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\begin{document}

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\section*{Fedosov&#39;s deformation quantization}

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

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

[[!include symplectic geometry - 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{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Fedosov's method (\hyperlink{Fedosov94}{Fedosov 94}) is a prescription for constructing [[formal deformation quantizations]] of the [[algebra of functions|algebra of]] [[smooth functions]] on any [[symplectic manifold]] $(X,\omega)$.

In particular this establishes the existence of formal deformation quantizations of all [[symplectic manifolds]]. While \href{deformation+quantization#Kontsevich97}{Kontsevich 97} more generally proves the existence of deformation quantization of all [[Poisson manifolds]] of finite dimension, Fedosov's method, in a variant applicable to [[almost Kähler structures]] (\hyperlink{KarabegovSchlichenmaier01}{Karabegov-Schlichenmaier 01}) generalizes to infinite-dimensional symplectic manifolds as they appear in [[local field theory]], where it yields the quantization to [[perturbative quantum field theory]] equivalent to the method of [[causal perturbation theory]] (\hyperlink{Collini16}{Collini 16}). (Kontsevich's deformation quantization is however not compatible with field theory (\hyperlink{HawkinsRejzner16}{Hawkins-Rejzner 16, section 5.3.2}).) For more on this see at \emph{[[locally covariant perturbative quantum field theory]]}.

Fedosov's method proceeds by the following steps:

\begin{enumerate}%
\item For each point $x \in X$ regard the [[tangent space]] $T_x X$ as a [[symplectic vector space]] $(T_x X, \omega_x)$ and consider its [[Moyal deformation quantization]] $\mathcal{W}_x \coloneqq A_{Moy}(T_x X, \omega_x)$. In the context of [[local field theory]] these Moyal algebras are the [[Wick algebras]] of the underlying [[free field theory]].

The union of these yelds an [[associative algebra]]-[[fiber bundle]] $\mathcal{W} \to X$. From the fiber-wise product in $A_{Moy}(T_x X, \omega_x)$ the [[space of sections]] $\Gamma_X(\mathcal{W})$ inherits itself the structure of an [[associative algebra]].


\item Find a [[flat connection]] $\nabla_{Fed}$ on $\mathcal{W}$ which respects the associative algebra structure, and such that there is a [[linear isomorphism]]

\begin{displaymath}
\Gamma_X(\mathcal{W}) \supset  ker(\nabla_{Fed}) \simeq C^\infty(X) [ [ \hbar ] ]
\end{displaymath}
between the [[covariantly constant sections]] of the algebra bundle and the algebra of smooth functions on $X$ with a [[formal power series|formal variable]] $\hbar$ adjoined.

Such $\nabla_{Fed}$ exists, (non-uniquely), it is called a \emph{Fedosov connection}.


\item By this linear isomorphism the algebra structure on $\Gamma_X(\mathcal{W})$ induces an [[associative algebra]] structure on $C^\infty(X)[ [ \hbar ] ]$, and this turns out to be a [[formal deformation quantization]] of $(X,\omega)$.

In the context of [[local field theory]], this deformation quantization is equivalent to the result of quantization via [[causal perturbation theory]] (\hyperlink{Collini16}{Collini 16}).



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

\begin{itemize}%
\item [[almost Kähler geometric quantization]]

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

The method is due to

\begin{itemize}%
\item [[Boris Fedosov]], \emph{A simple geometrical construction of deformation quantization} Journal of Differential Geometry, 40(2):213--238, 1994 (\href{https://projecteuclid.org/euclid.jdg/1214455536}{Euclid})

\end{itemize}
Its generalization to [[almost Kähler structures]] is due to

\begin{itemize}%
\item V. Karabegov, M. Schlichenmaier, \emph{Almost-K\"a{}hler deformation quantization}, Letters in Mathematical Physics, 57(2):135--148, 2001 (\href{http://www.atlantis-press.com/php/download_paper.php?id=540}{proceedings pdf})

\end{itemize}
The observation that the construction of [[perturbative quantum field theory]] via [[causal perturbation theory]] is equivalent to Fedosov quantization 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}
Discussion showing that this generalization to field theory is not given by Kontsevich deformation quantization is in

\begin{itemize}%
\item [[Eli Hawkins]], [[Kasia Rejzner]], section 5.3.2 of \emph{The Star Product in Interacting Quantum Field Theory} (\href{https://arxiv.org/abs/1612.09157}{arXiv:1612.09157})

\end{itemize}
[[!redirects Fedosov deformation quantization]] [[!redirects Fedosov deformation quantizations]]

[[!redirects Fedosov quantization]] [[!redirects Fedosov quantizations]]

[[!redirects Fedosov's formal deformation quantization]] [[!redirects Fedosov's formal deformation quantizations]]



\end{document}