A first idea of quantum field theory -- Free quantum fields

## Free quantum fields
 {#FreeQuantumFields}

In this chapter we discuss the following topics:

* _[Wick algebra](#WickAlgebraAbstract)_

* _[Time-ordered product](#AbstractTimeOrderedProduct)_

* _[Operator product notation](#OperatorProductAndNormalOrderedProduct)_

* _[Hadamard vacuum state](#HadamardVacuumStatesOnWickAlgebras)_

* _[Free quantum BV-differential](#FreeQuantumBVDifferential)_

* _[Schwinger-Dyson equation](#SchwingerDysonEquation)_

$\,$

In the [previous chapter](#Quantization) we discussed _[[quantization]]_ of linear [[phase spaces]], which turns the [[algebra of observables]] into a [[noncommutative algebra]] of [[quantum observables]]. Here we apply this to the [[covariant phase spaces]] of [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theories]] (as discussed in the chapter _[Gauge fixing](#GaugeFixing)_), obtaining genuine [[quantum field theory]] for [[free fields]].

For this purpose we first need to find a sub-algebra of all observables which is large enough to contain all [[local observables]] (such as the [[phi^n interaction]], example \ref{InWickAlgebraphinInteraction} below, and the [[electron-photon interaction]], example \ref{InWickAlgebraElectronPhotonInteraction} below) but small enough for the [[star product]] [[deformation quantization]] to meet [[Hörmander's criterion]] for absence of [[UV-divergences]] (remark \ref{UltravioletDivergencesFromPaleyWiener}). This does exist (example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal} below): It is called the algebra of _[[microcausal polynomial observables]]_ (def. \ref{MicrocausalObservable} below).

[[!include perturbative observables -- table]]

While the [[star product]] of the [[causal propagator]] still violates [[Hörmander's criterion]] for absence of [[UV-divergences]] on [[microcausal polynomial observables]], we have seen in the [previous chapter](#Quantization) that qantization freedom allows to shift this [[Poisson tensor]] by a symmetric contribution. By prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime} such a shift is provided by  passage from the [[causal propagator]] to the [[Wightman propagator]], and by prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski} this reduces the [[wave front set]] and hence the UV-singularities "by half".

This way the [[deformation quantization]] of the [[Peierls-Poisson bracket]] exists on  [[microcausal polynomial observables]] as the [[star product]] algebra induced by the [[Wightman propagator]]. The resulting [[non-commutative algebra|non-commutative]] [[algebra of observables]] is called the _[[Wick algebra]]_ (prop. \ref{MoyalStarProductOnMicrocausal} below).
Its algebra structure may be expressed in terms of a commutative "[[normal-ordered product]]" (def. \ref{NormalOrderedProductNotation} below) and
the [[vacuum expectation values]] of [[field observables]] in a canonically induced [[vacuum state]] (prop. \ref{WickAlgebraCanonicalState} below).

The analogous [[star product]] induced by the [[Feynman propagator]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} below) acts by first [[causal ordering]] its arguments and then multiplying them with the [[Wick algebra|Wick algebra product]] (prop. \ref{CausalOrderingTimeOrderedProductOnRegular} below) and hence is called the _[[time-ordered product]]_ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} below). This is the key structure in the discussion of [[interacting field theory]] discussed in the next chapter _[Interacting quantum fields](#InteractingQuantumFields)_.
Here we consider this on [[regular polynomial observables]] only, hence for averages of [[field observables]] that evaluate
at distinct [[spacetime]] points. The [[extension]] of the [[time-ordered product]] to
[[local observables]] is possible, but requires making choices: This is called _[[renormalization]]_, which we turn to in
the chapter _[Renormalization](#Renormalization)_ below.

[[!include Wick algebra -- table]]

While the [[Wick algebra]] with its [[vacuum state]] provides a [[quantization]] of the [[algebra of observables]]
of [[free field theory|free]] [[gauge fixing|gauge fixed]] [[Lagrangian field theories]], the possible existence of
[[infinitesimal gauge symmetries]] implies that the physically relevant observables are
just the [[gauge invariance|gauge invariant]] [[on-shell]] ones, exhibited by the [[cochain cohomology]] of the [[BV-BRST differential]] $\{-S' + S'_{BRST}, (-)\}$. Hence to complete [[quantization]]
of [[gauge theories]], the [[BV-BRST differential]] needs to be lifted to the [[noncommutative algebra]] of [[quantum observables]] --
this is called _[[BV-BRST quantization]]_.

To do so, we may regard the [[gauge fixing|gauge fixed]] [[BRST complex|BRST]]-[[action functional]] $S'_{BRST}$ as an [[interaction]] term,
to be dealt with later via [[scattering theory]], and hence consider quantization of just the free BV-differential $\{-S',(-)\}$.
One finds that this is equal to its [[time-ordered product|time-ordered]] version $\{-S',(-)\}_{\mathcal{T}}$ (prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket} below) plus a quantum correction, called the _[[BV-operator]]_ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator} below) or _[[BV-Laplacian]]_ (prop. \ref{ComponentsBVOperator} below).

Applied to [[observables]] this relation is the _[[Schwinger-Dyson equation]]_ (prop. \ref{DysonSchwinger} below), which expresses
the [[quantum physics|quantum]]-correction to the [[equations of motion]] of the [[free field theory|free]] [[gauge fixing|gauge field]] [[Lagrangian field theory]] as seen by [[time-ordered products]] of [[observables]] (example \ref{SchwingerDysonDistributional} below.)

After introducing [[field (physics)|field]]-[[interactions]] via [[scattering theory]] in the [next chapter](#InteractingQuantumFields)
the quantum correction to the [[BV-differential]] by the [[BV-operator]] becomes the "[[quantum master equation]]"
and the [[Schwinger-Dyson equation]] becomes the "[[master Ward identity]]". When choosing [[renormalization]] these
identities become _conditions_ to be satisfied by [[renormalization]] choices in order for the
interacting quantum BV-BRST differential, and hence for [[gauge invariance|gauge invariant]] quantum observables, to be well defined in
[[perturbative quantum field theory]] of [[gauge theories]]. This we discuss below in _[Renormalization](#Renormalization)_.

$\,$

**[[Wick algebra]]**
 {#WickAlgebraAbstract}

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]] (def. \ref{WickAlgebraOfAlmostKaehlerVectorSpace})  under the following identification of the [[Wightman propagator]] with the [[Kähler space]]-[[structure]]:

+-- {: .num_remark #WightmanPropagatorAsKaehlerVectorSpaceStructure}
###### Remark
**([[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:

| [[covariant phase space]]
 of [[free field theory|free]] [[Green hyperbolic differential equation|Green hyperbolic]]
 [[Lagrangian field theory]] | [[finite dimensional vector space|finite dimensional]]
 [[Kähler vector space]] |
|-------------------|-------------------------|
| [[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]]
  $\pi = \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}$ |

=--

([Fredenhagen-Rejzner 15, section 3.6](pAQFT#FredenhagenRejzner15), [Collini 16, table 2.1](pAQFT#Collini16))

+-- {: .num_defn #MicrocausalObservable}
###### Definition
**([[microcausal observable|microcausal]] [[polynomial observables]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] which is a [[vector bundle]], over some [[spacetime]] $\Sigma$.

A [[polynomial observable]] (def. \ref{PolynomialObservables})

$$
  \begin{aligned}
    A
     & = \phantom{+}
    \alpha^{(0)}
    \\
    &
    \phantom{=}
      +
    \int_{\Sigma} \mathbf{\Phi}^a(x) \alpha^{(1)}_a(x) \, dvol_\Sigma(x)
    \\
    &
    \phantom{=}
     +
    \int_{\Sigma^2}
      \mathbf{\Phi}^{a_1}(x_1) \cdot \mathbf{\Phi}^{a_2}(x_2)
      \alpha^{(2)}_{a_1 a_2}(x_1, x_2) \, dvol_\Sigma(x_1) dvol_\Sigma(x_2)
    \\
    &
    \phantom{=}
     +
    \int_{\Sigma^3}
      \mathbf{\Phi}^{a_1}(x_1)
        \cdot
      \mathbf{\Phi}^{a_2}(x_2)
        \cdot
      \mathbf{\Phi}^{a_3}(x_3)
      \alpha^{(3)}_{a_1 a_2 a_3}(x_1,x_2,x_3)
      \, dvol_\Sigma(x_1) dvol_\Sigma(x_2) dvol_\Sigma(x^3)
    \\
    &
    \phantom{=}
     +
    \cdots
  \,.
  \end{aligned}
$$

is called _[[microcausal polynomial observable|microcausal]]_ if each [[distribution|distributional]] [[coefficient]]

$$
  \alpha^{(k)}
  \;\in\;
  \Gamma'_{\Sigma^k}(E^{\boxtimes^k})
$$

as above has [[wave front set]] (def. \ref{WaveFrontSet}) _not_ containing those elements $(x_1, \cdots x_k, k_1, \cdots k_k)$ where the $k$ [[wave vectors]] are all in the [[closed future cone]] or all in the [[closed past cone]] (def. \ref{CausalPastAndFuture}).

We write

$$
  \array{
    PolyObs(E)_{mc}
     &\hookrightarrow&
    PolyObs(E)
    \\
    PolyObs(E,\mathbf{L})_{mc}
    \simeq
    PolyObs(E)_{mc}/im(P)
     &\hookrightarrow&
    PolyObs(E,\mathbf{L})
  }
$$

for the [[subspace]] of [[off-shell]]/[[on-shell]] [[microcausal polynomial observables]] inside all [[off-shell]]/[[on-shell]] [[polynomial observables]].

=--

The important point is that [[microcausal polynomial observables]] still contain all [[regular polynomial observables]] but also all polynomial [[local observables]]:

+-- {: .num_example #MicrocausalRegularObservables}
###### Example
**([[regular polynomial observables]] are [[microcausal observables|microcausal]])**

Every [[regular polynomial observable]] (def. \ref{PolynomialObservables}) is [[microcausal polynomial observable|microcausal]] (def. \ref{MicrocausalObservable}).

=--

+-- {: .proof}
###### Proof

By definition of regular polynomial observables, their [[coefficients]] are [[non-singular distributions]] and because the [[wave front set]] of [[non-singular distributions]] is [[empty set|empty]] (example \ref{NonSingularDistributionTrivialWaveFrontSet})

=--

+-- {: .num_example #PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}
###### Example
**(polynomial [[local observables]] are [[microcausal polynomial observables|microcausal]])**

Every polynomial [[local observable]] (def. \ref{LocalObservables}) is a [[microcausal polynomial observable]] (def. \ref{MicrocausalObservable}).

=--

+-- {: .proof}
###### Proof

For notational convenience, consider the case of the [[scalar field]] with $k = 2$; the general case is directly analogous. Then the [[local observable]] coming from $\phi^2$ (a [[phi^n interaction]]-term), has, regarded as a [[polynomial observable]], the [[delta distribution]] $\delta(x_1-x_2)$ as [[coefficient]] in degree 2:

$$
  \begin{aligned}
    A(\Phi)
    & =
    \underset{\Sigma}{\int} g(x) (\Phi(x))^2 \,dvol_\Sigma(x)
    \\
    & =
    \underset{\Sigma \times \Sigma}{\int}
      \underset{ = \alpha^{(2)}}{
      \underbrace{
        g(x_1)
        \delta(x_1 - x_2)
      }}
      \,
     \Phi(x_1) \Phi(x_2) \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
  \end{aligned}
  \,.
$$

Now for $(x_1, x_2) \in \Sigma \times \Sigma$ and $\mathbb{R}^{2n} \simeq U \subset X \times X$ a [[chart]] around this point,  the [[Fourier transform of distributions]] of $g \cdot \delta(-,-)$ restricted to this chart is proportional to the Fourier transform $\hat g$ of $g$ evaluated at the sum of the two covectors:

$$
  \begin{aligned}
    (k_1, k_2)
      & \mapsto
      \underset{\mathbb{R}^{2n}}{\int}
      g(x_1)
      \delta(x_1, x_2)
      e^{i (k_1 \cdot x_1 + k_2 \cdot x_2 )}
      \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2)
    \\
    & \propto \hat g(k_1 + k_2)
   \end{aligned}
  \,.
$$

Since $g$ is a plain [[bump function]], its [[Fourier transform]] $\hat g$ is quickly decaying (according to prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}) along $k_1 + k_2$, as long as $k_1 + k_2 \neq 0$. Only on the [[cone]] $k_1 + k_2 = 0$ the Fourier transform is [[constant function|constant]], and hence in particular not decaying.

{#WaveFrontOfdeltaxy} This means that the wave front set consists of the elements of the form $(x, (k, -k))$ with $k \neq 0$. Since $k$ and $-k$ are both in the [[closed future cone]] or both in the [[closed past cone]] precisely if $k = 0$, this situation is excluded in the wave front set and hence the distribution $g \cdot \delta(-,-)$ is [[microcausal observable|microcausal]].

> (graphics grabbed from [Khavkine-Moretti 14, p. 45](microcausal+polynomial+observable#KhavkineMoretti14))

=--

+-- {: .num_prop #MoyalStarProductOnMicrocausal}
###### Proposition
**([[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

$$
  \Delta_H
  \;=\;
  \tfrac{i}{2}\Delta + H
$$

be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]).

Then the [[star product]] induced by $\Delta_H$

$$
  A \star_H A
  \;\coloneqq\;
  prod
   \circ
  \exp\left(
    \int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \mathbf{\Phi}^a(x_1)}
    \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(x_2)} dvol_g
  \right)
  (P_1 \otimes P_2)
$$

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]] (prop. \ref{AssociativeAndUnitalStarProduct}) this yields a [[unital algebra|unital]] [[associative algebra]] [[structure]] on the space of [[formal power series]] in $\hbar$ of [[off-shell]] [[microcausal observables]]

$$
  \left(
    PolyObs(E)_{mc}[ [\hbar] ] \,,\, \star_H
  \right)
  \,.
$$

This is the _[[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 _[[on-shell]] [[Wick algebra]]_

$$
  \left(
    PolyObs(E,\mathbf{L})_{mc}[ [ \hbar ] ] \,,\, \star_H
  \right)
  \,.
$$

Finally, under [[complex conjugation]] $(-)^\ast$ these are [[star algebras]] in that

$$
  \left(
    A_1 \star_H A_2
  \right)^\ast
  =
  A_2^\ast \star_H A_1^\ast
  \,.
$$

=--

(e.g. [Collini 16, p. 25-26](Wick+algebra#Collini16))

+-- {: .proof}
###### Proof

By prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski} the [[wave front set]] of $\Delta_H$ has all cotangents 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]] (prop. \ref{HoermanderCriterionForProductOfDistributions}) for partial [[products of distributions|products of]] [[distributions of several variables]] (prop. \ref{PartialProductOfDistributionsOfSeveralVariables}).

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 (prop. \ref{WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets}, prop. \ref{PartialProductOfDistributionsOfSeveralVariables}).

Finally the [[star algebra]]-structure via [[complex conjugation]] follows via remark \ref{WightmanPropagatorAsKaehlerVectorSpaceStructure} as in prop. \ref{StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra}.

=--

+-- {: .num_remark #WickAlgebraIsFormalDeformationQuantization}
###### Remark
**([[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

$$
  A_1 \star_H A_2
  \;=\;
  A_1 \cdot A_2
  \;mod\;
  \hbar
$$

and

$$
  A_1 \star_H A_2
  -
  A_2 \star_H a_1
  \;=\;
  i \hbar \{A_1, A_2\}
  \;mod\;
  \hbar^2
  \,.
$$

=--

([Dito 90](Wick+algebra#Dito90), [Dütsch-Fredenhagen 00](Wick+algebra#DuetschFredenhagen00), [Dütsch-Fredenhagen 01](Wick+algebra#DuetschFredenhagen01), [Hirshfeld-Henselder 02](Wick+algebra#HirschfeldHenselder02))

+-- {: .proof}
###### Proof

By prop. \ref{MoyalStarProductOnMicrocausal} this is immediate from the general properties of the [[star product]] (example \ref{MoyalStarProductIsFormalDeformationQuantization}).

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

$$
  \begin{aligned}
    & A_1 \star_H A_2
    \\
    & =
    \phantom{+} A_1 A_2
    \\
    & \phantom{=}
    +
    \hbar
    \int
    \left(
      \tfrac{i}{2} \Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1,x_2)
     \right)
      \frac{\partial A_1}{\partial \mathbf{\Phi}^{a_1}(x_1)}
      \frac{\partial A_2}{\partial \mathbf{\Phi}^{a_2}(x_2)}
    \;mod\;
    \hbar^2
   \\
   & =
   \phantom{+} A_1 A_2
   \\
   & \phantom{=}
   +
   \hbar
   \left(
    \int
      (\alpha_1)_{a_1}(x_1)
      \left(
       \tfrac{i}{2}\Delta^{a_1 a_2}(x_1, x_2)
       + H^{a_1 a_2}(x_1, x_2)
      \right)
      (\alpha_2)_{a_2}(x_2)
   \right)
   \;mod\;
   \hbar^2
  \end{aligned}
$$

Now since $\Delta$ is skew-symmetric while $H$ is symmetric (prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) it follows that

$$
  \begin{aligned}
    A_1 \star_H A_2
    -
    A_2 \star_H A_1
    & =
    i \hbar
    \left(
     \int
       (\alpha_1)_{a_1}(x_1)
       \Delta^{a_1 a_2}(x_1, x_2)
       (\alpha_2)_{a_2}(x_2)
    \right)
    \;mod\;
    \hbar^2
    \\
    & =
    i \hbar \, \left\{ A_1, A_2\right\}
  \end{aligned}
  \,.
$$

The right hand side is the [[integral kernel]]-expression for the [[Poisson-Peierls bracket]], as shown in the second line.

=--

$\,$

**[[time-ordered product]]**
 {#AbstractTimeOrderedProduct}

+-- {: .num_defn #OnRegularPolynomialObservablesTimeOrderedProduct}
###### Definition
**([[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 _[[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 prop. \ref{PropagatorStarProduct}):

$$
  \array{
    PolyObs(E)_{reg}[ [\hbar] ]
    \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{}{\longrightarrow}&
    PolyObs(E)_{reg}[ [\hbar] ]
    \\
    (A_1, A_2)
    &\mapsto&
    \phantom{\coloneqq} A_1 \star_F A_2
  }
$$

hence

$$
  A_1 \star_F A_2
  \;
  \coloneqq
  \;
  ((-)\cdot(-))
  \circ
  \exp\left(
    \underset{\Sigma \times \Sigma}{\int}
    \Delta_F^{a b}(x,y)
    \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
    \otimes
    \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
    \,
    dvol_\Sigma(x)
    \,
    dvol_\Sigma(y)
  \right)
$$

(Notice that this does not descend to the [[on-shell]] observables, since the [[Feynman propagator]] is not a solution to the _homogeneous_ [[equations of motion]].)

=--

+-- {: .num_prop #CausalOrderingTimeOrderedProductOnRegular}
###### Proposition
**([[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]]

$$
  A_1, A_2 \in PolyObs(E)_{reg}[ [\hbar] ]
$$

with [[disjoint subset|disjoint]] [[spacetime]] [[support]] we have

$$
  A_1 \star_F A_2
  \;=\;
  \left\{
    \array{
      A_1 \star_H A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
      \\
      A_2 \star_H A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
    }
  \right.
  \,.
$$

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 $S_2 {\vee\!\!\!\wedge} S_1$.

=--

+-- {: .proof}
###### Proof

Recall the following facts:

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

   $$
     supp(\Delta_{\pm}) \subset \overline{V}^{\pm}
   $$

1. they turn into each other under exchange of their arguments (cor. \ref{CausalPropagatorIsSkewSymmetric}):

   $$
     \Delta_\pm(y,x) = \Delta_{\mp}(x,y)
     \,.
   $$

1. 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 prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}):

   $$
     H(x,y) = H(y,x)
   $$

Using this we compute as follows:

$$
  \begin{aligned}
    A_1 \underset{\Delta_{F}}{\star} A_2
    \;
    & =
    A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2
    \\
    & =
    \left\{
      \array{
        A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
    \\
    & =
    \left\{
      \array{
        A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
    \\
    & =
    \left\{
      \array{
        A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
    \\
    & =
    \left\{
      \array{
        A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
  \end{aligned}
$$

=--

+-- {: .num_prop #IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}
###### Proposition
**([[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]] (def. \ref{Observable}) via the [[linear isomorphism]]

$$
  \mathcal{T}
  \;\colon\;
  PolyObs(E)_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{reg}[ [\hbar] ]
$$

given by

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

in that

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

hence

$$
  \array{
    PolyObs(E)_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{(-)\cdot (-)}{\longrightarrow}&
    PolyObs(E)_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow
      &&
    \Big\downarrow{}^{\mathrlap{\mathcal{T}}}_\simeq
    \\
    PolyObs(E)_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{(-) \star_F (-)}{\longrightarrow}&
    PolyObs(E)_{reg}[ [\hbar] ]
  }
$$

=--

([Brunetti-Dütsch-Fredenhagen 09, (12)-(13)](time-ordered+product#BrunettiDuetschFredenhagen09), [Fredenhagen-Rejzner 11b, (14)](time-ordered+product#FredenhagenRejzner11b))

+-- {: .proof}
###### Proof

Since the [[Feynman propagator]] is symmetric (prop. \ref{SymmetricFeynmanPropagator}), the statement is a special case of prop. \ref{SymmetricContribution}.

=--

+-- {: .num_example #RegularObservablesExponentialTimeOrdered}
###### Example
**([[time-ordered product|time-ordered]] [[exponential]] of [[regular polynomial observables]])**

Let  $V \in PolyObs_{reg, deg = 0}[ [ \hbar ] ]$ be a [[regular polynomial observable]] (def. \ref{PolynomialObservables}) of degree zero, and write

$$
  \exp(V)
  =
  1 + V + \tfrac{1}{2!} V \cdot V + \tfrac{1}{3!} V \cdot V \cdot V + \cdots
$$

for the [[exponential]] of $V$ with respect to the pointwise product (eq:ObservablesPointwiseProduct).

Then the [[exponential]] $\exp_{\mathcal{T}}(V)$ of $V$ with respect to the [[time-ordered product]] $\star_F$ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is equal to the [[conjugation]] of the exponential with respect to the
pointwise product by the time-ordering isomorphism $\mathcal{T}$ from prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}:

$$
  \begin{aligned}
    \exp_{\mathcal{T}}(V)
    &
    \coloneqq
    1 + V + \tfrac{1}{2} V \star_F V + \tfrac{1}{3!} V \star_F V \star_F V + \cdots
    \\
    & =
    \mathcal{T} \circ \exp(-) \circ \mathcal{T}^{-1}(V)
    \,.
  \end{aligned}
$$

=--

+-- {: .num_remark }
###### Remark
**([[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]] (def. \ref{LocalObservables}), then taking values in [[microcausal observables]] (def. \ref{MicrocausalObservable}):

$$
  T
  \;\colon\;
  PolyLocObs(E)^{\otimes_n}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{mc}[ [\hbar] ]
  \,.
$$

This extension is not unique. A choice of such an extension, satisfying some evident compatibility conditions, is a choice of _[[renormalization scheme]]_ for the given [[perturbative quantum field theory]]. Every such choice corresponds to a choice of [[perturbative S-matrix]] for the theory, namely an extension of the time-ordered exponential $\exp_{\mathcal{T}}$ (example \ref{RegularObservablesExponentialTimeOrdered}) from regular to local observables.

This construction of [[perturbative quantum field theory]] is called _[[causal perturbation theory]]_.  We discuss this below in the chapters _[Interacting quantum fields](#InteractingQuantumFields)_ and _[Renormalization](#Renormalization)_.

=--

$\,$

**operator product notation**
 {#OperatorProductAndNormalOrderedProduct}

+-- {: .num_defn #NormalOrderedProductNotation}
###### Definition
**(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]]:

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

   $$
     \text{"operator product"}
     \phantom{AAA}
     A_1 A_2 \phantom{AA} \coloneqq \phantom{AA} A_1 \star_H A_1
     \phantom{AAAA}
     \array{
       \text{star product of}
       \\
       \text{Wightman propagator}
     }
     \,.
   $$

1. The pointwise product of observables (def. \ref{Observable})  $A_1 \cdot A_2$
   is equivalently written as plain juxtaposition enclosed by colons:

   $$
     \array{
       \text{"normal-ordered}
       \\
       \text{product"}
     }
     \phantom{AAAA}
     :A_1 A_2:
      \phantom{AA}\coloneqq\phantom{AA}
     A_1 \cdot A_2
     \phantom{AAAA} \phantom{AAa}\text{pointwise product}\phantom{AAa}
   $$

1. 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 "$T$"

   $$
     \array{
       \text{"time-ordered}
       \\
       \text{product"}
     }
     \phantom{AAAA}
     T(A_1 A_2)
      \phantom{AA}\coloneqq\phantom{AA}
     A_1 \star_F A_2
     \phantom{AAAA}
     \array{
       \text{star product of}
       \\
       \text{Feynman propagator}
     }
   $$

Under [[representation]] of the [[Wick algebra]] on a [[Fock space|Fock]] [[Hilbert space]] by [[linear operators]]
the first product becomes the _[[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.

=--

$\,$

+-- {: .num_example #InWickAlgebraphinInteraction}
###### Example
**([[phi^n interaction]])**

Consider [[phi^n theory]] from example \ref{phintheoryLagrangian}.
The [[adiabatic switching|adiabatically switched]] [[action functional]] (example \ref{ActionFunctional}) which is the [[transgression of variational differential forms|transgression]] of the [[phi^n interaction]] is the following [[local observable|local]] (hence, by example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}, [[microcausal polynomial observable|microcausal]]) observable:

$$
  \begin{aligned}
  S_{int}
  & =
  \underset{\Sigma}{\int}
  \underset{
     n \, \text{factors}
  }{
  \underbrace{
    \mathbf{\Phi}(x)
     \cdot
    \mathbf{\Phi}(x)
     \cdots
    \mathbf{\Phi}(x)
     \cdot
    \mathbf{\Phi}(x)
  }
  }
  \,
  dvol_\Sigma(x)
  \\
  & =
  \underset{\Sigma}{\int}
  :
  \underset{
     n \, \text{factors}
  }{
  \underbrace{
    \mathbf{\Phi}(x)
    \mathbf{\Phi}(x)
      \cdots
    \mathbf{\Phi}(x)
    \mathbf{\Phi}(x)
  }
  }
  :
  \,
  dvol_\Sigma(X)
  \end{aligned}
  \,,
$$

Here in the first line we have the [[integral]] over a pointwise product (def. \ref{Observable}) of $n$ [[field observables]] (example \ref{PointEvaluationObservables}), which in the second line we write equivalently as a [[normal ordered product]] by def. \ref{NormalOrderedProductNotation}.

=--

+-- {: .num_example #InWickAlgebraElectronPhotonInteraction}
###### Example
**([[electron-photon interaction]])**

Consider the [[Lagrangian field theory]] defining [[quantum electrodynamics]] from example \ref{LagrangianQED}.
The [[adiabatic switching|adiabatically switched]] [[action functional]] (example \ref{ActionFunctional}) which is the [[transgression of variational differential forms|transgression]] of the [[electron-photon interaction]] is the [[local observable|local]] (hence, by example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}, [[microcausal polynomial observable|microcausal]]) observable

$$
  \begin{aligned}
    S_{int}
    & \coloneqq
    i
    \underset{\Sigma}{\int}
      g_{sw}(x)
      \,
      (\Gamma^\mu)^\alpha{}_\beta
      \,
      \overline{\mathbf{\Psi}}_\alpha(x)
      \cdot
      \mathbf{\Psi}^\beta(x)
      \cdot
      \mathbf{A}_\mu(x)
      \,
     dvol_\Sigma(x)
     \\
     & =
    i
    \underset{\Sigma}{\int}
      g_{sw}(x)
      \,
      (\Gamma^\mu)^\alpha{}_\beta
      \,
      :
      \overline{\mathbf{\Psi}}_\alpha(x)
      \mathbf{\Psi}^\beta(x)
      \mathbf{A}_\mu(x)
      :
      \,
     dvol_\Sigma(x)
  \end{aligned}
  \,,
$$

Here in the first line we have the [[integral]] over a pointwise product (def. \ref{Observable}) of $n$ [[field observables]] (example \ref{PointEvaluationObservables}), which in the second line we write equivalently as a [[normal ordered product]] by def. \ref{NormalOrderedProductNotation}.

=--

(e.g. [Scharf 95, (3.3.1)](electron-photon+interaction#Scharf95))

$\,$

**[[Hadamard vacuum state]]**
 {#HadamardVacuumStatesOnWickAlgebras}

+-- {: .num_prop #WickAlgebraCanonicalState}
###### Proposition
**(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

$$
  \Phi_0 \in \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
$$

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 $\Phi_0$

$$
  \array{
    PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
    &\overset{\langle -\rangle_{\Phi_0}}{\longrightarrow}&
    \mathbb{C}[ [\hbar] ]
    \\
    A &\mapsto& A(\Phi_0)
  }
$$

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

$$
  \array{
    PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
    &\overset{\langle -\rangle_0}{\longrightarrow}&
    \mathbb{C}[ [\hbar] ]
    \\
    \left.
    \begin{aligned}
      A & = \alpha^{(0)}
      \\ & \phantom{=} + \underset{\Sigma}{\int} \alpha^{(1)}_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)
      \\
      & \phantom{=} + \cdots
    \end{aligned}
    \right\}
    &\mapsto&
    A(0) = \alpha^{(0)}
  }
$$

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

1. [[linear function|linear]] over $\mathbb{C}[ [\hbar] ]$;

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

   $$
     \langle A^\ast \rangle = \langle A \rangle^\ast
   $$

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

   $$
     \langle A^\ast \star_H A\rangle_{\Phi_0} = c_A^\ast \cdot c_A
     \,,
   $$

1. normalized, in that

   $$
     \langle 1\rangle_H = 1
   $$

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

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

* $\langle - \rangle_0$ is a _[[Hadamard vacuum state]]_;

and generally

* $\langle - \rangle_{\Phi_0}$ is called a _[[coherent state]]_.

=--

([Dütsch 18, def. 2.12, remark 2.20, def. 5.28, exercise 5.30 and equations (5.178)](#Duetsch18))

+-- {: .proof}
###### 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 _[[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]].

=--

+-- {: .num_example #HadamardMoyalStarProductOfTwoLinearObservables}
###### Example
**([[operator product]] of two [[linear observables]])**

Let

$$
  A_i \in LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc}
$$

for $i \in \{1,2\}$ be two [[linear observable|linear]] [[microcausal observables]] represented by [[distributions]] which in [[generalized function]]-notation are given by

$$
  A_i
  \;=\;
  \int (\alpha_i)_{a_i}(x_i) \mathbf{\Phi}^{a_i}(x_i) \, dvol_\Sigma(x_i)
  \,.
$$

Then their Hadamard-Moyal [[star product]] (prop. \ref{MoyalStarProductOnMicrocausal}) is the [[sum]] of their pointwise product
with their value

$$
  \label{EvaluatingLinearObservablesInWightmanPropagator}
  \langle A_1 \star_H A_2 \rangle_0
  \;\coloneqq\;
  i \hbar \int \int
   (\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2)
   \,dvol_\Sigma(x_1)
   \,dvol_\Sigma(x_2)
$$

in the [[Wightman propagator]], which is the value of the [[Hadamard vacuum state]] from prop. \ref{WickAlgebraCanonicalState}:

$$
  A_1 \star_H A_2
  \;=\;
  A_1 \cdot A_2
  \;+\;
  \langle A_1 \star_H A_2 \rangle_0
$$

In the [[operator product]]/[[normal-ordered product]]-notation of def. \ref{NormalOrderedProductNotation} this reads

$$
  A_1 A_2 \;=\; :A_1 A_2: \;+\; \langle A_1 A_2\rangle
  \,.
$$

=--

+-- {: .num_example #WeylRelations}
###### Example
**([[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

$$
  A_1, A_2
  \;\in\;
  LinObs(E,\mathbf{L})_{mc}
  \hookrightarrow
  PolyObs(E,\mathbf{L})_{mc}
$$

two [[linear observables|linear]] [[microcausal observables]], the Hadamard-Moyal star product (def. \ref{MoyalStarProductOnMicrocausal}) of their [[exponentials]] exhibits the [[Weyl relations]]:

$$
  e^{A_1}
  \star_H
  e^{A_2}
  \;=\;
  e^{A_1 + A_2} \; e^{\langle A_1 \star_H A_2\rangle_0}
$$

where on the right we have the [[exponential]] of the value of the [[Hadamard vacuum state]] (prop. \ref{WickAlgebraCanonicalState}) as in example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.

=--

(e.g. [Dütsch 18, exercise 2.3](Wick+algebra#Duetsch18))

+-- {: .num_example }
###### Example
**([[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 _[[2-point function]]_, namely the [[distribution|distributional]] [[vacuum expectation value]] of the operator product of two [[field observables]]:

$$
  \left\langle
    \mathbf{\Phi}^a(x) \star_H \mathbf{\Phi}^b(y)
  \right\rangle_0
  \;=\;
  \underset{ = 0 }{
  \underbrace{
    \left\langle
       \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
    \right\rangle
  }}
  +
  \underset{ = \hbar \Delta^{a b}_H(x,y) }{
  \underbrace{
    \left \langle
      \hbar
      \underset{\Sigma \times \Sigma}{\int}
       \delta(x-x') \Delta^{a b}_H(x,y) \delta(y-y')
    \right\rangle
  }}
$$

by example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.

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

$$
  \left\langle
    \mathbf{\Phi}^a(x)
    \mathbf{\Phi}^b(y)
  \right\rangle_0
  \;=\;
  \hbar \Delta_H(x,y)
  \,.
$$

=--

Similarly:

+-- {: .num_example }
###### Example
**([[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 _time-ordered [[2-point function]]_, namely the [[distribution|distributional]] [[vacuum expectation value]] of the [[time-ordered product]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) of two [[field observables]]:

$$
  \left\langle
    T\left(
      \mathbf{\Phi}^a(x) \star_F \mathbf{\Phi}^b(y)
    \right)
  \right\rangle_0
  \;=\;
  \underset{ = 0 }{
  \underbrace{
    \left\langle
       \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
    \right\rangle
  }}
  +
  \underset{ = \hbar \Delta^{a b}_H(x,y) }{
  \underbrace{
    \left \langle
      \hbar
      \underset{\Sigma \times \Sigma}{\int}
       \delta(x-x') \Delta^{a b}_F(x,y) \delta(y-y')
    \right\rangle
  }}
$$

analogous to example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.

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

$$
  \left\langle
    T\left(
     \mathbf{\Phi}^a(x)
     \mathbf{\Phi}^b(y)
    \right)
  \right\rangle_0
  \;=\;
  \hbar \Delta_F(x,y)
  \,.
$$

=--

[[!include propagators - table]]

$\,$

**[[BV-operator|free quantum BV-differential]]**
  {#FreeQuantumBVDifferential}

So far we have discussed the plain (graded-commutative) [[algebra of quantum observables]] of a [[gauge fixing|gauged fixed]] [[free field theory|free]] [[Lagrangian field theory]], [[deformation quantization|deforming]] the commutative pointwise product of [[observables]]. But after [[gauge fixing]], the algebra of observables is not just a (graded-commutative) algebra, but carries also a [[differential]] making it a [[differential graded-commutative superalgebra]]: the global [[BV-differential]] $\{-S' + S_{BRST}, -\}$ (def. \ref{ComplexBVBRSTGlobal}). The [[gauge invariance|gauge invariant]] [[on-shell]] [[observables]] are (only) the [[cochain cohomology]] of this differential. Here we discuss what becomes of this differential as we pass to the non-commutative [[Wick algebra|Wick]]-[[algebra of quantum observables]].

+-- {: .num_prop #OnMicrocausalObservablesGlobalBVDifferential}
###### Proposition
**(global [[BV-differential]] on [[Wick algebra]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded [[BV-BRST formalism|BV-BRST]] [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}). Let $\Delta_H$ be a compatible [[Wightman propagator]] (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}).

Then the global [[BV-differential]] $\{-S',(-)\}$ (def. \ref{ComplexBVBRSTGlobal}) restricts from [[polynomial observables]] to a linear map on [[microcausal polynomial observables]] (def. \ref{MicrocausalObservable})

$$
  \{-S',(-)\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ]
$$

and as such is a [[derivation]] not only for the pointwise product, but also for the product in the [[Wick algebra]] (the [[star product]] induced by the [[Wightman propagator]]):

$$
  \{-S', A_1 \star_H A_2\}
  \;=\;
  \{-S', A_1\} \star_H A_2
  +
  A_1 \star_H \{-S', A_2\}
  \,.
$$

We call $\{-S,(-)\}$ regarded as a nilpotent derivation on the [[Wick algebra]] this way the _free quantum [[BV-differential]]_.

=--

([Fredenhagen-Rejzner 11b, below (37)](BV-differential#FredenhagenRejzner11b), [Rejzner 11, below (5.28)](BV-differential#Rejzner11))

+-- {: .proof}
###### Proof

By example \ref{BVDifferentialGlobal} the action of $\{-S',(-)\}$ on polyomial observables is to replace [[antifield]] [[field observables]] by

$$
  \mathbf{\Phi}^\ddagger_a(x)
  \;\mapsto\;
  \pm
   (P_{A B}\mathbf{\Phi}^A)(x)
  \,,
$$

where $P$ is a [[differential operator]]. By [[partial integration]] this translates to $\{-S',(-)\}$ acting by the [[formally adjoint differential operator]] $P^\ast$ (def. \ref{FormallyAdjointDifferentialOperators}) via [[derivative of distributions|distributional derivative]] on the [[distribution|distributional]] [[coefficients]] of the given polynomial observable.

Now by prop. \ref{RetainsOrShrinksWaveFrontSetDifferentialOperator} the application of $P^\ast$ retains or shrinks the [[wave front set]] of the distributional coefficient, hence it preserves the microcausality condition (def. \ref{MicrocausalObservable}). This makes $\{-S',(-)\}$ restrict to microcausal polynomial observables.

To see that $\{-S',(-)\}$ thus restricted is a [[derivation]] of the Wick algebra product, it is sufficient to see that its [[commutators]] with the [[Wightman propagator]] vanish in each argument:

$$
  \left[
    \{-S',(-)\} \otimes id
    \;,\;
       \Delta_H
       \left(
         \frac{\delta}{\delta \mathbf{\Phi}}
           \otimes
         \frac{\delta}{\delta \mathbf{\Phi}}
       \right)
  \right]
  \;=\;
  0
$$

and

$$
  \left[
    id \otimes \{-S',(-)\}
    \;,\;
       \Delta_H
       \left(
         \frac{\delta}{\delta \mathbf{\Phi}}
           \otimes
         \frac{\delta}{\delta \mathbf{\Phi}}
       \right)
  \right]
  \;=\;
  0
  \,.
$$

Because with this we have:

$$
  \begin{aligned}
    \{-S', A_1 \star_H A_2\}
    & =
    \{-S',(-)\}
    \circ
     ((-)\cdot(-))
     \circ
     \exp\left(
       \hbar
        \Delta_H\left(
          \frac{\delta}{\delta \mathbf{\Phi}}
          \otimes
          \frac{\delta}{\delta \mathbf{\Phi}}
        \right)
     \right)
     (A_1 \otimes A_2)
     \\
     & =
     ((-)\cdot(-))
     \circ
     \left(
       \phantom{a \atop a}
       \{-S',-\} \otimes id
       +
       id \otimes \{-S',(-)\}
     \right)
     \circ
     \exp\left(
       \hbar
        \Delta_H\left(
          \frac{\delta}{\delta \mathbf{\Phi}}
          \otimes
          \frac{\delta}{\delta \mathbf{\Phi}}
        \right)
     \right)
     (A_1 \otimes A_2)
     \\
     & =
     ((-)\cdot(-))
     \circ
     \exp\left(
       \hbar
        \Delta_H\left(
          \frac{\delta}{\delta \mathbf{\Phi}}
          \otimes
          \frac{\delta}{\delta \mathbf{\Phi}}
        \right)
     \right)
     \circ
     \left(
       \phantom{a \atop b}
       \{-S',-\} \otimes id
       +
       id \otimes \{-S',(-)\}
     \right)
     (A_1 \otimes A_2)
     \\
     & =
     \{-S',A_1\} \star_H A_2
     +
     A_1 \star_H \{-S', A_2\}
  \end{aligned}
$$

Here in the first step we used that $\{-S',(-)\}$ is a derivation with respect to the pointwise product, by construction (def. \ref{ComplexBVBRSTGlobal}) and then we used the vanishing of the above commutators.

To see that these commutators indeed vanish, use that by example \ref{BVDifferentialGlobal} we have

$$
  \begin{aligned}
    &
    \left[
    \{-S',(-)\} \otimes id
    \;,\;
    \Delta_H\left( \frac{\delta }{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right)
    \right]
    \\
    & =
    \left[
      \underset{A}{\sum} (-1)^{deg(\phi^A)}
      \underset{\Sigma}{\int}
      (P_{A B}\mathbf{\Phi}^A)(x)
      \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \otimes id
      \,
      dvol_\Sigma(x)
      \;\,\;
      \underset{\Sigma \times \Sigma}{\int}
        \Delta_H^{A B}(x,y)
        \frac{\delta}{\delta \mathbf{\Phi}^A(x)}
        \otimes
        \frac{\delta}{\delta \mathbf{\Phi}^B(y)}
        \, dvol_\Sigma(x)
        \, dvol_\Sigma(y)
    \right]
    \\
    & =
    -\underset{a}{\sum} (-1)^{deg(\phi^A)}
    \underset{\Sigma \times \Sigma}{\int}
      \underset{
        = 0
      }{
      \underbrace{
        (P_x \Delta_H)_A{}^B(x,y)
      }
      }
      \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
      \otimes
      \frac{\delta}{\delta \mathbf{\Phi}^B(y)}
      \, dvol_\Sigma(x)
      \, dvol_\Sigma(y)
     \\
     & = 0
  \end{aligned}
$$

and similarly for the other order of the tensor products. Here the term over the brace vanishes by the fact that the Wightman propagator is a solution to the homogeneous equations of motion by prop. \ref{OnMinkowskiWightmanIsDistributionalSolutionToKleinGordon}.

=--

To analyze the behaviour of the free quantum BV-differential in general and specifically after passing to [[interacting field theory]] (below in chapter _[Interacting quantum fields](#InteractingQuantumFields)_) it is useful to re-express it in terms of the incarnation of the global [[antibracket]] with respect not to the pointwise product of observables, but the [[time-ordered product]]:

+-- {: .num_defn #AntibracketTimeOrdered}
###### Definition
**([[time-ordered product|time-ordered]] [[antibracket]])

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}).

Then the _time-ordered global [[antibracket]]_ on [[regular polynomial observables]]

$$
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
   \otimes
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \overset{\{-,-\}_{\mathcal{T}}}{\longrightarrow}
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

is the [[conjugation]] of the global [[antibracket]] (def. \ref{ComplexBVBRSTGlobal}) by the time-ordering operator $\mathcal{T}$ (from prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}):

$$
  \{-,-\}_{\mathcal{T}}
  \;\coloneqq\;
  \mathcal{T}\left(\left\{ \mathcal{T}^{-1}(-), \mathcal{T}^{-1}(-)\right\}\right)
$$

hence

$$
  \array{
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      &\overset{\{-,-\}}{\longrightarrow}&
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T}}}_{\mathllap{\simeq}}\Big\downarrow
      &&
    \Big\downarrow{}^{\mathrlap{\mathcal{T}}}_\simeq
    \\
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      &\overset{ \{-,-\}_{\mathcal{T}} }{\longrightarrow}&
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  }
$$

=--

([Fredenhagen-Rejzner 11, (27)](#FredenhagenRejzner11), [Rejzner 11, (5.14)](#Rejzner11))

+-- {: .num_prop #GaugeFixedActionFunctionalTimeOrderedAntibracket}
###### Proposition
**([[time-ordered product|time-ordered]] [[antibracket]] with [[gauge fixing|gauge fixed]] [[action functional]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}).

Then the [[time-ordered product|time-ordered]] [[antibracket]] (def. \ref{AntibracketTimeOrdered}) with the gauge fixed BV-[[action functional]] $-S'$ (def. \ref{ComplexBVBRSTGlobal}) equals the [[conjugation]] of the global [[BV-differential]] with the [[isomorphism]] $\mathcal{T}$ from the pointwise to the [[time-ordered product]] of [[observables]] (from prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise})

$$
  \{-S',-\}_{\mathcal{T}}
  \;=\;
  \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
  \,,
$$

hence

$$
  \array{
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
     &\overset{ \{-S',-\} }{\longrightarrow}&
    PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T}}}\Big\downarrow
      &&
    \Big\downarrow{}^{\mathrlap{\mathcal{T}}}
    \\
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
     &\overset{ \{-S',-\}_{\mathcal{T}} }{\longrightarrow}&
    PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  }
$$

=--

+-- {: .proof}
###### Proof

By the assumption that $(E,\mathbf{L})$ is a [[free field theory]] its [[Euler-Lagrange equations]] are linear in the fields, and hence $S'$ is quadratic in the fields. This means that

$$
  \mathcal{T}^{-1}S' = S' + const
  \,,
$$

where the second term on the right is independent of the fields, and hence that

$$
  \{\mathcal{T}^{-1}(-S'),-\}
  =
  \{-S', - \}
  \,.
$$

This implies the claim:

$$
  \begin{aligned}
    \{-S',-\}_{\mathcal{T}}
    & \coloneqq
    \mathcal{T}\left(\{ \mathcal{T}^{-1}(-S'), \mathcal{T}^{-1}(-) \}\right)
    \\
   & =
   \mathcal{T}\left(\{ -S', \mathcal{T}^{-1}(-) \}\right)
   \\
   & =
   \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
   \,.
  \end{aligned}
$$

=--

+-- {: .num_defn #ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}
###### Definition
**([[BV-operator]] for [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theory]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}) and with corresponding [[gauge fixing|gauge-fixed]] global [[BV-BRST differential]] on graded [[regular polynomial observables]]

$$
  \{-S' + S'_{BRST}, -\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

(def. \ref{GaugeFixingLagrangianDensity}).

Then the corresponding _[[BV-operator]]_

$$
  \Delta_{BV}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

on [[regular polynomial observables]] is, up to a factor of $i \hbar$, the difference between the free component $\{-S',-\}$ of the gauge fixed global BV differential and its time-ordered version (def. \ref{AntibracketTimeOrdered})

$$
  \Delta_{BV}
    \;\coloneqq\;
  \tfrac{1}{i \hbar}
  \left(
    \left\{
      -S',-
    \right\}_{\mathcal{T}}
    -
    \left\{ -S',(-) \right\}
  \right)
  \,,
$$

hence

$$
  \label{BVOperatorDefiningRelation}
  \{-S',-\}_{\mathcal{T}}
  \;=\;
  \{-S',-\} + i \hbar \Delta_{BV}
  \,.
$$

=--

+-- {: .num_prop #ComponentsBVOperator}
###### Proposition
**([[BV-operator]] in components)**

If the [[field bundles]] of all [[field (physics)|fields]], [[ghost fields]] and [[auxiliary fields]] are [[trivial vector bundles]], with field/ghost-field/auxiliary-field coordinates collectively denoted $(\phi^A)$ then
the [[BV-operator]] $\Delta_{BV}$ from prop. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator} is given explicitly by

$$
  \Delta_{BV}
  \;=\;
  \underset{a}{\sum} (-1)^{deg(\Phi^A)}
  \underset{\Sigma}{\int}
    \frac{\delta}{\delta \Phi^A(x)}
    \frac{\delta}{\delta \Phi^{\ddagger}_A(y)}
  dvol_\Sigma
$$

Since this formula exhibits a graded [[Laplace operator]], the BV-operator is also called the _BV-Laplace operator_ or _BV-Laplacian_, for short.

=--

([Fredenhagen-Rejzner 11, (29)](#FredenhagenRejzner11), [Rejzner 11, (5.20)](#Rejzner11))

+-- {: .proof}
###### Proof

By prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket} we have equivalently

$$
  i \hbar \Delta_{BV}
  \;=\;
  \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
  \,-\,
  \{-S',-\}
$$

and by example \ref{BVDifferentialGlobal} the second term on the right is

$$
  \begin{aligned}
    \left\{ -S', -\right\}
    & =
    \underset{\Sigma}{\int}
    j^{\infty}\left(\mathbf{\Phi}\right)^\ast
    \left(
      \frac{\overset{\leftarrow}{\delta}_{EL} L}{\delta \phi^A}
    \right)(x)
    \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
    \,
    dvol_\Sigma(x)
    \\
    & =
    \underset{a}{\sum}
      (-1)^{deg(\phi^A)}
    \underset{}{\int}
      (P\mathbf{\Phi})_A(x)
      \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
    \,
    dvol_\Sigma(x)
  \end{aligned}
$$

With this we compute as follows:

$$
  \label{AAA}
  \begin{aligned}
    \{-S',-\}_{\mathcal{T}}
    & =
    \mathcal{T} \circ \left\{ -S,-\right\} \circ \mathcal{T}^{-1}
    \\
    & =
    \exp\left(
     \left[
       \hbar \tfrac{1}{2} \Delta_F
       \left(
         \frac{\delta}{\delta \mathbf{\Phi}},
         \frac{\delta}{\delta \mathbf{\Phi}}
        \right)
        \,,\,
        -
      \right]
    \right)
    \left(
      \{-S',-\}
    \right)
    \\
    & =
    \{-S',-\}
    +
    \left[
      \hbar \tfrac{1}{2}
      \Delta_F
      \left(
        \frac{\delta}{\delta \mathbf{\Phi}},
        \frac{\delta}{\delta \mathbf{\Phi}}
      \right)
      \,,
      \{-S',-\}
    \right]
    + \underset{ = 0 }{\underbrace{\hbar^2(...)}}
    \\
    & =
    \phantom{+}
    \left\{ -S' , -\right\}
    \\
    &
    \phantom{=}
    +
    \left[
      \tfrac{1}{2}\hbar
      \underset{\Sigma \times \Sigma}{\int}
      \Delta_F^{A B}(x,y)
      \frac{\delta^2}{\delta \mathbf{\Phi}^A(x) \delta \mathbf{\Phi}^B(y)}
      \, dvol_\Sigma(x)
      \, dvol_\Sigma(y)
      \;,\;
      \underset{a}{\sum}
      (-1)^{deg(\phi^A)}
      \underset{\Sigma}{\int}
        (P\mathbf{\Phi})_A(x)
        \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
      \,
      dvol_\Sigma(x)
    \right]
   \\
   & =
   \left\{ -S', -\right\}
   \\
   & \phantom{=}
   +
   \underset{A}{\sum}
    (-1)^{deg(\phi^A)}
   \underset{\Sigma \times \Sigma}{\int}
     \underset{ =  i \delta(x-y) }{\underbrace{P_x \Delta_F(x,y)}}
     \frac{\delta}{\delta \mathbf{\Phi}^A(x)}
     \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(y)}
    \, dvol_\Sigma(x)
    \, dvol_\Sigma(y)
    \\
   & = \left\{ -S', -\right\}
   + i \hbar
   \underset{A}{\sum}
   (-1)^{deg(\phi^A)}
   \underset{\Sigma}{\int}
    \frac{\delta}{\delta \mathbf{\Phi}^A(x)}
    \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
    \,
    dvol_\Sigma(x)
  \end{aligned}
$$

Here we used

1. under the first brace that by assumption of a [[free field theory]], $\{-S',-\}$ is linear in the fields, so that the first [[commutator]] with the [[Feynman propagator]] is independent of the fields, and hence all the higher commutators vanish;

1.  under the second brace that the [[Feynman propagator]] is $+i$ times the [[Green function]] for the [[Green hyperbolic differential equation|Green hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] (cor. \ref{GreenFunctionFeynmanPropagator}).

=--

+-- {: .num_prop #AntibracketBVOperatorRelation}
###### Proposition
**(global [[antibracket]] exhibits failure of [[BV-operator]] to be a [[derivation]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$

The [[BV-operator]] $\Delta_{BV}$ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) and the global [[antibracket]] $\{-,-\}$ (def. \ref{ComplexBVBRSTGlobal}) satisfy for all [[polynomial observables]] (def. \ref{PolynomialObservables}) $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})[ [\hbar] ]$ the relation

$$
  \label{GlobalAntibracketInteractingWithBVOperator}
  \{A_1, A_2\}
  \;=\;
  (-1)^{deg(A_2)}
  \,
  \Delta_{BV}(A_1 \cdot A_2)
  -
  (-1)^{deg(A_2)}
  \,
  \Delta_{BV}(A_1) \cdot A_2
  -
  A_1 \cdot \Delta_{BV}(A_2)
$$

for $(-) \cdot (-)$ the pointwise product of observables (def. \ref{Observable}).

Moreover, it commutes on [[regular polynomial observables]] with the [[time-ordered product|time-ordering operator]] $\mathcal{T}$ (prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise})

$$
  \Delta_{BV} \circ \mathcal{T}
  =
  \mathcal{T} \circ \Delta_{BV}
  \phantom{AAA}
  \text{on}
  \,\,
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

and hence satisfies the analogue of relation (eq:GlobalAntibracketInteractingWithBVOperator) also for the time-ordered antibracket $\{-,-\}_{\mathcal{T}}$ (def. \ref{AntibracketTimeOrdered}) and the [[time-ordered product]] $\star_F$ on regular polynomial observables

$$
  \{A_1, A_2\}_{\mathcal{T}}
  \;=\;
  (-1)^{deg(A_2)}
  \,
  \Delta_{BV}(A_1 \star_F A_2)
  -
  (-1)^{dag(A_2)}
  \Delta_{BV}(A_1) \star_F A_2
  -
  A_1 \star_F \Delta_{BV}(A_2)
  \,.
$$

=--

(e.g. [Henneaux-Teitelboim 92, (15.105d)](antibracket#HenneauxTeitelboim92))

+-- {: .proof}
###### Proof

With prop. \ref{ComponentsBVOperator} the first statement is a graded version of the analogous relation for an ordinary [[Laplace operator]] $\Delta \coloneqq g^{a b} \partial_a \partial_b$ acting on [[smooth functions]] on [[Cartesian space]], which on [[smooth functions]] $f,g$ satisfies

$$
  \Delta(f \cdot g)
  \;=\;
  (\nabla f, \nabla g) - \Delta(f) g - f \Delta(g)
  \,,
$$

by the [[product law]] for [[differentiation]], where now $\nabla f \coloneqq (g^{a b} \partial_b f)$ is the [[gradient]] and $(v,w) \coloneqq g_{a b} v^a w b$ the [[inner product]]. Here one just needs to carefully record the relative signs that appear.

That the BV-operator commutes with the time-ordering operator is clear from the fact that both of these are given by [[partial derivative|partial]] [[functional derivatives]] with _[[constant function|constant]]_ [[coefficients]]. This immediately implies the last statement from the first.

=--

+-- {: .num_example #TimeOrderedExponentialBVOperator}
###### Example
**([[BV-operator]] on [[time-ordered product|time-ordered]] [[exponentials]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.

Let moreover $V \in PolyObs(E_{\text{BV-BRST}})_{reg, deg = 0}[ [\hbar] ]$  be a [[regular polynomial observable]] (def. \ref{PolynomialObservables}) of degree zero. Then the application of the [[BV-operator]] $\Delta_{BV}$ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) to the [[time-ordered product|time-ordered]] [[exponential]] $\exp_{\mathcal{T}}(V)$ (example \ref{RegularObservablesExponentialTimeOrdered}) is the [[time-ordered product]] of the time-ordered exponential with the sum of $\Delta_{BV}(V)$ and the global [[antibracket]] $\tfrac{1}{2}\{V,V\}$ of $V$ with itself:

$$
  \Delta_{BV}
  \left(
    \exp_{\mathcal{T}}(V)
  \right)
  \;=\;
  \left(
    \Delta_{BV}(V)
    +
    \tfrac{1}{2}\{V,V\}
  \right)
   \star_F
  \exp_{\mathcal{T}}(V)
$$

=--

+-- {: .proof}
###### Proof

By prop. \ref{AntibracketBVOperatorRelation} $\Delta_{BV}$ acts as a [[derivation]] on the [[time-ordered product]] up to a correction given by the antibracket of the two factors. This yields the result by the usual combinatorics of [[exponentials]].

$$
  \begin{aligned}
    & \Delta_{BV}
    \left(
      1 + V + \tfrac{1}{2}V \star_F V + \cdots
    \right)
    \\
    & =
    \Delta_{BV}(V)
      +
    \tfrac{1}{2}\left( \Delta_{BV}(V) \star_F V + V \star_F \Delta_{BV}(V) \right)
      +
    \tfrac{1}{2}\{V,V\}
      +
    \cdots
    \\
    & =
    \Delta_{BV}(V)
      +
    \tfrac{1}{2}\{V,V\}
      \;+\;
    \Delta_{BV}(V) \star_F V
     + \cdots
  \end{aligned}
$$

=--

$\,$

**[[Schwinger-Dyson equation]]**
  {#SchwingerDysonEquation}

A special case of the general occurrence of the [[BV-operator]] is the following important property of [[on-shell]] [[time-ordered products]]:

+-- {: .num_prop #DysonSchwinger}
###### Proposition
**([[Schwinger-Dyson equation]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (def. \ref{GaugeFixingLagrangianDensity}) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ (remark \ref{FieldBundleBVBRST}).

Let

$$
  \label{SchwingerDysonTestObservable}
  A
  \;\coloneqq\;
  \underset{\Sigma}{\int}
    A^a(x) \cdot \mathbf{\Phi}^\ddagger_a(x)
  \, dvol_\Sigma(x)
  \;\in\;
  PolyObs_{reg}(E_{\text{BV-BRST}})
$$

be an [[off-shell]] [[regular polynomial observable]] which is [[linear map|linear]] in the [[antifield]] [[field observables]] $\mathbf{\Phi}^\ddagger$. Then

$$
  \label{EquationSchwingerDyson}
  \mathcal{T}^{\pm 1}
  \left(
    \underset{\Sigma}{\int}
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
      \cdot
      A^a(x)
      \, dvol_\Sigma(x)
  \right)
  \;=\;
  \pm i \hbar
  \,
  \mathcal{T}^{\pm}
  \left(
    \underset{\Sigma}{\int}
    \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)}
    \,
    dvol_\Sigma(x)
  \right)
  \phantom{A}
  \in
  \underset{
    \text{on-shell}
  }{
  \underbrace{
    PolyObs_{reg}(E_{\text{BV-BRST}}, \mathbf{L'})
  }}
  \,.
$$

This is called the _[[Schwinger-Dyson equation]]_.

=--

The following proof is due to ([Rejzner 16, remark 7.7](#Rejzner16)) following the informal traditional argument ([Henneaux-Teitelboim 92, (15.108b)](#HenneauxTeitelboim92)).

+-- {: .proof}
###### Proof

Applying the inverse time-ordering map $\mathcal{T}^{-1}$ (prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}) to equation (eq:BVOperatorDefiningRelation) applied to $A$ yields

$$
  \underset{
    \mathcal{T}^{-1}
    \underset{\Sigma}{\int}
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
      \cdot
      A^a(x)
    dvol_\Sigma(x)
  }{
  \underbrace{
    \mathcal{T}^{-1}\left\{ -S', A\right\}
  }
  }
  \;=\;
  -
  \underset{
    i \hbar
    \mathcal{T}^{-1}
    \underset{\Sigma}{\int}
      \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)}
    \, dvol_\Sigma
  }{
  \underbrace{
    i \hbar \mathcal{T}^{-1}\Delta_{BV}(A)
  }
  }
  +
  \underset{
    \{-S',\mathcal{T}^{-1}(A)\}
  }{
  \underbrace{
    \mathcal{T}^{-1}\left\{ -S',A\right\}_{\mathcal{T}}
  }
  }
$$

where we have identified the terms under the braces by 1) the component expression for the BV-differential $\{-S',-\}$ from prop. \ref{BVDifferentialGlobal}, 2) prop. \ref{ComponentsBVOperator} and 3) prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket}.

The last term is manifestly in the [[image]] of the [[BV-differential]] $\{-S',-\}$ and hence vanishes when passing to [[on-shell]] observables along the [[isomorphism]] (eq:OnShellPolynomialObservablesAsBVCohomology)

$$
  \underset{
    \text{on-shell}
  }{
  \underbrace{
    PolyObs(E_{\text{BV-BRST}}, \mathbf{L}')
  }}
  \;\simeq\;
  \underset{
    \text{off-shell}
  }{
  \underbrace{
    PolyObs(E_{\text{BV-BRST}})_{def(af = 0)}
  }}/im(\{-S',-\})
$$

(by example \ref{BVDifferentialGlobal}).

The same argument with the replacement $\mathcal{T} \leftrightarrow \mathcal{T}^{-1}$ throughout yields the other version of the equation (with time-ordering instead of reverse time ordering and the sign of the $\hbar$-term reversed).

=--

+-- {: .num_remark }
###### Remark
**("Schwinger-Dyson operator")**

The proof of the [[Schwinger-Dyson equation]] in prop. \ref{DysonSchwinger} shows that, up to [[time-ordered product|time-ordering]], the [[Schwinger-Dyson equation]] is the on-shell vanishing of the "quantized" [[BV-differential]] (eq:BVOperatorDefiningRelation)

$$
  \{-S',-\}_{\mathcal{T}}
  \;=\;
  \{-S', -\}
  \,+\,
  i \hbar \, \Delta_{BV}
  \,,
$$

where the [[BV-operator]] is the quantum correction of order $\hbar$. Therefore this is also called the _Schwinger-Dyson operator_ ([Henneaux-Teitelboim 92, (15.111)](#HenneauxTeitelboim92)).

=--

+-- {: .num_example #SchwingerDysonDistributional}
###### Example
**([[distribution|distributional]] [[Schwinger-Dyson equation]])**

Often the [[Schwinger-Dyson equation]] (prop. \ref{DysonSchwinger}) is displayed before spacetime-smearing of [[field observables]] in terms of [[operator products]] of [[operator-valued distributions]], taking the observable $A$ in (eq:SchwingerDysonTestObservable) to be

$$
  A^a(x)
    \;\coloneqq\;
  \delta(x-x_0) \delta^a_{a_0}
  \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \,.
$$

This choice makes (eq:EquationSchwingerDyson) become the [[distribution|distributional]] [[Schwinger-Dyson equation]]

$$
  \begin{aligned}
  & T
  \left(
    \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
      \cdot
    \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \right)
  \\
  &
  \underset{\text{on-shell}}{=}
  - i \hbar
  \underset{k}{\sum}
  T
  \left(
    \mathbf{\Phi}^{a_1}(x_1)
      \cdots
    \mathbf{\Phi}^{a_{k-1}}(x_{k-1})
      \cdot
    \delta(x_0 - x_k) \delta^{a_0}_{a_k}
      \cdot
    \mathbf{\Phi}^{a_{k+1}}(x_{k+1})
      \cdots
    \mathbf{\Phi}^{a_n}(x_m)
  \right)
  \end{aligned}
$$

(e.q. [Dermisek 09](Schwinger-Dyson+equation#Dermisek09)).

In particular this means that if $(x_0,a_0) \neq (x_k, a_k)$ for all $k \in \{1,\cdots ,n\}$ then

$$
  T
  \left(
    \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
      \cdot
    \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
$$

Since by the [[principle of extremal action]] (prop. \ref{PrincipleOfExtremalAction}) the equation

$$
  \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
  \;=\;
  0
$$

is  the [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] (for the [[classical field theory]]) "at $x_0$", this may be interpreted as saying that the classical equations of motion for fields at $x_0$ still hold for [[time-ordered product|time-ordered]] [[quantum theory|quantum]] [[expectation values]], as long as all other observables are evaluated away from $x_0$; while if observables do coincide at $x_0$ then there is a correction measured by the [[BV-operator]].

=--

$\,$

This concludes our discussion of the [[algebra of quantum observables]] for [[free field theories]]. In the [next chapter](#InteractingQuantumFields) we discuss the [[perturbative QFT]] of [[interacting field theories]] as [[deformations]] of such free quantum field theories.

$\,$

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