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\section*{Wick algebra}

\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{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{AbstractWickAlgebra}{Abstract Wick algebra}\dotfill \pageref*{AbstractWickAlgebra} \linebreak
\noindent\hyperlink{AbstractTimeOrderedProduct}{Abstract time-ordered product}\dotfill \pageref*{AbstractTimeOrderedProduct} \linebreak
\noindent\hyperlink{OperatorProductAndNormalOrderedProduct}{Operator product notation}\dotfill \pageref*{OperatorProductAndNormalOrderedProduct} \linebreak
\noindent\hyperlink{HadamardVacuumStatesOnWickAlgebras}{Hadamard vacuum states}\dotfill \pageref*{HadamardVacuumStatesOnWickAlgebras} \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}

A \emph{Wick algebra} is an [[algebra of quantum observables]] of [[free fields|free]] [[quantum field theory|quantum fields]].

In the [[quantum mechanics]] of [[harmonic oscillators]] or in [[quantum field theory]] of [[free fields]] on [[Minkowski spacetime]] one encounters [[linear operators]] $\{a_k, a^\ast_k\}_{k \in K}$ that satisfy the [[canonical commutation relation]] $[a_i, a^\ast_j] = diag((c_k))_{i j}$. Then by a \emph{normal ordered polynomial} or \emph{Wick polynomial} (\hyperlink{Wick50}{Wick 50}) one means a [[polynomial]], denoted $:P:$, which is obtained from a polynomial $P((a_k, a^\ast_k))$ in these operators by ordering all ``creation operators'' $a_k^\ast$ to the left of all ``annihiliation operators''. For example focusing on a single mode $k$ we have:

\begin{displaymath}
\itexarray{
    :a^\ast: = a^\ast
    \\
    :a: = a
    \\
    :a^\ast a: = a^\ast a
    \\
    :a a^\ast: = a^\ast a
    \\
    :a a^\ast a: = a^\ast a a
    \\
    etc.
  }
  \,
\end{displaymath}
The intuitive idea is that these operators span a [[Hilbert space]] $\mathcal{H}$ of [[quantum states]] from a [[vacuum state]] $\vert vac \rangle \in \mathcal{H}$ characterized by the condition

\begin{displaymath}
a_k \vert vac \rangle = 0
  \phantom{AAA}
  \text{for all} \,\, k
\end{displaymath}
hence (if we think of $a_k$ as acting by ``removing a quantum in mode $k$'') by the condition that it contains no quanta. So the normal ordered Wick polynomials represent the [[quantum observables]] with vanishing [[vacuum expectation value]]. In [[quantum field theory]] they model [[scattering]] processes where quanta enter a reaction process (the modes corresponding to the ``annihilation'' operators $a_k$) and other particles come out of the reaction (the modes corresppnding to the ``creation'' operators $a^\ast_k$).

The product of two Wick polynomials, computed in the ambient operator algebra and then re-expressed as a Wick polynomial, is given by computing the relevant sequence of [[commutators]] by [[Wick's lemma]], for example

\begin{displaymath}
{:a^\ast a:} \, {:a^\ast a:}
  =
  :a^\ast a^\ast a a: + \hbar \, :a^\ast a:
  \,,
\end{displaymath}
where $\hbar = [a, a^\ast]$ is the value of the [[canonical commutation relations|canonical commutator]].

The [[associative algebra]] thus obtained is hence called the \emph{algebra of normal ordered operators} or \emph{Wick polynomial algebra} or just \emph{Wick algebra}.

This plays a central role in [[perturbative quantum field theory]], where the [[quantization]] of [[quantum observables]] of [[free fields]] is traditionally \emph{defined} as the corresponding Wick algebra.

But the Wick algebra in [[quantum field theory]] may also be understood more systematically from first principles of [[quantization]]. It turns out that it is [[Moyal deformation quantization]] of the canonical [[Poisson bracket]] on the [[covariant phase space]] of the [[free field]], which is the [[Peierls bracket]] modified to an [[almost Kähler structure]] by the [[2-point function]] of a [[quasi-free Hadamard state]] (\hyperlink{Dito90}{Dito 90}, \hyperlink{DutschFredenhagen01}{D\"u{}tsch-Fredenhagen 01}).

In [[free field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations]] of motion, the analog of the star product tensor (\href{star+product#eq:InStarProductTensorInvertingHermitianForm}{this equation})

\begin{displaymath}
\pi \;=\; \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}
\end{displaymath}
is the [[Wightman propagator]] according to \href{Hadamard+distribution#eq:DeompositionOfHadamardPropagatorOnMinkowkski}{this equation}

\begin{displaymath}
\Delta_H
  \;=\;
  \tfrac{i}{2}\Delta_S + H
  \,.
\end{displaymath}
Understood in this form the construction directly generalizes to [[quantum field theory on curved spacetimes]] (\hyperlink{BrunettiFredenhagen95}{Brunetti-Fredenhagen 95}, \hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00}, \hyperlink{HollandsWald01}{Hollands-Wald 01}).

Finally, the shift by the [[quasi-free Hadamard state]], which is the very source of the ``normal ordering'', was understood as an example of the almost-K\"a{}hler version of the quantization recipe of [[Fedosov deformation quantization]] (\hyperlink{Collini16}{Collini 16}). For more on this see at \emph{[[locally covariant perturbative quantum field theory]]}.

$\,$

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

Traditionally the Wick algebra is regarded as an [[operator algebra]] acting on a [[Fock space]]. However, it is useful to realize the Wick algebra directly as an [[associative algebra]] [[structure]] on the space of [[microcausal polynomial observables]]. This ``abstract'' Wick algebra (meaning: not [[representation|represented]] yet on a [[Hilbert space]]) we discuss in

\begin{itemize}%
\item \emph{\hyperlink{AbstractWickAlgebra}{Abstract Wick algebra}}.

\end{itemize}
That the abstract Wick algebra indeed has a [[faithful representation]] on [[Fock space]] is \emph{[[Wick's lemma]]}.

Similarly there is the

\begin{itemize}%
\item \emph{\hyperlink{AbstractTimeOrderedProduct}{Abstract time-ordered product}}

\end{itemize}
The traditional notation for the [[operator products]] on the [[Fock space]] may be carried across the [[representation]] map to the abstract Wick algebra:

\begin{itemize}%
\item \emph{\hyperlink{OperatorProductAndNormalOrderedProduct}{Operator product notation}}

\end{itemize}
The abstract Wick algebra carries a canonical [[state on a star-algebra]], whose [[2-point function]] is just the [[Wightman propagator]] that the abstract Wick algebra structure is constructed from. This we discuss in

\begin{itemize}%
\item \emph{\hyperlink{HadamardVacuumStatesOnWickAlgebras}{Hadamard vacuum states}}.

\end{itemize}
The only non-trivial part of the proof of the [[state on a star-algebra|state]]-property (prop. \ref{WickAlgebraCanonicalState}) below is positivity. This however is immediate from the [[representation]] on the [[Fock space]], observing that under this identification the state is represented by the [[inner product]] of the [[Hilbert space]] (\hyperlink{Duetsch18}{Dütsch 18, remark 2.20}). From the point of view of [[algebraic quantum field theory]], the [[Hilbert space]]-structure mainly just serves as a technical tool for establishing this positivity property.

\hypertarget{AbstractWickAlgebra}{}\subsubsection*{{Abstract Wick algebra}}\label{AbstractWickAlgebra}

The abstract [[Wick algebra]] of a [[free field theory]] with [[Green hyperbolic differential equation]] is directly analogous to the [[star product]]-algebra induced by a [[finite dimensional vector space|finite dimensional]] [[Kähler vector space]] (\href{star+product#WickAlgebraOfAlmostKaehlerVectorSpace}{this def.}) under the following identification of the [[Wightman propagator]] with the [[Kähler space]]-[[structure]]:

\begin{remark}
\label{WightmanPropagatorAsKaehlerVectorSpaceStructure}\hypertarget{WightmanPropagatorAsKaehlerVectorSpaceStructure}{}
\textbf{([[Wightman propagator]] as [[Kähler vector space]]-[[structure]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] is a [[Green hyperbolic differential equation]]. Then the corresponding [[Wightman propagator]] is analogous to the rank-2 tensor on a [[Kähler vector space]] as follows:

\newline | [[space of field histories]]  $\Gamma_\Sigma(E)$ | $\mathbb{R}^{2n}$ | | [[symplectic form]]  $\tau_{\Sigma_p} \Omega_{BFV}$ | [[Kähler form]] $\omega$ | | [[causal propagator]] $\Delta$ | $\omega^{-1}$ | | [[Peierls-Poisson bracket]]  $\{A_1,A_2\} = \int \Delta^{a_1 a_2}(x_1,x_2) \frac{\delta A_1}{\delta \mathbf{\Phi}^{a_1}(x_1)} \frac{\delta A_2}{\delta \mathbf{\Phi}^{a_2}(x_2)} dvol_\Sigma(x)$ | [[Poisson bracket]] | | [[Wightman propagator]]  $\Delta_H = \tfrac{i}{2} \Delta + H$ | [[Hermitian form]]  $\pi = \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}$ |

\end{remark}
(\href{pAQFT#FredenhagenRejzner15}{Fredenhagen-Rejzner 15, section 3.6}, \href{pAQFT#Collini16}{Collini 16, table 2.1})

\begin{defn}
\label{MicrocausalObservable}\hypertarget{MicrocausalObservable}{}
\textbf{([[microcausal observable|microcausal]] [[polynomial observables]])} Let $E \overset{fb}{\to}$ be [[field bundle]] which is a [[vector bundle]]. An [[off-shell]] \emph{[[polynomial observable]]} is a [[smooth function]]

\begin{displaymath}
A
  \;\colon\;
  \Gamma_\Sigma(E)
    \longrightarrow
  \mathbb{C}
\end{displaymath}
on the [[on-shell]] [[space of sections]] of the [[field bundle]] $E \overset{fb}{\to} \Sigma$ (space of field histories) which may be expressed as

\begin{displaymath}
\begin{aligned}
  A(\Phi)
  & =
  \phantom{+} \alpha^{(0)}
  \\
  & \phantom{=}
  +
  \int_\Sigma
    \alpha^{(1)}_a(x) \Phi^a(x)
   \,
  dvol_\Sigma(x)
  \\
  & \phantom{=}
  +
  \int_\Sigma \int_\Sigma
   \alpha^{(2)}_{a_1 a_2}(x_1, x_2)
   \Phi^{a_1}(x_1) \Phi^{a_2}(x_2)
  \,dvol_\Sigma(x_1)
  \, dvol_\Sigma(x_2)
  \\
  & \phantom{=}
  +
  \cdots
  \,,
  \end{aligned}
\end{displaymath}
where

\begin{displaymath}
\alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right)
\end{displaymath}
is a [[compactly supported distribution]] [[distribution of two variables|of k variables]] on the $k$-fold graded-symmetric [[external tensor product of vector bundles]] of the [[field bundle]] with itself.

Write

\begin{displaymath}
PolyObs(E) \hookrightarrow Obs(E)
\end{displaymath}
for the [[subspace]] of off-shell polynomial observables onside all off-shell [[observables]].

Let moreover $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[equations of motion]] are [[Green hyperbolic differential equations]]. Then an \emph{[[on-shell]] polynomial observable} is the [[restriction]] of an off-shell polynomial observable along the inclusion of the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Write

\begin{displaymath}
PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})
\end{displaymath}
for the subspace of all on-shell polynomial observables inside all on-shell [[observables]].

By \href{Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions}{this prop.} restriction yields an [[isomorphism]] between polynomial on-shell observables and polynomial off-shell observables modulo the image of the [[differential operator]] $P$:

\begin{displaymath}
PolyObs(E,\mathbf{L})
    \underoverset{\simeq}{\text{restriction}}{\longleftarrow}
  PolyObs(E)/im(P)
  \,.
\end{displaymath}
Finally a polynomial observable is a \emph{[[microcausal observable]]} if each [[coefficient]] $\alpha^{(k)}$ as above has [[wave front set]] away from those points where the $k$ [[wave vectors]] are all in the [[future cone]] or all in the [[past cone]]. We write

\begin{displaymath}
\itexarray{
    PolyObs(E)_{mc}
     &\hookrightarrow&
    PolyObs(E)
    \\
    PolyObs(E,\mathbf{L})_{mc}
    \simeq
    PolyObs(E)_{mc}/im(P)
     &\hookrightarrow&
    PolyObs(E,\mathbf{L})
  }
\end{displaymath}
for the [[subspace]] of [[off-shell]]/[[on-shell]] [[microcausal observables]] inside all [[off-shell]]/[[on-shell]] [[polynomial observables]].

\end{defn}
\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 \mathbf{\Phi}^a(x_1)}
    \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(x_2)} dvol_g
  \right)
  (P_1 \otimes P_2)
\end{displaymath}
on [[off-shell]] [[microcausal observables]] $A_1, A_2 \in \mathcal{F}_{mc}$ (def. \ref{MicrocausalObservable}) 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}
(\hyperlink{Dito90}{Dito 90}, \hyperlink{DuetschFredenhagen00}{Dütsch-Fredenhagen 00} \hyperlink{DuetschFredenhagen01}{Dütsch-Fredenhagen 01}, \hyperlink{HirschfeldHenselder02}{Hirshfeld-Henselder 02}, see \hyperlink{Collini16}{Collini 16, p. 25-26})

\begin{proof}
By definition of the [[Wightman propagator]] (or else by \href{Hadamard+distribution#WaveFronSetsForKGPropagatorsOnMinkowski}{this prop.}), the [[wave front set]] of powers of $\Delta_H$ has all cotangent [[wave vectors]] on the first variables in the [[closed future cone]] at the given base point (which itself is on the [[light cone]])

and hence all those on the second variables in the [[closed past cone]].

The first variables are integrated against those of $A_1$ and the second against $A_2$. By definition of [[microcausal observables]] (def. \ref{MicrocausalObservable}), the wave front sets of $A_1$ and $A_2$ are disjoint from the subsets where all components are in the [[closed future cone]] or all components are in the [[closed past cone]]. Therefore the relevant sum of of the wave front covectors never vanishes and hence [[Hörmander's criterion]] for partial [[products of distributions|products of]] [[distributions of several variables]] (\href{product+of+distributions#PartialProductOfDistributionsOfSeveralVariables}{this prop.}) is met and the star product is well defined.

It remains to see that the star product $A_1 \star_H A_2$ is itself again a [[microcausal observable]]. It is clear that it is again a [[polynomial observable]] and that it respects the ideal generated by the equations of motion. That it still satisfies the condition on the [[wave front set]] follows directly from the fact that the wave front set of a [[product of distributions]] is inside the fiberwise sum of elements of the factor wave front sets (\href{product+of+distributions#WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets}{this prop.}, \href{}{}).

Finally the [[star algebra]]-structure follows via remark \ref{WightmanPropagatorAsKaehlerVectorSpaceStructure} as in \href{star+product#StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra}{this prop.}.

\end{proof}
\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}, \hyperlink{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}
\hypertarget{AbstractTimeOrderedProduct}{}\subsubsection*{{Abstract time-ordered product}}\label{AbstractTimeOrderedProduct}

\begin{defn}
\label{OnRegularPolynomialObservablesTimeOrderedProduct}\hypertarget{OnRegularPolynomialObservablesTimeOrderedProduct}{}
\textbf{([[time-ordered product]] on [[regular polynomial observables]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H$ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].

Then the \emph{[[time-ordered product]]} on the space of [[off-shell]] [[regular polynomial observable]] $PolyObs(E)_{reg}$ is the [[star product]] induced by the [[Feynman propagator]] (via \href{star+product#PropagatorStarProduct}{this prop.}):

\begin{displaymath}
\itexarray{
    PolyObs(E)_{reg}[ [\hbar] ]
    \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{}{\longrightarrow}&
    PolyObs(E)_{reg}[ [\hbar] ]
    \\
    (A_1, A_2)
    &\mapsto&
    \phantom{\coloneqq} A_1 \star_F A_2
  }
\end{displaymath}
hence

\begin{displaymath}
A_1 \star_F A_2
  \;
  \coloneqq
  \;
  ((-)\cdot(-))
  \circ
  \exp\left(
    \underset{\Sigma \times \Sigma}{\int}
    \Delta_F^{a b}(x,y)
    \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
    \otimes
    \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
    \,
    dvol_\Sigma(x)
    \,
    dvol_\Sigma(y)
  \right)
\end{displaymath}
(Notice that this does not descend to the [[on-shell]] observables, since the [[Feynman propagator]] is not a solution to the \emph{homogeneous} [[equations of motion]].)

\end{defn}
\begin{prop}
\label{CausalOrderingTimeOrderedProductOnRegular}\hypertarget{CausalOrderingTimeOrderedProductOnRegular}{}
\textbf{([[time-ordered product]] is indeed causally ordered [[Wick algebra]] product)}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H$ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].

Then the [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is indeed a time-ordering of the [[Wick algebra]] product $\star_H$ in that for all [[pairs]] of [[regular polynomial observables]]

\begin{displaymath}
A_1, A_2 \in PolyObs(E)_{reg}[ [\hbar] ]
\end{displaymath}
with [[disjoint subset|disjoint]] [[spacetime]] [[support]] we have

\begin{displaymath}
A_1 \star_F A_2
  \;=\;
  \left\{
    \itexarray{
      A_1 \star_H A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
      \\
      A_2 \star_H A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
    }
  \right.
  \,.
\end{displaymath}
Here $S_1 {\vee\!\!\!\wedge} S_2$ is the [[causal order]] relation (``$S_1$ does not intersect the [[past cone]] of $S_2$''). Beware that for general [[pairs]] $(S_1, S-2)$ of subsets neither $S_1 {\vee\!\!\!\wedge} S_2$ nor $S_2 {\vee\!\!\!\wedge} S_1$.

\end{prop}
\begin{proof}
Recall the following facts:

\begin{enumerate}%
\item the [[advanced and retarded propagators]] $\Delta_{\pm}$ by definition are [[support|supported]] in the [[future cone]]/[[past cone]], respectively

\begin{displaymath}
supp(\Delta_{\pm}) \subset \overline{V}^{\pm}
\end{displaymath}

\item they turn into each other under exchange of their arguments (\href{causal+propagator#CausalPropagatorIsSkewSymmetric}{this cor.}):

\begin{displaymath}
\Delta_\pm(y,x) = \Delta_{\mp}(x,y)
  \,.
\end{displaymath}

\item the real part $H$ of the [[Feynman propagator]], which by definition is the real part of the [[Wightman propagator]] is symmetric (by definition or else by \href{Hadamard+distribution#SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}{this prop.}):

\begin{displaymath}
H(x,y) = H(y,x)
\end{displaymath}


\end{enumerate}
Using this we compute as follows:

\begin{equation}
\begin{aligned}
    A_1 \underset{\Delta_{F}}{\star} A_2
    \;
    & =
    A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2
    \\
    & =
    \left\{
      \itexarray{
        A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
    \\
    & =
    \left\{
      \itexarray{
        A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
    \\
    & =
    \left\{
      \itexarray{
        A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
    \\
    & =
    \left\{
      \itexarray{
        A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
  \end{aligned}
\label{CausallyOrderedWickProductViaFeynmanPropagator}\end{equation}
\end{proof}
\begin{prop}
\label{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}\hypertarget{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}{}
\textbf{([[time-ordered product]] on [[regular polynomial observables]] [[isomorphism|isomorphic]] to pointwise product)}

The [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is [[isomorphism|isomorphic]] to the pointwise product of [[observables]] (\href{A+first+idea+of+quantum+field+theory#Observable}{this def.}) via the [[linear isomorphism]]

\begin{displaymath}
\mathcal{T}
  \;\colon\;
  PolyObs(E)_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{reg}[ [\hbar] ]
\end{displaymath}
given by

\begin{equation}
\mathcal{T}A
  \;\coloneqq\;
  \exp\left(
    \tfrac{1}{2}
    \hbar 
    \underset{\Sigma}{\int} 
     \Delta_F(x,y)^{a b} 
     \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)}
  \right)
  A
\label{OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism}\end{equation}
in that

\begin{displaymath}
\begin{aligned}
    T(A_1 A_2)
    & \coloneqq
    A_1 \star_{F} A_2
    \\
    & =
    \mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2) )
  \end{aligned}
\end{displaymath}
hence

\begin{displaymath}
\itexarray{
    PolyObs(E)_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{(-)\cdot (-)}{\longrightarrow}&   
    PolyObs(E)_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow 
      && 
    \downarrow^{\mathrlap{\mathcal{T}}}_\simeq
    \\
    PolyObs(E)_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{(-) \star_F (-)}{\longrightarrow}&   
    PolyObs(E)_{reg}[ [\hbar] ]
  }
\end{displaymath}
\end{prop}
(\href{time-ordered+product#BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, (12)-(13)}, \href{time-ordered+product#FredenhagenRejzner11b}{Fredenhagen-Rejzner 11b, (14)})

\begin{proof}
Since the [[Feynman propagator]] is symmetric (\href{A+first+idea+of+quantum+field+theory#SymmetricFeynmanPropagator}{this prop.}), the statement is a special case of \href{star+product#SymmetricContribution}{this prop.}).

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{([[renormalization]] of [[time-ordered product]])}

The [[time-ordered product]] on [[regular polynomial observables]] from prop. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} extends to a product on [[polynomial observable|polynomial]] [[local observables]], then taking values in [[microcausal observables]]:

\begin{displaymath}
T
  \;\colon\;
  PolyLocObs(E)^{\otimes_n}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{mc}[ [\hbar] ]
  \,.
\end{displaymath}
This extension is not unique. A choice of such an extension, satisfying some evident compatibility conditions, is a choice of \emph{[[renormalization scheme]]} for the given [[perturbative quantum field theory]]. Every such choice corresponds to a choice of [[perturbative S-matrix]] for the theory. This construction is called \emph{[[causal perturbation theory]]}.

\end{remark}
\hypertarget{OperatorProductAndNormalOrderedProduct}{}\subsubsection*{{Operator product notation}}\label{OperatorProductAndNormalOrderedProduct}

\begin{defn}
\label{NormalOrderedProductNotation}\hypertarget{NormalOrderedProductNotation}{}
\textbf{(notation for [[operator product]] and [[normal-ordered product]])}

It is traditional to use the following alternative notation for the product structures on [[microcausal polynomial observables]]:

\begin{enumerate}%
\item The [[Wick algebra]]-product, hence the [[star product]] $\star_H$ for the [[Wightman propagator]] (def. \ref{MoyalStarProductOnMicrocausal}), is rewritten as plain juxtaposition:

\begin{displaymath}
\text{"operator product"}
  \phantom{AAA}
  A_1 A_2 \phantom{AA} \coloneqq \phantom{AA} A_1 \star_H A_1
  \phantom{AAAA}
  \itexarray{
    \text{star product of}
    \\
    \text{Wightman propagator}
  }
  \,.
\end{displaymath}

\item The pointwise product of observables (\href{A+first+idea+of+quantum+field+theory#Observable}{this def.}) $A_1 \cdot A_2$ is equivalently written as plain juxtaposition enclosed by colons:

\begin{displaymath}
\itexarray{
    \text{"normal-ordered}
    \\
    \text{product"}
  }
  \phantom{AAAA}
  :A_1 A_2:
   \phantom{AA}\coloneqq\phantom{AA}
  A_1 \cdot A_2
  \phantom{AAAA} \phantom{AAa}\text{pointwise product}\phantom{AAa}
\end{displaymath}

\item The [[time-ordered product]], hence the [[star product]] for the [[Feynman propagator]] $\star_F$ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is equivalently written as plain juxtaposition prefixed by a ``$T$''

\begin{displaymath}
\itexarray{
    \text{"time-ordered}
    \\
    \text{product"}
  }
  \phantom{AAAA}
  T(A_1 A_2)
   \phantom{AA}\coloneqq\phantom{AA}
  A_1 \star_F A_2
  \phantom{AAAA}
  \itexarray{
    \text{star product of}
    \\
    \text{Feynman propagator}
  }
\end{displaymath}


\end{enumerate}
Under [[representation]] of the [[Wick algebra]] on a [[Fock space|Fock]] [[Hilbert space]] by [[linear operators]] the first product become the \emph{[[operator product]]}, while the second becomes the operator poduct applied after suitable re-ordering, called ``[[normal-ordered product|normal odering]]'' of the factors.

Disregarding the [[Fock space]]-representation, which is [[faithful representation|faithful]], we may still refer to these ``abstract'' products as the ``operator product'' and the ``normal-ordered product'', respectively.

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

\hypertarget{HadamardVacuumStatesOnWickAlgebras}{}\subsubsection*{{Hadamard vacuum states}}\label{HadamardVacuumStatesOnWickAlgebras}

\begin{prop}
\label{WickAlgebraCanonicalState}\hypertarget{WickAlgebraCanonicalState}{}
\textbf{(canonical [[Hadamard vacuum state|vacuum]] [[state on a star-algebra|states]] on abstract [[Wick algebra]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]].

For

\begin{displaymath}
\Phi_0 \in \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
\end{displaymath}
any [[on-shell]] [[field history]] (i.e. solving the [[equations of motion]]), consider the function from the [[Wick algebra]] to [[formal power series]] in $\hbar$ with [[coefficients]] in the [[complex numbers]] which evaluates any [[microcausal polynomial observable]] on $\Phi_0$

\begin{displaymath}
\itexarray{
    PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
    &\overset{\langle -\rangle_{\Phi_0}}{\longrightarrow}&
    \mathbb{C}[ [\hbar] ]
    \\
    A &\mapsto& A(\Phi_0)
  }
\end{displaymath}
Specifically for $\Phi_0 = 0$ (which is a solution of the [[equations of motion]] by the assumption that $(E,\mathbf{L})$ defines a [[free field theory]]) this is the function

\begin{displaymath}
\itexarray{
    PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
    &\overset{\langle -\rangle_0}{\longrightarrow}&
    \mathbb{C}[ [\hbar] ]
    \\
    \left.
    \begin{aligned}
      A & = \alpha^{(0)}
      \\ & \phantom{=} + \underset{\Sigma}{\int} \alpha^{(1)}_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)
      \\
      & \phantom{=} + \cdots
    \end{aligned}
    \right\}
    &\mapsto&
    A(0) = \alpha^{(0)}
  }
\end{displaymath}
which sends each [[microcausal polynomial observable]] to its value $A(\Phi = 0)$ on the zero [[field history]], hence to the constant contribution $\alpha^{(0)}$ in its [[polynomial]] expansion.

The function $\langle -\rangle_0$ is

\begin{enumerate}%
\item [[linear function|linear]] over $\mathbb{C}[ [\hbar] ]$;


\item real, in that for all $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$

\begin{displaymath}
\langle A^\ast \rangle = \langle A \rangle^\ast
\end{displaymath}

\item positive, in that for every $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$ there exist a $c_A \in \mathbb{C}[ [\hbar] ]$ such that

\begin{displaymath}
\langle A^\ast \star_H A\rangle_{\Phi_0} = c_A^\ast \cdot c_A
  \,,
\end{displaymath}

\item normalized, in that

\begin{displaymath}
\langle 1\rangle_H = 1
\end{displaymath}


\end{enumerate}
where $(-)^\ast$ denotes componet-wise [[complex conjugation]].

This means that $\langle -\rangle_{0}$ is a [[state on a star-algebra|states]] on the [[Wick algebra|Wick]] [[star-algebra]] $\left( (PolyObs(E,\mathbf{L}))_{mc}[ [\hbar] ], \star_H\right)$ (prop. \ref{MoyalStarProductOnMicrocausal}). One says that

\begin{itemize}%
\item $\langle - \rangle_0$ is a \emph{[[Hadamard vacuum state]]};

\end{itemize}
and generally

\begin{itemize}%
\item $\langle - \rangle_{\Phi_0}$ is called a \emph{[[coherent state]]}.

\end{itemize}
\end{prop}
(\hyperlink{Duetsch18}{Dütsch 18, def. 2.12, remark 2.20, def. 5.28, exercise 5.30 and equations (5.178)})

\begin{proof}
The properties of linearity, reality and normalization are obvious, what requires proof is positivity. This is proven by exhibiting a [[representation]] of the Wick algebra on a [[Fock space|Fock]] [[Hilbert space]] (this algebra [[homomorphism]] is \emph{[[Wick's lemma]]}), with formal powers in $\hbar$ suitably taken care of, and showing that under this representation the function $\langle -\rangle_0$ is represented, degreewise in $\hbar$, by the [[inner product]] of the [[Hilbert space]].

\end{proof}
\begin{example}
\label{HadamardMoyalStarProductOfTwoLinearObservables}\hypertarget{HadamardMoyalStarProductOfTwoLinearObservables}{}
\textbf{([[operator product]] of two [[linear observables]])}

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 their value

\begin{equation}
\langle A_1 \star_H A_2 \rangle_0
  \;\coloneqq\;
  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)
\label{EvaluatingLinearObservablesInWightmanPropagator}\end{equation}
in the [[Wightman propagator]], which is the value of the [[Hadamard vacuum state]] from prop. \ref{WickAlgebraCanonicalState}

\begin{displaymath}
A_1 \star_H A_2
  \;=\;
  A_1 \cdot A_2
  \;+\;
  \langle A_1 \star_H A_2 \rangle_0
\end{displaymath}
In the [[operator product]]/[[normal-ordered product]]-notation of def. \ref{NormalOrderedProductNotation} this reads

\begin{displaymath}
A_1 A_2 \;=\; :A_1 A_2: \;+\; \langle A_1 A_2\rangle
  \,.
\end{displaymath}
\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 \star_H A_2\rangle_0}
\end{displaymath}
where on the right we have the [[exponential]] of the value of the [[Hadamard vacuum state]] (prop. \ref{WickAlgebraCanonicalState}) as in example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.

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

\begin{example}
\label{}\hypertarget{}{}
\textbf{([[Wightman propagator]] is [[2-point function]] in the [[Hadamard vacuum state]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]].

With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Wightman propagator]] $\Delta_H(x,y)$ itself is the \emph{[[2-point function]]}, namely the [[distribution|distributional]] [[vacuum expectation value]] of the operator product of two [[field observables]]:

\begin{displaymath}
\left\langle
    \mathbf{\Phi}^a(x) \star_H \mathbf{\Phi}^b(y)
  \right\rangle_0
  \;=\;
  \underset{ = 0 }{
  \underbrace{
    \left\langle
       \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
    \right\rangle
  }}
  +
  \underset{ = \hbar \Delta^{a b}_H(x,y) }{
  \underbrace{
    \left \langle
      \hbar
      \underset{\Sigma \times \Sigma}{\int}
       \delta(x-x') \Delta^{a b}_H(x,y) \delta(y-y')
    \right\rangle
  }}
\end{displaymath}
by example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.

Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads:

\begin{displaymath}
\left\langle
    \mathbf{\Phi}^a(x)
    \mathbf{\Phi}^b(y)
  \right\rangle_0
  \;=\;
  \hbar \Delta_H(x,y)
  \,.
\end{displaymath}
\end{example}
Similarly:

\begin{example}
\label{}\hypertarget{}{}
\textbf{([[Feynman propagator]] is time-ordered [[2-point function]] in the [[Hadamard vacuum state]])}

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]] with induced [[Feynman propagator]] $\Delta_F$.

With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Feynman propagator]] $\Delta_F(x,y)$ itself is the \emph{time-ordered [[2-point function]]}, namely the [[distribution|distributional]] [[vacuum expectation value]] of the [[time-ordered product]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) of two [[field observables]]:

\begin{displaymath}
\left\langle
    T\left(
      \mathbf{\Phi}^a(x) \star_F \mathbf{\Phi}^b(y)
    \right)
  \right\rangle_0
  \;=\;
  \underset{ = 0 }{
  \underbrace{
    \left\langle
       \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
    \right\rangle
  }}
  +
  \underset{ = \hbar \Delta^{a b}_H(x,y) }{
  \underbrace{
    \left \langle
      \hbar
      \underset{\Sigma \times \Sigma}{\int}
       \delta(x-x') \Delta^{a b}_F(x,y) \delta(y-y')
    \right\rangle
  }}
\end{displaymath}
analogous to example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.

Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads:

\begin{displaymath}
\left\langle
    T\left(
     \mathbf{\Phi}^a(x)
     \mathbf{\Phi}^b(y)
    \right)
  \right\rangle_0
  \;=\;
  \hbar \Delta_F(x,y)
  \,.
\end{displaymath}
\end{example}
[[!include propagators - table]]

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

[[!include products in pQFT -- table]]

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

The construction goes back to

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

\end{itemize}
Its realization as the [[Moyal deformation quantization]] of the [[Peierls bracket]] shifted by a [[quasi-free Hadamard state]] is due to

\begin{itemize}%
\item Joseph 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}
further amplified in

\begin{itemize}%
\item [[Michael Dütsch]], [[Klaus Fredenhagen]], section 5.1 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 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 (\href{https://arxiv.org/abs/hep-th/0101079}{arXiv:hep-th/0101079})


\item A. C. Hirshfeld, P. Henselder, \emph{Star Products and Perturbative Quantum Field Theory}, Annals Phys. 298 (2002) 382-393 (\href{https://arxiv.org/abs/hep-th/0208194}{arXiv:hep-th/0208194})


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



\end{itemize}
and the generalization to [[quantum field theory on curved spacetime]] is discussed in

\begin{itemize}%
\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], M. K\"o{}hler, \emph{The microlocal spectrum condition and Wick polynomials on curved spacetimes}, Commun. Math. Phys. 180, 633-652, 1996 (\href{https://arxiv.org/abs/gr-qc/9510056}{arXiv:gr-qc/9510056})


\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], \emph{Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds}, Commun. Math. Phys. 208 : 623-661, 2000 (\href{https://arxiv.org/abs/math-ph/9903028}{math-ph/9903028})


\item [[Stefan Hollands]], [[Robert Wald]], \emph{Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime}, Commun. Math. Phys. 223:289-326, 2001 (\href{https://arxiv.org/abs/gr-qc/0103074}{arXiv:gr-qc/0103074})



\end{itemize}
Review is in

\begin{itemize}%
\item [[Michael Dütsch]], section 2.1 of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018

\end{itemize}
[[!redirects Wick algebras]]

[[!redirects Wick polynomial]] [[!redirects Wick polynomials]]

[[!redirects normal ordering]]

[[!redirects normal-ordered product]] [[!redirects normal-ordered products]]

[[!redirects normal ordered product]] [[!redirects normal ordered products]]

[[!redirects abstract Wick algebra]] [[!redirects abstract Wick algebras]]



\end{document}