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\begin{document}

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\section*{star product}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

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

[[!include geometric quantization - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{on_finitedimensional_vector_spaces}{On finite-dimensional vector spaces}\dotfill \pageref*{on_finitedimensional_vector_spaces} \linebreak
\noindent\hyperlink{on_polynomial_observables_in_field_theory}{On polynomial observables in field theory}\dotfill \pageref*{on_polynomial_observables_in_field_theory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{equivalences_of_star_products}{Equivalences of star products}\dotfill \pageref*{equivalences_of_star_products} \linebreak
\noindent\hyperlink{integral_representations}{Integral representations}\dotfill \pageref*{integral_representations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{general_examples}{General examples}\dotfill \pageref*{general_examples} \linebreak
\noindent\hyperlink{wick_algebras_of_normal_ordered_products}{Wick algebras of normal ordered products}\dotfill \pageref*{wick_algebras_of_normal_ordered_products} \linebreak
\noindent\hyperlink{finite_dimensional}{Finite dimensional}\dotfill \pageref*{finite_dimensional} \linebreak
\noindent\hyperlink{infinitedimensional}{Infinite-dimensional}\dotfill \pageref*{infinitedimensional} \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}

In [[deformation quantization]] of [[Poisson manifolds]] the commutative product $\cdot$ of the [[commutative algebra|commutative]] [[algebra of functions]] is replaced by a noncommutative associative product. This is often called a \textbf{star product} and denoted ``$\star$''.

An [[archetypal example]] is the \emph{[[Moyal star product]]} (example \ref{MoyalStarProduct} below) that deforms the function algebra on a Poisson vector space, and often ``star product'' is by default understood to be a Moyal star product. Indeed every star product induced from a constant rank-2 tensor on a vector space is [[isomorphism|isomorphic]] to a Moyal star product (prop. \ref{SymmetricContribution} below).

More recently also nonassociative ``star products'' have been proposed to be of interest.

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

\hypertarget{on_finitedimensional_vector_spaces}{}\subsubsection*{{On finite-dimensional vector spaces}}\label{on_finitedimensional_vector_spaces}

Let $V$ be a [[finite dimensional vector space]] and let $\pi \in V \otimes V$ be an element of the [[tensor product]] (not necessarily skew symmetric at the moment).

We may canonically regard $V$ as a [[smooth manifold]], in which case $\pi$ is canonically regarded as a constant rank-2 [[tensor]]. As such it has a canonical [[action]] by forming [[derivatives]] on the tensor product of the space of [[smooth functions]]:

\begin{displaymath}
\pi
  \;\colon\;
  C^\infty(V) \otimes C^\infty(V)
  \longrightarrow
  C^\infty(V) \otimes C^\infty(V)
  \,.
\end{displaymath}
If $\{\partial_i\}$ is a [[linear basis]] for $V$, identified, as before, with a basis for $\Gamma(T V)$, then in this basis this operation reads

\begin{displaymath}
\pi(f \otimes g)
  \;=\;
  \pi^{i j}
  (\partial_i f) \otimes (\partial_j g)
  \,,
\end{displaymath}
where $\partial_i f \coloneqq \frac{\partial f}{\partial x^i}$ denotes the [[partial derivative]] of the [[smooth function]] $f$ along the $i$th [[coordinate]], and where we use the [[Einstein summation convention]].

For emphasis we write

\begin{displaymath}
\itexarray{
    C^\infty(V) \otimes C^\infty(V)
      &\overset{prod}{\longrightarrow}&
    C^\infty(V)
    \\
    f \otimes g &\mapsto& f \cdot g
  }
\end{displaymath}
for the pointwise product of smooth functions.

\begin{defn}
\label{StarPoduct}\hypertarget{StarPoduct}{}
\textbf{([[star product]] induced by constant rank-2 [[tensor]])}

Given $(V,\pi)$ as above, then the \emph{[[star product]]} induced by $\pi$ on the [[formal power series algebra]] $C^\infty(V) [ [\hbar] ]$ in a formal variable $\hbar$ (``[[Planck's constant]]'') with [[coefficients]] in the [[smooth functions]] on $V$ is the linear map

\begin{displaymath}
(-) \star_\pi (-)
  \;\colon\;
  C^\infty(V)[ [ \hbar ] ] \otimes C^\infty(V)[ [ \hbar ] ]
    \longrightarrow
  C^\infty(V)[ [\hbar] ]
\end{displaymath}
given by

\begin{displaymath}
(-) \star_\pi (-)
  \;\coloneqq\;
  prod
    \circ \exp\left(
       \hbar \pi^{i j} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}
  \right)
\end{displaymath}
Hence

\begin{displaymath}
f \star_\pi g
  \;\coloneqq\;
  1
  +
  \hbar \pi^{i j}
  \frac{\partial f}{\partial x^i} \cdot \frac{\partial g}{\partial x^j}
  +
  \hbar^2
  \tfrac{ 1 }{2}
  \pi^{i j}
  \pi^{k l}
  \frac{\partial^2 f}{\partial x^{i} \partial x^{k}}
    \cdot
  \frac{\partial^2 g}{\partial x^{j} \partial x^{l}}
  +
  \cdots
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{AssociativeAndUnitalStarProduct}\hypertarget{AssociativeAndUnitalStarProduct}{}
\textbf{([[star product]] is [[associativity|associative]] and [[unitality|unital]])}

Given $(V,\pi)$ as above, then the star product $(-) \star_\pi (-)$ from def. \ref{StarPoduct} is [[associativity|associative]] and [[unitality|unital]] with unit the [[constant function]] $1 \in C^\infty(V) \hookrightarrow C^\infty(V)[ [ \hbar ] ]$.

Hence the [[vector space]] $C^\infty(V)$ equipped with the [[star product]] $\pi$ is a [[unital algebra|unital]] [[associative algebra]].

\end{prop}
\begin{proof}
Observe that the [[product rule]] of [[differentiation]] says that

\begin{displaymath}
\partial_i \circ prod
    =
   prod \circ ( \partial_i \otimes id \;+\; id \otimes \partial_i  )
  \,.
\end{displaymath}
Using this we compute as follows:

\begin{displaymath}
\begin{aligned}
  (f \star_\pi  g) \star_\pi h
  & =
  prod \circ
  \exp( \pi^{i j} \partial_i \otimes \partial_j )  \circ
  \left(
    \left( prod \circ \exp( \pi^{k l} \partial_k \otimes \partial_l ) \right) \otimes id
  \right)
  (f \otimes g \otimes g)
  \\
  & =
  prod \circ
  \exp( \pi^{i j} \partial_i \otimes \partial_j ) \circ
  (prod \otimes id)
  \circ
  \left(
    \exp( \pi^{k l} \partial_k \otimes \partial_l )  \otimes id
  \right)
  (f \otimes g \otimes g)
  \\
  & =
  prod \circ
  (prod \otimes id) \circ
  \exp( \pi^{i j}  ( \partial_i \otimes id \otimes \partial_j +id \otimes \partial_i \otimes \partial_j   )
  \circ
  \exp( \pi^{k l} \partial_k \otimes \partial_l )  \otimes id
  (f \otimes g \otimes g)
  \\
  & =
  prod \circ
  (prod \otimes id) \circ
  \exp( \pi^{i j}   \partial_i \otimes id \otimes \partial_j   )
  \circ
  \exp( \pi^{i j}  id \otimes \partial_i \otimes \partial_j )
  \circ
  \exp( \pi^{k l} \partial_k \otimes \partial_l \otimes id )
  (f \otimes g \otimes g)
  \\
  & =
  prod_3
  \circ
  \exp( \pi^{i j} ( \partial_i \otimes \partial_j \otimes id  +  \partial_i \otimes id \otimes \partial_j + id \otimes \partial_i \otimes \partial_j)   )
  \end{aligned}
\end{displaymath}
In the last line we used that the ordinary pointwise product of functions is associative, and wrote $prod_3 \colon C^\infty(V) \otimes C^\infty(V) \otimes C^\infty(V) \to C^\infty(V)$ for the unique pointwise product of three functions.

The last expression above is manifestly independent of the choice of order of the arguments in the triple star product, and hence it is clear that an analogous computation yields

\begin{displaymath}
\cdots = f \star_\pi (g \star_\pi h)
  \,.
\end{displaymath}
\end{proof}
\hypertarget{on_polynomial_observables_in_field_theory}{}\subsubsection*{{On polynomial observables in field theory}}\label{on_polynomial_observables_in_field_theory}

\begin{defn}
\label{PropagatorStarProduct}\hypertarget{PropagatorStarProduct}{}
\textbf{([[star products]] on [[regular polynomial observables]] induced from [[propagators]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[field bundle]] $E \overset{fb}{\to} \Sigma$, and let $\Delta \in \Gamma'_\Sigma((E \boxtimes E)^\ast)$ be a [[distribution of two variables]] on [[field histories]].

On the [[off-shell]] [[regular polynomial observables]] with a [[formal power series|formal paramater]] $\hbar$ adjoined consider the bilinear map

\begin{displaymath}
PolyObs(E)_{reg}[ [\hbar] ]
    \otimes
  PolyObs(E)_{reg} [ [ \hbar ] ]
    \overset{\star_{\Delta}}{\longrightarrow} 
  PolyObs(E)_{reg}[ [\hbar] ]
\end{displaymath}
given as in def. \ref{StarPoduct}, but with [[partial derivatives]] replaced by [[functional derivatives]]

\begin{displaymath}
A_1 \star_{\Delta} A_2
  \;\coloneqq\;
  ((-)\cdot(-))
  \circ
  \exp\left(
    \int_\Sigma \Delta^{a b}(x,y)
    \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
      \otimes
    \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
  \right)
  (A_1 \otimes A_2)
\end{displaymath}
As in prop. \ref{AssociativeAndUnitalStarProduct} this defines a [[unital algebra|unital]] and [[associative algebra]] [[structure]].

If the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]] induced by the [[Lagrangian density]] $\mathbf{L}$ are [[Green hyperbolic differential equations]] and if $\Delta$ is a [[propagator]] for these [[differential equations]], then this [[star product]] algebra descends to the [[on-shell]] [[regular polynomial observables]]

\begin{displaymath}
PolyObs(E,\mathbf{L})_{reg}[ [\hbar] ]
    \otimes
  PolyObs(E, \mathbf{L})_{reg} [ [ \hbar ] ]
    \overset{\star_{\Delta}}{\longrightarrow} 
  PolyObs(E, \mathbf{L})_{reg}[ [\hbar] ]
  \,.
\end{displaymath}
\end{defn}
(\href{deformation+quantization#Dito90}{Dito 90}, \href{deformation+quantization#DuetschFredenhagen00}{Dütsch-Fredenhagen 00} \href{deformation+quantization#DuetschFredenhagen01}{Dütsch-Fredenhagen 01}, \href{deformation+quantization#HirschfeldHenselder02}{Hirshfeld-Henselder 02})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{equivalences_of_star_products}{}\subsubsection*{{Equivalences of star products}}\label{equivalences_of_star_products}

\begin{prop}
\label{SymmetricContribution}\hypertarget{SymmetricContribution}{}
\textbf{(shift by symmetric contribution is [[isomorphism]] of [[star products]])}

Let $V$ be a vector space, $\pi \in V \otimes V$ a rank-2 [[tensor]] and $\alpha \in Sym(V \otimes V)$ a \emph{symmetric} rank-2 tensor.

Then the [[linear map]]

\begin{displaymath}
\itexarray{
    C^\infty(V)
     &\overset{\exp\left(\tfrac{1}{2}\alpha \right)}{\longrightarrow}&
    C^\infty(V)
    \\
    f 
      &\mapsto& 
    \exp\left( \tfrac{1}{2}\hbar \alpha^{i j} \partial_i \partial_j \right) f
  }
\end{displaymath}
constitutes an [[isomorphism]] of star product algebras (prop. \ref{AssociativeAndUnitalStarProduct}) of the form

\begin{displaymath}
\exp\left(\hbar\tfrac{1}{2}\hbar\alpha \right)
   \;\colon\;
  (C^\infty(V)[ [\hbar] ], \star_{\pi})
    \overset{\simeq}{\longrightarrow}
  (C^\infty(V))[ [\hbar] ], \star_{\pi + \alpha})
  \,,
\end{displaymath}
hence identifying the star product induced from $\pi$ with that induced from $\pi + \alpha$.

In particular every star product algebra $(C^\infty(V)[ [\hbar] ],\star_\pi)$ is isomorphic to a Moyal star product algebra $\star_{\tfrac{1}{2}\pi}$ (example \ref{MoyalStarProduct}) with $\tfrac{1}{2}\pi_{skew}^{i j} = \tfrac{1}{2}(\pi^{i j} - \pi^{j i})$ the skew-symmetric part of $\pi$, this isomorphism being exhibited by the symmetric part $2\alpha^{i j} = \tfrac{1}{2}(\pi^{i j} + \pi^{j i})$.

\end{prop}
\begin{proof}
We need to show that

\begin{displaymath}
\itexarray{
    C^\infty(V)[ [\hbar] ] \otimes  C^\infty(V)[ [\hbar] ]
    &
     \overset{ 
       \exp\left( \tfrac{1}{2}\hbar \alpha \right)    
       \otimes
       \exp\left( \tfrac{1}{2}\hbar \alpha \right)    
     }{\longrightarrow}&
    C^\infty(V)[ [\hbar] ] \otimes  C^\infty(V)[ [\hbar] ]
     \\
     {}^{\mathllap{\star_{\pi}}}\downarrow 
       && 
    \downarrow^{\mathrlap{\star_{\pi + \alpha}}}
    \\
    C^\infty(V)[ [\hbar] ]
      &\underset{\exp\left( \tfrac{1}{2} \alpha \right) }{\longrightarrow}&
    C^\infty(V)[ [\hbar] ]
  }
\end{displaymath}
hence that

\begin{displaymath}
prod \circ
  \exp( \hbar(\pi + \alpha) )
  \circ
  \left(
    \exp\left( \tfrac{1}{2}\alpha\right)
    \otimes
    \exp\left( \tfrac{1}{2}\alpha \right)
  \right)
  \;=\;
  \exp\left( \tfrac{1}{2}\alpha \right)
  \circ
   prod \circ \exp( \pi )
  \,.
\end{displaymath}
To this end, observe that the [[product rule]] of [[differentiation]] applied twice in a row implies that

\begin{displaymath}
\partial_i \partial_j \circ prod
  \;=\;
  prod \circ
  \left(
     (\partial_i \partial_j)  \otimes id
     +
     id \otimes (\partial_i \partial_j)
     +
     \partial_i \otimes \partial_j
     +
     \partial_j \otimes \partial_i
  \right)
  \,.
\end{displaymath}
Using this we compute

\begin{displaymath}
\begin{aligned}
    \exp\left( \hbar\tfrac{1}{2}\alpha^{i j} \partial_i \partial_j \right)
    \circ
    prod
    \circ
    \exp( \hbar \pi^{i j} \partial_{i} \otimes \partial_j )
    & =
    prod
      \circ
      \exp\left( 
        \hbar \tfrac{1}{2}\alpha^{i j}
        \left(
           (\partial_i \partial_j) \otimes id
           +
           id \otimes (\partial_i \partial_j)
           +
           \partial_i \otimes \partial_j
           +
           \partial_j \otimes \partial_i
        \right)
      \right)
      \circ
    \exp( \hbar \pi^{i j} \partial_{k} \otimes \partial_l )
    \\
    &
    =
    prod
    \circ
    \exp\left(
       \hbar (\pi^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j
    \right)
    \circ
    \exp\left(
      \hbar \tfrac{1}{2} \alpha^{i j} (\partial_i \partial_j) \otimes id
      \hbar \tfrac{1}{2} \alpha^{i j} id \otimes (\partial_i \partial_j)
    \right)
    \\
    & =
    prod
    \circ
    \exp\left(
       \hbar (\pi^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j
    \right)
    \circ
    \left(
      \exp\left( \tfrac{1}{2} \hbar \alpha \right)
        \otimes
      \exp\left( \tfrac{1}{2} \hbar \alpha \right)
    \right)
  \end{aligned}
\end{displaymath}
\end{proof}
\hypertarget{integral_representations}{}\subsubsection*{{Integral representations}}\label{integral_representations}

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

If $\pi$ skew-symmetric and invertible, in that there exists $\omega \in V^\ast \otimes V^\ast$ with $\pi^{i j}\omega_{j k} = \delta^i_k$, and if the functions $f,g$ admit [[Fourier analysis]] (are [[functions with rapidly decreasing partial derivatives]]), then their [[star product]] (def. \ref{StarPoduct}) is equivalently given by the following [[integral]] expression:

\begin{displaymath}
\begin{aligned}
  \left(f \star_\pi 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}
(\href{Moyal+deformation+quantization#Baker58}{Baker 58})

\begin{proof}
We compute as follows:

\begin{displaymath}
\begin{aligned}
  \left(f \star_\pi g\right)(x)
  & \coloneqq
  prod
    \circ \exp\left(
       \hbar \pi^{i j} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}
  \right)(f, g)
  \\
  & =
  \frac{1}{(2 \pi)^{2n}}
  \frac{1}{(2 \pi)^{2n}}
  \int
  \int
    \underbrace{
      e^{ i \hbar \pi(k,q) }
    }
    \underbrace{
      e^{i k \cdot (x-y)}
      f(y)
    }
    \underbrace{
      e^{i q \cdot (x- \tilde y)}
      g(\tilde y)
    }
   \,
   d^{2 n} k
   \,
   d^{2 n} q
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
   \\
   & =
  \frac{1}{(2 \pi)^{2n}}
  \int
    \delta\left( x - \tilde y + \hbar \pi \cdot k \right)
    e^{i k \cdot (x-y)}
    f(y)
    g(\tilde y)
   \,
   d^{2 n} k
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
   \\
   & = 
  \frac{1}{(2 \pi)^{2n}}
  \int
    \delta\left( x - \tilde y + z \right)
    e^{ \tfrac{i}{\hbar} \omega(z, (x-y))}
    f(y)
    g(\tilde y)
   \,
   d^{2 n} z
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
  \\
  & =
  \frac{(det(\pi)^{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}
Here in the first step we expressed $f$ and $g$ both by their [[Fourier transform]] (inserting the Fourier expression of the [[delta distribution]] from \href{Dirac+distribution#FourierTransformOfDeltaDistribution}{this example}) and used that under this transformation the [[partial derivative]] $\pi^{a b} \frac{\partial}{\partial\phi^a}{\frac{\partial}{\phi^b}}$ turns into the product with $i \pi^{i j} k_i k_j$ (\href{Fourier+transform#BasicPropertiesOfFourierTransformOverCartesianSpaces}{this prop.}). Then we identified again the Fourier-expansion of a [[delta distribution]] and finally we applied the [[change of integration variables]] $k = \tfrac{1}{\hbar}\omega \cdot z$ and then evaluated the [[delta distribution]].

\end{proof}
Next we express this as the [[groupoid convolution product]] of polarized sections of the [[symplectic groupoid]]. To this end, we first need the following definnition:

\begin{defn}
\label{SymplecticGroupoidOfSymplecticVectorSpace}\hypertarget{SymplecticGroupoidOfSymplecticVectorSpace}{}
\textbf{([[symplectic groupoid]] of [[symplectic vector space]])}

Assume that $\pi$ is the inverse of a [[symplectic form]] $\omega$ on $\mathbb{R}^{2n}$. Then the [[Cartesian product]]

\begin{displaymath}
\itexarray{
    && \mathbb{R}^{2n} \times \mathbb{R}^{2n}
    \\
    & {}^{\mathllap{pr_1}}\swarrow && \searrow^{\mathrlap{pr_2}}
    \\
    \mathbb{R}^{2n} && && \mathbb{R}^{2n}
  }
\end{displaymath}
inherits the symplectic structure

\begin{displaymath}
\Omega
  \;\coloneqq\;
  \left(
    pr_1^\ast \omega - pr_2^\ast \omega
  \right)
\end{displaymath}
given by

\begin{displaymath}
\begin{aligned}
    \Omega 
    & = 
    \omega_{i j} d x^i \wedge d x^j - \omega_{i j} d y^i \wedge d y^j
    \\
    & =
    \omega_{i j} ( d x^i - d y^i ) \wedge ( d x^j +  d y^j )
  \end{aligned}
  \,.
\end{displaymath}
The [[pair groupoid]] on $\mathbb{R}^{2n}$ equipped with this [[symplectic form]] on its space of [[morphisms]] is a [[symplectic groupoid]].

A choice of potential form $\Theta$ for $\Omega$, hence with $\Omega = d \Theta$, is given by

\begin{displaymath}
\Theta 
   \coloneqq
   -\omega_{i j} ( x^i + y^i ) d (x^j - y^j) )
\end{displaymath}
Choosing the [[real polarization]] spanned by $\partial_{x^i} - \partial_{y^i}$ a polarized section is function $F = F(x,y)$ such that

\begin{displaymath}
\iota_{\partial_{x^j} - \partial_{y^j}}(d F - \tfrac{1}{i \hbar} \tfrac{1}{4} \Theta F) = 0
\end{displaymath}
hence

\begin{equation}
F(x,y) 
   = 
  f\left( \tfrac{x + y}{2} \right) e^{ \tfrac{1}{i \hbar} \omega\left(  \tfrac{x - y}{2} , \tfrac{x + y}{2} \right)}
  \,.
\label{PolarizedSectionOnMorphismsOfSymplecticGroupoid}\end{equation}
\end{defn}
\begin{prop}
\label{PolarizedSymplecticGroupoidConvolutionProductOfSymplecticVectorSpaceIsMoyalStarProduct}\hypertarget{PolarizedSymplecticGroupoidConvolutionProductOfSymplecticVectorSpaceIsMoyalStarProduct}{}
\textbf{(polarized [[symplectic groupoid|symplectic]] [[groupoid convolution product]] of [[symplectic vector space]] is given by [[Moyal star product]])}

Given a [[symplectic vector space]] $(\mathbb{R}^{2n}, \omega)$, then the [[groupoid convolution product]] on polarized sections \eqref{PolarizedSectionOnMorphismsOfSymplecticGroupoid} on its [[symplectic groupoid]] (def. \ref{SymplecticGroupoidOfSymplecticVectorSpace}), given by [[convolution product]] followed by averaging ([[integration]]) over the polarization [[fiber]], is given by the [[star product]] (def. \ref{StarPoduct}) for the corresponding [[Poisson tensor]] $\pi \coloneqq \omega^{-1}$, in that

\begin{displaymath}
\begin{aligned}
    \int \int 
      F(x,t) G(t,y) \, d^{2n} t \, d^{2n} (x-y)
    & = 
    (f \star_\pi g)((x+y)/2)
  \end{aligned} 
  \,.
\end{displaymath}
\end{prop}
(\href{Moyal+deformation+quantization#Weinstein91}{Weinstein 91, p. 446}, \href{Moyal+deformation+quantization#GBV}{Garcia-Bondia \& Varilly 94, section V})

\begin{proof}
We compute as follows:

\begin{displaymath}
\begin{aligned}
    \int \int 
      F(x,t) G(t,y) \, d^{2n} t \, d^{2n} (x-y)
    & \coloneqq
    \int \int f((x + t)/2) g( (t + y)/2 ) 
      e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( x-t, x+t ) + \tfrac{1}{i \hbar} \tfrac{1}{4} \omega(t-y, t + y)  }
    \, d^{2n} t
    \, d^{2n} (x-y)
    \\
    & = 
    \int \int 
      f(t/2) g( (t - (x - y))/2 ) 
      e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) + (x - y) - t, t ) + \tfrac{1}{i \hbar} \tfrac{1}{4} \omega(t-(x+y), t - (x-y))  }
    \, d^{2n} t
    \, d^{2n} (x-y)   
    \\
    & =  
    \int \int 
      f(t/2) g( \tilde t / 2) 
      e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) - \tilde t, t ) - \tfrac{1}{i \hbar} \tfrac{1}{4} \omega((x+y)-t, \tilde t)  }
    \, d^{2n} t
    \, d^{2n} \tilde t
    \\
    & = 
    \int \int 
      f(t) g( \tilde t ) 
      e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) - 2 \tilde t, 2 t ) - \tfrac{1}{ii \hbar} \tfrac{1}{4} \omega((x+y)- 2 t, 2 \tilde t)  }
    \, d^{2n} t
    \, d^{2n} \tilde t
    \\
    \\
    & = 
    \int \int
       f(t) g(\tilde t )
      e^{ \tfrac{1}{i \hbar} \omega\left( \tfrac{1}{2}(x+y) - \tilde t, \tfrac{1}{2}(x + y) - t  \right)}   
     \, d^{2n} t
     \, d^{2n} \tilde t
    \\
    & = 
    (f \star_\omega g)((x+y)/2)
  \end{aligned}
\end{displaymath}
The first line just unwinds the definition of polarized sections from def. \ref{SymplecticGroupoidOfSymplecticVectorSpace}, the following lines each implement a [[change of integration variables]] and fnally in the last line we used prop. \ref{IntegralRepresentationOfStarProduct}.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{general_examples}{}\subsubsection*{{General examples}}\label{general_examples}

Some examples of star products as in def. \ref{StarPoduct}:

\begin{example}
\label{}\hypertarget{}{}
If $\pi = 0$ in def. \ref{StarPoduct}, then the star product $\star_0 = \cdot$ is the plain pointwise product.

\end{example}
\begin{example}
\label{MoyalStarProduct}\hypertarget{MoyalStarProduct}{}
\textbf{([[Moyal star product]])}

If the tensor $\pi$ in def. \ref{StarPoduct} is skew-symmetric, it may be regarded as a constant [[Poisson tensor]] on the smooth manifold $V$. In this case $\star_{\tfrac{1}{2}\pi}$ is called a \emph{[[Moyal star product]]} and the star-product algebra $C^\infty(V)[ [\hbar] ], \star_\pi)$ is called the \emph{[[Moyal deformation quantization]]} of the [[Poisson manifold]] $(V,\pi)$.

\end{example}
\hypertarget{wick_algebras_of_normal_ordered_products}{}\subsubsection*{{Wick algebras of normal ordered products}}\label{wick_algebras_of_normal_ordered_products}

\hypertarget{finite_dimensional}{}\paragraph*{{Finite dimensional}}\label{finite_dimensional}

\begin{defn}
\label{AlmostKaehlerVectorSpace}\hypertarget{AlmostKaehlerVectorSpace}{}
\textbf{([[Kähler vector space]])}

An \emph{[[Kähler vector space]]} is a [[real vector space]] $V$ equipped with a [[linear complex structure]] $J$ as well as two [[bilinear forms]] $\omega, g \;\colon\; V \otimes_{\mathbb{R}} V \longrightarrow \mathbb{R}$ such that the following equivalent conditions hold:

\begin{enumerate}%
\item $\omega(J v, J w) = \omega(v,w)$ and $g(v,w) = \omega(v,J w)$;


\item with $V$ regarded as a [[smooth manifold]] and with $\omega, g$ regarded as constant [[tensors]], then $(V, \omega, g)$ is an [[almost Kähler manifold]].



\end{enumerate}
\end{defn}
\begin{example}
\label{StandardAlmostKaehlerVectorSpaces}\hypertarget{StandardAlmostKaehlerVectorSpaces}{}
\textbf{(standard [[Kähler vector spaces]])}

Let $V \coloneqq \mathbb{R}^2$ equipped with the [[complex structure]] $J$ which is given by the canonical identification $\mathbb{R}^2 \simeq \mathbb{C}$, hence, in terms of the canonical [[linear basis]] $(e_i)$ of $\mathbb{R}^2$, this is

\begin{displaymath}
J = (J^i{}_j) 
  \coloneqq 
  \left( 
    \itexarray{
      0 & -1
      \\
      1 & 0
    }
  \right)
  \,.
\end{displaymath}
Moreover let

\begin{displaymath}
\omega = (\omega_{i j}) \coloneqq \left( \itexarray{0 & 1 \\ -1 & 0} \right)
\end{displaymath}
and

\begin{displaymath}
g = (g_{i j}) \coloneqq \left( \itexarray{ 1 & 0 \\ 0 & 1}  \right)
  \,.
\end{displaymath}
Then $(V, J, \omega, g)$ is a [[Kähler vector space]] (def. \ref{AlmostKaehlerVectorSpace}).

The corresponding [[Kähler manifold]] is $\mathbb{R}^2$ regarded as a [[smooth manifold]] in the standard way and equipped with the [[bilinear forms]] $J, \omega g$ extended as constant rank-2 [[tensors]] over this manifold.

If we write

\begin{displaymath}
x,y \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}
\end{displaymath}
for the standard [[coordinate functions]] on $\mathbb{R}^2$ with

\begin{displaymath}
z \coloneqq x + i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
\end{displaymath}
and

\begin{displaymath}
\overline{z} \coloneqq x - i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
\end{displaymath}
for the corresponding complex coordinates, then this translates to

\begin{displaymath}
\omega
  \in
  \Omega^2(\mathbb{R}^2)
\end{displaymath}
being the [[differential 2-form]] given by

\begin{displaymath}
\begin{aligned}
    \omega
    & =
    d x \wedge d y
    \\
    & =
    \tfrac{1}{2i} d z \wedge d \overline{z}
  \end{aligned}
\end{displaymath}
and with [[Riemannian metric]] [[tensor]] given by

\begin{displaymath}
g = d x \otimes d x + d y \otimes d y
  \,.
\end{displaymath}
The [[Hermitian form]] is given by

\begin{displaymath}
\begin{aligned}
    h 
    & = 
    g - i \omega
    \\
    & = 
    d z \otimes d \overline{z}
  \end{aligned}
\end{displaymath}
\end{example}
(for more see at \emph{[[Kähler vector space]]} \href{Kähler+vector+space#StandardAlmostKaehlerVectorSpaces}{this example}).

\begin{defn}
\label{WickAlgebraOfAlmostKaehlerVectorSpace}\hypertarget{WickAlgebraOfAlmostKaehlerVectorSpace}{}
\textbf{([[Wick algebra]] of a [[Kähler vector space]])}

Let $(\mathbb{R}^{2n},\sigma, g)$ be a [[Kähler vector space]] (def. \ref{AlmostKaehlerVectorSpace}). Then its \emph{Wick algebra} is the [[formal power series]] vector space $\mathbb{C}[ [ \mathbb{R}^{2n}  ] ] [ [ \hbar ] ]$ equipped with the [[star product]] (def. \ref{StarPoduct}) which is given by the [[bilinear form]]

\begin{equation}
\pi \coloneqq \tfrac{i}{2} \omega^{-1} + \tfrac{1}{2} g^{-1}
  \,,
\label{InStarProductTensorInvertingHermitianForm}\end{equation}
hence:

\begin{displaymath}
\begin{aligned}
    A_1 \star_\pi A_2
    & \coloneqq
    ((-)\cdot (-)) \circ \exp \left( \hbar\underoverset{k_1, k_2 = 1}{2 n}{\sum}\pi^{a b} \partial_a \otimes  \partial_b \right)  (A_1 \otimes A_2)
    \\
    & =
    A_1 \cdot A_2 + \hbar \underoverset{k_1, k_2 = 1}{2n}{\sum}\pi^{k_1 k_2}(\partial_{k_1} A_1) \cdot (\partial_{k_2} A_2)
    + \cdots
  \end{aligned}
\end{displaymath}
\end{defn}
(e.g. \hyperlink{Collini16}{Collini 16, def. 1})

\begin{prop}
\label{StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra}\hypertarget{StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra}{}
\textbf{([[star product]] [[associative algebra|algebra]] of [[Kähler vector space]] is [[star-algebra]])}

Under [[complex conjugation]] the [[star product]] $\star_\pi$ of a [[Kähler vector space]] structure (def. \ref{WickAlgebraOfAlmostKaehlerVectorSpace}) is a [[star algebra]] in that for all $A_1, A_2 \in \mathbb{C}[ [\mathbb{R}^{2n}] ][ [\hbar] ]$ we have

\begin{displaymath}
\left( A_1 \star_\pi A_2 \right)^\ast
  \;=\;
  A_2^\ast \star_\pi A_1^\ast
\end{displaymath}
\end{prop}
\begin{proof}
This follows directly from that fact that in $\pi = \tfrac{i}{2} \omega^{-1} + \tfrac{1}{2} g^{-1}$ the [[imaginary part]] coincides with the skew-symmetric part, so that

\begin{displaymath}
\begin{aligned}
    (\pi^\ast)^{a b}
    & =
    -\tfrac{i}{2} (\omega^{-1})^{a b}
    + \tfrac{1}{2} (g^{-1})^{a b}
    \\
    & = 
    \tfrac{i}{2} (\omega^{-1})^{b a}
    +   
    \tfrac{1}{2} (g^{-1})^{b a}
    \\
    & = \pi^{b a}
    \,.
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{example}
\label{WickAlgebraOfASingleMode}\hypertarget{WickAlgebraOfASingleMode}{}
\textbf{([[Wick algebra]] of a single mode)}

Let $V \coloneqq \mathbb{R}^2 \simeq Span(\{x,y\})$ be the standard [[Kähler vector space]] according to example \ref{StandardAlmostKaehlerVectorSpaces}, with canonical coordinates denoted $x$ and $y$. We discuss its Wick algebra according to def. \ref{WickAlgebraOfAlmostKaehlerVectorSpace} and show that this reproduces the traditional definition of products of ``normal ordered'' operators as \hyperlink{Idea}{above}.

To that end, consider the complex linear combination of the coordinates to the canonical complex coordinates

\begin{displaymath}
z \;\coloneqq\; x + i y
  \phantom{AAA} 
    \text{and} 
  \phantom{AAA}
  \overline{z} \coloneqq x - i y
\end{displaymath}
which we use in the form

\begin{displaymath}
a^\ast \;\coloneqq\; \tfrac{1}{\sqrt{2}}(x + i y)
  \phantom{AAA}
  \text{and}
  \phantom{AAA}
  a \;\coloneqq\; \tfrac{1}{\sqrt{2}}(x - i y)
\end{displaymath}
(with ``$a$'' the traditional symbol for the \emph{[[amplitude]]} of a field mode).

Now

\begin{displaymath}
\omega^{-1}
  =
  \frac{\partial}{\partial y} \otimes \frac{\partial}{\partial x}  
  -
  \frac{\partial}{\partial x} \otimes \frac{\partial}{\partial y}
\end{displaymath}
\begin{displaymath}
g^{-1} 
  =
  \frac{\partial}{\partial x} 
  \otimes
  \frac{\partial}{\partial x} 
  +
  \frac{\partial}{\partial y} 
  \otimes
  \frac{\partial}{\partial y}
\end{displaymath}
so that with

\begin{displaymath}
\frac{\partial}{\partial z}
  =
  \tfrac{1}{2}
  \left(
     \frac{\partial}{\partial x}
     -i
     \frac{\partial}{\partial y}
  \right)
  \phantom{AAAA}
  \frac{\partial}{\partial \overline{z}}
  =
  \frac{1}{2}
  \left(
    \frac{\partial}{\partial x}
    +
    i
    \frac{\partial}{\partial y}
  \right)
\end{displaymath}
we get

\begin{displaymath}
\begin{aligned}
    \tfrac{i \hbar}{2}\omega^{-1} + \tfrac{\hbar}{2} g^{-1}
    & =
    2 \hbar
    \frac{\partial}{\partial \overline{z}}
      \otimes 
    \frac{\partial}{\partial z}
    \\
    & =
    \hbar \frac{\partial}{\partial a} \otimes \frac{\partial }{\partial a^\ast}
  \end{aligned}
\end{displaymath}
Using this, we find the [[star product]]

\begin{displaymath}
A \star_\pi B
  \;=\;
  prod \circ \exp\left( \hbar \frac{\partial}{\partial a} \otimes \frac{\partial }{\partial a^\ast}  \right)
\end{displaymath}
to be as follows (where we write $(-)\cdot (-)$ for the plain commutative product in the [[formal power series algebra]]):

\begin{displaymath}
\begin{aligned}
    a \star_\pi a & = a \cdot a
    \\
    a^\ast \star_\pi a^\ast & = a^\ast \cdot a^\ast
    \\
    a^\ast \star_\pi a & = a^\ast \cdot a
    \\
    a \star_\pi a^\ast
    & =
    a \cdot a^\ast + \hbar
  \end{aligned}
\end{displaymath}
and so forth, for instance

\begin{displaymath}
\itexarray{
    (a \cdot a ) \star_\pi (a^\ast \cdot a^\ast)
    & =
    a^\ast \cdot a^\ast \cdot a \cdot a
    +
    4 \hbar a^\ast \cdot a
    +   
    \hbar^2
  }
\end{displaymath}
If we instead indicate the commutative pointwise product by colons and the star product by plain juxtaposition

\begin{displaymath}
:f g: \;\coloneqq\; f \cdot g
  \phantom{AAAA}
  f g \;\coloneqq\; f \star_\pi
\end{displaymath}
then this reads

\begin{displaymath}
\itexarray{
    :a a: \, :a^\ast a^\ast:
    & =
    : a^\ast a^\ast a  a :
    +
    4 \hbar \, : a^\ast a :
    +   
    \hbar^2
  }
\end{displaymath}
This is the way the \emph{[[Wick algebra]]} with its \emph{[[operator product]]} $\star_\pi$ and \emph{[[normal-ordered product]]} $:-:$ is traditionally presented.

\end{example}
\hypertarget{infinitedimensional}{}\paragraph*{{Infinite-dimensional}}\label{infinitedimensional}

\begin{example}
\label{WickAlgebras}\hypertarget{WickAlgebras}{}
\textbf{([[normal-ordered products]] and [[time-ordered products]])}

In cases where $V$ is \emph{not} [[finite number|finite]] [[dimension|dimensional]], but for instance a [[space of sections]] of a linear [[field bundle]], the expression involved in the definition of the star product in def. \ref{StarPoduct} still may be interpreted as [[products of distributions]], but these only exist if the [[wavefront sets]] satisfy [[Hörmander's criterion]].

In these situations it happens that the star product for some Poisson tensor $\tfrac{1}{2}\pi$ is not defined due to [[wavefront set]] collision (violation of [[Hörmander's criterion]]), but the star product $\star_{\tfrac{1}{2}\pi + \alpha}$ for the tensor $\tfrac{1}{2}\pi + \alpha$ shifted by a symmetric contribition $\alpha$ as in prop. \ref{SymmetricContribution} is defined, and serves as the proper stand-in for the non-existing $\star_{\tfrac{1}{2}\pi}$.

This is the case notably for [[Wick algebras]] ([[algebras of quantum observables]] of [[free fields]]): here $\tfrac{1}{2}\pi$ is the [[causal propagator]] for which $\star_\pi$ does not exist, and $\pi = \tfrac{1}{2}\pi + \alpha$ is a [[Wightman propagator]], so that $\star_{\pi}$ does exist on [[microcausal functionals]]: it is the ``[[normal-ordered product]]'' of quantum observables.

\end{example}
[[!include Wick algebra -- table]]

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

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

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

\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}
In [[perturbative quantum field theory]]:

\begin{itemize}%
\item [[Michael Dütsch]], section 2.1 of \emph{[[From classical field theory to perturbative quantum field theory]]}

\end{itemize}
[[!redirects star products]]

[[!redirects star-product]]



\end{document}