SymmetricContribution}. \end{proof} \begin{example} \label{RegularObservablesExponentialTimeOrdered}\hypertarget{RegularObservablesExponentialTimeOrdered}{} \textbf{([[time-ordered product|time-ordered]] [[exponential]] of [[regular polynomial observables]])} Let $V \in PolyObs_{reg, deg = 0}[ [ \hbar ] ]$ be a [[regular polynomial observable]] (def. \ref{PolynomialObservables}) of degree zero, and write \begin{displaymath} \exp(V) = 1 + V + \tfrac{1}{2!} V \cdot V + \tfrac{1}{3!} V \cdot V \cdot V + \cdots \end{displaymath} for the [[exponential]] of $V$ with respect to the pointwise product \eqref{ObservablesPointwiseProduct}. Then the [[exponential]] $\exp_{\mathcal{T}}(V)$ of $V$ with respect to the [[time-ordered product]] $\star_F$ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is equal to the [[conjugation]] of the exponential with respect to the pointwise product by the time-ordering isomorphism $\mathcal{T}$ from prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}: \begin{displaymath} \begin{aligned} \exp_{\mathcal{T}}(V) & \coloneqq 1 + V + \tfrac{1}{2} V \star_F V + \tfrac{1}{3!} V \star_F V \star_F V + \cdots \\ & = \mathcal{T} \circ \exp(-) \circ \mathcal{T}^{-1}(V) \,. \end{aligned} \end{displaymath} \end{example} \begin{remark} \label{}\hypertarget{}{} \textbf{([[renormalization]] of [[time-ordered product]])} The [[time-ordered product]] on [[regular polynomial observables]] from prop. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} extends to a product on [[polynomial observable|polynomial]] [[local observables]] (def. \ref{LocalObservables}), then taking values in [[microcausal observables]] (def. \ref{MicrocausalObservable}): \begin{displaymath} T \;\colon\; PolyLocObs(E)^{\otimes_n}[ [\hbar] ] \longrightarrow PolyObs(E)_{mc}[ [\hbar] ] \,. \end{displaymath} This extension is not unique. A choice of such an extension, satisfying some evident compatibility conditions, is a choice of \emph{[[renormalization scheme]]} for the given [[perturbative quantum field theory]]. Every such choice corresponds to a choice of [[perturbative S-matrix]] for the theory, namely an extension of the time-ordered exponential $\exp_{\mathcal{T}}$ (example \ref{RegularObservablesExponentialTimeOrdered}) from regular to local observables. This construction of [[perturbative quantum field theory]] is called \emph{[[causal perturbation theory]]}. We discuss this below in the chapters \emph{\hyperlink{InteractingQuantumFields}{Interacting quantum fields}} and \emph{\hyperlink{Renormalization}{Renormalization}}. \end{remark} $\,$ \textbf{operator product notation} \begin{defn} \label{NormalOrderedProductNotation}\hypertarget{NormalOrderedProductNotation}{} \textbf{(notation for [[operator product]] and [[normal-ordered product]])} It is traditional to use the following alternative notation for the product structures on [[microcausal polynomial observables]]: \begin{enumerate}% \item The [[Wick algebra]]-product, hence the [[star product]] $\star_H$ for the [[Wightman propagator]] (def. \ref{MoyalStarProductOnMicrocausal}), is rewritten as plain juxtaposition: \begin{displaymath} \text{"operator product"} \phantom{AAA} A_1 A_2 \phantom{AA} \coloneqq \phantom{AA} A_1 \star_H A_1 \phantom{AAAA} \itexarray{ \text{star product of} \\ \text{Wightman propagator} } \,. \end{displaymath} \item The pointwise product of observables (def. \ref{Observable}) $A_1 \cdot A_2$ is equivalently written as plain juxtaposition enclosed by colons: \begin{displaymath} \itexarray{ \text{"normal-ordered} \\ \text{product"} } \phantom{AAAA} :A_1 A_2: \phantom{AA}\coloneqq\phantom{AA} A_1 \cdot A_2 \phantom{AAAA} \phantom{AAa}\text{pointwise product}\phantom{AAa} \end{displaymath} \item The [[time-ordered product]], hence the [[star product]] for the [[Feynman propagator]] $\star_F$ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is equivalently written as plain juxtaposition prefixed by a ``$T$'' \begin{displaymath} \itexarray{ \text{"time-ordered} \\ \text{product"} } \phantom{AAAA} T(A_1 A_2) \phantom{AA}\coloneqq\phantom{AA} A_1 \star_F A_2 \phantom{AAAA} \itexarray{ \text{star product of} \\ \text{Feynman propagator} } \end{displaymath} \end{enumerate} Under [[representation]] of the [[Wick algebra]] on a [[Fock space|Fock]] [[Hilbert space]] by [[linear operators]] the first product becomes the \emph{[[operator product]]}, while the second becomes the operator poduct applied after suitable re-ordering, called ``[[normal-ordered product|normal odering]]'' of the factors. Disregarding the [[Fock space]]-representation, which is [[faithful representation|faithful]], we may still refer to these ``abstract'' products as the ``operator product'' and the ``normal-ordered product'', respectively. \end{defn} $\,$ \begin{example} \label{InWickAlgebraphinInteraction}\hypertarget{InWickAlgebraphinInteraction}{} \textbf{([[phi{\tt \symbol{94}}n interaction]])} Consider [[phi{\tt \symbol{94}}n theory]] from example \ref{phintheoryLagrangian}. The [[adiabatic switching|adiabatically switched]] [[action functional]] (example \ref{ActionFunctional}) which is the [[transgression of variational differential forms|transgression]] of the [[phi{\tt \symbol{94}}n interaction]] is the following [[local observable|local]] (hence, by example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}, [[microcausal polynomial observable|microcausal]]) observable: \begin{displaymath} \begin{aligned} S_{int} & = \underset{\Sigma}{\int} \underset{ n \, \text{factors} }{ \underbrace{ \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(x) \cdots \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(x) } } \, dvol_\Sigma(x) \\ & = \underset{\Sigma}{\int} : \underset{ n \, \text{factors} }{ \underbrace{ \mathbf{\Phi}(x) \mathbf{\Phi}(x) \cdots \mathbf{\Phi}(x) \mathbf{\Phi}(x) } } : \, dvol_\Sigma(X) \end{aligned} \,, \end{displaymath} Here in the first line we have the [[integral]] over a pointwise product (def. \ref{Observable}) of $n$ [[field observables]] (example \ref{PointEvaluationObservables}), which in the second line we write equivalently as a [[normal ordered product]] by def. \ref{NormalOrderedProductNotation}. \end{example} \begin{example} \label{InWickAlgebraElectronPhotonInteraction}\hypertarget{InWickAlgebraElectronPhotonInteraction}{} \textbf{([[electron-photon interaction]])} Consider the [[Lagrangian field theory]] defining [[quantum electrodynamics]] from example \ref{LagrangianQED}. The [[adiabatic switching|adiabatically switched]] [[action functional]] (example \ref{ActionFunctional}) which is the [[transgression of variational differential forms|transgression]] of the [[electron-photon interaction]] is the [[local observable|local]] (hence, by example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}, [[microcausal polynomial observable|microcausal]]) observable \begin{displaymath} \begin{aligned} S_{int} & \coloneqq i \underset{\Sigma}{\int} g_{sw}(x) \, (\Gamma^\mu)^\alpha{}_\beta \, \overline{\mathbf{\Psi}}_\alpha(x) \cdot \mathbf{\Psi}^\beta(x) \cdot \mathbf{A}_\mu(x) \, dvol_\Sigma(x) \\ & = i \underset{\Sigma}{\int} g_{sw}(x) \, (\Gamma^\mu)^\alpha{}_\beta \, : \overline{\mathbf{\Psi}}_\alpha(x) \mathbf{\Psi}^\beta(x) \mathbf{A}_\mu(x) : \, dvol_\Sigma(x) \end{aligned} \,, \end{displaymath} Here in the first line we have the [[integral]] over a pointwise product (def. \ref{Observable}) of $n$ [[field observables]] (example \ref{PointEvaluationObservables}), which in the second line we write equivalently as a [[normal ordered product]] by def. \ref{NormalOrderedProductNotation}. \end{example} (e.g. \href{electron-photon+interaction#Scharf95}{Scharf 95, (3.3.1)}) $\,$ \textbf{[[Hadamard vacuum state]]} \begin{prop} \label{WickAlgebraCanonicalState}\hypertarget{WickAlgebraCanonicalState}{} \textbf{(canonical [[Hadamard vacuum state|vacuum]] [[state on a star-algebra|states]] on abstract [[Wick algebra]])} Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]]. For \begin{displaymath} \Phi_0 \in \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \end{displaymath} any [[on-shell]] [[field history]] (i.e. solving the [[equations of motion]]), consider the function from the [[Wick algebra]] to [[formal power series]] in $\hbar$ with [[coefficients]] in the [[complex numbers]] which evaluates any [[microcausal polynomial observable]] on $\Phi_0$ \begin{displaymath} \itexarray{ PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ] &\overset{\langle -\rangle_{\Phi_0}}{\longrightarrow}& \mathbb{C}[ [\hbar] ] \\ A &\mapsto& A(\Phi_0) } \end{displaymath} Specifically for $\Phi_0 = 0$ (which is a solution of the [[equations of motion]] by the assumption that $(E,\mathbf{L})$ defines a [[free field theory]]) this is the function \begin{displaymath} \itexarray{ PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ] &\overset{\langle -\rangle_0}{\longrightarrow}& \mathbb{C}[ [\hbar] ] \\ \left. \begin{aligned} A & = \alpha^{(0)} \\ & \phantom{=} + \underset{\Sigma}{\int} \alpha^{(1)}_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x) \\ & \phantom{=} + \cdots \end{aligned} \right\} &\mapsto& A(0) = \alpha^{(0)} } \end{displaymath} which sends each [[microcausal polynomial observable]] to its value $A(\Phi = 0)$ on the zero [[field history]], hence to the constant contribution $\alpha^{(0)}$ in its [[polynomial]] expansion. The function $\langle -\rangle_0$ is \begin{enumerate}% \item [[linear function|linear]] over $\mathbb{C}[ [\hbar] ]$; \item real, in that for all $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$ \begin{displaymath} \langle A^\ast \rangle = \langle A \rangle^\ast \end{displaymath} \item positive, in that for every $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$ there exist a $c_A \in \mathbb{C}[ [\hbar] ]$ such that \begin{displaymath} \langle A^\ast \star_H A\rangle_{\Phi_0} = c_A^\ast \cdot c_A \,, \end{displaymath} \item normalized, in that \begin{displaymath} \langle 1\rangle_H = 1 \end{displaymath} \end{enumerate} where $(-)^\ast$ denotes componet-wise [[complex conjugation]]. This means that $\langle -\rangle_{0}$ is a [[state on a star-algebra|state]] on the [[Wick algebra|Wick]] [[star-algebra]] $\left( (PolyObs(E,\mathbf{L}))_{mc}[ [\hbar] ], \star_H\right)$ (prop. \ref{MoyalStarProductOnMicrocausal}). One says that \begin{itemize}% \item $\langle - \rangle_0$ is a \emph{[[Hadamard vacuum state]]}; \end{itemize} and generally \begin{itemize}% \item $\langle - \rangle_{\Phi_0}$ is called a \emph{[[coherent state]]}. \end{itemize} \end{prop} (\hyperlink{Duetsch18}{Dütsch 18, def. 2.12, remark 2.20, def. 5.28, exercise 5.30 and equations (5.178)}) \begin{proof} The properties of linearity, reality and normalization are obvious, what requires proof is positivity. This is proven by exhibiting a [[representation]] of the Wick algebra on a [[Fock space|Fock]] [[Hilbert space]] (this algebra [[homomorphism]] is \emph{[[Wick's lemma]]}), with formal powers in $\hbar$ suitably taken care of, and showing that under this representation the function $\langle -\rangle_0$ is represented, degreewise in $\hbar$, by the [[inner product]] of the [[Hilbert space]]. \end{proof} \begin{example} \label{HadamardMoyalStarProductOfTwoLinearObservables}\hypertarget{HadamardMoyalStarProductOfTwoLinearObservables}{} \textbf{([[operator product]] of two [[linear observables]])} Let \begin{displaymath} A_i \in LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc} \end{displaymath} for $i \in \{1,2\}$ be two [[linear observable|linear]] [[microcausal observables]] represented by [[distributions]] which in [[generalized function]]-notation are given by \begin{displaymath} A_i \;=\; \int (\alpha_i)_{a_i}(x_i) \mathbf{\Phi}^{a_i}(x_i) \, dvol_\Sigma(x_i) \,. \end{displaymath} Then their Hadamard-Moyal [[star product]] (prop. \ref{MoyalStarProductOnMicrocausal}) is the [[sum]] of their pointwise product with their value \begin{equation} \langle A_1 \star_H A_2 \rangle_0 \;\coloneqq\; i \hbar \int \int (\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2) \,dvol_\Sigma(x_1) \,dvol_\Sigma(x_2) \label{EvaluatingLinearObservablesInWightmanPropagator}\end{equation} in the [[Wightman propagator]], which is the value of the [[Hadamard vacuum state]] from prop. \ref{WickAlgebraCanonicalState}: \begin{displaymath} A_1 \star_H A_2 \;=\; A_1 \cdot A_2 \;+\; \langle A_1 \star_H A_2 \rangle_0 \end{displaymath} In the [[operator product]]/[[normal-ordered product]]-notation of def. \ref{NormalOrderedProductNotation} this reads \begin{displaymath} A_1 A_2 \;=\; :A_1 A_2: \;+\; \langle A_1 A_2\rangle \,. \end{displaymath} \end{example} \begin{example} \label{WeylRelations}\hypertarget{WeylRelations}{} \textbf{([[Weyl relations]])} Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] and with [[Wightman propagator]] $\Delta_H$. Then for \begin{displaymath} A_1, A_2 \;\in\; LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc} \end{displaymath} two [[linear observables|linear]] [[microcausal observables]], the Hadamard-Moyal star product (def. \ref{MoyalStarProductOnMicrocausal}) of their [[exponentials]] exhibits the [[Weyl relations]]: \begin{displaymath} e^{A_1} \star_H e^{A_2} \;=\; e^{A_1 + A_2} \; e^{\langle A_1 \star_H A_2\rangle_0} \end{displaymath} where on the right we have the [[exponential]] of the value of the [[Hadamard vacuum state]] (prop. \ref{WickAlgebraCanonicalState}) as in example \ref{HadamardMoyalStarProductOfTwoLinearObservables}. \end{example} (e.g. \href{Wick+algebra#Duetsch18}{Dütsch 18, exercise 2.3}) \begin{example} \label{}\hypertarget{}{} \textbf{([[Wightman propagator]] is [[2-point function]] in the [[Hadamard vacuum state]])} Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]]. With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Wightman propagator]] $\Delta_H(x,y)$ itself is the \emph{[[2-point function]]}, namely the [[distribution|distributional]] [[vacuum expectation value]] of the operator product of two [[field observables]]: \begin{displaymath} \left\langle \mathbf{\Phi}^a(x) \star_H \mathbf{\Phi}^b(y) \right\rangle_0 \;=\; \underset{ = 0 }{ \underbrace{ \left\langle \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y) \right\rangle }} + \underset{ = \hbar \Delta^{a b}_H(x,y) }{ \underbrace{ \left \langle \hbar \underset{\Sigma \times \Sigma}{\int} \delta(x-x') \Delta^{a b}_H(x,y) \delta(y-y') \right\rangle }} \end{displaymath} by example \ref{HadamardMoyalStarProductOfTwoLinearObservables}. Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads: \begin{displaymath} \left\langle \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) \right\rangle_0 \;=\; \hbar \Delta_H(x,y) \,. \end{displaymath} \end{example} Similarly: \begin{example} \label{}\hypertarget{}{} \textbf{([[Feynman propagator]] is time-ordered [[2-point function]] in the [[Hadamard vacuum state]])} Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]] with induced [[Feynman propagator]] $\Delta_F$. With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Feynman propagator]] $\Delta_F(x,y)$ itself is the \emph{time-ordered [[2-point function]]}, namely the [[distribution|distributional]] [[vacuum expectation value]] of the [[time-ordered product]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) of two [[field observables]]: \begin{displaymath} \left\langle T\left( \mathbf{\Phi}^a(x) \star_F \mathbf{\Phi}^b(y) \right) \right\rangle_0 \;=\; \underset{ = 0 }{ \underbrace{ \left\langle \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y) \right\rangle }} + \underset{ = \hbar \Delta^{a b}_H(x,y) }{ \underbrace{ \left \langle \hbar \underset{\Sigma \times \Sigma}{\int} \delta(x-x') \Delta^{a b}_F(x,y) \delta(y-y') \right\rangle }} \end{displaymath} analogous to example \ref{HadamardMoyalStarProductOfTwoLinearObservables}. Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads: \begin{displaymath} \left\langle T\left( \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) \right) \right\rangle_0 \;=\; \hbar \Delta_F(x,y) \,. \end{displaymath} \end{example} [[!include propagators - table]] $\,$ \textbf{[[BV-operator|free quantum BV-differential]]} So far we have discussed the plain (graded-commutative) [[algebra of quantum observables]] of a [[gauge fixing|gauged fixed]] [[free field theory|free]] [[Lagrangian field theory]], [[deformation quantization|deforming]] the commutative pointwise product of [[observables]]. But after [[gauge fixing]], the algebra of observables is not just a (graded-commutative) algebra, but carries also a [[differential]] making it a [[differential graded-commutative superalgebra]]: the global [[BV-differential]] $\{-S' + S_{BRST}, -\}$ (def. \ref{ComplexBVBRSTGlobal}). The [[gauge invariance|gauge invariant]] [[on-shell]] [[observables]] are (only) the [[cochain cohomology]] of this differential. Here we discuss what becomes of this differential as we pass to the non-commutative [[Wick algebra|Wick]]-[[algebra of quantum observables]]. \begin{prop} \label{OnMicrocausalObservablesGlobalBVDifferential}\hypertarget{OnMicrocausalObservablesGlobalBVDifferential}{} \textbf{(global [[BV-differential]] on [[Wick algebra]])} Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded [[BV-BRST formalism|BV-BRST]] [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}). Let $\Delta_H$ be a compatible [[Wightman propagator]] (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}). Then the global [[BV-differential]] $\{-S',(-)\}$ (def. \ref{ComplexBVBRSTGlobal}) restricts from [[polynomial observables]] to a linear map on [[microcausal polynomial observables]] (def. \ref{MicrocausalObservable}) \begin{displaymath} \{-S',(-)\} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ] \end{displaymath} and as such is a [[derivation]] not only for the pointwise product, but also for the product in the [[Wick algebra]] (the [[star product]] induced by the [[Wightman propagator]]): \begin{displaymath} \{-S', A_1 \star_H A_2\} \;=\; \{-S', A_1\} \star_H A_2 + A_1 \star_H \{-S', A_2\} \,. \end{displaymath} We call $\{-S,(-)\}$ regarded as a nilpotent derivation on the [[Wick algebra]] this way the \emph{free quantum [[BV-differential]]}. \end{prop} (\href{BV-differential#FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b, below (37)}, \href{BV-differential#Rejzner11}{Rejzner 11, below (5.28)}) \begin{proof} By example \ref{BVDifferentialGlobal} the action of $\{-S',(-)\}$ on polyomial observables is to replace [[antifield]] [[field observables]] by \begin{displaymath} \mathbf{\Phi}^\ddagger_a(x) \;\mapsto\; \pm (P_{A B}\mathbf{\Phi}^A)(x) \,, \end{displaymath} where $P$ is a [[differential operator]]. By [[partial integration]] this translates to $\{-S',(-)\}$ acting by the [[formally adjoint differential operator]] $P^\ast$ (def. \ref{FormallyAdjointDifferentialOperators}) via [[derivative of distributions|distributional derivative]] on the [[distribution|distributional]] [[coefficients]] of the given polynomial observable. Now by prop. \ref{RetainsOrShrinksWaveFrontSetDifferentialOperator} the application of $P^\ast$ retains or shrinks the [[wave front set]] of the distributional coefficient, hence it preserves the microcausality condition (def. \ref{MicrocausalObservable}). This makes $\{-S',(-)\}$ restrict to microcausal polynomial observables. To see that $\{-S',(-)\}$ thus restricted is a [[derivation]] of the Wick algebra product, it is sufficient to see that its [[commutators]] with the [[Wightman propagator]] vanish in each argument: \begin{displaymath} \left[ \{-S',(-)\} \otimes id \;,\; \Delta_H \left( \frac{\delta}{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right) \right] \;=\; 0 \end{displaymath} and \begin{displaymath} \left[ id \otimes \{-S',(-)\} \;,\; \Delta_H \left( \frac{\delta}{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right) \right] \;=\; 0 \,. \end{displaymath} Because with this we have: \begin{displaymath} \begin{aligned} \{-S', A_1 \star_H A_2\} & = \{-S',(-)\} \circ ((-)\cdot(-)) \circ \exp\left( \hbar \Delta_H\left( \frac{\delta}{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right) \right) (A_1 \otimes A_2) \\ & = ((-)\cdot(-)) \circ \left( \phantom{a \atop a} \{-S',-\} \otimes id + id \otimes \{-S',(-)\} \right) \circ \exp\left( \hbar \Delta_H\left( \frac{\delta}{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right) \right) (A_1 \otimes A_2) \\ & = ((-)\cdot(-)) \circ \exp\left( \hbar \Delta_H\left( \frac{\delta}{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right) \right) \circ \left( \phantom{a \atop b} \{-S',-\} \otimes id + id \otimes \{-S',(-)\} \right) (A_1 \otimes A_2) \\ & = \{-S',A_1\} \star_H A_2 + A_1 \star_H \{-S', A_2\} \end{aligned} \end{displaymath} Here in the first step we used that $\{-S',(-)\}$ is a derivation with respect to the pointwise product, by construction (def. \ref{ComplexBVBRSTGlobal}) and then we used the vanishing of the above commutators. To see that these commutators indeed vanish, use that by example \ref{BVDifferentialGlobal} we have \begin{displaymath} \begin{aligned} & \left[ \{-S',(-)\} \otimes id \;,\; \Delta_H\left( \frac{\delta }{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right) \right] \\ & = \left[ \underset{A}{\sum} (-1)^{deg(\phi^A)} \underset{\Sigma}{\int} (P_{A B}\mathbf{\Phi}^A)(x) \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \otimes id \, dvol_\Sigma(x) \;\,\; \underset{\Sigma \times \Sigma}{\int} \Delta_H^{A B}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^A(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^B(y)} \, dvol_\Sigma(x) \, dvol_\Sigma(y) \right] \\ & = -\underset{a}{\sum} (-1)^{deg(\phi^A)} \underset{\Sigma \times \Sigma}{\int} \underset{ = 0 }{ \underbrace{ (P_x \Delta_H)_A{}^B(x,y) } } \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^B(y)} \, dvol_\Sigma(x) \, dvol_\Sigma(y) \\ & = 0 \end{aligned} \end{displaymath} and similarly for the other order of the tensor products. Here the term over the brace vanishes by the fact that the Wightman propagator is a solution to the homogeneous equations of motion by prop. \ref{OnMinkowskiWightmanIsDistributionalSolutionToKleinGordon}. \end{proof} To analyze the behaviour of the free quantum BV-differential in general and specifically after passing to [[interacting field theory]] (below in chapter \emph{\hyperlink{InteractingQuantumFields}{Interacting quantum fields}}) it is useful to re-express it in terms of the incarnation of the global [[antibracket]] with respect not to the pointwise product of observables, but the [[time-ordered product]]: \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]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}). 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]] (def. \ref{ComplexBVBRSTGlobal}) by the time-ordering operator $\mathcal{T}$ (from prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}): \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}}\Big\downarrow && \Big\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]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}). 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 prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}) \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}}}\Big\downarrow && \Big\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]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}) and with corresponding [[gauge fixing|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} (def. \ref{GaugeFixingLagrangianDensity}). 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 example \ref{BVDifferentialGlobal} 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]] (cor. \ref{GreenFunctionFeynmanPropagator}). \end{enumerate} \end{proof} \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]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) 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]] $\{-,-\}$ (def. \ref{ComplexBVBRSTGlobal}) satisfy for all [[polynomial observables]] (def. \ref{PolynomialObservables}) $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})[ [\hbar] ]$ 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]] $\mathcal{T}$ (prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}) \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)^{dag(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]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) 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]] $\exp_{\mathcal{T}}(V)$ (example \ref{RegularObservablesExponentialTimeOrdered}) is the [[time-ordered product]] of the time-ordered exponential with the sum of $\Delta_{BV}(V)$ and the global [[antibracket]] $\tfrac{1}{2}\{V,V\}$ of $V$ with itself: \begin{displaymath} \Delta_{BV} \left( \exp_{\mathcal{T}}(V) \right) \;=\; \left( \Delta_{BV}(V) + \tfrac{1}{2}\{V,V\} \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\} + \cdots \\ & = \Delta_{BV}(V) + \tfrac{1}{2}\{V,V\} \;+\; \Delta_{BV}(V) \star_F V + \cdots \end{aligned} \end{displaymath} \end{proof} $\,$ \textbf{[[Schwinger-Dyson equation]]} A special case of the general occurrence 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]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}). 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 following proof 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 $\mathcal{T}^{-1}$ (prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}) to equation \eqref{BVOperatorDefiningRelation} applied to $A$ yields \begin{displaymath} \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}} } } \end{displaymath} where we have identified the terms under the braces by 1) the component expression for the BV-differential $\{-S',-\}$ from prop. \ref{BVDifferentialGlobal}, 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]] \eqref{OnShellPolynomialObservablesAsBVCohomology} \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 example \ref{BVDifferentialGlobal}). 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{example} \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]] (prop. \ref{PrincipleOfExtremalAction}) 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{example} $\,$ This concludes our discussion of the [[algebra of quantum observables]] for [[free field theories]]. In the \hyperlink{InteractingQuantumFields}{next chapter} we discuss the [[perturbative QFT]] of [[interacting field theories]] as [[deformations]] of such free quantum field theories. $\,$ $\,$ \end{document}