\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*{Moyal deformation quantization}

[[!redirects Moyal quantization]]

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

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

\hypertarget{geometric_quantization}{}\paragraph*{{Geometric quantization}}\label{geometric_quantization}

[[!include geometric quantization - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{integral_representation}{Integral representation}\dotfill \pageref*{integral_representation} \linebreak
\noindent\hyperlink{ViaGeometricQuantization}{Via geometric quantization}\dotfill \pageref*{ViaGeometricQuantization} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

The \emph{Moyal product} is a [[formal deformation quantization]] of a linear [[Poisson manifold]], hence of a [[vector space]] $V$ equipped with a [[Poisson bivector]] $\pi \in V \wedge V$, regarded as a constant (translation invariant) [[tensor field|bivector field]].

Moyal quantization serves as an intermediate step in quantization of more general situations:

Given a [[symplectic manifold]], then Moyal quantization applies in each [[fiber]] of the [[tangent bundle]]. The resulting [[fiber bundle]] of Moyal algebras admits a [[flat connection]] (non-uniquely) compatible with the algebra structure. The [[covariantly constant sections]] of this Moyal-algebra bundle constitute a [[formal deformation quantization]] of the symplectic manifold, see at \emph{[[Fedosov's deformation quantization]]}.

With a little care, the Moyal construction applies also to infinite-dimensional Poisson vector spaces such as appear in [[local field theory]]. Here the Moyal quantization yields [[formal deformation quantization]] of [[free field theories]] to [[perturbative quantum field theories]], the result are the \emph{[[Wick algebras]]} of free field theory (\hyperlink{Dito90}{Dito 90}, \hyperlink{DutschFredenhagen01}{D\"u{}tsch-Fredenhagen 01}). Combining this this with [[Fedosov's deformation quantization]] as above yields [[interaction|interacting]] [[perturbative quantum field theories]] as constructed via [[causal perturbation theory]] (\hyperlink{Collini16}{Collini 16}), see at \emph{[[locally covariant perturbative quantum field theory]]} for more on this.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The Moyal [[star product]] on [[smooth functions]] $C^\infty(V)$ is given on $f,g \in C^\infty(V)$ by

\begin{displaymath}
f \star g \coloneqq prod \circ \exp(\hbar \pi)(f , g)
  \,,
\end{displaymath}
where in the exponent we regard $\pi$ as an [[endomorphism]] on the [[tensor product]] $C^\infty(V) \otimes C^\infty(V)$ by [[differentiation]] in each argument, where the [[exponential]] denotes the corresponding [[formal power series]] of iterated applications of this endomorphism, and where $prod \colon C^\infty(V) \otimes C^\infty(V) \to C^\infty(V)$ is the usual pointwise product of functions.

This means that given a choice of [[basis]] $\{x^i\}_i$ of $V$ such that $\pi$ has components $\{\pi^{i j}\}_{i j}$ in this basis, the resulting [[formal power series]] in the formal parameter $\hbar$ (``[[Planck's constant]]'') starts out as

\begin{displaymath}
(f \star g) =
  f \cdot g
  + 
  \hbar 
  \sum_{i,j} \pi^{i j} \frac{\partial f}{\partial x^i}\cdot \frac{\partial g}{\partial x^j}
  + 
  \frac{1}{2}
  \hbar^2
  \sum_{i,j,k, l} \pi^{k l} \pi^{i j} \frac{\partial^2 f}{\partial x^k\partial x^i}\cdot \frac{\partial^2 g}{\partial x^k \partial x^j}
   + 
  \cdots
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{integral_representation}{}\subsubsection*{{Integral representation}}\label{integral_representation}

\begin{prop}
\label{IntegralRepresentationOfStarProduct}\hypertarget{IntegralRepresentationOfStarProduct}{}
\textbf{([[integral]] representation of [[star product]])}

If the functions $f,g$ admit [[Fourier analysis]] (are [[functions with rapidly decreasing partial derivatives]]), then their [[star product]] is equivalently given by the following [[integral]] expression:

\begin{displaymath}
\begin{aligned}
  \left(f \star_\omega g\right)(x)
   &= 
  \frac{(det(\omega)^{2n})}{(2 \pi \hbar)^{2n} }
  \int
    e^{ - \tfrac{1}{i \hbar} \omega((x - \tilde y),(x-y))}
    f(y)
    g(\tilde y)
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
  \end{aligned}
\end{displaymath}
\end{prop}
(\hyperlink{Baker58}{Baker 58}, see at \emph{[[star product]]} \href{star+product#IntegralRepresentationOfStarProduct}{this prop}).

\hypertarget{ViaGeometricQuantization}{}\subsubsection*{{Via geometric quantization}}\label{ViaGeometricQuantization}

The Moyal quantization of a Poisson vector space $(V,\pi)$ arises equivalently as the canonical [[geometric quantization of symplectic groupoids]] of the [[symplectic groupoid]] which is the [[Lie integration]] of the corresponding [[Poisson Lie algebroid]] (\hyperlink{Weinstein91}{Weinstein 91, p. 446}, \hyperlink{GBV}{Garcia-Bondia \& Varilly 94, section V}, \hyperlink{EH}{Hawkins 06}).

See at \emph{[[star product]]} \href{star+product#PolarizedSymplecticGroupoidConvolutionProductOfSymplecticVectorSpaceIsMoyalStarProduct}{this prop.} for the \textbf{proof}; and see at \emph{\href{geometric+quantization+of+symplectic+groupoids#MoyalQuantizationofPoissonVectorSpace}{geometric quantization of symplectic groupoids -- Examples -- Moyal quantization}} for more.

\hypertarget{References}{}\subsection*{{References}}\label{References}

The Moyal product was introduced independently in

\begin{itemize}%
\item [[Hilbrand Groenewold]], \emph{On the Principles of elementary quantum mechanics}, Physica,12 (1946) pp. 405-460.


\item [[José Moyal]], \emph{Quantum mechanics as a statistical theory}. Mathematical Proceedings of the Cambridge Philosophical Society 45: 99 (1949)



\end{itemize}
The integral expression (prop. \ref{IntegralRepresentationOfStarProduct}) is apparently due to

\begin{itemize}%
\item George A. Baker, \emph{Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space}, Jr. Phys. Rev. 109, 2198 – Published 15 March 1958 (\href{https://doi.org/10.1103/PhysRev.109.2198}{doi:10.1103/PhysRev.109.2198})

\end{itemize}
General accounts include

\begin{itemize}%
\item D. B. Fairlie, \emph{Moyal Brackets, Star Products and the Generalised Wigner Function} (\href{https://arxiv.org/abs/hep-th/9806198}{arXiv:hep-th/9806198})


\item Maciej Blaszak, Ziemowit Domanski, \emph{Maciej Blaszak, Ziemowit Domanski} (\href{https://arxiv.org/abs/1009.0150}{arXiv:1009.0150})



\end{itemize}
The understanding of the Moyal product as the [[polarization|polarized]] [[groupoid convolution algebra]] of the corresponding [[symplectic groupoid]], hence as an example of [[geometric quantization of symplectic groupoids]] had been suggested without proof in

\begin{itemize}%
\item [[Alan Weinstein]], p. 446 in P. Donato et al. (eds.) \emph{Symplectic Geometry and Mathematical Physics, (Birkh\"a{}user, Basel, 1991);}

\end{itemize}
and was proven in detail in

\begin{itemize}%
\item [[José Gracia-Bondia]], [[Joseph Varilly]], \emph{From geometric quantization to Moyal quantization}, J. Math. Phys. 36 (1995) 2691-2701 (\href{http://arxiv.org/abs/hep-th/9406170}{arXiv:hep-th/9406170})

\end{itemize}
In a broader context this was reconsidered in

\begin{itemize}%
\item [[Eli Hawkins]], example 6.2 of \emph{A groupoid approach to quantization}, J. Symplectic Geom. Volume 6, Number 1 (2008), 61-125. (\href{http://arxiv.org/abs/math.SG/0612363}{arXiv:math.SG/0612363})

\end{itemize}
The observation that Moyal deformation quantization applied to the [[Peierls-Poisson bracket]] yields the [[Wick algebra]] quantization of [[free field theories]] is due to

\begin{itemize}%
\item J. Dito, \emph{Star-product approach to quantum field theory: The free scalar field}. Letters in Mathematical Physics, 20(2):125--134, 1990 (\href{https://inspirehep.net/record/303898/}{spire})

\end{itemize}
and was amplified in the broader context of [[perturbative AQFT]] in

\begin{itemize}%
\item [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Perturbative algebraic field theory, and deformation quantization}, in [[Roberto Longo]] (ed.), \emph{Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects}, volume 30 of Fields Institute Communications, pages 151--160. American Mathematical Society, 2001

\end{itemize}
That moreover the corresponding [[Fedosov deformation quantization]] based on this free field theory star product yields the [[causal perturbation theory]] quantization of interacting field theories is due to

\begin{itemize}%
\item [[Giovanni Collini]], \emph{Fedosov Quantization and Perturbative Quantum Field Theory} (\href{https://arxiv.org/abs/1603.09626}{arXiv:1603.09626})

\end{itemize}
[[!redirects Moyal product]] [[!redirects Moyal products]]

[[!redirects Moyal star-product]] [[!redirects Moyal star-products]]

[[!redirects Moyal star product]] [[!redirects Moyal star products]]

[[!redirects Moyal \emph{-product]] [[!redirects Moyal}-products]]

[[!redirects Moyal star]]

[[!redirects Moyal deformation quantization]]



\end{document}