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\begin{document}

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\section*{Wick&#39;s lemma}

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

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - 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{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\emph{Wick's lemma} is a [[combinatorics|combinatorial]] formula for the product in [[associative algebras]] that appear in the context of [[free field theory]].

From the point of view of [[path integral quantization]], Wick's lemma is about the [[moments]] of [[Gaussian probability distributions]]. See at \emph{[[Feynman diagram]]} for more on this.

From the point of view of [[causal perturbation theory]] Wick's lemma expresses the [[Moyal deformation quantization]] of a [[free field theory]] ([[Wick algebras]]) in terms of [[operator products]] on.

[[!include Wick algebra -- table]]

From the point of view of [[BV-quantization]] Wick's lemma arises as a consequence of the [[homological perturbation lemma]] (\hyperlink{Gwilliam}{Gwilliam, section 2.3}).

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

Let $\mathcal{W}$ be the [[Wick algebra]] of the [[free field theory|free]] [[scalar field]], hence the space of [[microcausal observables]] with product the [[star product]] induced by the [[Wightman propagator]]:

\begin{displaymath}
\mathcal{W}
  \;\coloneqq\;
  \left(
    PolyObs(E,\mathbf{L})_{mc}, \star_{\Delta_H}
  \right)
\end{displaymath}
Then the evident map from $\mathcal{W}$ to [[linear operators]] on the [[Fock space]] equipped with their [[operator product]] $\circ$ is an [[associative algebra]] [[isomorphism]] onto its image:

\begin{displaymath}
\left(
    PolyObs(E,\mathbf{L})_{mc}, \star_{\Delta_H}
  \right)
    \underoverset{\simeq}{\text{Wick's lemma}}{\hookrightarrow}  
  \left(
    End(FockSpace), \circ
  \right)
\end{displaymath}
(\hyperlink{Duetsch18}{Dütsch 18, theorem 2.17}, following \hyperlink{DuetschFredenhagen00}{Dütsch-Fredenhagen 00,, pages 10-11}, \hyperlink{DuetschFredenhagen01}{Dütsch-Fredenhagen 01})

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

\begin{itemize}%
\item [[perturbation theory]]


\item [[Feynman diagram]]



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

Original articles include

\begin{itemize}%
\item [[Gian-Carlo Wick]], \emph{The evaluation of the collision matrix}, Phys. Rev. 80, 268-272 (1950)


\item [[Klaus Hepp]], \emph{Th\'e{}orie de la Renormalisation} Lect. Notes in Physics, Springer 1969


\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Rainer Verch]], theorem 2.4 in \emph{The generally covariant locality principle -- A new paradigm for local quantum physics}, Commun.Math.Phys.237:31-68, 2003 (\href{https://arxiv.org/abs/math-ph/0112041}{arXiv:math-ph/0112041})



\end{itemize}
The interpretation as an algebra isomorphism to the [[star product]] with respect to the [[Wightman propagator]] is made explicit in

\begin{itemize}%
\item [[Michael Dütsch]], [[Klaus Fredenhagen]], pages 10-11 of \emph{Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion}, Commun.Math.Phys. 219 (2001) 5-30 (\href{https://arxiv.org/abs/hep-th/0001129}{arXiv:hep-th/0001129})


\item [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Perturbative algebraic quantum field theory and deformation quantization}, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) (\href{http://xxx.uni-augsburg.de/abs/hep-th/0101079}{arXiv:hep-th/0101079})



\end{itemize}
and \hyperlink{Duetsch18}{Dütsch 18, theorem 2.17}

Textbook accounts include

\begin{itemize}%
\item [[Günter Scharf]], theorem 3.1 in \emph{[[Finite Quantum Electrodynamics -- The Causal Approach]]}, Berlin: Springer-Verlag, 1995, 2nd edition


\item [[Michael Dütsch]], E.26 - E.30 in \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Wick%27s_theorem}{Wick's theorem}}

\end{itemize}
Discussion of Wick's lemma as a consequence of the [[homological perturbation lemma]] for [[BV-complexes]] is in

\begin{itemize}%
\item [[Owen Gwilliam]], section 2.3 \emph{Factorization algebras and free field theories} PhD thesis (\href{http://math.berkeley.edu/~gwilliam/thesis.pdf}{pdf})

\end{itemize}
[[!redirects Wick lemma]] [[!redirects Wick's theorem]] [[!redirects Wick theorem]]



\end{document}