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\begin{document}

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\section*{Weyl relation}

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

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_the_wick_algebra_of_free_quantum_fields}{In the Wick algebra of free quantum fields}\dotfill \pageref*{in_the_wick_algebra_of_free_quantum_fields} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

What are called \emph{Weyl relations} is the incarnation of [[canonical commutation relations]] under passing to [[exponentials]].

For example if $a,a^\ast$ are two elements of an [[associative algebra]] with [[commutator]]

\begin{displaymath}
[a,a^\ast] = \hbar
\end{displaymath}
then the corresponding Weyl relation is, by the [[Baker-Campbell-Hausdorff formula]],

\begin{displaymath}
e^{z a}e^{z^\ast a^\ast} = e^{z^\ast a^\ast} e^{z a} e^{\hbar z z^\ast}
\end{displaymath}
for $z,z^\ast \in \mathbb{C}$

\hypertarget{in_the_wick_algebra_of_free_quantum_fields}{}\subsection*{{In the Wick algebra of free quantum fields}}\label{in_the_wick_algebra_of_free_quantum_fields}

\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 \Phi^a(x_1)}
    \otimes \frac{\delta}{\delta \Phi^b(x_2)} dvol_g
  \right)
  (P_1 \otimes P_2)
\end{displaymath}
on [[off-shell]] [[microcausal observables]] $A_1, A_2 \in \mathcal{F}_{mc}$ 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]] (\href{star+product#AssociativeAndUnitalStarProduct}{this prop.}) 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}
For \textbf{proof} see at \emph{[[Wick algebra]]} \href{Wick+algebra#MoyalStarProductOnMicrocausal}{this prop.}.

\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}
(\hyperlink{Dito90}{Dito 90}, \href{Wick+algebra#DutschFredenhagen01}{Dütsch-Fredenhagen 01})

\begin{proof}
By prop. \ref{MoyalStarProductOnMicrocausal} this is immediate from the general properties of the [[star product]] (\href{A+first+idea+of+quantum+field+theory+--+Quantization#MoyalStarProductIsFormalDeformationQuantization}{this example}).

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
    & =
    A_1 A_2 
    +
    \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  
   \\
   & = 
   A_1 A_2 
   + 
   \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 is 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}
\begin{example}
\label{}\hypertarget{}{}
\textbf{([[Hadamard vacuum state]] [[2-point function]])}

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 $\tfrac{1}{2} i \hbar$ times the evaluation

\begin{displaymath}
\begin{aligned}
    \langle A_1 A_2\rangle
    & \coloneqq
    \int \int
      (\alpha_1)_{a_1}(x_1) 
      \,
      \left\langle \mathbf{\Phi}^{a_1}(x_1) \mathbf{\Phi}^{a_2}(x_2)\right\rangle
     \,  
     (\alpha_2)_{a_2}(x_2)
     \, dvol_\Sigma(x_1)
     \, dvol_\Sigma(x_2)
    \\
    & \coloneqq
      \tfrac{1}{2} 
     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)
  \end{aligned}
\end{displaymath}
of the [[Wightman propagator]] $\Delta_H$:

\begin{equation}
A_1 \star_H A_2
  =
  A_1 \cdot A_2
  +
  \langle A_1 A_2\rangle
\label{StarProductOfTwoLinearObservables}\end{equation}
Further \hyperlink{HadamardVacuumStatesOnWickAlgebras}{below} we see that this evaluation is the [[2-point function]] of a [[state on a star-algebra|state]] on the [[Wick algebra]].

\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 A_2\rangle}
\end{displaymath}
where on the right we have the [[exponential]] [[Wightman 2-point function]] \eqref{StarProductOfTwoLinearObservables}.

\end{example}
(e.g. \hyperlink{Duetsch18}{Dütsch 18, exercise 2.3})

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

\begin{itemize}%
\item [[Michael Dütsch]], exercise 2.3 in \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018

\end{itemize}
[[!redirects Weyl relations]]



\end{document}