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

## Interacting quantum fields
 {#QuantumObservables}

In this chapter we discuss the following topics:

* _[Free field vacua](#FreeFieldVacua)_

* _[Perturbative S-matrices](#PerturbativeSMatrixAndTimeOrderedProducts)_

* _[Conceptual remarks](#RemarksOnCausalPerturbationTheoryAxioms)_

* _[Interacting field observables](#LocalNetsOfInteractingFieldObservables)_

* _[Time-ordered products](#TimeOrderedProducts)_

* _[("Re"-)Normalization](#ExistenceAndRenormalization)

* _[Feynman perturbation series](#FeynmanDiagrams)_

* _[Effective action](#EffectiveAction)_

* _[Vacuum diagrams](#VacuumDiagrams)_

* _[Interacting quantum BV-differential](#InteractingQantumBVDifferential)_

* _[Ward identities](#WardIdentities)_

$\,$

In the [previous chapter](#FreeQuantumFields) we have found the [[quantization]] of _[[free field theories|free]]_ [[Lagrangian field theories]] by first choosing a [[gauge fixing|gauge fixed]] [[BV-BRST complex|BV-BRST]]-[[homological resolution|resolution]] of the [[algebra of observables|algebra of]] [[gauge invariance|gauge invariant]] [[on-shell]] observabes, then applying [[algebraic deformation quantization]] induced by the resulting [[Peierls-Poisson bracket]] on the graded [[covariant phase space]] to pass to a [[non-commutative algebra]] of quantum observables, such that, finally, the [[BV-BRST differential]] is respected.

Of course most [[quantum field theories]] of interest are non-[[free field theories|free]]; they are _[[interacting field theories]]_ whose [[equations of motion]] is a _non-linear_ differential equation. The archetypical example is the coupling of the [[Dirac field]] to the [[electromagnetic field]] via the [[electron-photon interaction]], corresponding to the [[interacting field theory]] called _[[quantum electrodynamics]]_ (discussed [below](#QuantumElectrodynamics)).

In principle the [[perturbative quantum field theory|perturbative]] [[quantization]] of such non-[[free field theory]] [[interacting field theories]] proceeds the same way: One picks a [[BV-BRST complex|BV-BRST]]-[[gauge fixing]], computes the [[Peierls-Poisson bracket]] on the resulting [[covariant phase space]] ([Khavkine 14](Peierls+bracket#Khavkine14)) and then finds a [[formal deformation quantization]] of this [[Poisson structure]] to obtain the quantized [[non-commutative algebra]] of [[quantum observables]], as [[formal power series]] in [[Planck's constant]] $\hbar$.

It turns out ([Collini 16](perturbative+algebraic+quantum+field+theory#Collini16), [Hawkins-Rejzner 16](perturbative+algebraic+quantum+field+theory#HawkinsRejzner16), prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization} below)
that the resulting [[interacting field theory|interacting]] [[formal deformation quantization]] may equivalently be expressed in terms of _[[scattering amplitudes]]_ (example \ref{ScatteringAmplitudeFromInteractingFieldObservables} below): These are the [[probability amplitudes]] for [[plane waves]] of [[free fields]] to come in from the far [[past]], then [[interaction|interact]] in a compact region of [[spacetime]] via the given [[interaction]] ([[adiabatic switching|adiabatically switched-off]] outside that region) and to emerge again as [[free fields]] into the far [[future]].

The collection of all these [[scattering amplitudes]], as the [[types]] and  [[wave vectors]] of the incoming and outgoing [[free fields]] varies, is called the _[[perturbative S-matrix|perturbative scattering matrix]]_ of the [[interacting field theory]], or just _[[S-matrix]]_ for short. It may equivalently be expressed as the [[exponential]] of [[time-ordered products]] of the [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]] with itself (def. \ref{LagrangianFieldTheoryPerturbativeScattering} below). The [[combinatorics]] of the terms in this exponential is captured by _[[Feynman diagrams]]_ (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} below), which, with some care (remark \ref{WorldlineFormalism} below), may be thought of as [[finite multigraphs]] (def. \ref{Graphs} below) whose [[edges]] are [[worldlines]] of [[virtual particles]] and whose [[vertices]] are the [[interactions]] that these particles undergo (def. \ref{FeynmanDiagram} below).

The [[axiom|axiomatic]] definition of [[S-matrices]] for [[relativistic field theory|relativistic]] [[Lagrangian field theories]] and their rigorous construction via [[renormalization|("re"-)normalization]] of [[time-ordered products]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} below) is called _[[causal perturbation theory]]_, due to ([Epstein-Glaser 73](causal+perturbation+theory#EpsteinGlaser73)). This makes precise and well-defined the would-be [[path integral quantization]] of [[interacting field theories]] (remark \ref{InterpretationOfPerturbativeSMatrix} below) and removes the errors (remark \ref{TheTraditionalErrorThatLeadsToTheNotoriouDivergencies} below) and ensuing puzzlements (expressed in [Feynman 85](Schwinger-Tomonaga-Feynman-Dyson#Feynman85SuchABunchOfWords)) that plagued the original informal conception of [[perturbative quantum field theory]] due to [[Schwinger-Tomonaga-Feynman-Dyson]] (remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences} below).

The equivalent re-formulation of the [[formal deformation quantization]] of [[interacting field theories]] in terms of [[scattering amplitudes]] (prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization} below) has the advantage that it gives a direct handle on those [[observables]] that are measured in [[scattering]] [[experiments]], such as the [[LHC]]-experiment. The bulk of mankind's knowledge about realistic [[perturbative quantum field theory]] -- such as notably the [[standard model of particle physics]] -- is reflected in such [[scattering amplitudes]] given via their [[Feynman perturbation series]] in [[formal power series|formal powers]] of [[Planck's constant]] and the [[coupling constant]].

Moreover, the mathematical passage from [[scattering amplitudes]] to the actual [[interacting field algebra]] [[algebra of quantum observables|of quantum observables]] (def. \ref{QuntumMollerOperator} below) corresponding to the [[formal deformation quantization]] is well understood, given via "[[Bogoliubov's formula]]" by the _[[quantum Møller operators]]_ (def. \ref{InteractingFieldObservables} below).

Via [[Bogoliubov's formula]] every perturbative [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) induces for every choice of [[adiabatic switching|adiabatically switched]]
[[interaction]] [[action functional]] a notion of [[perturbative QFT|perturbative]] [[interacting field observables]] (def. \ref{InteractingFieldObservables}). These generate an algebra (def. \ref{QuntumMollerOperator} below).
By [[Bogoliubov's formula]], in general this algebra depends on the choice of [[adiabatic switching]];
which however is not meant to be part of the [[physics]], but just a mathematical device for grasping global field structures locally.

But this spurious dependence  goes away (prop. \ref{IsomorphismFromChangeOfAdiabaticSwitching} below) when restricting attention to
observables whose spacetime support is inside a compact [[causally closed subsets]] $\mathcal{O}$ of spacetime (def. \ref{PerturbativeGeneratingLocalNetOfObservables} below). This is a sensible condition for an [[observable]] in [[physics]],
where any realistic [[experiment]] nessecarily probes only a compact subset of spacetime, see also remark \ref{AdiabaticLimit}.

The resulting system (a "[[co-presheaf]]") of well-defined perturbative [[interacting field algebras of observables]] (def. \ref{SystemOfAlgebrasOfQuantumObservables} below)

$$
  \mathcal{O} \mapsto IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
$$

is in fact [[causal locality|causally local]] (prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below).
This fact was presupposed without proof already in [Il'in-Slavnov 78](perturbative+algebraic+quantum+field+theory#IlinSlavnov78);
because this is one of two key properties that the [[Haag-Kastler axioms]] ([Haag-Kastler 64](Haag-Kastler+axioms#HaagKastler64)) demand of an intrinsically defined [[quantum field theory]]
(i.e. defined without necessarily making recourse to the geometric backdrop of [[Lagrangian field theory]]). The only other key property demanded by the [[Haag-Kastler axioms]]
is that the [[algebras of observables]] be [[C*-algebras]]; this however must be regarded as the axiom encoding
[[non-perturbative quantum field theory]] and hence is necessarily violated in the present context of [[perturbative QFT]].
Since quantum field theory following the full [[Haag-Kastler axioms]] is commonly known as _[[AQFT]]_, this perturbative version,
with [[causally local nets of observables]] but without the [[C*-algebra]]-condition on them, has come to be called
_[[perturbative AQFT]]_ ([Dütsch-Fredenhagen 01](perturbative+algebraic+quantum+field+theory#DuetschFredenhagen01),
[Fredenhagen-Rejzner 12](perturbative+algebraic+quantum+field+theory#FredenhagenRejzner12)).

In this terminology the content of prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below is that
_while the input of [[causal perturbation theory]] is a [[gauge fixing|gauge fixed]] [[Lagrangian field theory]],
the output is a [[perturbative algebraic quantum field theory]]_:

$$
  \array{
    \array{
      \text{gauge-fixed}
      \\
      \text{Lagrangian}
      \\
      \text{field theory}
    }
    &
    \overset{
      \array{
        \text{causal}
        \\
        \text{perturbation theory}
        \\
      }
    }{\longrightarrow}&
    \array{
      \text{perturbative}
      \\
      \text{algebraic}
      \\
      \text{quantum}
      \\
      \text{field theory}
    }
    \\
    \underset{
      \array{
        \text{(Becchi-Rouet-Stora 76,}
        \\
        \text{Batalin-Vilkovisky 80s)}
      }
    }{\,}
    &
    \underset{
      \array{
        \text{(Bogoliubov-Shirkov 59,}
        \\
        \text{Epstein-Glaser 73)}
      }
    }{\,}
    &
    \underset{
    \array{
      \text{ (Il'in-Slavnov 78,  }
      \\
      \text{Brunetti-Fredenhagen 99,}
      \\
      \text{Dütsch-Fredenhagen 01)}
    }
    }{\,}
  }
$$

The independence of the [[causally local net]] of localized [[interacting field algebras of observables]]
$IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int} )(\mathcal{O})$ from the choice of [[adiabatic switching]]
implies a well-defined spacetime-global [[algebra of observables]] by forming the [[inductive limit]]

$$
  IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})
    \;\coloneqq\;
  \underset{\underset{\mathcal{O}}{\longrightarrow}}{\lim}
  \left(
    {\, \atop \,}
    IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
    {\, \atop \,}
  \right)
  \,.
$$

This is also called the _[[algebraic adiabatic limit]]_, defining the [[algebras of observables]] of [[perturbative QFT]]
"in the infrared". The only remaining step in the construction of a [[perturbative QFT]] that remains is then
to find an [[interacting vacuum state]]

$$
  \left\langle
    -
  \right\rangle_{int}
  \;\colon\;
  IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})
   \longrightarrow
  \mathbb{C}[ [ \hbar, g] ]
$$

on the global [[interacting field algebra]] $Obs_{\mathbf{L}_{int}}$.
This is related to the actual _[[adiabatic limit]]_, and it is by and large an open problem, see remark \ref{AdiabaticLimit} below.

In conclusion so far, the [[algebraic adiabatic limit]] yields, starting with a [[BV-BRST formalism|BV-BRST]] [[gauge fixing|gauge fixed]] [[free field]] [[vacuum]],
the perturbative construction of [[interacting field algebras of observables]] (def. \ref{QuntumMollerOperator}) and their organization
in increasing powers of $\hbar$ and $g$ ([[loop order]], prop. \ref{FeynmanDiagramLoopOrder}) via the [[Feynman perturbation series]] (example \ref{FeynmanPerturbationSeries}, example \ref{SMatrixVacuumContribution}).

But this [[interacting field algebra of observables]] still involves all the [[auxiliary fields]] of the [[BV-BRST formalism|BV-BRST]] [[gauge fixing|gauge fixed]] [[free field]] [[vacuum]] (as in example \ref{FieldSpeciesQED} for QED), while the
actual physical [[gauge invariance|gauge invariant]] [[on-shell]] observables should be (just) the [[cochain cohomology]] of the
[[BV-BRST differential]] on this enlarged space of observables. Hence for the construction of [[perturbative QFT]] to conclude,
it remains to pass the [[BV-BRST differential]] of the [[free field]] [[Wick algebra]] of observables to
a [[differential]] on the [[interacting field algebra]], such that its [[cochain cohomology]] is well defined.

Since the [[time-ordered products]] away from coinciding interaction points are uniquely fixed (prop. \ref{TimeOrderedProductAwayFromDiagonal} below), one finds that also this _interacting quantum BV-differential_
is uniquely fixed, on [[regular polynomial observables]], by [[conjugation]] with the [[quantum Møller operators]] (def. \ref{BVDifferentialInteractingQuantum}).
The formula that characterizes it there is called the _[[quantum master equation]]_ or equivalently the _[[quantum master Ward identity]]_ (prop. \ref{QuantumMasterEquation} below).

In its incarnation as the [[master Ward identity]], this expresses the difference between the [[shell]] of the free classical field theory
and that of the interacting quantum field theory, thus generalizing the [[Schwinger-Dyson equation]] to [[interacting field theory]] (example \ref{SchwingerDysonReductionOfQuantumMasterWardIdentity} below). Applied to [[Noether's theorem]] it expresses the possible
failure of [[conserved currents]] associated with [[infinitesimal symmetries of the Lagrangian]] to still be conserved
in the [[interacting field theory|interacting]] [[perturbative QFT]] (example \ref{NoetherCurrentConservationQuantumCorrection} below).

As one [[extension of distributions|extends]] the [[time-ordered products]] to coinciding interaction points in [[renormalization|("re"-)normalization]]
of the [[perturbative QFT]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} below), the [[quantum master equation]]/[[master Ward identity]] becomes a _[[renormalization condition]]_ (prop. \ref{BasicConditionsRenormalization} below). If this condition fails
one says that the [[interacting field theory|interacting]] [[perturbative QFT]] has a _[[quantum anomaly]]_, specifically a _[[gauge anomaly]]_
if the [[Ward identity]] of an [[infinitesimal gauge symmetry]] is violated.

These issues of [[renormalization|"(re)-"normalization]] we discuss in detail in the [next chapter](#Renormalization).

$\,$

**Free field vacua**
 {#FreeFieldVacua}

In considering [[perturbative QFT]], we are considering [[perturbation theory]] in formal [[deformation]] parameters around a fixed [[free field theory|free]]
[[Lagrangian field theory|Lagrangian]] [[quantum field theory]] in a chosen [[Hadamard vacuum state]].

For convenient referencing we collect all the structure and notation that goes into this in the following definitions:

+-- {: .num_defn #VacuumFree}
###### Definition
**([[free field theory|free]] [[relativistic field theory|relativistic]] [[Lagrangian field theory|Lagrangian]] [[quantum field theory|quantum field]] [[vacuum]])**

Let

1. $\Sigma$ be a [[spacetime]] (e.g. [[Minkowski spacetime]]);

1. $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] (def. \ref{FreeFieldTheory}), with [[field bundle]] $E \overset{fb}{\to} \Sigma$;

1. $\mathcal{G} \overset{fb}{\to} \Sigma$ a [[gauge parameter bundle]] for $(E,\mathbf{L})$ (def. \ref{GaugeParameters}), with induced [[BRST-complex|BRST]]-[[reduced phase space|reduced]] [[Lagrangian field theory]] $\left( E \times_\Sigma \mathcal{G}[1], \mathbf{L} - \mathbf{L}_{BRST}\right)$ (example \ref{LocalOffShellBRSTComplex});

1. $(E_{\text{BV-BRST}}, \mathbf{L}' - \mathbf{L}'_{BRST})$ a [[gauge fixing]] (def. \ref{GaugeFixingLagrangianDensity}) with [[graded manifold|graded]] [[BV-BRST formalism|BV-BRST]] [[field bundle]] $E_{\text{BV-BRST}} = T^\ast_{\Sigma}[-1]\left( E\times_\Sigma \mathcal{G}[1] \times_\Sigma A \times_\Sigma A[-1]\right)$ (remark \ref{FieldBundleBVBRST});

1. $\Delta_H \in \Gamma'( E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}} )$ a [[Wightman propagator]] $\Delta_H = \tfrac{i}{2} \Delta + H$ compatible with the [[causal propagator]] $\Delta$ which corresponds to the [[Green hyperbolic partial differential equation|Green hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] induced by the [[gauge fixing|gauge-fixed]] [[Lagrangian density]] $\mathbf{L}'$.

Given this, we write

$$
  \left(
    {\, \atop \,}
    PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ] \;,\;
   \star_H
   {\, \atop \,}
  \right)
$$

for the corresponding [[Wick algebra]]-[[structure]] on [[formal power series]] in $\hbar$ ([[Planck's constant]]) of [[microcausal polynomial observables]] (def. \ref{MicrocausalObservable}). This is a [[star algebra]] with respect to ([[coefficient]]-wise) [[complex conjugation]] (prop. \ref{MoyalStarProductOnMicrocausal}).

Write

$$
  \label{HadamardVacuumStateForFreeFieldTheory}
  \array{
    PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ]
     &\overset{\langle - \rangle}{\longrightarrow}&
    \mathbb{C}[ [\hbar] ]
    \\
    A &\mapsto& A(\Phi = 0)
  }
$$

for the induced [[Hadamard vacuum state]] (prop. \ref{WickAlgebraCanonicalState}), hence the [[state on a star-algebra|state]] whose [[distribution|distributional]] [[2-point function]] is the chosen [[Wightman propagator]]:

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

Given any [[microcausal polynomial observable]] $A \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ then its value in this state is called its _free [[vacuum expectation value]]_

$$
  \left\langle
    A
  \right\rangle
  \;\in\;
  \mathbb{C}[ [ \hbar, g, j] ]
  \,.
$$

Write

$$
  \label{NormalOrderingLocalObservables}
  \array{
    LocObs(E_{\text{BV-BRST}})
      &\overset{\phantom{A}:(-):\phantom{A}}{\hookrightarrow}&
    PolyObs(E_{\text{BV-BRST}})_{mc}
    \\
    A &\mapsto& :A:
  }
$$

for the inclusion of [[local observables]] (def. \ref{LocalObservables}) into [[microcausal polynomial observables]] (example \ref{PointwiseProductsOfFieldObservablesAdiabaticallySwitchedIsMicrocausal}), thought of as forming [[normal-ordered products]] in the [[Wick algebra]] (by def. \ref{NormalOrderedProductNotation}).

We denote the [[Wick algebra]]-product (the [[star product]] $\star_H$ induced by the [[Wightman propagator]] $\Delta_H$ according to prop. \ref{PropagatorStarProduct}) by juxtaposition (def. \ref{NormalOrderedProductNotation})

$$
  A_1 A_2 \;\coloneqq\; A_1 \star_H A_2
  \,.
$$

If an element $A \in PolyObs(E_{\text{BV-BRST}})$ has an [[inverse]] with respect to this product, we denote that by
$A^{-1}$:

$$
  A^{-1} A = 1
  \,.
$$

Finally, for $A \in LocObs(E_{\text{BV-BRST}})$ we write $supp(A) \subset \Sigma$ for its spacetime support (def. \ref{SpacetimeSupport}).
For $S_1, S_2 \subset \Sigma$ two [[subsets]] of [[spacetime]] we write

$$
  S_1 {\vee\!\!\!\wedge} S_2
  \phantom{AAA}
  \left\{
    \array{
      \text{"}S_1 \, \text{does not intersect the past of} \, S_2\text{"}
      \\
      \Updownarrow
      \\
      \text{"}S_2 \, \text{does not intersect the future of} \, S_1\text{"}
    }
  \right.
$$

for the  [[causal order]]-[[relation]] (def. \ref{CausalOrdering}) and

$$
  S_1 {\gt\!\!\!\!\lt} S_2
  \phantom{AAA}
  \text{for}
  \phantom{AAA}
  \array{
    S_1 {\vee\!\!\!\wedge} S_2
    \\
    \text{and}
    \\
    S_2 {\vee\!\!\!\wedge} A_1
  }
$$

for _[[spacelike]] separation_.

=--

Being concerned with [[perturbative QFT|perturbation theory]] means mathematically that we consider _[[formal power series]]_
in [[deformation]] parameters $\hbar$ ("[[Planck's constant]]") and $g$ ("[[coupling constant]]"), also in $j$ ("[[source field]]"),
see also remark \ref{AsymptoticSeriesObservables}. The following collects our notational conventions for these matters:

+-- {: .num_defn #FormalParameters}
###### Definition
**([[formal power series]] of [[observables]] for [[perturbative QFT]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

Write

$$
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
  \;\coloneqq\;
  \underset{ k_1, k_2, k_3 \in \mathbb{N}}{\prod}
  LocObs(E_{\text{BV-BRST}})\langle \hbar^{k_1} g^{k_2} j^{k^3}\rangle
$$

for the space of [[formal power series]] in three formal [[variables]]

1. $\hbar$ ("[[Planck's constant]]"),

1. $g$ ("[[coupling constant]]")

1. $j$ ("[[source field]]")

with  [[coefficients]] in the [[topological vector spaces]] of the [[off-shell]] polynomial [[local observables]] of the [[free field]] theory (def. \ref{LocalObservables}); similarly for the [[off-shell]] [[microcausal polynomial observables]] (def. \ref{MicrocausalObservable}):

$$
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j ] ]
  \;\coloneqq\;
  \underset{ k_1, k_2, k_3 \in \mathbb{N}}{\prod}
  PolyObs(E_{\text{BV-BRST}})_{mc}\langle \hbar^{k_1} g^{k_2} j^{k^3}\rangle
  \,.
$$

Similary

$$
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g  ] ]
  \,,
   \phantom{AAA}
  PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g  ] ]
$$

denotes the subspace for which no powers of $j$ appear, etc.

Accordingly

$$
  C^\infty_{cp}(\Sigma) \langle g \rangle
$$

denotes the vector space of [[bump functions]] on [[spacetime]] tensored with the vector space spanned by a single copy of $g$.
The elements

$$
  g_{sw} \in C^\infty_{cp}(\Sigma)\langle g \rangle
$$

may be regarded as [[spacetime]]-dependent "[[coupling constants]]" with compact support,
called _[[adiabatic switching|adiabatically switched]] couplings_.

Similarly then

$$
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g , j \rangle
$$

is the subspace of those formal power series that are at least linear in $g$ or $j$ (hence those that vanish if one sets $g,j = 0$ ).
Hence every element of this space may be written in the form

$$
  O
    =
  g S_{int} + j A
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g , j \rangle
  \,,
$$

where the notation is to suggest that we will think of the coefficient of $g$ as an ([[adiabatic switching|adiabatically switched]])
[[interaction]] [[action functional]] and of the coefficient of $j$ as an external [[source field]] (reflected by internal and external
vertices, respectively, in [[Feynman diagrams]], see def. \ref{VerticesAndFieldSpecies} below).

In particular for

$$
  \mathbf{L}_{int}
  \;\in\;
  \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g] ]
$$

a [[formal power series]] in $\hbar$ and $g$ of [[local Lagrangian densities]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), thought of as a local [[interaction]] Lagrangians, and if

$$
  g_{sw}
  \;\in\;
  C^\infty_{cp}(\Sigma) \langle g \rangle
$$

is an [[adiabatic switching|adiabatically switched]] coupling as before, then the [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) of the product

$$
  g_{sw} \mathbf{L}_{int}
  \;\in\;
  \Omega^{p+1,0}_{\Sigma,cp}(E_{\text{BV-BRST}})[ [ \hbar ,g ] ]\langle g \rangle
$$

is such an [[adiabatic switching|adiabatically switched]] [[interaction]]

$$
  g S_{int}
  \;=\;
  \tau_\Sigma( g_{sw} \mathbf{L}_{int} )
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]\langle g \rangle
  \,.
$$

We also consider the space of [[off-shell]] [[microcausal polynomial observables]] of the [[free field theory]] with formal parameters adjoined

$$
  PolyObs(E_{\text{BV-BRST}})_{mc} ((\hbar)) [ [ g , j] ]
  \,,
$$

which, in its $\hbar$-dependent, is the space of _[[Laurent series]]_ in $\hbar$, hence the space exhibiting also [[negative number|negative]] formal powers of $\hbar$.

=--

$\,$

**Perturbative S-Matrices**
 {#PerturbativeSMatrixAndTimeOrderedProducts}

We introduce now the [[axioms]] for perturbative [[scattering matrices]] relative to a fixed [[relativistic field theory|relativistic]] [[free field theory|free]] [[Lagrangian field theory|Lagrangian]] [[quantum field theory|quantum field]] [[vacuum]] (def. \ref{VacuumFree} below) according to _[[causal perturbation theory]]_ (def. \ref{LagrangianFieldTheoryPerturbativeScattering} below).
Since the first of these axioms requires the S-matrix to be a formal sum of [[multilinear map|multi-]][[linear continuous functionals]], it is convenient to impose axioms on these directly: this is the axiomatics for _[[time-ordered products]]_
in def. \ref{TimeOrderedProduct} below. That these latter axioms already imply the former
is the statement of prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix}, prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} below . Its proof requires a close look at the "[[reverse-time ordered products]]" for the inverse S-matrix (def. \ref{ReverseTimeOrderedProduct} below)
and their induced reverse-causal factorization (prop. \ref{ReverseCausalFactorizationOfReverseTimeOrderedProducts} below).

+-- {: .num_defn #LagrangianFieldTheoryPerturbativeScattering}
###### Definition
**([[S-matrix]] [[axioms]] -- [[causal perturbation theory]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

Then a _perturbative [[S-matrix]] [[renormalization scheme|scheme]]_ for [[perturbative QFT]] around this [[free field|free]] [[vacuum]] is a [[function]]

$$
  \mathcal{S}
    \;\;\colon\;\;
  LocObs(E_{\text{BV-BRST}})[ [\hbar , g, j] ]\langle g, j \rangle
    \overset{\phantom{AAA}}{\longrightarrow}
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ]
$$

from [[local observables]] to [[microcausal polynomial observables]] of the free vacuum theory,
with formal parameters adjoined as indicated (def. \ref{FormalParameters}),
such that the following two conditions "perturbation" and "causal additivity (jointly: "[[causal perturbation theory]]") hold:

1. ([[perturbative quantum field theory|perturbation]])

   There exist [[multilinear map|multi-]][[linear continuous functionals]] (over $\mathbb{C}[ [\hbar, g, j] ]$) of the form

   $$
     \label{TimeOrderedProductsInSMatrix}
     T_k
       \;\colon\;
     \left(
       {\, \atop \,}
       LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]\langle g, j \rangle
       {\, \atop \,}
     \right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}
       \longrightarrow
     PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ]
   $$

   for all $k \in \mathbb{N}$, such that:

   1. The nullary map is [[constant function|constant]] on the [[neutral element|unit]] of the [[Wick algebra]]

      $$
        T_0( g S_{int} + j A) = 1
      $$

   1. The unary map is the inclusion of [[local observables]] as [[normal-ordered products]] (eq:NormalOrderingLocalObservables)

      $$
        T_1(g S_{int} + j A) = g :S_{int}: + j :A:
      $$

   1. The perturbative S-matrix is the [[exponential series]] of these maps in that for
      all $g S_{int} + j A  \in LocObs(E_{\text{BV-BRST}})[ [\hbar , g, j] ]\langle g,j\rangle $

      $$
        \label{ExponentialSeriesScatteringMatrix}
        \begin{aligned}
          \mathcal{S}( g S_{int} + j A)
          & =
          T
          \left(
            \exp_{\otimes}
            \left(
              \tfrac{ 1 }{i \hbar}
              \left(
                g S_{int} + j A
              \right)
            \right)
          \right)
          \\
          & \coloneqq
          \underoverset{k = 0}{\infty}{\sum}
          \frac{1}{k!}
          \left( \frac{1}{i \hbar} \right)^k
          T_k
          \left(
            {\, \atop \,}
            \underset{k\,\text{arguments}}{\underbrace{ (g S_{int}  + jA) , \cdots,  (g S_{int} + j A)  }}
          {\, \atop \,}
          \right)
        \end{aligned}
      $$

1. ([[causal additivity]])

   For all perturbative [[local observables]] $ O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]$ we have

   $$
     \label{CausalAdditivity}
     \left(
        {\, \atop \,}
        supp( O_1 ) {\vee\!\!\!\wedge} supp( O_2 )
        {\, \atop \,}
     \right)
     \;\; \Rightarrow \;\;
     \left(
       {\, \atop \,}
       \mathcal{S}( O_0 + O_1 + O_2 )
       \;\,
       \mathcal{S}( O_0 + O_1 )
       \,
       \mathcal{S}( O_0 )^{-1}
       \,
       \mathcal{S}(O_0 + O_2)
       {\, \atop \,}
     \right)
     \,.
   $$

(The [[inverse]] $\mathcal{S}(O)^{-1}$ of $\mathcal{S}(O)$ with respect to the [[Wick algebra]]-[[structure]]
is implied to exist by the axiom "perturbation", see remark \ref{PerturbativeSMatrixInverse} below.)

=--

Def. \ref{LagrangianFieldTheoryPerturbativeScattering} is due to ([Epstein-Glaser 73 (1)](causal+perturbation+theory#EpsteinGlaser73)),
following ([Stückelberg 49-53](causal+perturbation+theory#Stueckelberg49), [Bogoliubov-Shirkov 59](causal+perturbation+theory#BogoliubovShirkov59)).
That the [[domain]] of an S-matrix scheme is indeed the space of [[local observables]] was made explicit (in terms of axioms for the [[time-ordered products]], see def. \ref{TimeOrderedProduct} below), in ([Brunetti-Fredenhagen 99, section 3](S-matrix#BrunettiFredenhagen99), [Dütsch-Fredenhagen 04, appendix E](S-matrix#DuetschFredenhagen04), [Hollands-Wald 04,around (20)](S-matrix#HollandsWald04)). Review includes ([Rejzner 16, around def. 6.7](S-matrix#Rejzner16), [Dütsch 18, section 3.3](S-matrix#Duetsch18)).

+-- {: .num_remark #PerturbativeSMatrixInverse}
###### Remark
**([[inverse|invertibility]] of the [[S-matrix]])**

The mutliplicative inverse $S(-)^{-1}$ of the perturbative [[S-matrix]] in def. \ref{LagrangianFieldTheoryPerturbativeScattering}
with respect to the [[Wick algebra]]-product indeed exists, so that the list of axioms is indeed well defined: By the axiom "perturbation" this follows with the usual formula for the multiplicative inverse of [[formal power series]] that are non-vanishing in degree 0:

If we write

$$
  \mathcal{S}(g S_{int} + j A) = 1 + \mathcal{D}(g S_{int} + j A)
$$

then

$$
  \label{InfverseOfPerturbativeSMatrix}
  \begin{aligned}
    \left(
      {\, \atop \,}
      \mathcal{S}(g S_{int} + j A)
      {\, \atop \,}
    \right)^{-1}
    &=
    \left(
      {\, \atop \,}
      1 + \mathcal{D}(g S_{int} + j A)
      {\, \atop \,}
    \right)^{-1}
    \\
    & =
    \underoverset{r = 0}{\infty}{\sum}
    \left(
      {\, \atop \,}
      -\mathcal{D}(g S_{int} + j A)
      {\, \atop \,}
    \right)^r
  \end{aligned}
$$

where the sum does exist in $PolyObs(E_{\text{BV-BRST}})((\hbar))[ [[ g,j ] ]$, because (by
the axiom "perturbation")
$\mathcal{D}(g S_{int} + j A)$ has vanishing coefficient in zeroth order in the formal parameters $g$ and $j$, so that
only a finite sub-sum of the formal infinite sum contributes in each order in $g$ and $j$.

This expression for the inverse of S-matrix may usefully be re-organized in terms of "rever-time ordered products" (def. \ref{ReverseTimeOrderedProduct} below), see prop. \ref{ReverseTimOrderedProductsGiveReverseSMatrix} below.

Notice that $\mathcal{S}(-g S_{int} - j A )$ is instead the inverse with respect to the [[time-ordered products]] (eq:TimeOrderedProductsInSMatrix) in that

$$
  T( \mathcal{S}(-g S_{int} -  j A ) \,,\, \mathcal{S}(g S_{int} + j A)  )
  \;=\;
  1
  \;=\;
  T( \mathcal{S}(g S_{int} + j A )  \,,\,  \mathcal{S}(-g S_{in} - j A ) )
  \,.
$$

(Since the time-ordered product is, by definition, symmetric in its arguments, the usual formula for the multiplicative inverse of
an [[exponential series]] applies).

=--

+-- {: .num_remark #MoreDeformationParameters}
###### Remark
**(adjoining further [[deformation]] parameters)

The definition of [[S-matrix]] schemes in def. \ref{LagrangianFieldTheoryPerturbativeScattering} has immediate variants where arbitrary countable sets $\{g_n\}$ and $\{j_m\}$ of formal [[deformation]] parameters are considered, instead of just a single [[coupling constant]] $g$ and a single [[source field]] $j$. The more such constants are considered, the "more perturbative" the theory becomes and the stronger the implications.

=--

Given a perturbative [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) it immediately
induces a corresponding concept of [[observables]]:

+-- {: .num_defn #SchemeGeneratingFunction}
###### Definition
**([[generating function]] scheme for [[interacting field observables]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}.

The corresponding _[[generating function]] [[renormalization scheme|scheme]]_
(for [[interacting field observables]], def. \ref{InteractingFieldObservables} below) is the functional

$$
  \mathcal{Z}_{(-)}(-)
    \;\colon\;
    LocObs(E_{\text{BV-BRST}})[ [\hbar, g] ]\langle g \rangle
    \;\times\;
    LocObs(E_{\text{BV-BRST}})[ [\hbar, j] ]\langle j \rangle
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [g , j] ]
$$

given by

$$
  \label{GeneratingFunctionInducedFromSMatrix}
  \mathcal{Z}_{g S_{int}}(j A)
    \;\coloneqq\;
  \mathcal{S}(g S_{int})^{-1} \mathcal{S}( g S_{int} + j A )
  \,.
$$

=--

+-- {: .num_prop #ZCausalAdditivity}
###### Proposition
**([[causal additivity]] in terms of [[generating functions]])**

In terms of the [[generating functions]] $\mathcal{Z}$ (def. \ref{SchemeGeneratingFunction}) the axiom "[[causal additivity]]"
on the [[S-matrix]] [[renormalization scheme|scheme]] $\mathcal{S}$ (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) is equivalent to:

* ([[causal additivity]] in terms of $\mathcal{Z}$)

  For all [[local observables]] $O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]\otimes\mathbb{C}\langle g,j\rangle$ we have

  $$
    \label{GeneratingFunctionCausalAdditivity}
    \begin{aligned}
    \left(
       {\, \atop \,}
       supp(O_1) {\vee\!\!\!\wedge} supp(O_2)
       {\, \atop \,}
    \right)
    & \;\; \Rightarrow \;\;
    \left(
      {\, \atop \,}
      \mathcal{Z}_{O_0}( O_1 ) \, \mathcal{Z}_{O_0}( O_2)
      =
      \mathcal{Z}_{  O_0 }( O_1 + O_2 )
      {\, \atop \,}
    \right)
    \\
    & \;\; \Leftrightarrow \;\;
    \left(
      {\, \atop \,}
      \mathcal{Z}_{ O_0 + O_1 }( O_2 )
      =
      \mathcal{Z}_{ O_0 }( O_2 )
      {\, \atop \,}
    \right)
    \end{aligned}
    \,.
  $$

(Whence "additivity".)

=--

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

This follows by elementary manipulations:

Multiplying both sides of (eq:CausalAdditivity) by $\mathcal{S}(O_0)^{-1}$ yields

$$
  \underset{
    \mathcal{Z}_{ O_0 }( O_1 + O_2 )
  }{
  \underbrace{
    \mathcal{S}( O_0 )^{-1}
    \mathcal{S}( O_0 + O_1 + O_2 )
  }
  }
  \;=\;
  \underset{
    \mathcal{Z}_{ O_0 }( O_1 )
  }{
  \underbrace{
    \mathcal{S}( O_0 )^{-1} \mathcal{S}( O_0 + O_1 )
  }
  }
  \underset{
    \mathcal{Z}_{ O_0 }( O_2 )
  }{
  \underbrace{
    \mathcal{S}( O_0 )^{-1} \mathcal{S}( O_0 + O_2 )
  }
  }
$$

This is the first line of (eq:GeneratingFunctionCausalAdditivity).

Multiplying both sides of (eq:CausalAdditivity) by $\mathcal{S}( O_0 + O_1 )^{-1}$ yields

$$
  \underset{
    = \mathcal{Z}_{ O_0 + O_1 }(  O_2 )
  }{
  \underbrace{
    \mathcal{S}( O_0 + O_1 )^{-1}
    \mathcal{S}( O_0 + O_1 + O_2 )
  }
  }
  \;=\;
  \underset{
    = \mathcal{Z}_{ O_0 }( O_2 )
  }{
  \underbrace{
    \mathcal{S}( O_0 )^{-1} \mathcal{S}( O_0 + O_2 )
  }
  }
  \,.
$$

This is the second line of (eq:GeneratingFunctionCausalAdditivity).

=--

+-- {: .num_defn #InteractingFieldObservables}
###### Definition
**([[interacting field observables]] -- [[Bogoliubov's formula]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle$ be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]].

Then for $A \in LocObs(E_{\text{BV-BRST}})[ [\hbar , g] ]$ a [[local observable]] of the [[free field theory]],
we say that the corresponding [[local interacting field observable]]

$$
  A_{int} \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar, g] ]
$$

is the [[coefficient]] of $j^1$ in the [[generating function]] (eq:GeneratingFunctionInducedFromSMatrix):

$$
  \label{BogoliubovsFormula}
  \begin{aligned}
    A_{int}
    &\coloneqq
    i \hbar \frac{d}{d j}
    \left(
      {\, \atop \,}
      \mathcal{Z}_{ g S_{int} }( j A )
      {\, \atop \,}
    \right)_{\vert_{j = 0}}
    \\
    & \coloneqq
    i \hbar \frac{d}{d j}
    \left(
      {\, \atop \,}
      \mathcal{S}(g S_{int})^{-1} \, \mathcal{S}( g S_{int} + j A )
      {\, \atop \,}
    \right)_{\vert_{j = 0}}
    \\
    & =
    \mathcal{S}(g S_{int})^{-1}
    T\left(
      \mathcal{S}(g S_{int}), A
    \right)
    \,.
  \end{aligned}
$$

This expression is called _[[Bogoliubov's formula]]_, due to ([Bogoliubov-Shirkov 59](S-matrix#BogoliubovShirkov59)).

One thinks of $A_{int}$ as the [[deformation]] of the [[local observable]] $A$ as the [[interaction]] $S_{int}$ is turned on; and speaks of an element of the _[[interacting field algebra of observables]]_. Their value ("[[expectation value]]") in the given free [[Hadamard vacuum state]]
$\langle  -\rangle$ (def. \ref{VacuumFree}) is a [[formal power series]] in [[Planck's constant]] $\hbar$ and in the [[coupling constant]] $g$, with [[coefficients]] in the [[complex numbers]]

$$
  \left\langle
    A_{int}
  \right\rangle
  \;\in\;
  \mathbb{C}[ [\hbar, g] ]
$$

which express the [[probability amplitudes]] that reflect the predictions of the [[perturbative QFT]], which may be compared to [[experiment]].

=--

([Epstein-Glaser 73, around (74)](S-matrix#EpsteinGlaser73)); review includes ([Dütsch-Fredenhagen 00, around (17)](S-matrix#DuetschFredenhagen00), [Dütsch 18, around (3.212)](pAQFT#Duetsch18)).

+-- {: .num_example #FormalPowerSeriesInteractingFieldObservables}
###### Remark
**([[interacting field observables]] are [[formal deformation quantization]])**

The [[interacting field observables]] in def. \ref{InteractingFieldObservables} are indeed [[formal power series]] in the formal parameter $\hbar$ ([[Planck's constant]]), as opposed to being more general [[Laurent series]], hence they involve no [[negative number|negative]] powers of $\hbar$ ([Dütsch-Fredenhagen 00, prop. 2 (ii)](interacting+field+observable#DuetschFredenhagen00), [Hawkins-Rejzner 16, cor. 5.2](interacting+field+observable#HawkinsRejzner16)).  This is not immediate, since by def. \ref{LagrangianFieldTheoryPerturbativeScattering} the [[S-matrix]] that they are defined from does involve negative powers of $\hbar$.

It follows in particular that the [[interacting field observables]] have a [[classical limit]] $\hbar \to 0$, which
is not the case for the [[S-matrix]] itself (due to it involving negative powers of $\hbar$).
Indeed the [[interacting field observables]] constitute a _[[formal deformation quantization]]_ of the [[covariant phase space]]
of the [[interacting field theory]] (prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization} below)
and are thus the more fundamental concept.

=--

As the name suggests, the [[S-matrices]] in def. \ref{LagrangianFieldTheoryPerturbativeScattering} serve to express [[scattering amplitudes]] (example \ref{ScatteringAmplitudeFromInteractingFieldObservables} below). But by remark \ref{FormalPowerSeriesInteractingFieldObservables}
the more fundamental concept is that of the [[interacting field observables]]. Their perspective reveals that
consistent interpretation of [[scattering amplitudes]] requires the following condition on the relation between the
[[vacuum state]] and the [[interaction]] term:

+-- {: .num_defn #VacuumStability}
###### Definition
**([[vacuum stability]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]\langle g \rangle$ be a [[local observable]], regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

We say that the given [[Hadamard vacuum state|Hadamard]] [[vacuum state]] (prop. \ref{WickAlgebraCanonicalState})

$$
  \langle - \rangle
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar , g, j ] ]
  \longrightarrow
  \mathbb{C}[ [ \hbar, g, j ] ]
$$

is _[[vacuum stability|stable]]_ with respect to the [[interaction]] $S_{int}$, if for all elements of the [[Wick algebra]]

$$
  A \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g] ]
$$

we have

$$
  \left\langle
    A \mathcal{S}(g S_{int})
  \right\rangle
  \;=\;
  \left\langle
     \mathcal{S}(g S_{int})
  \right\rangle
  \,
  \left\langle
    A
  \right\rangle
  \phantom{AA}
  \text{and}
  \phantom{AA}
  \left\langle
    \mathcal{S}(g S_{int})^{-1}
    A
  \right\rangle
  \;=\;
  \frac{1}
  {
    \left\langle
      \mathcal{S}(g S_{int})
    \right\rangle
   }
   \left\langle
     A
   \right\rangle
$$

=--

+-- {: .num_example #InteractinFieldTimeOrderedProduct}
###### Example
**([[time-ordered product]] of [[interacting field observables]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle$ be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]].

Consider two [[local observables]]

$$
  A_1, A_2
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar , g] ]
$$

with [[causal ordering|causally ordered]] spacetime support

$$
  supp(A_1) {\vee\!\!\!\!\wedge} supp(A_2)
$$

Then [[causal additivity]] according to prop. \ref{ZCausalAdditivity} implies that the
[[Wick algebra]]-product of the corresponding [[interacting field observables]]
$(A_1)_{int}, (A_2)_{int} \in PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ] $ (def. \ref{InteractingFieldObservables})
is

$$
  \begin{aligned}
    (A_1)_{int}
    (A_2)_{int}
    & =
    \left(
      \frac{\partial}{\partial j} \mathcal{Z}(j A_1 )
    \right)_{\vert j = 0}
    \left(
      \frac{\partial}{\partial j} \mathcal{Z}( j A_2 )
    \right)_{\vert j = 0}
    \\
    & =
    \frac{\partial^2}{\partial j_1 \partial j_2}
    \left(
      {\, \atop \,}
      \mathcal{Z}( j_1 A_1 )
      \mathcal{Z}( j_2 A_2 )
      {\, \atop \,}
    \right)_{
       \left\vert { {j_1 = 0}, \atop {j_2 = 0} } \right.
    }
    \\
    & =
    \frac{\partial^2}{\partial j_1 \partial j_2}
    \left(
      {\, \atop \,}
      \mathcal{Z}( j_1 A_1 + j_2 A_2 )
      {\, \atop \,}
    \right)_{
       \left\vert { {j_1 = 0}, \atop {j_2 = 0} } \right.
    }
  \end{aligned}
$$

Here the last line makes sense if one extends the axioms on the [[S-matrix]] in prop. \ref{LagrangianFieldTheoryPerturbativeScattering}
from formal power series in $\hbar, g, j$ to formal power series in $\hbar, g, j_1, j_2, \cdots$ (remark \ref{MoreDeformationParameters}).
Hence in this generalization, the [[generating functions]] $\mathcal{Z}$ are not just generating functions
for [[interacting field observables]] themselves, but in fact for _[[time-ordered products]]_ of interacting field observables.

=--

An important special case of [[time-ordered products]] of [[interacting field observables]] as in example \ref{InteractinFieldTimeOrderedProduct}
is the following special case of _[[scattering amplitudes]]_, which is the example that gives the _[[scattering matrix]]_ in def. \ref{LagrangianFieldTheoryPerturbativeScattering} its name:

+-- {: .num_example #ScatteringAmplitudeFromInteractingFieldObservables}
###### Example
**([[scattering amplitudes]] as [[vacuum expectation values]] of [[interacting field observables]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle$ be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]],
such that the [[vacuum state]] is [[vacuum stability|stable]] with respect to $g S_{int}$ (def. \ref{VacuumStability}).

Consider [[local observables]]

$$
  \array{
    A_{in,1}, \cdots, A_{in , n_{in}},
    \\
    A_{out,1}, \cdots, A_{out, n_{out}}
  }
  \;\;\in\;\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
$$

whose spacetime support satisfies the following [[causal ordering]]:

$$
  A_{out, i_{out} }
  {\gt\!\!\!\!\lt}
  A_{out, j_{out}}
  \phantom{AAA}
  A_{out, i_{out} }
    {\vee\!\!\!\wedge}
  S_{int}
    {\vee\!\!\!\wedge}
  A_{in, i_{in}}
  \phantom{AAA}
  A_{in, i_{in} }
  {\gt\!\!\!\!\lt}
  A_{in, j_{in}}
$$

for all $1 \leq i_{out} \lt j_{out} \leq n_{out}$ and $1 \leq  i_{in} \lt j_{in} \leq n_{in}$.

Then the [[vacuum expectation value]] of the [[Wick algebra]]-product of the corresponding [[interacting field observables]] (def. \ref{InteractingFieldObservables})
is

$$
  \begin{aligned}
   &
   \left\langle
     {\, \atop \,}
     (A_{out, 1})_{int}
     \cdots
     (A_{out,n_{out}})_{int}
     \,
     (A_{in, 1})_{int}
     \cdots
     (A_{in,n_{in}})_{int}
     {\, \atop \,}
   \right\rangle
   \\
   & =
   \left\langle
     {\, \atop \,}
       A_{out,1}
       \cdots
       A_{out,n_{out}}
    \right|
       \;
       \mathcal{S}(g S_{int})
       \;
    \left|
       A_{in,1}
       \cdots
       A_{in, n_{in}}
     {\, \atop \,}
   \right\rangle
   \\
   & \coloneqq
   \frac{1}{
     \left\langle \mathcal{S}(g S_{int}) \right\rangle
   }
   \left\langle
     {\, \atop \,}
       A_{out,1}
       \cdots
       A_{out,n_{out}}
       \;
       \mathcal{S}(g S_{int})
       \;
       A_{in,1}
       \cdots
       A_{in, n_{in}}
     {\, \atop \,}
   \right\rangle
   \,.
  \end{aligned}
$$

These [[vacuum expectation values]] are interpreted, in the [[adiabatic limit]] where $g_{sw} \to 1$, as _[[scattering amplitudes]]_ (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes} below).

=--

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

For notational convenience, we spell out the argument for $n_{in} =  1 = n_{out}$. The general case
is directly analogous.

So assuming the [[causal order]] (def. \ref{CausalOrdering})

$$
  supp(A_{out})
   {\vee\!\!\!\wedge}
  supp(S_{int})
   {\vee\!\!\!\wedge}
  supp(A_{in})
$$

we compute with [[causal additivity]] via prop. \ref{ZCausalAdditivity} as follows:

$$
  \begin{aligned}
    (A_{out})_{int} (A_{in})_{int}
    & =
    \frac{d^2 }{\partial j_{out} \partial j_{in}}
    \left(
       \mathcal{Z}( j_{out} A_{out} )
       \mathcal{Z}( j_{in} A_{in} )
    \right)_{\left\vert { { j_{out} = 0 } \atop { j_{in} = 0 } }  \right.}
    \\
    & =
    \frac{\partial^2 }{\partial j_{out} \partial j_{in}}
    \left(
       \mathcal{S}(g S_{int})^{-1}
       \underset{
         =
         \mathcal{S}(j_{out} A_{out})
         \mathcal{S}(g S_{int})
       }{
       \underbrace{
         \mathcal{S}(g S_{int} + j_{out} A_{out})
       }
       }
       \mathcal{S}(g S_{int})^{-1}
       \underset{
         = \mathcal{S}(g S_{int}) \mathcal{S}(j_{in} A_{in})
       }{
       \underbrace{
         \mathcal{S}(g S_{int} + j_{in}A_{in})
       }
       }
    \right)_{\left\vert { { j_{out} = 0 } \atop { j_{in} = 0 } }  \right.}
    \\
    & =
    \frac{\partial^2 }{\partial j_{out} \partial j_{in}}
    \left(
       \mathcal{S}(g S_{int})^{-1}
       \mathcal{S}(j_{out} A_{out})
       \underset{
         = \mathcal{S}(g S_{int})
       }{
       \underbrace{
         \mathcal{S}(g S_{int})
         \mathcal{S}(g S_{int})^{-1}
         \mathcal{S}(g S_{int})
       }
       }
       \mathcal{S}(j_{in} A_{in})
    \right)_{\left\vert { { j_{out} = 0 } \atop { j_{in} = 0 } }  \right.}
    \\
    & =
    \mathcal{S}(g S_{int})^{-1}
    \,
    \left(
      {\, \atop \,}
      A_{out}
      \mathcal{S}(g S_{int})
      A_{in}
      {\, \atop \,}
    \right)
    \,.
  \end{aligned}
$$

With this the statement follows by the definition of [[vacuum stability]] (def. \ref{VacuumStability}).

=--

+-- {: .num_remark}
###### Remark
**(computing [[S-matrices]] via [[Feynman perturbation series]])**

For practical computation of [[vacuum expectation values]] of [[interacting field observables]] (example \ref{InteractinFieldTimeOrderedProduct})
and hence in particular, via example \ref{ScatteringAmplitudeFromInteractingFieldObservables}, of [[scattering amplitudes]],
one needs some method for collecting all the contributions to the [[formal power series]] in increasing order in $\hbar$ and $g$.

Such a method is provided by the _[[Feynman perturbation series]]_ (example \ref{FeynmanPerturbationSeries} below)
and the _[[effective action]]_ (def. \ref{InPerturbationTheoryActionEffective}), see example \ref{SMatrixVacuumContribution} below.

=--

$\,$

**Conceptual remarks**
 {#RemarksOnCausalPerturbationTheoryAxioms}

The simple axioms for [[S-matrix]] [[renormalization scheme|schemes]] in [[causal perturbation theory]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) and hence for [[interacting field observables]] (def. \ref{InteractingFieldObservables}) have a wealth of implications and consequences. Before discussing these formally below, we here make a few informal remarks meant to put various relevant concepts into perspective:

+-- {: .num_remark #AsymptoticSeriesObservables}
###### Remark
**([[perturbative quantum field theory|perturbative QFT]] and [[asymptotic expansion]] of [[probability amplitudes]])**

Given a [[perturbative quantum field theory|perturbative]] [[S-matrix]] [[renormalization scheme|scheme]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}), then by remark \ref{FormalPowerSeriesInteractingFieldObservables} the
[[expectation values]] of [[interacting field observables]] (def. \ref{InteractingFieldObservables}) are [[formal power series]]
in the formal parameters $\hbar$ and $g$ (which are interpreted as [[Planck's constant]], and as the [[coupling constant]], respectively):

$$
 \left\langle A_{int} \right\rangle
 \;\in\;
 \mathbb{C}[ [\hbar, g] ]
 \,.
$$

This means that there is _no_ guarantee that these series _[[convergence|converge]]_ for any [[positive real number|positive]]
value of $\hbar$ and/or $g$.
In terms of [[synthetic differential geometry]] this means that in [[perturbative QFT]] the [[deformation]] of the [[classical field theory|classical]] [[free field theory]] by quantum effects (measured by $\hbar$) and [[interactions]] (meaured by $g$) is so very tiny as to actually be [[infinitesimal]]: formal power series may be read as functions on the [[infinitesimal neighbourhood]] in a space of [[Lagrangian field theories]] at the point $\hbar = 0$, $g = 0$.

In fact, a simple argument (due to [Dyson 52](perturbative+quantum+field+theory#Dyson52))
suggests that in realistic field theories these series _never_ converge for _any_ [[positive real number|positive]] value of $\hbar$ and/or $g$. Namely convergence for $g$ would imply a [[positive real number|positive]] _[[radius of convergence]]_ around $g = 0$, which would imply convergence also for $-g$ and even for [[imaginary number|imaginary]] values of $g$, which would however correspond to unstable [[interactions]] for which no converging field theory is to be expected. (See [Helling, p. 4](perturbative+quantum+field+theory#Helling) for the example of [[phi^4 theory]].)

In physical practice one tries to interpret these non-converging [[formal power series]] as _[[asymptotic expansions]]_ of
actual but hypothetical functions in $\hbar, g$, which reflect the actual but hypothetical _[[non-perturbative quantum field theory]]_
that one imagines is being approximated by [[perturbative QFT]] methods. An _[[asymptotic expansion]]_ of a function is a [[power series]] which may not converge, but which has for every $n \in \mathbb{N}$ an estimate for how far the [[sum]] of the first $n$ terms in the series may differ from the function being approximated.

For examples such as [[quantum electrodynamics]] and [[quantum chromodynamics]], as in the [[standard model of particle physics]], the truncation of these [[formal power series]] [[scattering amplitudes]] to the first handful of [[loop orders]] in $\hbar$ happens to agree with [[experiment]] (such as at the [[LHC]] collider) to high precision (for [[QED]]) or at least decent precision (for [[QCD]]), at least away from infrared phenomena (see remark \ref{AdiabaticLimit}).

In summary this says that [[perturbative QFT]] is an extremely _coarse_ and restrictive approximation to what should be genuine [[non-perturbative quantum field theory]], while at the same time it happens to match certain experimental observations to remarkable degree, albeit only if some ad-hoc truncation of the resulting power series is considered.

This is strong motivation for going beyond [[perturbative QFT]] to understand and construct genuine [[non-perturbative quantum field theory]]. Unfortunately, this is a wide-open problem, away from toy examples. Not a single [[interacting field theory]] in [[spacetime]] [[dimension]] $\geq 4$ has been non-perturbatively quantized. Already a single aspect of the [[non-perturbative quantum field theory|non-perturbative]] [[quantization of Yang-Mills theory]] (as in [[QCD]]) has famously been advertized as one of the _[Millennium Problems](http://www.claymath.org/millennium-problems/yang%E2%80%93mills-and-mass-gap)_ of our age; and speculation about [[non-perturbative quantum field theory|non-perturbative]] [[quantum gravity]] is the subject of much activity.

Now, as the name indicates, the [[axioms]] of _[[causal perturbation theory]]_ (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) do
_not_ address [[non-perturbative effect|non-perturbative aspects]] of [[non-perturbative field theory]]; the convergence or non-convergence of the [[formal power series]] that are axiomatized by [[Bogoliubov's formula]] (def. \ref{InteractingFieldObservables}) is _not_ addressed by the theory.
The point of the axioms of [[causal perturbation theory]] is to give rigorous mathematical meaning to _everything else_ in [[perturbative QFT]].

=--

+-- {: .num_remark #DysonCausalFactorization}
###### Remark
**([[Dyson series]] and [[Schrödinger equation]] in [[interaction picture]])**

The axiom "[[causal additivity]]" (eq:CausalAdditivity) on an [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering})
implies immediately this seemingly weaker condition (which turns out to be equivalent, this is prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} below):

* ([[causal factorization]])

  For all [[local observables]] $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [\hbar, h, j] ]\langle g , j\rangle $ we have

  $$
    \left(
       {\, \atop \,}
       supp(O_1) {\vee\!\!\!\wedge} supp(O_2)
       {\, \atop \,}
    \right)
    \;\; \Rightarrow \;\;
    \left(
      {\, \atop \,}
      \mathcal{S}( O_1 + O_2 )
      =
      \mathcal{S}( O_1 ) \, \mathcal{S}( O_2 )
      {\, \atop \,}
   \right)
  $$

(This is the special case of "causal additivity" for $O_0 = 0$, using that  by the axiom "perturbation" (eq:ExponentialSeriesScatteringMatrix) we have $\mathcal{S}(0) = 1$.)

If we now think of $O_1 = g S_{1}$ and $O_2 = g S_2$ themselves as [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functionals]], then this becomes

  $$
    \left(
       {\, \atop \,}
       supp(S_1) {\vee\!\!\!\wedge} supp(S_2)
       {\, \atop \,}
    \right)
    \;\; \Rightarrow \;\;
    \left(
      {\, \atop \,}
      \mathcal{S}( g S_1 + g S_2 )
      =
      \mathcal{S}( g S_1) \, \mathcal{S}( g sS_2)
      {\, \atop \,}
   \right)
  $$

This exhibits the [[S-matrix]]-scheme as a "[[causal ordering|causally ordered]] [[exponential]]"
or "[[Dyson series]]" of the [[interaction]], hence as a refinement to [[relativistic field theory]] of what in
[[quantum mechanics]] is the "integral version of the [[Schrödinger equation]] in the [[interaction picture]]" (see [this equation](S-matrix#IntegralVersionSchroedingerEquationInInteractionPicture) at _[[S-matrix]]_; see also [Scharf 95, second half of 0.3](S-matrix#Scharf95)).

The relevance of manifest [[causal additivity]] of the [[S-matrix]], over just [[causal factorization]] (even though both conditions happen to be equivalent, see prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} below), is that it directly implies that
the induced [[interacting field algebra of observables]] (def. \ref{InteractingFieldObservables}) forms a [[causally local net]] (prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below).

=--

+-- {: .num_remark #InterpretationOfPerturbativeSMatrix}
###### Remark
**([[path integral]]-intuition)**

In informal discussion of [[perturbative QFT]] going back to informal ideas of [[Schwinger-Tomonaga-Feynman-Dyson]], the
perturbative [[S-matrix]] is thought of in terms of a would-be _[[path integral]]_, symbolically written

$$
  \mathcal{S}\left(
    g S_{int} + j A
  \right)
  \;\overset{\text{not really!}}{=}\;
  \!\!\!\!\!
  \underset{\Phi \in \Gamma_\Sigma(E_{\text{BV-BRST}})_{asm}}{\int}
  \!\!\!\!\!\!
  \exp\left(
    \tfrac{1}{i \hbar}
    \int_\Sigma
    \left(
      g L_{int}(\Phi) + j A(\Phi)
    \right)
  \right)
  \,
  \exp\left(
    \tfrac{1}{i \hbar}\int_\Sigma L_{free}(\Phi)
  \right) D[\Phi]
  \,.
$$

Here the would-be [[integration]] is thought to be over the [[space of field histories]] $\Gamma_\Sigma(E_{\text{BV-BRST}})_{asm}$ (the [[space of sections]] of the given [[field bundle]], remark \ref{PossibleFieldHistories}) for [[field histories]] which satisfy given asymptotic conditions at $x^0 \to \pm \infty$; and as these boundary conditions vary the above is regarded as a would-be [[integral kernel]] that defines the required operator in the [[Wick algebra]] (e.g. [Weinberg 95, around (9.3.10) and (9.4.1)](S-matrix#Weinberg95)). This is related to the intuitive picture of the [[Feynman perturbation series]] (example \ref{FeynmanPerturbationSeries} below) expressing a sum over all possible interactions of [[virtual particles]] (remark \ref{WorldlineFormalism}).

Beyond toy examples, it is not known how to define the would-be [[measure]] $D[\Phi]$ and it is not known how to make sense of this expression as an actual [[integral]].

The analogous path-integral intuition for [[Bogoliubov's formula]] for [[interacting field observables]] (def. \ref{InteractingFieldObservables}) symbolically reads

$$
  \begin{aligned}
  A_{int}
  & \overset{\text{not really!}}{=}
  \frac{d}{d j}
  \ln
  \left(
  \underset{\Phi \in \Gamma_\Sigma(E)_{asm}}{\int}
  \!\!\!\!
  \exp\left(
    \underset{\Sigma}{\int}
      g L_{int}(\Phi) + j A(\Phi)
  \right)
  \,
  \exp\left( \underset{\Sigma}{\int} L_{free}(\Phi) \right)
  D[\Phi]
  \right)
  \vert_{j = 0}
  \end{aligned}
$$

If here we were to regard the expression

$$
  \mu(\Phi)
  \;\overset{\text{not really!}}{\coloneqq}\;
  \frac{
    \exp\left( \underset{\Sigma}{\int} L_{free}(\Phi) \right)\, D[\Phi]
  }
  {
    \underset{\Phi \in \Gamma_\Sigma(E_{\text{BV-BRST}})_{asm}}{\int}
    \!\!\!\!
    \exp\left( \underset{\Sigma}{\int} L_{free}(\Phi) \right)\, D[\Phi]
  }
$$

as a would-be [[Gaussian measure]] on the [[space of field histories]], normalized to a would-be [[probability measure]], then this formula
would express interacting field observables as ordinary [[expectation values]]

$$
  A_{int}
   \overset{\text{not really!}}{=}
   \!\!\!
  \underset{\Phi \in \Gamma_\Sigma(E_{\text{BV-BRST}})_{asm}}{\int}
  \!\!\!\!\!\!
     A(\Phi)
  \,\mu(\Phi)
  \,.
$$

As before, beyond toy examples it is not known how to make sense of this as an actual [[integration]].

But we may think of the axioms for the [[S-matrix]] in [[causal perturbation theory]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) as rigorously _defining_ the [[path integral]], not analytically as an actual [[integration]], but _[[synthetic mathematics|synthetically]]_ by axiomatizing the properties of the desired _outcome_ of the would-be integration:

The analogy with a well-defined [[integral]] and the usual properties of an [[exponential]] vividly _suggest_
that the would-be [[path integral]] should obey [[causal factorization]]. Instead of trying to make sense of
[[path integral|path integration]] so that this factorization property could then be appealed to as a _consequence_ of general properties
of [[integration]] and [[exponentials]], the axioms of [[causal perturbation theory]] directly prescribe the
desired factorization property, without insisting that it derives from an actual integration.

The great success of [[path integral]]-intuition in the development of [[quantum field theory]], despite the dearth of
actual constructions, indicates that it is not the would-be integration process as such that actually matters in field
theory, but only the resulting properties that this _suggests_ the S-matrix should have; which is what
[[causal perturbation theory]] axiomatizes. Indeed, the simple [[axioms]] of [[causal perturbation theory]]
rigorously _imply_ finite (i.e. [[renormalization|("re"-)normalized]]) [[perturbative quantum field theory]] (see remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences}).

$$
  \array{
  \array{
    \text{would-be}
    \\
    \text{path integral}
    \\
    \text{intuition}
  }
  &
  \overset{
    \array{
      \text{informally}
      \\
      \text{suggests}
    }
  }{\longrightarrow}
  &
  \array{
    \text{causally additive}
    \\
    \text{scattering matrix}
  }
  &
  \overset{
    \array{
      \text{rigorously}
      \\
      \text{implies}
    }
  }{\longrightarrow}
  &
  \array{
    \text{UV-finite}
    \\
    \text{(i.e. (re-)normalized)}
    \\
    \text{perturbative QFT}
  }
  }
$$

=--

+-- {: .num_remark #FromAxiomaticSMatrixScatteringAmplitudes}
###### Remark
**([[scattering amplitudes]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let

$$
  S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]
$$

be a [[local observable]], regarded
as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Then for

$$
  A_{in}, A_{out} \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [\hbar] ]
$$

two [[microcausal polynomial observables]], with [[causal ordering]]

$$
  supp(A_{out}) {\vee\!\!\!\wedge} supp(A_{int})
$$

the corresponding _[[scattering amplitude]]_ (as in example \ref{ScatteringAmplitudeFromInteractingFieldObservables}) is the value (called "[[expectation value]]" when referring to $A^\ast_{out} \, \mathcal{S}(S_{int}) \, A_{in}$, or "matrix element" when referring to $\mathcal{S}(S_{int})$, or "transition amplitude" when referring to $\left\langle A_{out} \right\vert$ and $\left\vert A_{in} \right\rangle$)

$$
  \left\langle
    A_{out}
    \,\vert\,
    \mathcal{S}(S_{int})
    \,\vert\,
    A_{in}
  \right\rangle
  \;\coloneqq\;
  \left\langle
    A^\ast_{out} \, \mathcal{S}(S_{int}) \, A_{in}
  \right\rangle
  \;\in\;
  \mathbb{C}[ [ \hbar, g ] ]
  \,.
$$

for the [[Wick algebra]]-product $A^\ast_{out} \, \mathcal{S}(S_{int})\, A_{in} \in PolyObs(E_{\text{BV-BRST}})[ [\hbar, g ] ]$
in the given [[Hadamard vacuum state]] $\langle -\rangle \colon PolyObs(E_{\text{BV-BRST}})[ [\hbar, g] ] \to \mathbb{C}[ [\hbar,g] ]$.

If here $A_{in}$ and $A_{out}$ are monomials in [[Wick algebra]]-products of the [[field observables]] $\mathbf{\Phi}^a(x) \in Obs(E_{\text{BV-BRST}})[ [\hbar] ]$,
then this [[scattering amplitude]] comes from the [[integral kernel]]

$$
  \begin{aligned}
  &
  \left\langle
    \mathbf{\Phi}^{a_{out,1}}(x_{out,1}) \cdots \mathbf{\Phi}^{a_{out,s}}(x_{out,s})
    \vert
    \,
    \mathcal{S}(S_{int})
    \,
    \vert
    \mathbf{\Phi}^{a_{in,1}}(x_{in,1})
    \cdots
    \mathbf{\Phi}^{a_{in,r}}(x_{in,r})
  \right\rangle
  \\
  & \coloneqq
  \left\langle
     \left(\mathbf{\Phi}^{a_{out,1}}(x_{out,1})\right)^\ast
      \cdots
      \left(\mathbf{\Phi}^{a_{out,s}}(x_{out,s})\right)^\ast
    \;\mathcal{S}(S_{int})\;
      \mathbf{\Phi}^{a_{in,1}}(x_{in,1})
      \cdots
      \mathbf{\Phi}^{a_{in,r}}(x_{in,r})
  \right\rangle
  \end{aligned}
$$

or similarly, under [[Fourier transform of distributions]],

$$
  \label{ScatteringPlaneWaves}
  \begin{aligned}
  &
  \left\langle
    \widehat{\mathbf{\Phi}}^{a_{out,1}}(k_{out,1}) \cdots \widehat{\mathbf{\Phi}}^{a_{out,s}}(k_{out,s})
    \vert
    \,
    \mathcal{S}(S_{int})
    \,
    \vert
    \widehat{\mathbf{\Phi}}^{a_{in,1}}(k_{in,1})
    \cdots
    \widehat{\mathbf{\Phi}}^{a_{in,r}}(k_{in,r})
  \right\rangle
  \\
  & \coloneqq
  \left\langle
     \left(\widehat{\mathbf{\Phi}}^{a_{out,1}}(k_{out,1})\right)^\ast
      \cdots
      \left(\widehat{\mathbf{\Phi}}^{a_{out,s}}(k_{out,s})\right)^\ast
    \;\mathcal{S}(S_{int})\;
      \widehat{\mathbf{\Phi}}^{a_{in,1}}(k_{in,1})
      \cdots
      \widehat{\mathbf{\Phi}}^{a_{in,r}}(k_{in,r})
  \right\rangle
  \end{aligned}
  \,.
$$

These are interpreted as the (distributional) _[[probability amplitudes]]_ for [[plane waves]] of field species $a_{in,\cdot}$
with [[wave vector]] $k_{in,\cdot}$ to come in from the far past, ineract with each other via $S_{int}$,
and emerge in the far future as [[plane waves]] of field species $a_{out,\cdot}$ with [[wave vectors]] $k_{out,\cdot}$.

=--

Or rather:

+-- {: .num_remark #AdiabaticLimit}
###### Remark
**([[adiabatic limit]], [[infrared divergences]] and [[interacting vacuum]])**

Since a [[local observable]] $S_{int} \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]$  by definition
has compact spacetime support, the [[scattering amplitudes]] in remark \ref{FromAxiomaticSMatrixScatteringAmplitudes}
describe [[scattering]] processes for [[interactions]] that vanish (are "[[adiabatic switching|adiabatically switched off]]") outside a compact subset of [[spacetime]]. This constraint is crucial for [[causal perturbation theory]] to work.

There are several aspects to this:

* ([[adiabatic limit]]) On the one hand, real physical interactions $\mathbf{L}_{int}$ (say the [[electron-photon interaction]])
are not _really_ supposed to vanish outside a compact region of spacetime. In order to reflect this mathematically, one may consider a [[sequence]] of [[adiabatic switchings]] $g_{sw} \in C^\infty_{cp}(\Sigma)\langle g \rangle$ (each of [[compact support]]) whose [[limit of a sequence|limit]] is the [[constant function]] $g \in C^\infty(\Sigma)\langle g\rangle$ (the actual [[coupling constant]]), then consider the corresponding [[sequence]] of [[interaction]] [[action functionals]] $S_{int,sw} \coloneqq \tau_\Sigma( g_{sw} \mathbf{L}_{int} )$ and finally consider:

   1. as the true [[scattering amplitude]] the corresponding [[limit of a sequence|limit]]

      $$
        \left\langle
          A_{out} \vert \mathcal{S}(S_{int}) \vert A_{int}
        \right\rangle
        \;\coloneqq\;
        \underset{g_{sw} \to 1}{\lim}
        \left\langle
          A_{out} \vert \mathcal{S}(S_{int,sw}) \vert A_{int}
        \right\rangle
      $$

      of adiabatically switched [[scattering amplitudes]] (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes}) --  if it exists. This is called the _[[strong adiabatic limit]]_.

  1. as the true [[n-point functions]] the corresponding [[limit of a sequence|limit]]

     $$
       \begin{aligned}
       &
       \left\langle
         \mathbf{\Phi}^{a_1}_{int}(x_1)
         \mathbf{\Phi}^{a_2}_{int}(x_2)
           \cdots
         \mathbf{\Phi}^{a_{n-1}}_{int}(x_{n-1})
         \mathbf{\Phi}^{a_n}_{int,sw}(x_n)
       \right\rangle
       \\
       & =
       \underset{\underset{g_{sw} \to 1}{\longrightarrow}}{\lim}
       \left\langle
         \mathbf{\Phi}^{a_1}_{int,sw}(x_1)
         \mathbf{\Phi}^{a_2}_{int,sw}(x_2)
           \cdots
         \mathbf{\Phi}^{a_{n-1}}_{int,sw}(x_{n-1})
         \mathbf{\Phi}^{a_n}_{int,sw}(x_n)
       \right\rangle
       \end{aligned}
     $$

     of [[tempered distribution|tempered distributional]] [[expectation values]] of products of [[interacting field algebra|interacting]] [[field observables]] (def. \ref{InteractingFieldObservables}) -- if it exists. (Similarly for [[time-ordered products]].)  This is called the _[[weak adiabatic limit]]_.

  Beware that the left hand sides here are symbolic: Even if the limit exists in [[expectation values]], in general there is no actual observable whose expectation value is that limit.

  The strong and weak adiabatic limits have been shown to exist if all [[field (physics)|fields]] are [[mass|massive]] ([Epstein-Glaser 73](S-matrix#EpsteinGlaser73)). The weak adiabatic limit has been shown to exists for [[quantum electrodynamics]] and for [[mass]]-less [[phi^4 theory]] ([Blanchard-Seneor 75](adiabatic+switching#BlanchardSeneor75)) and for larger classes of field theories in ([Duch 17, p. 113, 114](adiabatic+switching#Duch17)).

  If these limits do not exist, one says that the [[perturbative QFT]] has an _[[infrared divergence]]_.

* ([[algebraic adiabatic limit]]) On the other hand, it is equally unrealistic that an actual [[experiment]] _detects_ phenomena outside a given compact subset of spacetime. Realistic scattering [[experiments]] (such as the [[LHC]]) do not really prepare or measure [[plane waves]] filling all of [[spacetime]] as described by the [[scattering amplitudes]] (eq:ScatteringPlaneWaves). Any [[observable]] that is realistically measurable must have compact spacetime support. We see below in prop. \ref{IsomorphismFromChangeOfAdiabaticSwitching} that such [[interacting field observables]] with compact spacetime support may be computed without taking the [[adiabatic limit]]: It is sufficient to use any [[adiabatic switching]] which is constant on the support of the observable.

  This way one obtains for each [[causally closed subset]] $\mathcal{O}$ of spacetime an  algebra of observables $\mathcal{A}_{int}(\mathcal{O})$ whose support is in $\mathcal{O}$, and for each inclusion of subsets a corresponding inclusion of algebras of observables (prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below). Of this system of observables one may form the [[category theory|category-theoretic]] [[inductive limit]] to obtain a single global algebra of observables.

  $$
    \mathcal{A}_{int}
      \;\coloneqq\;
    \underset{\underset{\mathcal{O}}{\longrightarrow}}{\lim}
      \mathcal{A}_{int}(\mathcal{O})
  $$

  This always exists. It is called the _[[algebraic adiabatic limit]]_ (going back to [Brunetti-Fredenhagen 00, section 8](perturbative+algebraic+quantum+field+theory#BrunettiFredenhagen00)).

  For [[quantum electrodynamics]] the [[algebraic adiabatic limit]] was worked out in ([Dütsch-Fredenhagen 98](quantum+electrodynamics#DuetschFredenhagen98), reviewed in [Dütsch 18, 5,3](QED+Duetsch18)).

* ([[interacting vacuum]]) While, via the above [[algebraic adiabatic limit]], [[causal perturbation theory]] yields the correct [[interacting field algebra of quantum observables]] independent of choices of [[adiabatic switching]], a theory of _[[quantum probability]]_ requires, on top of the [[algebra of observables]], also a _[[state on a star-algebra|state]]_

  $$
    \langle - \rangle_{int} \;\colon\; \mathcal{A}_{int} \longrightarrow \mathbb{C}[ [\hbar] ]
  $$

  Just as the [[interacting field algebra of observables]] $\mathcal{A}_{int}$ is a [[deformation]] of the free field algebra of observables ([[Wick algebra]]), there ought to be a corresponding deformation of the free [[Hadamard vacuum state]] $\langle- \rangle$ into an "[[interacting vacuum state]]" $\langle - \rangle_{int}$.

  Sometimes the [[weak adiabatic limit]] serves to define the [[interacting vacuum]] (see [Duch 17, p. 113-114](adiabatic+switching#Duch17)).

A stark example of these infrared issues is the phenomenon of _[[confinement]]_ of [[quarks]] to [[hadron]] [[bound states]] (notably to [[protons]] and [[neutrons]]) at large [[wavelengths]]. This is paramount in [[experiment|observation]] and reproduced in numerical [[lattice gauge theory]] simulation, but is invisible to [[perturbative QFT|perturbative]] [[quantum chromodynamics]] in its [[free field]] [[vacuum state]], due to [[infrared divergences]].
It is expected that this should be rectified by the proper [[interacting vacuum]] of [[QCD]] ([Rafelski 90, pages 12-16](confinement#Rafelski90)), which is possibly a "[[theta-vacuum]]" exhibiting [[superposition]] of [[instanton in QCD|QCD instantons]] ([Schäfer-Shuryak 98, section III.D](instanton+in+QCD#SchaeferShuryak98)). This remains open, closely related to the _[Millennium Problem](http://www.claymath.org/millennium-problems/yang%E2%80%93mills-and-mass-gap)_ of [[quantization of Yang-Mills theory]].

=--

In contrast to the above subtleties about the [[infrared divergences]], any would-be [[UV-divergences]] in [[perturbative QFT]] are
dealt with by [[causal perturbation theory]]:

+-- {: .num_remark #TheTraditionalErrorThatLeadsToTheNotoriouDivergencies}
###### Remark
**(the traditional error leading to [[UV-divergences]])**

Naively it might seem that (say over [[Minkowski spacetime]], for simplicity)
examples of [[time-ordered products]] according to def. \ref{TimeOrderedProduct}
might simply be obtained by multiplying [[Wick algebra]]-products with [[step functions]] $\Theta$ of the time coordinates, hence to write, in the notation as [[generalized functions]] (remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}):

$$
  T(x_1, x_2)
  \overset{\text{no!}}{=}
  \Theta(x_1^0 - x_2^0) \, T(x_1) \, T(x_2)
  +
  \Theta(x_2^0 - x_1^0) \, T(x_2) \, T(x_1)
$$

and analogously for time-ordered products of more arguments (for instance [Weinberg 95, p. 143, between (3.5.9) and (3.5.10)](S-matrix#Weinberg95)).

This however is simply a mathematical error (as amplified in [Scharf 95, below (3.2.4), below (3.2.44) and in fig. 3](causal+perturbation+theory#Scharf95)):

Both $T$ as well as $\Theta$ are [[distributions]] and their [[product of distributions]] is in general not defined, as [[Hörmander's criterion]] (prop. \ref{HoermanderCriterionForProductOfDistributions}), which is exactly what guarantees absence of [[UV-divergences]] (remark \ref{UltravioletDivergencesFromPaleyWiener}), may be violated. The notorious [[ultraviolet divergences]] which plagued ([Feynman 85](Schwinger-Tomonaga-Feynman-Dyson#Feynman85SuchABunchOfWords)) the original conception of [[perturbative QFT]] due to [[Schwinger-Tomonaga-Feynman-Dyson]] are the signature of this ill-defined product (see remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences}).

On the other hand, when both distributions are [[restriction of distributions|restricted]] to the [[complement]] of the [[diagonal]]
(i.e. restricted away from coinciding points $x_1 = x_2$), then the [[step function]] becomes a [[non-singular distribution]]
so that the above expression happens to be well defined and does solve the axioms for time-ordered products.

Hence what needs to be done to properly define the [[time-ordered product]] is to choose an [[extension of distributions]] of the above product expression back from the complement of the diagonal to the whole space of [[tuples]] of points. Any such extension will produce time-ordered products.

There are in general several different such [[extension of distributions|extensions]]. This freedom of choice is the freedom
of _[[renormalization|"re-"normalization]]_; or equivalently, by the [[main theorem of perturbative renormalization theory]] (theorem \ref{PerturbativeRenormalizationMainTheorem} below),
this is the freedom of choosing "[[counterterms]]" (remark \ref{TermCounter} below) for the [[local observable|local]] [[interactions]]. This we discuss [below](#ExistenceAndRenormalization) and in more detail in the [next chapter](#Renormalization).

=--

+-- {: .num_remark #CausalPerturbationTheoryAbsenceOfUVDivergences}
###### Remark
**(absence of [[ultraviolet divergences]] and [[renormalization|re-normalization]])**

The simple axioms of [[causal perturbation theory]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering})
do fully capture [[perturbative quantum field theory]] "in the ultraviolet": A solution to these axioms
induces, by definition, well-defined [[perturbative QFT|perturbative]] [[scattering amplitudes]] (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes})
and well-defined [[perturbative QFT|perturbative]] [[probability amplitudes]] of [[interacting field observables]] (def. \ref{InteractingFieldObservables}) induced by _[[local observables|local]]_ [[action functionals]] (describing point-interactions such as the [[electron-photon interaction]]). By the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}) such solutions exist. This means that, while these are necessarily [[formal power series]] in $\hbar$ and $g$ (remark \ref{AsymptoticSeriesObservables}), all the [[coefficients]] of these formal power series ("[[loop order]] contributions") are well defined.

This is in contrast to the original informal conception of [[perturbative QFT]] due to [[Schwinger-Tomonaga-Feynman-Dyson]],
which in a first stage produced ill-defined [[divergence|diverging]] expressions for the [[coefficients]] (due to the mathematical error discussed in remark \ref{TheTraditionalErrorThatLeadsToTheNotoriouDivergencies} below), which were then "[[renormalization|re-normalized]]" to finite values, by further informal arguments.

Here in [[causal perturbation theory]] no [[divergences]] in the [[coefficients]] of the [[formal power series]] are considered in the first place, all coefficients are well-defined, hence "finite". In this sense [[causal perturbation theory]] is about "finite" perturbative QFT, where instead of "re-normalization" of ill-defined expressions one just encounters "normalization" (prominently highlighted in [Scharf 95, see title, introduction, and section 4.3](causal+perturbation+theoryscatt#Scharf95)), namely compatible choices of these finite values. The actual "re-normalization" in the sense of "change of normalization" is expressed by the [[Stückelberg-Petermann renormalization group]].

This refers to those [[divergences]] that are known as _[[UV-divergences]]_, namely short-distance effects,
which are mathematically reflected in the fact that the perturbative [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering})
is defined on _[[local observables]]_, which, by their very locality, encode point-[[interactions]].
See also remark \ref{AdiabaticLimit} on _[[infrared divergences]]_.

=--

+-- {: .num_remark #WorldlineFormalism}
###### Remark
**([[virtual particles]], [[worldline formalism]] and [[perturbative string theory]])**

It is suggestive to think of the [[edges]] in the [[Feynman diagrams]] (def. \ref{FeynmanDiagram}) as [[worldlines]] of "[[virtual particles]]" and of the [[vertices]] as the points where they collide and transmute. (Care must be exercised not to confuse this with concepts of real [[particles]].) With this interpretation prop. \ref{FeynmanDiagramAmplitude} may be read as saying that the [[scattering amplitude]] for given external [[source fields]] (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes}) is the [[superposition]] of the [[Feynman amplitudes]] of all possible ways that these may interact; which is closely related to the intuition for the [[path integral]] (remark \ref{InterpretationOfPerturbativeSMatrix}).

This intuition is made precise by the _[[worldline formalism]]_ of [[perturbative quantum field theory]] ([Strassler 92](worldline+formalism#Strassler92)). This is the perspective on [[perturbative QFT]] which directly relates [[perturbative QFT]] to [[perturbative string theory]] ([Schmidt-Schubert 94](worldline+formalism#SchmidtSchubert94)). In fact the [[worldline formalism]] for [[perturbative QFT]] was originally found by taking thre point-particle limit of [[string scattering amplitudes]] ([Bern-Kosower 91](worldline+formalism#BernKosower91), [Bern-Kosower 92](worldline+formalism#BernKosower92)).

=--

+-- {: .num_remark #calSFunctionIsRenormalizationScheme}
###### Remark
**([[renormalization scheme]])**

Beware the terminology in def. \ref{LagrangianFieldTheoryPerturbativeScattering}: A _single_ S-matrix is one single observable

$$
  \mathcal{S}(S_{int})
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [g,j] ]
$$

for a fixed ([[adiabatic switching|adiabatically switched]] [[local observable|local]]) [[interaction]] $S_{int}$, reflecting the [[scattering amplitudes]] (remark \ref{FromAxiomaticSMatrixScatteringAmplitudes}) with respect to that particular interaction.  Hence the function

$$
  \mathcal{S}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [\hbar, g,j] ]\langle g, j
  \rangle
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})((\hbar))[ [g,j] ]
$$

axiomatized in def. \ref{LagrangianFieldTheoryPerturbativeScattering} is really a whole _scheme_ for constructing compatible S-matrices for _all_ possible (adiabatically switched, local) interactions at once.

Since the usual proof of the construction of such schemes of S-matrices involves _[[renormalization|("re"-)normalization]]_, the function $\mathcal{S}$ axiomatized by def. \ref{LagrangianFieldTheoryPerturbativeScattering} may also be referred to as a _[[renormalization scheme|("re"-)normalization scheme]]_.

This perspective on $\mathcal{S}$ as a [[renormalization scheme]] is amplified by the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem})
wich states that the space of choices for $\mathcal{S}$ is a [[torsor]] over the [[Stückelberg-Petermann renormalization group]].

=--

+-- {: .num_remark}
###### Remark
**([[quantum anomalies]])**

The [[axioms]] for the [[S-matrix]] in def. \ref{LagrangianFieldTheoryPerturbativeScattering}
(and similarly that for the [[time-ordered products]] below in def. \ref{TimeOrderedProduct})
are sufficient to imply a [[causally local net]] of perturbative [[interacting field algebras of quantum observables]] (prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below), and thus its [[algebraic adiabatic limit]] (remark \ref{AdiabaticLimit}).

It does not guarantee, however, that the [[BV-BRST differential]] passes to those
[[algebras of quantum observables]], hence it does not guarantee that the [[infinitesimal symmetries of the Lagrangian]]
are respected by the [[quantization]] process (there may be "[[quantum anomalies]]").
The extra condition that does ensure this is the _[[quantum master Ward identity]]_ or _[[quantum master equation]]_.
This we discuss elsewhere.

Apart from [[gauge symmetries]] one also wants to require that rigid symmetries  are preserved by the S-matrix, notably [[Poincare group]]-symmetry for scattering on [[Minkowski spacetime]].

=--

$\,$

**Interacting field observables**
 {#LocalNetsOfInteractingFieldObservables}

We now discuss how the perturbative [[interacting field observables]] which are induced from an [[S-matrix]]
enjoy good properties expected of any abstractly defined [[perturbative algebraic quantum field theory]].

$\,$

+-- {: .num_defn #QuntumMollerOperator}
###### Definition
**([[interacting field algebra of observables]] -- [[quantum Møller operator]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]$ be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]].

We write

$$
  LocIntObs_{\mathcal{S}}(E_{\text{BV-BRST}}, g S_{int})
  \;\coloneqq\;
  \left\{
    {\, \atop \,}
    A_{int} \;\vert\; A \in LocObs(E_{BV-BRST})[ [ \hbar, g ] ]
    {\, \atop \,}
  \right\}
    \hookrightarrow
  PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
$$

for the subspace of [[interacting field observables]] $A_{int}$ (def. \ref{InteractingFieldObservables}) corresponding to [[local observables]] $A$,
the _[[local interacting field observables]]_.

Furthermore we write

$$
  \array{
    LocObs(E_{\text{BV-BRST}})[ [ \hbar , g] ]
    &
    \underoverset{\simeq}{\phantom{A}\mathcal{R}^{-1}\phantom{A}}{\longrightarrow}
    &
    IntLocObs(E_{\text{BV-BRST}}, g S_{int})[ [ \hbar , g ] ]
    \\
    A
      &\mapsto&
    A_{int}
      \coloneqq
    \mathcal{S}(g S_{int})^{-1} T( \mathcal{S}(g S_{int}), A )
  }
$$

for the factorization of the function $A \mapsto A_{int}$ through its image, which, by remark \ref{PerturbativeSMatrixInverse},
is a [[linear isomorphism]] with [[inverse]]

$$
  \array{
    IntLocObs(E_{\text{BV-BRST}}, g S_{int})[ [ \hbar , g ] ]
    &
    \underoverset{\simeq}{\phantom{A}\mathcal{R}\phantom{A}}{\longrightarrow}
    &
    LocObs(E_{\text{BV-BRST}})[ [ \hbar , g] ]
    \\
    A_{int}
      &\mapsto&
    A
      \coloneqq
    T\left(
      \mathcal{S}(-g S_{int})
      ,
      \left( \mathcal{S}(g S_{int}) A_{int} \right)
    \right)
  }
$$

This may be called the _[[quantum Møller operator]]_ ([Hawkins-Rejzner 16, (33)](perturbative+algebraic+quantum+field+theory#HawkinsRejzner16)).

Finally we write

$$
  \begin{aligned}
    IntObs(E_{\text{BV-BRST}}, S_{int})
    & \coloneqq
    \left\langle
    {\, \atop \,}
      IntLocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
    {\, \atop \,}
    \right\rangle
    \\
    &
     \phantom{\coloneqq}
     \hookrightarrow
    PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]
  \end{aligned}
$$

for the smallest subalgebra of the [[Wick algebra]] containing the [[interacting local observables]].
This is the _perturbative [[interacting field algebra of observables]]_.

=--

The definition of the [[interacting field algebra of observables]] from the data of a [[scattering matrix]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) via [[Bogoliubov's formula]] (def. \ref{InteractingFieldObservables}) is physically well-motivated, but is not immediately recognizable as the result of applying a systematic concept of [[quantization]] (such as [[formal deformation quantization]]) to the given [[Lagrangian field theory]]. The following
proposition \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization} says that this is nevertheless the case.
(The special case of this statement for [[free field theory]] is discussed at _[[Wick algebra]]_, see remark \ref{WickAlgebraIsFormalDeformationQuantization}).

+-- {: .num_prop #InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization}
###### Proposition
**([[interacting field algebra of observables]] is [[formal deformation quantization]] of [[interacting field theory|interacting]] [[Lagrangian field theory]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g_{sw} \mathbf{L}_{int} \in \Omega^{p+1,0}_{\Sigma,cp}(E_{\text{BV-BRST}})[ [\hbar, g ] ]\langle g\rangle$ be an [[adiabatic switching|adiabatically switched]] [[interaction]] [[Lagrangian density]] with corresponding
[[action functional]] $g S_{int} \coloneqq \tau_\Sigma( g_{sw} \mathbf{L}_{int} )$.

Then, at least on [[regular polynomial observables]], the construction of perturbative [[interacting field algebras of observables]] in def. \ref{QuntumMollerOperator} is a [[formal deformation quantization]] of the [[interacting field theory|interacting]] [[Lagrangian field theory]]
$(E_{\text{BV-BRST}}, \mathbf{L}'  + g_{sw} \mathbf{L}_{int})$.

=--

([Hawkins-Rejzner 16, prop. 5.4](perturbative+algebraic+quantum+field+theory#HawkinsRejzner16), [Collini 16](pAQFT#Collini16))

The following definition collects the system (a [[co-presheaf]]) of [[generating functions]] for [[interacting field observables]]
which are localized in spacetime as the spacetime localization region varies:

+-- {: .num_defn #PerturbativeGeneratingLocalNetOfObservables}
###### Definition
**(system of spacetime-localized [[generating functions]] for [[interacting field observables]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let

$$
  \mathbf{L}_{int}
  \;\in\;
  \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g  ] ]
$$

be a [[Lagrangian density]], to be thought of as an [[interaction]], so that for $g_{sw} \in C^\infty_{sp}(\Sigma)\langle g \rangle$
an [[adiabatic switching]] the [[transgression of variational differential forms|transgression]]

$$
  S_{int,sw}
    \;\coloneqq\;
  \tau_\Sigma(g_{sw} \mathbf{L}_{int})
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$

is a [[local observable]], to be thought of as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

For $\mathcal{O} \subset \Sigma$ a [[causally closed subset]] of [[spacetime]]
(def. \ref{CausalComplementOfSubsetOfLorentzianManifold})
and for $g_{sw} \in Cutoffs(\mathcal{O})$ an [[adiabatic switching]] function (def. \ref{CutoffFunctions})
which is constant on a [[neighbourhood]] of $\mathcal{O}$, write

$$
  Gen(E_{\text{BV-BRST}}, S_{int,sw} )(\mathcal{O})
   \;\coloneqq\;
  \left\langle
    \mathcal{Z}_{S_{int,sw}}(j A)
    \;\vert\;
    A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
    \,\text{with}\,
    supp(A) \subset \mathcal{O}
  \right\rangle
   \;\subset\;
  PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$

for the smallest subalgebra of the [[Wick algebra]] which contains the
[[generating functions]] (def. \ref{SchemeGeneratingFunction})
with respect to $S_{int,sw}$ for all those [[local observables]] $A$ whose spacetime support is in $\mathcal{O}$.

Moreover, write

$$
  Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
    \;\subset\;
  \underset{g_{sw} \in Cutoffs(\mathcal{O})}{\prod}
  Gen(E_{\text{BV-BRST}}, S_{int,sw})(\mathcal{O})
$$

be the subalgebra of the [[Cartesian product]] of all these algebras as $g_{sw}$ ranges over cutoffs, which is generated by the
[[tuples]]

$$
  \mathcal{Z}_{\mathbf{L}_{int}}(A)
    \;\coloneqq\;
  \left(
    \mathcal{Z}_{S_{int,sw}}(j A)
  \right)_{g_{sw} \in Cutoffs(\mathcal{O})}
$$

for $A$ with $supp(A) \subset \mathcal{O}$.

We call $Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int} )(\mathcal{O})$ the _algebra of [[generating functions]] for [[interacting field observables]] localized in $\mathcal{O}$_.

Finally, for $\mathcal{O}_1 \subset \mathcal{O}_2$ an inclusion of two [[causally closed subsets]], let

$$
  i_{\mathcal{O}_1, \mathcal{O}_2}
  \;\colon\;
  Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_1)
    \longrightarrow
  Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_2)
$$

be the algebra [[homomorphism]] which is given simply by restricting the index set of [[tuples]].

This construction defines a [[functor]]

$$
  Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})
    \;\colon\;
  CausClsdSubsets(\Sigma)
    \longrightarrow
  Algebras
$$

from the [[poset]] of [[causally closed subsets]] of [[spacetime]] to the [[category]] of [[algebras]].

> (extends to [[star algebras]] if scattering matrices are chosen unitary...)

=--

([Brunetti-Fredenhagen 99, (65)-(67)](S-matrix#BrunettiFredenhagen99))

The key technical fact is the following:

+-- {: .num_prop #IsomorphismFromChangeOfAdiabaticSwitching}
###### Proposition
**(localized [[interacting field observables]] independent of [[adiabatic switching]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let

$$
  \mathbf{L}_{int}
  \;\in\;
  \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g  ] ]
$$

be a [[Lagrangian density]], to be thought of as an [[interaction]], so that for $g_{sw} \in C^\infty_{sp}(\Sigma)\langle g \rangle$
an [[adiabatic switching]] the [[transgression of variational differential forms|transgression]]

$$
  g S_{int,sw} \;\coloneqq\; \tau_\Sigma(g_{sw} \mathbf{L}_{int})
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$

is a [[local observable]], to be thought of as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

If two such [[adiabatic switchings]] $g_{sw,1}, g_{sw,2} \in C^\infty_{cp}(\Sigma)$ agree on a
[[causally closed subset]]

$$
  \mathcal{O} \;\subset\; \Sigma
$$

in that

$$
  g_{sw,1}\vert_{\mathcal{O}} = g_{sw,2}\vert_{\mathcal{O}}
$$

then there exists a [[microcausal polynomial observable]]

$$
  K \;\in\; PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j ] ]
$$

such that for every [[local observable]]

$$
  A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]
$$

with spacetime support in $\mathcal{O}$

$$
  supp(A) \;\subset\; \mathcal{O}
$$

the corresponding two [[generating functions]] (eq:GeneratingFunctionInducedFromSMatrix)
are related via [[conjugation]] by $K$:

$$
  \label{AdiabaticSwitchingRelationGeneratingFunctions}
  \mathcal{Z}_{S_{int,sw_2}}
  \left(
    j A
  \right)
  \;=\;
  K^{-1}
  \,
  \left(
    \mathcal{Z}_{S_{int,sw_1}}
    \left(
      j A
    \right)
  \right)
  \,
  K
  \,.
$$

In particular this means that for every choice of [[adiabatic switching]] $g_{sw} \in Cutoffs(\mathcal{O})$
the algebra $Gen_{S_{int,sw}}(\mathcal{O})$ of [[generating functions]] for [[interacting field observables]]
computed with $g_{sw}$ is canonically [[isomorphism|isomorphic]] to the
abstract algebra $Gen_{\mathbf{L}_{int}}(\mathcal{O})$  (def. \ref{PerturbativeGeneratingLocalNetOfObservables}),
by the evident map on generators:

$$
  \label{AbstractGeneratingFunctionAlgebraIsomorphicToAnyAdiabaticSwitching}
  \array{
    Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{o})
      &\overset{\simeq}{\longrightarrow}&
    Gen(E_{\text{BV-BRST}}, S_{int,sw})(\mathcal{O})
    \\
    \left(
      \mathcal{Z}_{S_{int,sw'}}
    \right)_{g_{sw'} \in Cutoffs(\mathcal{O})}
    &\mapsto&
    \mathcal{Z}_{S_{int,sw}}
  }
  \,.
$$

=--

([Brunetti-Fredenhagen 99, prop. 8.1](S-matrix#BrunettiFredenhagen99))

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

By causal closure of $\mathcal{O}$, lemma \ref{CausalPartition}
says that there are [[bump functions]]

$$
  a, r \in C^\infty_{cp}(\Sigma)\langle g \rangle
$$

which decompose the difference of [[adiabatic switchings]]

$$
  g_{sw,2} - g_{sw,1} = a + r
$$

subject to the [[causal ordering]]

$$
  supp(a) \,{\vee\!\!\!\wedge}\, \mathcal{O} \,{\vee\!\!\!\wedge}\, supp(r)
  \,.
$$

With this the result follows from repeated use of [[causal additivity]] in its various equivalent
incarnations from prop. \ref{ZCausalAdditivity}:

$$
  \begin{aligned}
    & \mathcal{Z}_{g S_{int,sw_2}}(j A)
    \\
    & =
    \mathcal{Z}_{
      \left(
        \tau_\Sigma
        \left(
          g_{sw,2}
          \mathbf{L}_{int}
        \right)
      \right)
    }
    \left(
      j A
    \right)
    \\
    & =
    \mathcal{Z}_{
      \left(
        \tau_\Sigma
        \left(
          (g_{sw,1} + a + r)\mathbf{L}_{int}
        \right)
      \right)
    }
    \left(
      j A
    \right)
    \\
    & =
    \mathcal{Z}_{
      \left(
        g S_{int,sw_1}
        +
        \tau_\Sigma
        \left(
          r \mathbf{L}_{int}
        \right)
        +
        \tau_\Sigma
        \left(
          a \mathbf{L}_{int}
        \right)
      \right)
    }
    \left(
      j A
    \right)
    \\
    & =
    \mathcal{Z}_{
      \left(
        g S_{int,sw_1}
        +
        \tau_\Sigma
        \left(
          r \mathbf{L}_{int}
        \right)
      \right)
    }
    \left(
      j A
    \right)
    \\
    & =
    \mathcal{S}
    \left(
      g S_{int,sw_1}
      +
      \tau_\Sigma
      \left(
        r \mathbf{L}_{int}
      \right)
    \right)^{-1}
    \,
    \mathcal{S}
    \left(
      g S_{int,sw_1}
      +
      j A
      +
      \tau_\Sigma
      \left(
        r \mathbf{L}_{int}
      \right)
    \right)
    \\
    & =
    \mathcal{S}
    \left(
      g S_{int,sw_1}
      +
      \tau_\Sigma
      \left(
        r \mathbf{L}_{int}
      \right)
    \right)^{-1}
    \,
    \mathcal{S}
    \left(
      g S_{int,sw_1}
      +
      j A
    \right)
    \,
    \mathcal{S}
    \left(
      g S_{int,sw_1}
    \right)^{-1}
    \,
    \mathcal{S}
    \left(
      j A
      +
      \tau_\Sigma
      \left(
        r \mathbf{L}_{int}
      \right)
    \right)
    \\
    & =
    \mathcal{S}
    \left(
      g S_{int,sw_1}
      +
      \tau_\Sigma
      \left(
        r\mathbf{L}_{int}
      \right)
    \right)^{-1}
    \,
    \underset{
      = id
    }{
    \underbrace{
    \mathcal{S}
    \left(
      g S_{int,sw_1}
    \right)
    \,
    \mathcal{S}
    \left(
      g S_{int,sw_1}
    \right)^{-1}
    }
    }
    \,
    \mathcal{S}
    \left(
      g S_{int,sw_1}
      +
      j A
    \right)
    \,
    \mathcal{S}
    \left(
      g S_{int , sw_1}
    \right)^{-1}
    \,
    \mathcal{S}
    \left(
      j A
      +
      \tau_\Sigma
      \left(
        r \mathbf{L}_{int}
      \right)
    \right)
    \\
    & =
    \underset{
      K^{-1}
    }{
      \underbrace{
        \left(
          \mathcal{Z}_{
            g S_{int,sw_1}
          }
          \left(
            \tau_\Sigma
            \left(
              r \mathbf{L}_{int}
            \right)
          \right)
        \right)^{-1}
      }
    }
    \,
    \mathcal{Z}_{
      g S_{int,sw_1}
    }
    \left(
      j A
    \right)
    \,\,
    \underset{
      K
    }{
    \underbrace{
    \mathcal{Z}_{
      g S_{int,sw_1}
    }
    \left(
      \tau_\Sigma
      \left(
        r \mathbf{L}_{int}
      \right)
    \right)
    }}
  \end{aligned}
$$

This proves the existence of elements $K$ as claimed.

It is clear that conjugation induces an algebra homomorphism,
and since the map is a linear isomorphism on the space of generators,
it is an algebra isomorphism on the algebras being generated (eq:AbstractGeneratingFunctionAlgebraIsomorphicToAnyAdiabaticSwitching).

(While the elements $K$ in (eq:AdiabaticSwitchingRelationGeneratingFunctions) are far from being unique themselves,
equation (eq:AdiabaticSwitchingRelationGeneratingFunctions) says that the map on generators induced
by conjugation with $K$ is independent of this choice.)

=--

+-- {: .num_prop #GeneratingAlgebrasIsLocalNet}
###### Proposition
**(system of generating algebras is [[causally local net]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let

$$
  \mathbf{L}_{int}
  \;\in\;
  \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g  ] ]
$$

be a [[Lagrangian density]], to be thought of as an [[interaction]].

Then the system

$$
  Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})
  \;\colon\;
  CausCldSubsets(\Sigma)
    \longrightarrow
  Algebra
$$

of localized [[generating functions]] for [[interacting field observables]] (def. \ref{PerturbativeGeneratingLocalNetOfObservables})
is a _[[causally local net]]_ in that it satisfies the following conditions:

1. (isotony) For every inclusion $\mathcal{O}_1 \subset \mathcal{O}_2$ of [[causally closed subsets]] of [[spacetime]]
   the corresponding algebra homomorphism is a [[monomorphism]]

   $$
     i_{\mathcal{O}_1, \mathcal{O}_2}
     \;\colon\;
     Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_1) \hookrightarrow Gen(E_{\text{BV-BRST}},\mathbf{L}_{int})(\mathcal{O}_2)
   $$

1. ([[causal locality]]) For $\mathcal{O}_1, \mathcal{O}_2 \subset X$ two [[causally closed subsets]]
   which are [[spacelike]] separated, in that their [[causal ordering]] (def. \ref{CausalOrdering}) satisfies

   $$
     \mathcal{O}_1 {\vee\!\!\!\wedge} \mathcal{O}_2
       \;\text{and}\;
     \mathcal{O}_2 {\vee\!\!\!\wedge} \mathcal{O}_1
   $$

   and for $\mathcal{O} \subset \Sigma$ any further [[causally closed subset]] which contains both

   $$
     \mathcal{O}_1 , \mathcal{O}_2 \subset \mathcal{O}
   $$

   then the corresponding images of the generating function algebras of interacting field observables
   localized in  $\mathcal{O}_1$ and in $\mathcal{O}_2$,
   respectively, commute with each other as subalgebras of the generating function algebras of interacting
   field observables localized in $\mathcal{O}$:

   $$
     \left[
       i_{\mathcal{O}_1,\mathcal{O}}(Gen_{L_{int}}(\mathcal{O}_1))
       \;,\;
       i_{\mathcal{O}_2,\mathcal{O}}(Gen_{L_{int}}(\mathcal{O}_2))
     \right]
     \;=\;
     0
     \;\;\;
     \in Gen(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
     \,.
   $$

=--

([Dütsch-Fredenhagen 00, section 3](S-matrix#DuetschFredenhagen00), following [Brunetti-Fredenhagen 99, section 8](S-matrix#BrunettiFredenhagen99),
[Il'in-Slavnov 78](S-matrix#IlinSlavnov78))

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

Isotony is immediate from the definition of the algebra homomorphisms in def. \ref{PerturbativeGeneratingLocalNetOfObservables}.

By the isomorphism (eq:AbstractGeneratingFunctionAlgebraIsomorphicToAnyAdiabaticSwitching) we may check causal localizy with respect to any
choice of [[adiabatic switching]] $g_{sw} \in Cautoff(\mathcal{O})$ constant over $\mathcal{O}$. For this
the statement follows, with the assumption of spacelike separation, by [[causal additivity]] (prop. \ref{ZCausalAdditivity}):

For $supp(A_1) \subset \mathcal{O}_1$ and $supp(A_2) \subset \mathcal{O}_2$ we have:

$$
  \begin{aligned}
    \mathcal{Z}_{g S_{int,sw}}( j A_1 )
    \mathcal{Z}_{g S_{int,sw}}( j A_2 )
    & =
    \mathcal{S}_{g S_{int,sw}}( j A_1 + j A_2)
    \\
    & =
    \mathcal{S}_{g S_{int,sw}}( j A_2 + j A_1)
    \\
    & =
    \mathcal{Z}_{g S_{int,sw}}( j A_2 )
    \mathcal{Z}_{g S_{int,sw}}( j A_1 )
  \end{aligned}
$$

=--

With the [[causally local net]] of localized [[generating functions]] for
[[interacting field observables]] in hand, it is now immediate to get the

+-- {: .num_defn #SystemOfAlgebrasOfQuantumObservables}
###### Definition
**(system of [[interacting field algebras of observables]])

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let

$$
  \mathbf{L}_{int}
  \;\in\;
  \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g  ] ]
$$

be a [[Lagrangian density]], to be thought of as an [[interaction]], so that for $g_{sw} \in C^\infty_{sp}(\Sigma)\langle g \rangle$
an [[adiabatic switching]] the [[transgression of variational differential forms|transgression]]

$$
  g S_{int,sw} \;\coloneqq\; g \tau_\Sigma(g_{sw} \mathbf{L}_{int})
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle
$$

is a [[local observable]], to be thought of as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

For $\mathcal{O} \subset \Sigma$ a [[causally closed subset]]
of [[spacetime]] (def. \ref{CausalComplementOfSubsetOfLorentzianManifold})
and for $g_{sw} \in Cutoffs(\mathcal{O})$ an compatible [[adiabatic switching]] function (def. \ref{CutoffFunctions}) write

$$
  IntObs(E_{\text{BV-BRST}}, S_{int,sw})(\mathcal{O})
   \coloneqq
  \left\langle
     i \hbar \frac{d}{d j}
     \mathcal{Z}_{S_{int}}(j A)\vert_{j = 0} \;\vert\; supp(A) \subset \mathcal{O}
  \right\rangle
   \;\subset\;
  PolyObs((\hbar))[ [ g ] ]
$$

for the [[interacting field algebra of observables]] (def. \ref{QuntumMollerOperator}) with spacetime support in $\mathcal{O}$.

Let then

$$
  IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
    \subset
  \underset{g_{sw} \in Cutoffs(\mathcal{O})}{\prod} IntObs(E_{\text{BV-BRST}}, S_{int,sw})(\mathcal{O})
$$

be the subalgebra of the [[Cartesian product]] of all these algebras as $g_{sw}$ ranges, which is generated by the
[[tuples]]

$$
   i \hbar
   \frac{d}{d j } \mathcal{Z}_{\mathbf{L}_{int}}\vert_{j = 0}
    \;\coloneqq\;
  \left(
      i \hbar
      \frac{d}{d j } \mathcal{Z}_{S_{int,sw}} (j A)\vert_{j  = 0}
  \right)_{g_{sw} \in Cutoffs(\mathcal{O})}
$$

for $supp(A) \subset \mathcal{O}$.

Finally, for $\mathcal{O}_1 \subset \mathcal{O}_2$ an inclusion of two [[causally closed subsets]], let

$$
  i_{\mathcal{O}_1, \mathcal{O}_2}
  \;\colon\;
  IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_1)
    \longrightarrow
  IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_2)
$$

be the algebra [[homomorphism]] which is given simply by restricting the index set of [[tuples]].

This construction defines a [[functor]]

$$
  IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int}) \;\colon\; CausClsdSubsets(\Sigma) \longrightarrow Algebras
$$

from the [[poset]] of [[causally closed subsets]] in the [[spacetime]] $\Sigma$ to the [[category]] of [[star algebras]].

=--

Finally, as a direct corollary of prop. \ref{GeneratingAlgebrasIsLocalNet}, we obtain the key result:

+-- {: .num_prop #PerturbativeQuantumObservablesIsLocalnet}
###### Proposition
**(system of [[interacting field algebras of observables]] is [[causally local net|causally local]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}, and let

$$
  \mathbf{L}_{int}
  \;\in\;
  \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})[ [ \hbar , g ] ]
  \,.
$$

be a [[Lagrangian density]], to be thought of as an [[interaction]],
then the system of [[algebras of observables]] $Obs_{L_{int}}$ (def. \ref{SystemOfAlgebrasOfQuantumObservables})
is a [[local net of observables]] in that

1. (isotony) For every inclusion $\mathcal{O}_1 \subset \mathcal{O}_2$ of [[causally closed subsets]]
   the corresponding algebra homomorphism is a [[monomorphism]]

   $$
     i_{\mathcal{O}_1, \mathcal{O}_2}
     \;\colon\;
     IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_1) \hookrightarrow IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O}_2)
   $$

1. ([[causal locality]]) For $\mathcal{O}_1, \mathcal{O}_2 \subset X$ two [[causally closed subsets]]
   which are [[spacelike]] separated, in that their [[causal ordering]] (def. \ref{CausalOrdering}) satisfies

   $$
     \mathcal{O}_1 {\vee\!\!\!\wedge} \mathcal{O}_2
       \;\text{and}\;
     \mathcal{O}_2 {\vee\!\!\!\wedge} \mathcal{O}_1
   $$

   and for $\mathcal{O} \subset \Sigma$ any further causally closed subset which contains both

   $$
     \mathcal{O}_1 , \mathcal{O}_2 \subset \mathcal{O}
   $$

   then the corresponding images of the generating algebras of $\mathcal{O}_1$ and $\mathcal{O}_2$,
   respectively, commute with each other as subalgebras of the generating algebra of $\mathcal{O}$:

   $$
     \left[
       i_{\mathcal{O}_1,\mathcal{O}}(Obs_{\mathbf{L}_{int}}(\mathcal{O}_1))
       \;,\;
       i_{\mathcal{O}_2,\mathcal{O}}(Obs_{\mathbf{L}_{int}}(\mathcal{O}_2))
     \right]
     \;=\;
     0
     \;\;\;
     \in IntObs(E_{\text{BV-BRST}}, \mathbf{L}_{int})(\mathcal{O})
     \,.
   $$

=--

([Dütsch-Fredenhagen 00, below (17)](S-matrix#DuetschFredenhagen00), following [Brunetti-Fredenhagen 99, section 8](S-matrix#BrunettiFredenhagen99),
[Il'in-Slavnov 78](S-matrix#IlinSlavnov78))

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

The first point is again immediate from the definition (def. \ref{SystemOfAlgebrasOfQuantumObservables}).

For the second point it is sufficient to check the commutativity relation on generators. For these
the statement follows with prop. \ref{GeneratingAlgebrasIsLocalNet}:

$$
  \begin{aligned}
    & \left[
      i \hbar \frac{d}{d j} \mathcal{Z}_{S_{int,sw}}(j A_1)\vert_{j = 0}
      \;,\;
      i \hbar  \frac{d}{d j} \mathcal{Z}_{S_{int,sw}}(j J_2)\vert_{j = 0}
    \right]
    \\
    & =
     (i \hbar)^2
     \frac{
       \partial^2
     }{
       \partial j_1 \partial j_2
     }
     \underset{ = 0}{
     \underbrace{
    \left[
       \mathcal{Z}_{S_{int,sw}}(j_1 A_1)
      \;,\;
       \mathcal{Z}_{S_{int,sw}}(j_1 A_2)
    \right]}}_{ \left\vert  { {j_1 = 0} \atop {j_2 = 0} } \right. }
    \\
    & = 0
  \end{aligned}
$$

=--

$\,$

**[[time-ordered products]]**
 {#TimeOrderedProducts}

Definition \ref{LagrangianFieldTheoryPerturbativeScattering} suggests to focus on the multilinear operations $T(...)$ which define the perturbative [[S-matrix]] order-by-order in $\hbar$. We impose [[axioms]] on these _[[time-ordered products]]_ directly (def. \ref{TimeOrderedProduct}) and then prove that these axioms imply the axioms for the corresponding [[S-matrix]] (prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix} below).

+-- {: .num_defn #TimeOrderedProduct}
###### Definition
**([[time-ordered products]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

A _[[time-ordered product]]_ is a sequence of [[multilinear map|multi-]][[linear continuous functionals]] for all $k \in \mathbb{N}$ of the form

$$
  T_k
    \;\colon\;
  \left(
    {\, \atop \,}
    LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]\langle g,j \rangle
    {\, \atop \,}
  \right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))[ [g , j] ]
$$

(from [[tensor products]] of [[local observables]] to [[microcausal polynomial observables]], with formal parameters adjoined
according to def. \ref{FormalParameters})
such that the following conditions hold for all possible arguments:

1. (normalization)

   $$
     T_0(O) = 1
   $$

1. (perturbation)

   $$
     T_1(O) =  :O:
   $$

1. (symmetry) each $T_k$ is symmetric in its arguments, in that for every [[permutation]]
   $\sigma \in \Sigma(k)$ of $k$ elements

   $$
     T_k(O_{\sigma(1)}, O_{\sigma(2)}, \cdots, O_{\sigma(k)})
     \;=\;
     T_k(O_1, O_2, \cdots, O_k)
   $$

1. ([[causal factorization]]) If the spacetime support (def. \ref{SpacetimeSupport}) of [[local observables]] satisfies the [[causal ordering]] (def. \ref{CausalOrdering})

   $$
     \left(
       {\, \atop \,}
       supp(O_1)
         \cup
         \cdots
         \cup
       supp(O_r)
       {\, \atop \,}
     \right)
       \;{\vee\!\!\!\wedge}\;
     \left(
       {\, \atop \,}
       supp(O_{r+1})
         \cup
         \cdots
         \cup
       supp(O_k)
       {\, \atop \,}
     \right)
   $$

   then the time-ordered product of these $k$ arguments factors as
   the [[Wick algebra]]-product of the time-ordered product of the first $r$ and that of the second $k-r$ arguments:

   $$
     T(O_1, \cdots, O_k)
     \; = \;
     T( O_1,   \cdots , O_r )
     \,
     T( O_{r+1},   \cdots , O_k )
     \,.
   $$

=--

+-- {: .num_example #TimeOrderedProductsFromSMatrixScheme}
###### Example
**([[S-matrix]] scheme implies [[time-ordered products]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree} and let

$$
  \mathcal{S}
  \;=\;
  \underset{k \in \mathbb{N}}{\sum}
  \frac{1}{k!}\frac{1}{(i \hbar)^k}
  T_k
$$

be a corresponding [[S-matrix]] scheme according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}.

Then the $\{T_k\}_{k \in \mathbb{N}}$ are [[time-ordered products]] in the sense of def. \ref{TimeOrderedProduct}.

=--

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

We need to show that the $\{T_k\}_{k \in \mathbb{N}}$ satisfy [[causal factorization]].

For

$$
  O_j\;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
$$

a local observable, consider the continuous linear function that muliplies this by any [[real number]]

$$
  \array{
    \mathbb{R}
      &\longrightarrow&
    LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
    \\
    \kappa_j &\mapsto& \kappa_j O_j
  }
  \,.
$$

Since the $T_k$ by definition are [[continuous linear functionals]], they are in particular
[[differentiable maps]], and hence so is the S-matrix $\mathcal{S}$. We may extract $T_k$ from
$\mathcal{S}$ by [[differentiation]] with respect to the parameters $\kappa_j$ at $\kappa_j = 0$:

$$
  T_k(O_1, \cdots, O_k)
  \;=\;
  \frac{\partial^k}{
    \partial \kappa_1
    \cdots
    \partial \kappa_k
  }
  \mathcal{S}\left( \kappa_1 O_1 + \cdots + \kappa_k O_k \right)\vert_{\kappa_1, \cdots, \kappa_k = 0}
$$

for all $k \in \mathbb{N}$.

Now the [[causal additivity]] of the S-matrix $\mathcal{S}$ implies its [[causal factorization]] (remark \ref{DysonCausalFactorization})
and this implies the causal factorization of the $\{T_k\}$ by the [[product law]] of [[differentiation]]:

$$
  \begin{aligned}
    T_k(O_1, \cdots, O_k)
    & =
    (i \hbar)^k
    \frac{\partial^k}{
      \partial \kappa_1
      \cdots
      \partial \kappa_k
    }
    \mathcal{S}\left( \kappa_1 O_1 + \cdots + \kappa_k O_k \right)\vert_{\kappa_1, \cdots, \kappa_k = 0}
    \\
    & =
    (i \hbar)^k
    \frac{\partial^k}{
      \partial \kappa_1
      \cdots
      \partial \kappa_k
    }
    \left(
      {\, \atop \,}
      \mathcal{S}(\kappa_1 O_1 + \cdots + \kappa_r O_r)
      \,
      \mathcal{S}(\kappa_{r+1} O_{r+1} + \cdots + \kappa_k O_k)
      {\, \atop \,}
    \right)
    \vert_{\kappa_1, \cdots, \kappa_k = 0}
    \\
    & =
    (i \hbar)^r
    \frac{\partial^r}{
      \partial \kappa_1
      \cdots
      \partial \kappa_r
    }
    \mathcal{S}(\kappa_1 O_1 + \cdots + \kappa_r O_r)
    \vert_{\kappa_1, \cdots, \kappa_r = 0}
    \;
    (i \hbar)^{k-r}
    \frac{\partial^{k-r}}{
      \partial \kappa_{r+1}
      \cdots
      \partial \kappa_k
    }
    \mathcal{S}(\kappa_{r+1} O_{r+1} + \cdots + \kappa_k O_k)
    \vert_{\kappa_{r+1}, \cdots, \kappa_k = 0}
    \\
    & =
    T_{r}( O_1, \cdots, O_{r} )
    \,
    T_{k-r}( O_{r+1}, \cdots, O_{k} )
  \end{aligned}
  \,.
$$

=--

The converse implication, that [[time-ordered products]] induce an [[S-matrix]] scheme involves
more work (prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix} below).

+-- {: .num_remark #NotationForTimeOrderedProductsAsGeneralizedFunctions}
###### Remark
**([[time-ordered products]] as [[generalized functions]])**

It is convenient (as in [Epstein-Glaser 73](S-matrix#EpsteinGlaser73)) to think of [[time-ordered products]] (def. \ref{TimeOrderedProduct}), being
[[Wick algebra]]-valued [[distributions]] (hence [[operator-valued distributions]] if we were to choose a [[representation]] of the
[[Wick algebra]] by [[linear operators]] on a [[Hilbert space]]), as [[generalized functions]] depending on spacetime points:

If

$$
  \left\{
    \alpha_i \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})\langle g \rangle
  \right\}
  \cup
  \left\{
    \beta_j \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})\langle j \rangle
  \right\}
$$

is a [[finite set]] of [[horizontal differential forms]], and

$$
  \left\{
    g_i, j_{j} \in C^\infty_{cp}(\Sigma)
  \right\}
$$

is a corresponding set of [[bump functions]] on [[spacetime]] ([[adiabatic switchings]]), so that

$$
  \left\{
    S_j
      \colon
    \Phi
      \mapsto
    \underset{\Sigma}{\int} g_i(x) \, \left(j^\infty_\Sigma(\Phi)^\ast \alpha_i\right)(x)\, dvol_\Sigma(x)
  \right\}
  \;\cup\;
  \left\{
    A_j
      \colon
    \Phi
      \mapsto
    \underset{\Sigma}{\int} j_i(x) \, \left(j^\infty_\Sigma(\Phi)^\ast \beta_i\right)(x)\, dvol_\Sigma(x)
  \right\}
$$

is the corresponding set of [[local observables]], then we may write the [[time-ordered product]]
of these observables as the [[integration]] of these [[bump functions]] against a [[generalized function]]
$T_{(\alpha_i)}$ with values in the [[Wick algebra]]:

$$
  \begin{aligned}
    &
    \underset{\Sigma^n}{\int}
      T_{(\alpha_i), (\beta_j)}(x_1, \cdots, x_{r}, x_{r+1}, \cdots x_{n})
      g_1(x_1) \cdots g_r(x_r)
      \,
      j_1(x_{r+1}) \cdots j_n(x_n)
    \, dvol_{\Sigma^n}(x_1, \cdots x_n)
    \\
    & \coloneqq
      T( S_1, \cdots, S_r, A_{r+1}, \cdots, A_n )
  \end{aligned}
  \,.
$$

Moreover, the subscripts on these [[generalized functions]] will always be
clear from the context, so that in computations we may notationally suppress these.

Finally, due to the "symmetry" axiom in def. \ref{TimeOrderedProduct}, a time-ordered product
depends, up to signs, only on its [[set]] of arguments, not on the order of the arguments. We will write
$\mathbf{X} \coloneqq \{x_1, \cdots, x_r\}$ and $\mathbf{Y} \coloneqq \{y_1, \cdots y_r\}$
for sets of spacetime points, and hence abbreviate the expression for the "value" of the
generalized function in the above as $T(\mathbf{X}, \mathbf{Y})$ etc.

In this condensed notation the above reads

$$
    \underset{\Sigma^{r+s}}{\int}
      T(\mathbf{X}, \mathbf{Y})
      \,
      g_1(x_1) \cdots g_r(x_r)
      j_{r+1}(x_{r+1}) \cdots j_n(x_n)
    \,
    dvol_{\Sigma^{r+s}}(\mathbf{X})
  \,.
$$

=--

This condensed notation turns out to be greatly simplify computations, as it absorbs all the "relative"
combinatorial prefactors:

+-- {: .num_example #ProductOfPerturbationSeriesInGenealizedFunctionNotation}
###### Example
**(product of perturbation series in [[generalized function]]-notation)**

Let

$$
  U(g)
  \coloneqq
  \underoverset{n = 0}{\infty}{\sum}
   \frac{1}{n!}
   \int U(x_1, \cdots, x_n)
   \,
   g(x_1) \cdots g(x_n) \, dvol
$$

and

$$
  V(g)
  \coloneqq
  \underoverset{n = 0}{\infty}{\sum}
   \frac{1}{n!}
   \int V(x_1, \cdots, x_n)
   \,
   g(x_1) \cdots g(x_n) \, dvol
$$

be power series of [[Wick algebra]]-valued [[distributions]] in the [[generalized function]]-notation of remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}.

Then their product $W(g) \coloneqq U(g) V(g)$ with [[generalized function]]-representation

$$
  W(g)
  \coloneqq
  \underoverset{n = 0}{\infty}{\sum}
   \frac{1}{n!}
   \int W(x_1, \cdots, x_n)
   \,
   g(x_1) \cdots g(x_n) \, dvol
$$

is given simply by

$$
  W(\mathbf{X})
    \;=\;
  \underset{\mathbf{I} \subset \mathbf{X}}{\sum} U(\mathbf{I}) V(\mathbf{X} \setminus \mathbf{I})
  \,.
$$

=--

([Epstein-Glaser 73 (5)](S-matrix#EpsteinGlaser73))

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

For fixed [[cardinality]] ${\vert \mathbf{I} \vert} = n_1$
the sum over all subsets $\mathbf{I} \subset \mathbf{X}$ overcounts the sum over
[[partitions]] of the coordinates as $(x_1, \cdots x_{n_1}, x_{n_1 + 1}, \cdots x_n)$ precisely by the
[[binomial coefficient]] $\frac{n!}{n_1! (n - n_1) !}$. Here the factor of $n!$ cancels
against the "global" combinatorial prefactor in the above expansion of $W(g)$, while the remaining
factor $\frac{1}{n_1! (n - n_1) !}$ is just the "relative" combinatorial prefactor seen at total order $n$
when expanding the product $U(g)V(g)$.

=--

In order to prove that the axioms for [[time-ordered products]] do imply those for a perturbative [[S-matrix]]
(prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix} below) we need to consider
the corresponding reverse-time ordered products:

+-- {: .num_defn #ReverseTimeOrderedProduct}
###### Definition
**([[reverse-time ordered products]])**

Given a [[time-ordered product]] $T = \{T_k\}_{k \in \mathbb{N}}$ (def. \ref{TimeOrderedProduct}),
its _[[reverse-time ordered product]]_

$$
  \overline{T}_k
    \;\colon\;
  \left(
    {\, \atop \,}
    LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
    {\, \atop \,}
  \right)
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})((\hbar))[ [g, j] ]
$$

for $k \in \mathbb{N}$ is defined by

$$
  \overline{T}( A_1 \cdots A_n )
  \;\coloneqq\;
  \left\{
    \array{
      \underoverset{r = 1}{n}{\sum}
      (-1)^r
      \underset{\sigma \in Unshuffl(n,r)}{\sum}
      T( A_{\sigma(1)} \cdots A_{\sigma(k_1)} )
      \,
      T( A_{\sigma(k_1 + 1)} \cdots A_{\sigma(k_2)} )
      \cdots
      T( A_{\sigma(k_{r-1}+1)} \cdots A_{\sigma_{k_r}}  )
      &\vert& k \geq 1
      \\
      1 &\vert& k = 0
    }
  \right.
  \,,
$$

where the sum is over all [[unshuffles]] $\sigma$ of $(1 \leq \cdots \leq n)$ into $r$ non-empty
ordered subsequences. Alternatively, in the [[generalized function]]-notation of remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}, this reads

$$
  \overline{T}( \mathbf{X} )
  =
  \underoverset{r = 1}{{\vert \mathbf{X} \vert}}{\sum}
  (-1)^r
  \underset{ \array{
    \mathbf{I}_1, \cdots, \mathbf{I}_r \neq \emptyset
    \\
    \underset{j \neq k}{\forall}\left( \mathbf{I}_j \cap \mathbf{I}_k = \emptyset  \right)
    \\
    \mathbf{I}_1 \cup \cdots \cup \mathbf{I}_r = \mathbf{X}
  } }{\sum}  T( \mathbf{I}_1 ) \cdots T(\mathbf{I}_r)
$$

=--

([Epstein-Glaser 73, (11)](S-matrix#EpsteinGlaser73))

+-- {: .num_prop #ReverseTimOrderedProductsGiveReverseSMatrix}
###### Proposition
**([[reverse-time ordered products]] express [[inverse]] [[S-matrix]])

Given [[time-ordered products]] $T(-)$ (def. \ref{TimeOrderedProduct}), then the
corresponding reverse time-ordered product $\overline{T}(-)$ (def. \ref{ReverseTimeOrderedProduct})
expresses the [[inverse]] $S(-)^{-1}$ (according to remark \ref{PerturbativeSMatrixInverse}) of the corresponding
perturbative [[S-matrix]] scheme $\mathcal{S}(S_{int}) \coloneqq \underset{k \in \mathbb{N}}{\sum} \tfrac{1}{k!} T(\underset{k\,\text{args}}{\underbrace{S_{int}, \cdots , S_{int}}})$ (def. \ref{LagrangianFieldTheoryPerturbativeScattering}):

$$
  \left(
    {\, \atop \,}
    \mathcal{S}(g S_{int} + j A )
    {\, \atop \,}
  \right)^{-1}
  \;=\;
  \underset{k \in \mathbb{N}}{\sum}
    \frac{1}{k!}
    \left(
      \frac{1}{i \hbar}
    \right)^k
    \overline{T}( \underset{k \, \text{arguments}}{\underbrace{ (g S_{int} + j A), \cdots, (g S_{int} + j A)}} )
  \,.
$$

=--

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

For brevity we write just "$A$" for $\tfrac{1}{i \hbar}(g S_{int} + j A)$.
(Hence we assume without restriction that $A$ is not independent of
powers of $g$ and $j$; this is just for making all sums in the following be order-wise finite sums.)

By definition we have

$$
  \begin{aligned}
  &
  \underset{k \in \mathbb{N}}{\sum}
    \frac{1}{k!}
    \overline{T}( \underset{k \, \text{args}}{\underbrace{A,  \cdots , A}} )
  \\
  & =
  \underset{ k \in \mathbb{N}}{\sum}
    \frac{1}{k!}
    \underoverset{r = 1}{k}{\sum}
    (-1)^r
    \!\!\!\underset{\sigma \in Unshuffl(k,r)}{\sum}\!\!\!
    T( A_{\sigma(1)} \cdots A_{\sigma(k_1)} )
    T( A_{\sigma(k_1 + 1)} \cdots A_{\sigma(k_2)} )
    \cdots
    T( A_{\sigma(k_{r-1}+1)} \cdots A_{\sigma_{k_r}} )
  \end{aligned}
$$

where all the $A_k$ happen to coincide: $A_k = A$.

If instead of [[unshuffles]] (i.e. [[partitions]] into non-empty subsequences preserving the original order) we took
partitions into arbitrarily ordered subsequences,
we would be overcounting by the [[factorial]] of the length of the subsequences, and hence the
above may be equivalently written as:

$$
  \cdots
  =
  \underset{k \in \mathbb{N}}{\sum}
    \tfrac{1}{k!}
    \underoverset{r = 1}{k}{\sum}
    (-1)^r
    \!\!\!
    \underset{ {\sigma \in \Sigma(k)} \atop { { k_1 + \cdots + k_r = k } \atop { \underset{i}{\forall} (k_i \geq 1) } } }{\sum}
    \!\!\!
    \tfrac{1}{k_1!} \cdots \tfrac{1}{k_r !}
    \,
    T( A_{\sigma(1)} \cdots A_{\sigma(k_1)} )
    \,
    T( A_{\sigma(k_1 + 1)} \cdots A_{\sigma(k_2)} )
    \cdots
    T( A_{\sigma(k_{r-1}+1)} \cdots A_{\sigma_{k_r}} )
    \,,
$$

where $\Sigma(k)$ denotes the [[symmetric group]] (the set of all [[permutations]] of $k$ elements).

Moreover, since all the $A_k$ are equal, the sum is in fact independent of $\sigma$, it only depends
on the length of the subsequences. Since there are $k!$ permutations of $k$ elements
the above reduces to

$$
  \begin{aligned}
    \cdots
    & =
    \underset{k \in \mathbb{N}}{\sum}
      \underoverset{r = 1}{k}{\sum}
      (-1)^r
      \!\!\!
      \underset{  k_1 + \cdots + k_r = k }{\sum}
      \tfrac{1}{k_1!} \cdots \tfrac{1}{k_r !}
      T( \underset{k_1 \, \text{factors}}{\underbrace{ A, \cdots , A  }}  )
      T( \underset{k_2 \, \text{factors}}{\underbrace{ A, \cdots , A }}  )
      \cdots
      T( \underset{k_r \, \text{factors}}{\underbrace{ A,  \cdots , A }}  )
    \\
    & =
      \underoverset{r = 0}{\infty}{\sum}
      \left(
        - \underoverset{k = 0}{\infty}{\sum} T ( \underset{k\,\text{factors}}{\underbrace{A,  \cdots , A}}   )
      \right)^r
    \\
    & =
      \mathcal{S}(A)^{-1}
      \,,
  \end{aligned}
$$

where in the last line we used (eq:InfverseOfPerturbativeSMatrix).

=--

In fact prop. \ref{ReverseTimOrderedProductsGiveReverseSMatrix} is a special case of the
following more general statement:

+-- {: .num_prop #InversionFormulaForTimeOrderedProducts}
###### Proposition
**(inversion relation for [[reverse-time ordered products]])

Let $\{T_k\}_{k \in \mathbb{N}}$ be [[time-ordered products]] according to def. \ref{TimeOrderedProduct}.
Then the [[reverse-time ordered products]] according to def. \ref{ReverseTimeOrderedProduct}
satisfies the following inversion relation for all $\mathbf{X} \neq \emptyset$
(in the condensed notation of remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}):

$$
  \underset{\mathbf{J} \subset \mathbf{X}}{\sum}
    T(\mathbf{J}) \overline{T}(\mathbf{X} \setminus \mathbf{J})
  \;=\;
  0
$$

and

$$
  \underset{\mathbf{J} \subset \mathbf{X}}{\sum}
    \overline{T}(\mathbf{X} \setminus \mathbf{J})  T(\mathbf{J})
  \;=\;
  0
$$

=--

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

This is immediate from unwinding the definitions.

=--

+-- {: .num_prop #ReverseCausalFactorizationOfReverseTimeOrderedProducts}
###### Proposition
**(reverse [[causal factorization]] of [[reverse-time ordered products]])**

Let $\{T_k\}_{k \in \mathbb{N}}$ be [[time-ordered products]] according to def. \ref{TimeOrderedProduct}.
Then the reverse-time ordered products according to def. \ref{ReverseTimeOrderedProduct} satisfies
reverse-[[causal factorization]].

=--

([Epstein-Glaser 73, around (15)](S-matrix#EpsteinGlaser73))

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

In the condensed notation of remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions},
we need to show that for $\mathbf{X} = \mathbf{P} \cup \mathbf{Q}$ with $\mathbf{P} \cap \mathbf{Q} = \emptyset$
then

$$
  \left(
    \mathbf{P} {\vee\!\!\!\wedge} \mathbf{Q}
  \right)
  \;\Rightarrow\;
  \left(
    \overline{T}(\mathbf{X})
     =
    \overline{T}(\mathbf{Q}) \overline{T}(\mathbf{P})
  \right)
  \,.
$$

We proceed by [[induction]]. If ${\vert \mathbf{X}\vert} = 1$ the statement is immediate.
So assume that the statement is true for sets of [[cardinality]] $n \geq 1$ and consider
$\mathbf{X}$ with ${\vert \mathbf{X}\vert} = n+1$.

We make free use of the condensed notation as in example \ref{ProductOfPerturbationSeriesInGenealizedFunctionNotation}.

From the formal inversion

$$
  \underset{\mathbf{J} \subset \mathbf{X}}{\sum} \overline{T}(\mathbf{J}) T(\mathbf{X}\setminus \mathbf{J})
  = 0
$$

(which uses the induction assumption that ${\vert \mathbf{X}\vert} \geq 1$) it follows that

$$
  \begin{aligned}
    \overline{T}(\mathbf{X})
      & =
    - \underset{ { \mathbf{J} \subset \mathbf{X} } \atop { \mathbf{J} \neq \mathbf{X} }  }{\sum}
     \overline{T}(\mathbf{J}) T( \mathbf{X} \setminus \mathbf{J} )
     \\
     & =
    - \underset{
        { \mathbf{J} \cup \mathbf{J}' = \mathbf{X} }
        \atop
        {
          { \mathbf{J} \cap \mathbf{J}' = \emptyset }
          \atop
          { \mathbf{J}' \neq \emptyset }
        }
     }{\sum}
     \overline{T}( \mathbf{Q} \cap \mathbf{J} )
     \overline{T}( \mathbf{P} \cap \mathbf{J} )
               T ( \mathbf{P} \cap ( \mathbf{J}' ) )
               T ( \mathbf{Q} \cap ( \mathbf{J}' ) )
     \\
     & =
    - \underset{
      { \mathbf{L} \cup \mathbf{L}' = \mathbf{Q} \,,\, \mathbf{L} \cap \mathbf{L}' = \emptyset }
      \atop
      { \mathbf{L}' \neq \emptyset }
    }{\sum}
     \!\!\!
     \overline{T}( \mathbf{L} )
     \underset{ = 0}{
       \underbrace{
         \left(
          \underset{
              \mathbf{K} \subset \mathbf{P}
           }{\sum}
           \overline{T}( \mathbf{K} ) T( \mathbf{P} \setminus \mathbf{K})
         \right)
       }
     }
     T(\mathbf{L'})
    -
    \overline{T}(\mathbf{Q})
    \underset{
      = - \overline{T}(\mathbf{P})
    }{
    \underbrace{
      \underset{
        {\mathbf{K} \subset \mathbf{P}}
        \atop
        { \mathbf{K} \neq \emptyset }
      }{\sum}
      \overline{T}(\mathbf{K})
                T (\mathbf{P} \setminus \mathbf{K} )
    }}
    \\
    & =
    \overline{T}(\mathbf{Q}) \overline{T}(\mathbf{P})
  \end{aligned}
   \,.
$$

Here

1. in the second line we used that $\mathbf{X} = \mathbf{Q} \sqcup \mathbf{P}$, together with the
[[causal factorization]] property of $T(-)$ (which holds by def. \ref{TimeOrderedProduct}) and that of $\overline{T}(-)$
(which holds by the induction assumption, using that $\mathbf{J} \neq \mathbf{X}$ hence that
${\vert \mathbf{J}\vert} \lt {\vert \mathbf{X}\vert}$).

1. in the third line we decomposed the sum over $\mathbf{J}, \mathbf{J}' \subset \mathbf{X}$
into two sums over subsets of $\mathbf{Q}$ and $\mathbf{P}$:

   1. The first summand in the third line is the contribution where $\mathbf{J}'$ has a non-empty intersection with $\mathbf{Q}$. This makes $\mathbf{K}$ range without constraint, and therefore the sum in the middle vanishes, as indicated, as it is the contribution at order ${\vert \mathbf{Q}\vert}$ of the inversion formula from prop. \ref{InversionFormulaForTimeOrderedProducts}.

   1. The second summand in the third line is the contribution where $\mathbf{J}'$ does not intersect $\mathbf{Q}$.
     Now the sum over $\mathbf{K}$ is the inversion formula from prop. \ref{InversionFormulaForTimeOrderedProducts} except for one term, and so it equals that term.

=--

Using these facts about the reverse-time ordered products,
we may finally prove that [[time-ordered products]] indeed do induced a perturbative S-matrix:

+-- {: .num_prop #TimeOrderedProductInducesPerturbativeSMatrix}
###### Proposition
**([[time-ordered products]] induce [[S-matrix]])**

Let $\{T_k\}_{k \in \mathbb{N}}$ be a system of [[time-ordered products]] according to def. \ref{TimeOrderedProduct}.
Then

$$
  \begin{aligned}
    \mathcal{S}(-)
    & \coloneqq
    T
    \left(
      \exp_\otimes
      \left(
        \tfrac{1}{i \hbar}(-)
      \right)
    \right)
    \\
    &
    \coloneqq
    \underset{k \in \mathbb{N}}{\sum}
    \tfrac{1}{k!}
    \tfrac{1}{(i \hbar)^k}  T( \underset{k \, \text{factors}}{\underbrace{-,  \cdots , -}} )
  \end{aligned}
$$

is indeed a perturbative S-matrix according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}.

=--

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

The axiom "perturbation" of the S-matrix is
immediate from the axioms "perturbation" and "normalization" of the time-ordered products.
What requires proof is that [[causal additivity]] of the S-matrix follows from
the [[causal factorization]] property of the time-ordered products.

Notice that also the weaker [[causal factorization]] property of the S-matrix (remark \ref{DysonCausalFactorization})
is immediate from the causal factorization condition on the time-ordered products.

But [[causal additivity]] is stronger. It is remarkable that this, too,
follows from just the time-ordering ([Epstein-Glaser 73, around (73)](S-matrix#EpsteinGlaser73)):

To see this, first expand the
generating function $\mathcal{Z}$ (eq:GeneratingFunctionInducedFromSMatrix)  into
powers of $g$ and $j$

$$
  \mathcal{Z}_{g S_{int}}(j A)
  \;=\;
  \underoverset{n,m = 0}{\infty}{\sum}
   \frac{1}{n! m!}
   R\left(
      {\, \atop \,}
      \underset{n\, \text{factors}}{\underbrace{g S_{int},  \cdots ,g S_{int}}},
      ( \underset{m \, \text{factors}}{ \underbrace{ j A ,  \cdots , j A } }  )
      {\, \atop \,}
   \right)
$$

and then compare order-by-order with the given time-ordered product $T$ and
its induced reverse-time ordered product (def. \ref{ReverseTimeOrderedProduct}) via prop. \ref{ReverseTimOrderedProductsGiveReverseSMatrix}.
(These $R(-,-)$ are also called the "generating [[retarded products]], discussed in their own right around def. \ref{RetardedProductFromPerturbativeSMatrix} below.)

In the condensed notation of
remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}
and its way of absorbing combinatorial prefactors as in example \ref{ProductOfPerturbationSeriesInGenealizedFunctionNotation}
this
yields at order $(g/\hbar)^{\vert \mathbf{Y}\vert} (j/\hbar)^{\vert \mathbf{X}\vert}$ the coefficient

$$
  \label{CoefficientOfgeneratingRetardedProduct}
  R(\mathbf{Y}, \mathbf{X})
   \;=\;
  \underset{\mathbf{I} \subset \mathbf{Y}}{\sum}
    \overline{T}(\mathbf{I})
    T( (\mathbf{Y} \setminus \mathbf{I}) , \mathbf{X} )
  \,.
$$

We claim now that the [[support of a distribution|support]] of $R$ is inside the subset for which
$\mathbf{Y}$ is in the [[causal past]] of $\mathbf{X}$. This will imply the claim, because by
multi-linearity of $R(-,-)$ it then follows that

$$
  \left(supp(A_1) {\vee\!\!\!\wedge} supp(A_2)\right)
  \Rightarrow
  \left( Z_{(g S_{int}  + j A_1)}(j A_2) = Z_{S_{int}}(A_2) \right)
$$

and by prop. \ref{ZCausalAdditivity} this is equivalent to [[causal additivity]] of the S-matrix.

It remains to prove the claim:

Consider $\mathbf{X}, \mathbf{Y} \subset \Sigma$ such that the subset $\mathbf{P} \subset \mathbf{Y}$
of points not in the past of $\mathbf{X}$, hence the maximal subset with [[causal ordering]]

$$
  \mathbf{P} {\vee\!\!\!\wedge} \mathbf{X}
  \,,
$$

is non-empty. We need to show that in this case $R(\mathbf{Y}, \mathbf{X}) = 0$ (in the sense of generalized functions).

Write $\mathbf{Q} \coloneqq \mathbf{Y} \setminus \mathbf{P}$
for the complementary set of points, so that all points of $\mathbf{Q}$ are in the past of $\mathbf{X}$.
Notice that this implies that $\mathbf{P}$ is also not in the past of $\mathbf{Q}$:

$$
  \mathbf{P} {\vee\!\!\!\wedge} \mathbf{Q}
  \,.
$$

With this decomposition of $\mathbf{Y}$, the sum in (eq:CoefficientOfgeneratingRetardedProduct)
over subsets $\mathbf{I}$ of $\mathbf{Y}$ may be decomposed into a sum over
subsets $\mathbf{J}$ of $\mathbf{P}$ and $\mathbf{K}$ of $\mathbf{Q}$, respectively.
These subsets inherit the above causal ordering, so that by the causal factorization property of
$T(-)$ (def. \ref{TimeOrderedProduct}) and $\overline{T}(-)$ (prop. \ref{ReverseCausalFactorizationOfReverseTimeOrderedProducts})
the time-ordered and reverse time-ordered products
factor on these arguments:

$$
  \begin{aligned}
    R(\mathbf{Y}, \mathbf{X})
    & =
    \underset{ {\mathbf{J} \subset \mathbf{P}} \atop { \mathbf{K} \subset \mathbf{Q} } }{\sum}
     \,
     \overline{T}( \mathbf{J} \cup \mathbf{K} )
     T( (\mathbf{P} \setminus \mathbf{J}) \cup (\mathbf{Q} \setminus \mathbf{K}), \mathbf{X} )
     \\
     & =
     \underset{ {\mathbf{J} \subset \mathbf{P}} \atop { \mathbf{K} \subset \mathbf{Q} } }{\sum}
     \,
     \overline{T}( \mathbf{K} )
     \overline{T}( \mathbf{J} )
     T( \mathbf{P} \setminus \mathbf{J} )
     T( \mathbf{Q} \setminus \mathbf{K}, \mathbf{X} )
     \\
     & =
     \underset{ \mathbf{K} \subset \mathbf{Q} }{\sum}
     \overline{T}(\mathbf{K})
     \underset{= 0}{
     \underbrace{
     \left(
       \underset{\mathbf{J} \subset \mathbf{P}}{\sum}
       \overline{T}(\mathbf{J})
       T( \mathbf{P} \setminus \mathbf{J} )
     \right)
     }}
     T(\mathbf{Q} \setminus \mathbf{K}, \mathbf{X})
  \end{aligned}
  \,.
$$

Here the sub-sum in brackets vanishes by the inversion formula, prop. \ref{InversionFormulaForTimeOrderedProducts}.

=--

In conclusion:

+-- {: .num_prop #CausalFactorizationAlreadyImpliesSMatrix}
###### Proposition
**([[S-matrix]] scheme via [[causal factorization]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree} and consider a function

$$
  \mathcal{S}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j \rangle
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j] ]
$$

from [[local observables]] to [[microcausal polynomial observables]] which satisfies the condition "perturbation"
from def. \ref{LagrangianFieldTheoryPerturbativeScattering}. Then the following two conditions
on $\mathcal{S}$ are equivalent

1. [[causal additivity]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering})

1. [[causal factorization]] (remark \ref{DysonCausalFactorization})

and hence either of them is necessary and sufficient for $\mathcal{S}$ to be a perturbative [[S-matrix]] scheme
according to def. \ref{LagrangianFieldTheoryPerturbativeScattering}.

=--

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

That causal factorization follows from causal additivity is immediate (remark \ref{DysonCausalFactorization}).

Conversely, causal factorization of $\mathcal{S}$ implies that its expansion coefficients $\{T_k\}_{k \in \mathbb{N}}$
are [[time-ordered products]] (def. \ref{TimeOrderedProduct}), via the proof of example \ref{TimeOrderedProductsFromSMatrixScheme},
and this implies causal additivity by prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix}.

=--

$\,$

**("Re"-)Normalization**
 {#ExistenceAndRenormalization}

We discuss now that [[time-ordered products]] as in def. \ref{TimeOrderedProduct}, hence, by prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix}, perturbative [[S-matrix]] schemes (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) exist in fact uniquely away from coinciding interaction points (prop. \ref{TimeOrderedProductAwayFromDiagonal} below).

This means that the construction of full [[time-ordered products]]/[[S-matrix]] schemes may be phrased as an [[extension of distributions]] of time-ordered products to the [[diagonal]] locus of coinciding spacetime arguments (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} below).
This choice in their definition is called the choice of _[[renormalization|("re"-)normalization]]_ of the [[time-ordered products]] (remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences}), and hence of the [[interacting field theory|interacting]] [[pQFT]] that these define (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} below).

The space of these choices may be accurately characterized, it is a [[torsor]] over a [[group]] of re-definitions of the
[[interaction]]-terms, called the "[[Stückelberg-Petermann renormalization group]]". This is called the
_[[main theorem of perturbative renormalization]]_, theorem \ref{PerturbativeRenormalizationMainTheorem} below.

Here we discuss just enough of the ingredients needed to _state_ this theorem. We give the proof in the [next chapter](#Renormalization).

$\,$

+-- {: .num_defn #TuplesOfCompactlySupportedPolynomialLocalFunctionalsWithPairwiseDisjointSupport}
###### Definition
**([[tuples]] of [[local observables]] with pairwise disjoint spacetime support)**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

For $k \in \mathbb{N}$, write

$$
  \left(
    {\, \atop \,}
    LocPoly(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
    {\, \atop \,}
  \right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j ] ]}}_{pds}
  \hookrightarrow
  \left(
    {\, \atop \,}
    LocPoly(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
    {\, \atop \,}
  \right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j ] ]}}
$$

for the linear subspace of the $k$-fold [[tensor product]] of [[local observables]] (as in def. \ref{LagrangianFieldTheoryPerturbativeScattering}, def. \ref{TimeOrderedProduct}) on those tensor products $A_1 \otimes \cdots A_k$ of [[tuples]] with disjoint spacetime [[support]]:

$$
  supp(A_j) \cap supp(A_k) = \emptyset
  \phantom{AAA}
  \text{for} \, i \neq j \in \{1, \cdots, k\}
  \,.
$$

=--

+-- {: .num_prop #TimeOrderedProductAwayFromDiagonal}
###### Proposition
**([[time-ordered product]] unique away from coinciding spacetime arguments)**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $T = \{T_k\}_{k \in \mathbb{N}}$ be a sequence of [[time-ordered products]] (def. \ref{TimeOrderedProduct})

$$
  \array{
    \left(
      {\, \atop \,}
      LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
      {\, \atop \,}
    \right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}
      & \longrightarrow &
    PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))[ [g , j] ]
    \\
    \uparrow & \nearrow_{(-) \star_F \cdots \star_F (-)}
    \\
    \left(
      {\, \atop \,}
      LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
      {\, \atop \,}
    \right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}_{pds}
  }
$$

Then their [[restriction]] to the subspace of [[tuples]] of [[local observables]] of pairwise disjoint spacetime support (def. \ref{TuplesOfCompactlySupportedPolynomialLocalFunctionalsWithPairwiseDisjointSupport}) is unique (independent of the [[renormalization|"re-"normalization]] freedom in choosing $T$) and is given by the [[star product]]

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

that is induced (def. \ref{PropagatorStarProduct}) by the [[Feynman propagator]] $\Delta_F \coloneqq \tfrac{i}{2}(\Delta_+ + \Delta_- + H)$ (corresponding to the [[Wightman propagator]] $\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H$ which is given by the choice of [[free field|free]] [[vacuum]]), in that

$$
  T
  \left(
    {\, \atop \,}
    A_1 , \cdots, A_k
    {\, \atop \,}
  \right)
  \;=\;
  A_1
    \star_F
    \cdots
    \star_F
  A_k
  \,.
$$

In particular the time-ordered product extends from the restricted domain of tensor products of local observables to a restricted domain of [[microcausal polynomial observables]], where it becomes an [[associativity|associative]] product:

$$
  \label{RestrictedTimeOrderedProductAssociative}
  \begin{aligned}
    T(A_1, \cdots, A_{k_n})
    & =
    T(A_1, \cdots, A_{k_1})
      \star_F
    T(A_{k_1 + 1}, \cdots, A_{k_2})
      \star_F
      \cdots
      \star_F
    T(A_{k_{n-1} + 1}, \cdots, A_{k_n})
    \\
    & =
    A_1 \star_F \cdots \star_F A_{k_n}
  \end{aligned}
$$

for all tuples of local observables $A_1, \cdots, A_{k_1}, A_{k_1+1}, \cdots, A_{k_2}, \cdots, \cdots A_{k_n}$ with pairwise disjoint spacetime support.

=--

The idea of this statement goes back at least to [Epstein-Glaser 73](S-matrix#EpsteinGlaser73), as in remark \ref{TheTraditionalErrorThatLeadsToTheNotoriouDivergencies}. One formulation appears as ([Brunetti-Fredenhagen 00, theorem 4.3](perturbative+algebraic+quantum+field+theory#BrunettiFredenhagen00)). The above formulation in terms of the [[star product]] is stated in ([Fredenhagen-Rejzner 12, p. 27](perturbative+algebraic+quantum+field+theory#FredenhagenRejzner12), [Dütsch 18, lemma 3.63 (b)](perturbative+algebraic+quantum+field+theory#Duetsch18)).

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

By [[induction]] over the number of arguments, it is sufficient to see that, more generally, for $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ two [[microcausal polynomial observables]] with disjoint spacetime support the star product $A_1 \star_F A_2$ is well-defined and satisfies causal factorization.

Consider two [[partitions of unity]]

$$
  (\chi_{1,i} \in C^\infty_{cp}(\Sigma))_{i}
  \phantom{AAA}
  (\chi_{1,j} \in C^\infty_{cp}(\Sigma))_{j}
$$

and write $(A_{1,i})_i$ and $(A_{2,j})_{j}$ for the collection of [[microcausal polynomial observables]] obtained by multiplying all the [[distribution|distributional]] [[coefficients]] of $A_1$ and of $A_2$ with $\chi_{1,i}$ and with $\chi_{2,j}$, respectively, for all $i$ and $j$, hence such that

$$
  A_1 \;=\; \underset{i}{\sum} A_{1,i}
  \phantom{AAA}
  A_2 \;=\; \underset{j}{\sum} A_{1,j}
  \,.
$$

By linearity, it is sufficient to prove that $A_{1,i} \star_F A_{2,j}$ is well defined for all $i,j$ and satisfies causal factorization.

Since the spacetime supports of $A_1$ and $A_2$ are assumed to be disjoint

$$
  supp(A_1) \cap supp(A_2) \;=\; \emptyset
$$

we may find partitions such that each resulting pair of smaller supports is in fact in [[causal order]]-relation:

$$
  \array{
    \left(
      supp(A_1) \cap supp(\chi_{1,i})
    \right)
       {\vee\!\!\!\wedge}
    \left(
      supp(A_2) \cap supp(\chi_{2,j})
    \right)
    \\
    \text{or}
    \\
    \left(
      supp(A_2) \cap supp(\chi_{2,j})
    \right)
       {\vee\!\!\!\wedge}
    \left(
      supp(A_1) \cap supp(\chi_{1,u})
    \right)
  }
  \phantom{AAAAA}
  \text{for all}\,\, i,j
  \,.
$$

But now it follows as in the proof of prop. \ref{CausalOrderingTimeOrderedProductOnRegular}) via (eq:CausallyOrderedWickProductViaFeynmanPropagator) that

$$
  A_{1,i}
   \star_F
  A_{2,j}
  \;=\;
  \left\{
    \array{
      A_{1,i} \star_H A_{2,j} &\vert& supp(A_{1,i}) {\vee\!\!\!\wedge} supp(A_{2,j})
      \\
      A_{2,j} \star_H A_{1,i} &\vert& supp(A_{2,j}) {\vee\!\!\!\wedge} supp(A_{1,i})
  }
  \right.
$$

Finally the [[associativity]]-statement follows as in prop. \ref{AssociativeAndUnitalStarProduct}.

=--

Before using the unqueness of the [[time-ordered products]] away from coinciding spacetime arguments (prop. \ref{TimeOrderedProductAwayFromDiagonal}) to characterize the freedom in [[renormalization|("re"-)normalizing]] [[time-ordered products]], we pause to observe that in the same vein the [[time-ordered products]] have a unique extension of their domain also to [[regular polynomial observables]]. This is in itself a trivial statement (since all [[star products]] are defined on [[regular polynomial observables]], def. \ref{PropagatorStarProduct}) but for understanding the behaviour under [[renormalization|("re"-)normalization]] of other structures, such as the interacting [[BV-differential]] (def. \ref{BVDifferentialInteractingQuantum} below) it is useful to understand renormalization as a process that starts extending awa from [[regular polynomial observables]].

By prop. \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}, on [[regular polynomial observables]] the
[[S-matrix]] is given as follows:

+-- {: .num_defn #OnRegularObservablesPerturbativeSMatrix}
###### Definition
**([[perturbative S-matrix]] on [[regular polynomial observables]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

Recall that the _[[time-ordered product]] on [[regular polynomial observables]]_
is the [[star product]] $\star_F$ induced by the [[Feynman propagator]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) and that, due to the [[non-singular distribution|non-singular]]
nature of [[regular polynomial observables]], this is given by [[conjugation]] of the pointwise product (eq:ObservablesPointwiseProduct) with $\mathcal{T}$ (eq:OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism) as

$$
  T(A_1, A_2)
  \;=\;
  A_1 \star_F A_2
  \;=\;
  \mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2))
$$

(prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}).

We say that the _[[perturbative S-matrix]] scheme_ on [[regular polynomial observables]]
is the [[exponential]] with respect to $\star_F$:

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

given by

$$
  \mathcal{S}(S_{int})
  =
  \exp_{\star_F}
  \left(
    \tfrac{1}{i \hbar} S_{int})
  \right)
  \coloneqq
  1
    +
  \tfrac{1}{\i \hbar} S_{int}
    +
  \tfrac{1}{2}
  \tfrac{1}{(i \hbar)^2}
  S_{int} \star_F S_{int}
  +
  \cdots
  \,.
$$

We think of $S_{int}$ here as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]].

We write $\mathcal{S}(S_{int})^{-1}$ for the [[inverse]] with respect to the [[Wick algebra|Wick product]] (which exists by remark \ref{PerturbativeSMatrixInverse})

$$
  \mathcal{S}(S_{int})^{-1} \star_H \mathcal{S}(S_{int})
  =
  1
  \,.
$$

Notice that this is in general different form the inverse with respect to the [[time-ordered product]] $\star_F$, which is $\mathcal{S}(-S_{int})$:

$$
  \mathcal{S}(-S_{int})
    \star_F
  \mathcal{S}(S_{int})
  =
  1
  \,.
$$

=--

Similarly, by def. \ref{QuntumMollerOperator}, on [[regular polynomial observables]] the [[quantum Møller operator]] is given as follows:

+-- {: .num_defn #MollerOperatorOnRegularPolynomialObservables}
###### Definition
**([[quantum Møller operator]] on [[regular polynomial observables]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.
Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] of degree 0

$$
  S_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar , g, j] ]
$$

then the corresponding _[[quantum Møller operator]]_ on [[regular polynomial observables]]

$$
  \mathcal{R}^{-1}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar , g, j] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, j] ]
$$

is given by the [[derivative]] of [[Bogoliubov's formula]]

$$
  \mathcal{R}^{-1}
    \;\coloneqq\;
  \mathcal{S}(S_{int})^{-1}
    \star_H
  (\mathcal{S}(S_{int}) \star_F (-))
  \,,
$$

where $\mathcal{S}(S_{int}) = \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)$ is the [[perturbative S-matrix]] from  def. \ref{OnRegularObservablesPerturbativeSMatrix}.

This indeed lands in [[formal power series]] in [[Planck's constant]] $\hbar$ (by remark \ref{PowersInPlancksConstant}),  instead of in more general [[Laurent series]] as the [[perturbative S-matrix]] does (def. \ref{OnRegularObservablesPerturbativeSMatrix}).

Hence the inverse map is

$$
  \mathcal{R}
   \;=\;
  \mathcal{S}(-S_{int}) \star_F ( \mathcal{S}(S_{int}) \star(-) )
  \,.
$$

=--

([Bogoliubov-Shirkov 59](Bogoliubov's+formula#BogoliubovShirkov59); the above terminology follows [Hawkins-Rejzner 16, below def. 5.1](Møller+operator#HawkinsRejzner16))

(Beware that compared to Fredenhagen, Rejzner et. al. we change notation conventions $\mathcal{R} \leftrightarrow \mathcal{R}^{-1}$ in order to bring out the analogy to (the conventions for the) [[time-ordered product]]  $A_1 \star_F A_2 = \mathcal{T}(\mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2))$ on regular polynomial observables.)

Still by def. \ref{QuntumMollerOperator}, on [[regular polynomial observables]] the [[interacting field algebra of observables]] is given as follows:

+-- {: .num_defn #FieldAlgebraObservablesInteracting}
###### Definition
**([[interacting field algebra]] [[structure]] on [[regular polynomial observables]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}. Given an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] in degree 0

$$
  S_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, j] ]
  \,,
$$

then the _[[interacting field algebra]]_ [[structure]] on [[regular polynomial observables]]

$$
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, j] ]
    \otimes
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, h] ]
    \overset{ \star_{int} }{\longrightarrow}
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar, g, j] ]
$$

is the [[conjugation]] of the [[Wick algebra]]-[[structure]] by the [[quantum Møller operator]] (def. \ref{MollerOperatorOnRegularPolynomialObservables}):

$$
  A_1 \star_{int} A_2
  \;\coloneqq\;
  \mathcal{R}
  \left(
    \mathcal{R}^{-1}(A_1)
     \star_H
    \mathcal{R}^{-1}(A_2)
  \right)
$$

=--

(e.g. [Fredenhagen-Rejzner 11b, (19)](quantum+master+equation#FredenhagenRejzner11b))

Notice the following dependencies of these defnitions, which we leave notationally implicit:

| [[endomorphism]] of <br/> [[regular polynomial observables]] | meaning | depends on choice of |
|--------|---------|----------------------|
| $\phantom{AA}\mathcal{T}$ | [[time-ordered product|time-ordering]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] |
| $\phantom{AA}\mathcal{S}$ | [[S-matrix]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] |
| $\phantom{AA}\mathcal{R}$ | [[quantum Møller operator]] | [[free field theory|free]] [[Lagrangian density]] and [[Wightman propagator]] and [[interaction]] |

$\,$

After having discussed the uniqueness of the [[time-ordered products]] away from coinciding spacetime arguments (prop. \ref{TimeOrderedProductAwayFromDiagonal}) we now phrase and then discuss the freedom in defining these products at coinciding arguments, thus [[renormalization|("re"-)normalizing]] them.

+-- {: .num_defn #ExtensionOfTimeOrderedProoductsRenormalization}
###### Definition
**([[Epstein-Glaser renormalization|Epstein-Glaser ("re"-)normalization]] of [[perturbative QFT]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

Prop. \ref{TimeOrderedProductAwayFromDiagonal} implies that the
problem of constructing a sequence of [[time-ordered products]] (def. \ref{TimeOrderedProduct}), hence, by prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix}, an [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) for [[perturbative quantum field theory]] around the given [[free field]] [[vacuum]], is equivalently a problem of a sequence of compatible _[[extensions of distributions]]_ of the [[star products]] $\underset{k \; \text{arguments}}{\underbrace{(-)\star_F \cdots \star_F (-)}}$ of the [[Feynman propagator]] on $k$ arguments from the [[complement]] of coinciding [[events]] inside the [[Cartesian products]] $\Sigma^k$ of [[spacetime]] $\Sigma$, along the canonical inclusion

$$
  \Sigma^k \setminus \left\{ (x_i) \,\vert\, \underset{i \neq j}{\exists} (x_i = x_j) \right\}
    \overset{\phantom{AAA}}{\hookrightarrow}
  \Sigma^k
  \,.
$$

Via the [[associativity]] (eq:RestrictedTimeOrderedProductAssociative) of the restricted [[time-ordered product]] thesese choices are naturally made by [[induction]] over $k$, choosing the $(k+1)$-ary [[time-ordered product]] $T_{k+1}$ as an [[extension of distributions]] of $T_k(\underset{k \, \text{args}}{\underbrace{-, \cdots, -}}) \star_F (-)$.

This [[induction|inductive]] choice of [[extension of distributions]] of the [[time-ordered product]] to coinciding interaction points deserves to be called a choice of _normalization_ of the [[time-ordered product]] (e.g. [Scharf 94, section 4.3](#Scharf95)), but for historical reasons (see remark \ref{TheTraditionalErrorThatLeadsToTheNotoriouDivergencies} and remark \ref{CausalPerturbationTheoryAbsenceOfUVDivergences}) it is known as _[[renormalization|re-normalization]]_. Specifically the inductive construction by extension to coinciding interaction points is known as _[[Epstein-Glaser renormalization]]_.

=--

In ([Epstein-Glaser 73](S-matrix#EpsteinGlaser73)) this is phrased in terms of splitting of distributions. In ([Brunetti-Fredenhagen 00, sections 4 and 7](perturbative+algebraic+quantum+field+theory#BrunettiFredenhagen00)) the perspective via [[extension of distributions]] is introduced, following ([Stora 93](perturbative+algebraic+quantum+field+theory#Stora93)). Review is in ([Dütsch 18, section 3.3.2](perturbative+algebraic+quantum+field+theory#Duetsch18)).

Proposition \ref{TimeOrderedProductAwayFromDiagonal} already shows that the freedom in choosing the [[renormalization|("re"-)normalization]] of [[time-ordered products]] is at most that of [[extensions of distributions|extending]] them to the "[[fat diagonal]]", where at least one pair of interaction points coincides. The following proposition \ref{RenormalizationIsInductivelyExtensionToDiagonal} says that when making these choices [[induction|inductively]] in the arity of the [[time-ordered products]] as in def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} then the available choice of [[renormalization|("re"-)normalization)]] at each stage is in fact only that of extension to the actual [[diagonal]], where _all_ interaction points coincide:

+-- {: .num_prop #RenormalizationIsInductivelyExtensionToDiagonal}
###### Proposition
**([[renormalization|("re"-)normalization]] is [[induction|inductive]] [[extension of distributions|extension]] of [[time-ordered products]] to [[diagonal]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

Assume that for $n \in \mathbb{N}$, [[time-ordered products]]
$\{T_{k}\}_{k \leq n}$ of arity $k \leq n$ have been constructed
in the sense of def. \ref{TimeOrderedProduct}.
Then the time-ordered product $T_{n+1}$ of arity $n+1$ is uniquely fixed on the [[complement]]

$$
  \Sigma^{n+1} \setminus diag(n)
  \;=\;
  \left\{
     (x_i \in \Sigma)_{i = 1}^n
     \;\vert\;
     \underset{i,j}{\exists} (x_i \neq x_j)
  \right\}
$$

of the [[image]] of the [[diagonal]] inclusion $\Sigma \overset{diag}{\longrightarrow} \Sigma^{n}$
(where we regarded $T_{n+1}$ as a [[generalized function]] on $\Sigma^{n+1}$ according to remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}).

=--

This statement appears in ([Popineau-Stora 82](renormalization#PopineauStora82)), with (unpublished) details in ([Stora 93](renormalization#Stora93)), following personal communication by [[Henri Epstein]] (according to [Dütsch 18, footnote 57](S-matrix#Duetsch18)). Following this, statement and detailed proof appeared in ([Brunetti-Fredenhagen 99](S-matrix#BrunettiFredenhagen99)).

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

We will construct an [[open cover]] of $\Sigma^{n+1} \setminus \Sigma$ by subsets $\mathcal{C}_I \subset \Sigma^{n+1}$ which are [[disjoint unions]]
of [[inhabited set|non-empty]] sets that are in [[causal order]], so that by [[causal factorization]] the
time-ordered products $T_{n+1}$ on these subsets are uniquely given by $T_{k}(-) \star_H T_{n-k}(-)$.
Then we show that these unique products on these special subsets do coincide on [[intersections]].
This yields the claim by a [[partition of unity]].

We now say this in detail:

For $I \subset \{1, \cdots, n+1\}$  write $\overline{I} \coloneqq \{1, \cdots, n+1\} \setminus I$.
For $I, \overline{I} \neq \emptyset$, define the subset

$$
  \mathcal{C}_I
  \;\coloneqq\;
  \left\{
    (x_i)_{i \in \{1, \cdots, n+1\}} \in \Sigma^{n+1}
    \;\vert\;
    \{x_i\}_{i \in I} {\vee\!\!\!\wedge} \{x_j\}_{j \in \{1, \cdots, n+1\} \setminus I}
  \right\}
  \;\subset\;
  \Sigma^{n+1}
  \,.
$$

Since the [[causal order]]-relation involves the [[closed future cones]]/[[closed past cones]], respectively, it is clear that these are [[open subsets]]. Moreover it is immediate that they form an [[open cover]] of the [[complement]] of the [[diagonal]]:

$$
  \underset{ { I \subset \{1, \cdots, n+1\} \atop { I, \overline{I} \neq \emptyset } }   }{\cup}
  \mathcal{C}_I
  \;=\;
  \Sigma^{n+1} \setminus diag(\Sigma)
  \,.
$$

(Because any two distinct points in the [[globally hyperbolic spacetime]] $\Sigma$ may be causally separated by a [[Cauchy surface]],
and any such may be deformed a little such as not to intersect any of a given finite set of points. )

Hence the condition of [[causal factorization]] on $T_{n+1}$ implies that
[[restriction of distributions|restricted]] to any $\mathcal{C}_{I}$ these have to be given (in the condensed [[generalized function]]-notation from remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions} on
any unordered tuple $\mathbf{X} = \{x_1, \cdots, x_{n+1}\} \in \mathcal{C}_I$ with corresponding induced tuples
$\mathbf{I} \coloneqq \{x_i\}_{i \in I}$ and $\overline{\mathbf{I}} \coloneqq \{x_i\}_{i \in \overline{I}}$ by

$$
  \label{InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal}
  T_{n+1}( \mathbf{X} )
  \;=\;
  T(\mathbf{I}) T(\overline{\mathbf{I}})
  \phantom{AA}
  \text{for}
  \phantom{A}
  \mathcal{X} \in \mathcal{C}_I
  \,.
$$

This shows that $T_{n+1}$ is unique on $\Sigma^{n+1} \setminus diag(\Sigma)$ if it exists at all,
hence if these local identifications glue to a global definition of $T_{n+1}$. To see that this is the case,
we have to consider any two such subsets

$$
  I_1, I_2
  \subset
  \{1, \cdots, n+1\}
  \,,
  \phantom{AA}
  I_1, I_2, \overline{I_1}, \overline{I_2} \neq \emptyset
  \,.
$$

By definition this implies that for

$$
  \mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}
$$

a tuple of spacetime points which decomposes into causal order with respect to both these subsets,
the corresponding mixed intersections of tuples  are spacelike separated:

$$
  \mathbf{I}_1 \cap \overline{\mathbf{I}_2}
    \;
    {\gt\!\!\!\!\lt}
    \;
  \overline{\mathbf{I}_1} \cap \mathbf{I}_2
  \,.
$$

By the assumption that the $\{T_k\}_{k \neq n}$ satisfy causal factorization, this implies that the corresponding
time-ordered products commute:

$$
  \label{TimeOrderedProductsOfMixedIntersectionsCommute}
  T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2})
  \,
  T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2)
  \;=\;
  T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2)
  \,
  T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2})
  \,.
$$

Using this we find that the identifications of $T_{n+1}$
on $\mathcal{C}_{I_1}$ and on $\mathcal{C}_{I_2}$, accrding to (eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal),
agree on the intersection: in that for $  \mathbf{X}  \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}$ we have

$$
  \begin{aligned}
    T( \mathbf{I}_1 ) T( \overline{\mathbf{I}_1} )
    & =
    T( \mathbf{I}_1 \cap \mathbf{I}_2 )
    T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} )
    \,
    T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 )
    T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} )
    \\
    & =
    T( \mathbf{I}_1 \cap \mathbf{I}_2 )
    \underbrace{
    T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 )
    T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} )
    }
    T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} )
    \\
    & =
    T( \mathbf{I}_2 )
    T( \overline{\mathbf{I}_2} )
  \end{aligned}
$$

Here in the first step we expanded out the two factors using (eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal) for $I_2$,
then under the brace we used (eq:TimeOrderedProductsOfMixedIntersectionsCommute) and in the last step we used again
(eq:InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal), but now for $I_1$.

To conclude, let

$$
  \left(
    \chi_I
    \in
    C^\infty_{cp}(\Sigma^{n+1})
  \right)_{ { I \subset \{1, \cdots, n+1\} } \atop { I, \overline{I} \neq \emptyset } }
$$

be a [[partition of unity]] subordinate to the [[open cover]] formed by the $\mathcal{C}_I$. Then the above implies that
setting for any $\mathbf{X} \in \Sigma^{n+1} \setminus diag(\Sigma)$

$$
  T_{n+1}(\mathbf{X})
  \;\coloneqq\;
  \underset{
    { I \in \{1, \cdots, n+1\} }
    \atop
    { I, \overline{I} \neq \emptyset }
  }{\sum}
  \chi_i(\mathbf{X}) T( \mathbf{I} ) T( \overline{\mathbf{I}} )
$$

is well defined and satisfies causal factorization.

=--

Since [[renormalization|("re"-)normalization]] involves making choices, there is the freedom to impose
further conditions that one may want to have satisfied. These are called _[[renormalization conditions]]_.

+-- {: .num_defn #RenormalizationConditions}
###### Definition
**([[renormalization conditions]], [[protection from quantum corrections]] and [[quantum anomalies]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

Then a condition $P$ on $k$-ary functions of the form

$$
  T_k
    \;\colon\;
  \left(
    {\, \atop \,}
    LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
    {\, \atop \,}
  \right)^{\otimes^k_{\mathbb{C}[ [\hbar, g, j] ]}}
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))[ [g , j] ]
$$

is called a _renormalization condition_ if

1. it holds for the unique [[time-ordered products]] away from coinciding spacetime arguments (according to prop. \ref{TimeOrderedProductAwayFromDiagonal});

1. whenever it holds for all unrestricted $T_{k \leq n}$ for some $n \in \mathbb{N}$, then it also holds
   for $T_{n+1}$ restricted away from the diagonal:

   $$
     P(T_k)_{k \leq n}
     \;\Rightarrow\;
     P\left(
       T_{n+1}\vert_{\Sigma^{n+1} \setminus diag(\Sigma)}
     \right)
     \,.
   $$

This means that a renormalization condition is a condition that may consistently be imposed degreewise
in an [[induction|inductive]] construction of [[time-ordered products]] by degreewise [[extension of distributions|extension]]
to the [[diagonal]], according to prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}.

If specified renormalization conditions $\{P_i\}$ completely remove any freedom in the choice of time-ordered products for a given [[quantum observable]], one says that the renormalization conditions _[[protection from quantum corrections|protects the observable against quantum corrections]]_.

If for specified renormalization conditions $\{P_i\}$ there is _no_ choice of [[time-ordered products]] $\{T_k\}_{k \in \mathbb{N}}$
(def. \ref{TimeOrderedProduct}) that satisfies all these conditions, then one says that an
[[interacting field theory|interacting]] [[perturbative QFT]] satisfying $\{P_i\}$ fails to exist due to a _[[quantum anomaly]]_.

=--

+-- {: .num_prop #BasicConditionsRenormalization}
###### Proposition
**(basic [[renormalization conditions]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

Then the following conditions are [[renormalization conditions]] (def. \ref{RenormalizationConditions}):

1. (field independence) The [[functional derivative]] of a [[polynomial observable]] arising as a [[time-ordered product]] takes contributions only from the arguments, not from the product operation itself; in [[generalized function]]-notation:

   $$
     \label{FieldIndependenceRenormalizationCondition}
     \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
     T(A_1, \cdots, A_n)
     \;=\;
     \underset{1 \leq k \leq n}{\sum}
     T\left(
       A_1, \cdots, A_{k-1}, \frac{\delta}{\delta \mathbf{\Phi}^a(x)}A_k, A_{k+1}, \cdots, A_n
     \right)
   $$

1. ([[translation group|translation]] equivariance) If the underlying [[spacetime]] is [[Minkowski spacetime]], $\Sigma = \mathbb{R}^{p,1}$,
   with the induced [[action]] of the [[translation group]] on [[polynomial observables]]

   $$
     \rho
       \;\colon\;
     \mathbb{R}^{p,1} \times PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
       \longrightarrow
     PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
   $$

   then

   $$
     \rho_v \left( {\, \atop \,} T(A_1, \cdots, A_n) {\, \atop \,}\right)
     \;=\;
     T(\rho_{v}(A_1), \cdots, \rho_v(A_n))
   $$

1. ([[quantum master equation]], [[master Ward identity]]) see prop. \ref{QuantumMasterEquation}

   (if this condition fails, the corresponding [[quantum anomaly]] (def. \ref{RenormalizationConditions}) is called a _[[gauge anomaly]]_)

=--

([Dütsch 18, p. 150 and section 4.2](S-matrix#Duetsch18))

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

For the first two statements this is obvious from prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} and prop. \ref{TimeOrderedProductAwayFromDiagonal}, which imply that $T_{n+1}\vert_{\Sigma^{n+1} \setminus diag(\Sigma)}$ is
uniquely specified from $\{T_k\}_{k \leq n}$ via the [[star product]] induced by the [[Feynman propagator]],
and the fact that, on [[Minkowski spacetime]], this is manifestly translation invariant and independent of the fields
(e.q. prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue}).

The third statement requires work. That the [[quantum master equation]]/([[master Ward identity]]
always holds on [[regular polynomial observables]] is prop. \ref{QuantumMasterEquation} below.
That it holds for $T_{n+1}\vert_{\Sigma^{n+1} \setminus diag(\Sigma)}$ if it holds for $\{T_k\}_{k \leq n}$
is shown in ([Duetsch 18, section 4.2.2](S-matrix#Duetsch18)).

=--

$\,$

We discuss methods for [[renormalization|normalization]] (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal})
and [[renormalization|re-normalization]] in detail in the [next chapter](#Renormalization).

$\,$

**Feynman perturbation series**
 {#FeynmanDiagrams}

By def \ref{ExtensionOfTimeOrderedProoductsRenormalization} and the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}), the construction of perturbative [[S-matrix]] schemes/[[time-ordered products]] may be phrased as [[renormalization|("re-")normalization]] of the [[star product]] induced by the [[Feynman propagator]], namely as a choice of [[extension of distributions]] of the this star-product to the locus of coinciding interaction points.

Since the [[star product]] is the [[exponential]] of the binary contraction with the [[Feynman propagator]], it is naturally expanded as a [[sum]] of [[products of distributions]] labeled by [[finite multigraphs]] (def. \ref{Graphs} below),
where each [[vertex]] corresponds to an [[interaction]] or [[source field]] insertion, and where each [[edge]]
corresponds to one contractions of two of these with the [[Feynman propagator]]. The [[products of distributions]]
arising this way are the _[[Feynman amplitudes]]_ (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} below).

If the [[free field]] [[vacuum]] is decomposed as a [[direct sum]] of distinct [[free field]] [[types]]/species (def. \ref{VerticesAndFieldSpecies} below), then
in addition to the [[vertices]] also the edges in these [[graphs]] receive labels, now by the field species whose particular [[Feynman propagator]] is being used in the contraction at that edges.
These labeled graphs are now called _[[Feynman diagrams]]_ (def. \ref{FeynmanDiagram} below) and the [[products of distributions]] which they encode are their _[[Feynman amplitudes]]_ built by the _[[Feynman rules]]_ (prop. \ref{FeynmanDiagramAmplitude} below).

The choice of [[renormalization|("re"-)normalization]] of the [[time-ordered products]]/[[S-matrix]] is thus equivalently a choice of [[renormalization|("re"-)normalization]] of the [[Feynman amplitudes]] for all possible [[Feynman diagrams]].
These are usefully organized in powers of $\hbar$ by their _[[loop order]]_ (prop. \ref{FeynmanDiagramLoopOrder} below).

In conclusion, the [[Feynman rules]] make the perturbative [[S-matrix]] be equal to a [[formal power series]]
of [[Feynman amplitudes]] labeled by [[Feynman graphs]]. As such it is known as the _[[Feynman perturbation series]]_ (example \ref{FeynmanPerturbationSeries} below).

Notice how it is therefore the [[combinatorics]] of [[star products]] that governs both [[Wick's lemma]] in [[free field theory]] as well as [[Feynman diagram|Feynman diagrammatics]] in [[interacting field theory]]:

[[!include Wick algebra -- table]]

$\,$

{#FeynmanDiagramInTwoStages} We now discuss [[Feynman diagrams]] and their [[Feynman amplitudes]] in two stages:
First we consider plain [[finite multigraphs]] with [[linear order|linearly ordered]] vertices
but no other labels (def. \ref{Graphs} below) and discuss how these generally organize an expansion
of the [[time-ordered products]] as a sum of [[products of distributions|distributional products]]
of the given [[Feynman propagator]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} below).
These summands (or their [[vacuum expectation values]]) are called the _[[Feynman amplitudes]]_ if
one thinks of the underlying [[free field]] [[vacuum]] as having a single "field species"
and of the chosen [[interaction]] to be a single "interaction vertex".

But often it is possible and useful to identify different field species and different interaction vertices.
In fact in applications this choice is typically evident and not highlighted as a choice. We
make it explicit below as def. \ref{VerticesAndFieldSpecies}.  Such a choice makes
both the [[interaction]] term as well as the [[Feynman propagator]] decompose as sums (remark \ref{FeynmanPropagatorFieldSpecies} below).
Accordingly then, after "multiplying out" the products of these sums that appear in the Feynman amplitudes, these, too, decompose
further as as sums indexed by multigraphs whose edges are labeled by field species, and whose vertices are labeled
by interactions. These labeled multigraphs are the _[[Feynman diagrams]]_ (def. \ref{FeynmanDiagram} below) and the
corresponding summands are the [[Feynman amplitudes]] proper (prop. \ref{FeynmanDiagramAmplitude} below).

+-- {: .num_defn #Graphs}
###### Definition
**([[finite graph|finite]] [[multigraphs]])**

A _[[finite graph|finite]] [[multigraph]]_ is

1. a [[finite set]] $V$ ("of [[vertices]]");

1. a [[finite set]] $E$ ("of [[edges]]");

1. a [[function]] $E \overset{p}{\to} \left\{ {\,\atop \,} \{v_1, v_2\} = \{v_2, v_1\} \;\vert\; v_1, v_2 \in V \,,\; v_1 \neq v_2 {\, \atop \,} \right\}$

   (sending any [[edge]] to the unordered pair of distinct [[vertices]] that it goes between).

A choice of [[linear order]] on the set of vertices of a finite multigraph is a choice of [[bijection]] of the form

$$
  V \simeq \{1, 2, \cdots, \nu\}
  \,.
$$

Hence the [[isomorphism classes]] of a [[finite graph|finite]] [[multigraphs]] with [[linear order|linearly ordered]]
[[vertices]] are characterized by

1. a [[natural number]]

   $$
     \nu \coloneqq {\vert V\vert} \in \mathbb{N}
   $$

   (the number of [[vertices]]);

1. for each $i \lt j \in \{1, \cdots, \nu\}$ a natural number

   $$
     e_{i,j} \coloneqq {\vert p^{-1}(\{v_i,v_j\})\vert} \in \mathbb{N}
   $$

   (the number of [[edges]] between the $i$th and the $j$th vertex).

We write $\mathcal{G}_\nu$ for the set of such [[isomorphism classes]] of finite multigraphs with linearly ordered vertices
identified with $\{1, 2, \cdots, \nu\}$; and we write

$$
  \mathcal{G} \;\coloneqq\; \underset{\nu \in \mathbb{N}}{\sqcup} \mathcal{G}_\nu
$$

for the set of [[isomorphism classes]] of finite multigraphs with linearly ordered vertices of any number.

=--

+-- {: .num_prop #FeynmanPerturbationSeriesAwayFromCoincidingPoints}
###### Proposition
**([[Feynman amplitudes]] of [[finite multigraphs]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

For $\nu \in \mathbb{N}$, the $\nu$-fold [[time-ordered product]] away from coinciding interaction points, given by prop. \ref{TimeOrderedProductAwayFromDiagonal}

$$
  T_\nu
    \;\colon\;
  \left(
    {\, \atop \,}
    LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]
    {\, \atop \,}
  \right)^{\otimes^\nu_{\mathbb{C}[ [\hbar, g, j] ]}}_{pds}
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}((\hbar))[ [g , j] ]
$$

is equal to the following [[formal power series]] labeled by [[isomorphism classes]] of [[finite multigraphs]] with $\nu$ [[linear order|linearly ordered]] [[vertices]], $\Gamma \in \mathcal{G}_\nu$ (def. \ref{Graphs}):

$$
  \label{FeynmanAmplitudeExpansionOfTimeOrderedProductAwayFromDiagonal}
  \begin{aligned}
    & T_\nu(O_1, \cdots , O_\nu)
    \\
    & =
    \underset{\Gamma \in \mathcal{G}_\nu}{\sum}
     \Gamma\left(O_i)_{i = 1}^\nu\right)
    \\
    & \coloneqq
    \underset{ \Gamma \in \mathcal{G}_\nu }{\sum}
    prod
    \circ
    \underset{ r \lt s \in \{1, \cdots, \nu\} }{\prod}
    \frac{\hbar^{e_{r,s}}}{e_{r,s}!}
    \left\langle
      (\Delta_{F})^{e_{r,s}}
      ,
      \frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
      \frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_s^{e_{r,s}}}
    \right\rangle
    \left(
      O_1 \otimes \cdots \otimes O_{\nu}
    \right)
    \\
    & \coloneqq
    \underset{ \Gamma \in \mathcal{G}_\nu  }{\sum}
    ((-) \cdot \cdots \cdot (-))
    \circ
    \underset{ r \lt s \in \{1, \cdots,\nu\} }{\prod}
    \frac{\hbar^{e_{r,s}}}{e_{r,s}!}
    \\
    &
    \phantom{AAA}
      \underset{i = 1, \cdots e_{r,s}}{\prod}
        \underset{\Sigma \times \Sigma}{\int}
         dvol_\Sigma(x_i)
         dvol_\Sigma(y_i)
         \,
         \Delta_F^{a_i b_i}(x_i,y_i)
     \\
     & \phantom{AAAAAA}
    \left(
      O_1
        \otimes
        \cdots
        \otimes
      O_{r-1}
        \otimes
       \frac{
         \delta^{e_{r,s}} O_r
       }{
         \delta \mathbf{\Phi}^{a_1}(x_1)
         \cdots
         \delta \mathbf{\Phi}^{a_{e_{r,s}}}(x_{e_{r,s}})
       }
        \otimes
      O_{r+1}
        \otimes
        \cdots
        \otimes
      O_{s-1}
        \otimes
      \frac{
          \delta^{e_{r,s}} O_s
       }{
         \delta \mathbf{\Phi}^{b_1}(y_1)
         \cdots
         \delta \mathbf{\Phi}^{b_{e_{r,s}}}(y_{e_{r,s}})
       }
        \otimes
       O_{s+1}
        \otimes
        \cdots
        \otimes
      O_\nu
    \right)
    \,,
  \end{aligned}
$$

where $e_{r,s} \coloneqq e_{r,s}(\Gamma)$ is, for short, the number of [[edges]] between vertex $r$ and vertex $s$ in the
[[finite multigraph]] $\Gamma$ of the outer sum, according to def. \ref{Graphs}.

Here the summands of the expansion (eq:FeynmanAmplitudeExpansionOfTimeOrderedProductAwayFromDiagonal)

$$
  \label{FeynmanAmplitude}
  \Gamma\left( (O_i)_{i = 1}^\nu\right)
  \;\coloneqq\;
    prod
    \circ
    \underset{ r \lt s \in \{1, \cdots,\nu\} }{\prod}
    \frac{\hbar^{e_{r,s}}}{e_{r,s}!}
    \left\langle
      (\Delta_{F})^{e_{r,s}}
      ,
      \frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
      \frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_s^{e_{r,s}}}
    \right\rangle
    \left(
      O_1 \otimes \cdots \otimes O_{\nu}
    \right)
  \;\in\;
  PolyObs(E_{\text{BV-BRT}})((\hbar))[ [g,j ] ]
$$

and/or their [[vacuum expectation values]]

$$
  \left\langle
    \Gamma\left((V_i)_{i = 1}^v\right)
  \right\rangle
  \;\in\;
  \mathbb{C}((\hbar))[ [ h, j] ]
$$

are called the _[[Feynman amplitudes]]_ for scattering processes in the given [[free field]] [[vacuum]] of shape $\Gamma$  with [[interaction]] [[vertices]] $O_i$. Their expression as [[products of distributions]] via algebraic expression on the right hand side of (eq:FeynmanAmplitude) is also called the _[[Feynman rules]]_.

=--

([Keller 10, IV.1](S-matrix#Keller10))

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

We proceed by [[induction]] over the number $v$ of [[vertices]].
The statement is trivially true for a single vertex.
So assume that it is true for $v \geq 1$ vertices. It follows that

$$
  \begin{aligned}
    & T(O_1, \cdots, O_\nu, O_{\nu+1})
    \\
    & =
    T( T(O_1, \cdots ,O_\nu),  O_{\nu+1} )
    \\
    &=
    prod
      \circ
    \exp\left(
      \left\langle
        \hbar \Delta_F,
        \frac{\delta}{\delta \mathbf{\Phi}}
          \otimes
        \frac{\delta}{\delta \mathbf{\Phi}}
      \right\rangle
    \right)
    \left(
    \left(
      prod
        \circ
      \!\!\!\!
      \underset{\Gamma \in \mathcal{G}_\nu }{\sum}
      \underset{ { r \lt s } \atop { \in \{1, \cdots, \nu\} }  }{\prod}
      \frac{1}{e_{r,s}!}
      \left\langle
          (\hbar \Delta_F)^{e_{r,s}}
          \,,\,
          \frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
          \frac{\delta^{e_{r,s}}}{ \delta \mathbf{\Phi}_s^{e_{r,s}} }
      \right\rangle
      (O_1 \otimes \cdots \otimes O_\nu)
    \right)
      \,\otimes\,
    O_{\nu+1}
    \right)
    \\
    & =
    prod
      \circ
    \underset{\Gamma \in \mathcal{G}_\nu }{\sum}
    \\
    & \phantom{=}
      \underset{ {  r \lt s } \atop {  \in \{1,\cdots, \nu\}}  }{\prod}
      \frac{1}{e_{r,s}!}
    \left\langle
        (\hbar \Delta_F)^{e_{r,s}}
        \,,\,
        \frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
        \frac{\delta^{e_{r,s}}}{ \delta \mathbf{\Phi}_s^{e_{r,s}} }
    \right\rangle
    \\
    & \phantom{=}
    \underset{
       { e_{\nu+1} =}
       \atop
       { e_{1,{\nu+1}} + \cdots + e_{\nu,\nu + 1} }
    }{\sum}
      \underset{
        = (e_{1,\nu + 1}) \cdots (e_{\nu,\nu+1}))
      }{
      \underbrace{
      \frac{
        \left(
          { e_{\nu + 1} }
          \atop
          { (e_{1, \nu + 1}), \cdots, (e_{\nu , \nu+1}) }
        \right)
      }{
        ( e_{\nu+1} ) !
      }
      }
      }
    \left\langle
      (\hbar \Delta_F)^{e_{\nu+1}}
    \left(
      \frac{\delta^{e_{1,\nu+1}} O_1 }{\delta \mathbf{\Phi}^{e_{1,\nu+1}}}
        \otimes
        \cdots
        \otimes
      \frac{
        \delta^{e_{\nu,\nu+1}} O_\nu
      }{
        \delta \mathbf{\Phi}^{e_{\nu,\nu+1}}
      }
    \;\otimes\;
    \frac{
      \delta^{ e_{\nu + 1} } O_{\nu+1}
    }{
      \delta \mathbf{\Phi}^{e_{1,\nu+1} + \cdots + e_{\nu,\nu+1}}
    }
    \right\rangle
    \right)
    \\
    &=
    prod
      \circ
    \underset{\Gamma \in \mathcal{G}_{\nu+1} }{\sum}
    \underset{ { r \lt s } \atop { \in \{1, \cdots, \nu+1\} } }{\prod}
    \tfrac{1}{e_{r,s}!}
    \left\langle
        (\hbar \Delta_F)^{e_{r,s}}
        \,,\,
        \frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_r^{e_{r,s}}}
        \frac{\delta^{e_{r,s}}}{\delta \mathbf{\Phi}_s^{e_{r,s}}}
    \right\rangle
    (O_1 \otimes \cdots \otimes O_{\nu+1})
  \end{aligned}
$$

The combinatorial factor over the brace is the [[multinomial coefficient]] expressing the number of ways of distributing $e_{\nu+1}$-many functional derivatives to $v$ factors, via the [[product rule]], and quotiented by the [[factorial]] that comes from the [[exponential]] in the definition of the [[star product]].

Here in the first step we used the [[associativity]] (eq:RestrictedTimeOrderedProductAssociative) of the restricted time-ordered product, in the second step we used the induction assumption,
in the third we passed the outer functional derivatives through the pointwise product using the [[product rule]], and in the fourth step we recognized that this amounts to summing
in addition over all possible choices of sets of edges from the first $v$ vertices to the new $\nu+1$st vertex,
which yield in total the sum over all diagrams with $\nu+1$ vertices.

=--

If the [[free field theory]] is decomposed as a [[direct sum]] of free field theories (def. \ref{VerticesAndFieldSpecies} below), we obtain a more fine-grained concept of [[Feynman amplitudes]], associated not just with a [[finite multigraph]], but also with a
labelling of this graph by field species and interaction types. These labeled multigraphs are the genuine _[[Feynman diagrams]]_ (def. \ref{FeynmanDiagram} below):

+-- {: .num_defn #VerticesAndFieldSpecies}
###### Definition
**(field species and interaction vertices)**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Then

1. a choice of _field species_ is a choice of decomposition of the  [[BV-BRST formalism|BV-BRST]] [[field bundle]] $E_{\text{BV-BRST}}$
as a [[fiber product]] over [[finite set]] $Spec = \{sp_1, sp_2, \cdots, sp_n\}$ of ([[graded manifold|graded]] [[supermanifold|super-]]) [[field bundles]]

   $$
     E_{\text{BV-BRST}}
     \;\simeq\;
     E_{sp_1} \times_{\Sigma} \cdots \times_\Sigma E_{sp_n}
     \,,
   $$

   such that the [[gauge fixing|gauge fixed]] [[free field|free]] [[Lagrangian density]] $\mathbf{L}'$ is the [[sum]]

   $$
     \mathbf{L}' \;=\; \mathbf{L}'_{sp_1} + \cdots + \mathbf{L}'_{sp_n}
   $$

   of [[free field theory|free]] [[Lagrangian densities]]

   $$
     \mathbf{L}'_{sp_i} \in \Omega^{p+1,0}_\Sigma(E_i)
   $$

  on these separate field bundles.

1. a choice of _interaction vertices and external vertices_ is a choice of sum decomposition

   $$
     g S_{int} + j A
     \;=\;
     \underset{i \in Ext}{\sum} g S_{int,i}
     +
     \underset{j \in Int}{\sum} j A_j
   $$

   parameterized by [[finite sets]] $Int$ and $Ext$, to be called the sets of _internal vertex labels_ and _external vertex labels_,
respectively.

=--

+-- {: .num_remark #FeynmanPropagatorFieldSpecies}
###### Remark
**(Feynman propagator for separate field species)**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}.

Then a choice of field species as in def. \ref{VerticesAndFieldSpecies}
induces a corresponding decomposition of the [[Feynman propagator]] of the gauge fixed free field theory

$$
  \Delta_F
  \;\in\;
  \Gamma'_{\Sigma \times \Sigma}( E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}} )
$$

as the sum of Feynman propagators for each of the chosen field species:

$$
  \Delta_F
  \;=\;
  \Delta_{F,1} + \cdots + \Delta_{F,n}
  \;\in\;
  \underoverset{i = 1}{n}{\oplus}
  \Gamma'_{\Sigma \times \Sigma}( E_{sp_i} \boxtimes E_{sp_i} )
    \;\subset\;
  \Gamma'_{\Sigma \times \Sigma}( E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}} )
$$

hence in components, with $(\phi^A$ the collective field coordinates on $E_{\text{BV-BRST}}$,
this decomposition is of the form

$$
  \left(
    \Delta_F^{A, B}
  \right)
  \;=\;
  \left(
    \array{
      (\Delta_{F,1}^{a b}) & 0 & 0 &  \cdots & 0
      \\
      0 & (\Delta_{F,2}^{\alpha \beta}) & 0 & \cdots & 0
      \\
      \vdots & & & & \vdots
      \\
      0 & \cdots & \cdots & 0 & (\Delta_{F,n}^{i j})
    }
  \right)
$$

=--

+-- {: .num_example #FieldSpeciesQED}
###### Example
**(field species in [[quantum electrodynamics]])**

The [[field bundle]] for [[Gaussian-averaged Lorenz gauge|Lorenz]] [[gauge fixing|gauge fixed]] [[quantum electrodynamics]]
on [[Minkowski spacetime]] $\Sigma$ admits a decomposition into field species, according to def. \ref{VerticesAndFieldSpecies}, as

$$
  E_{\text{BV-BRST}}
  \;=\;
   \underset{
      \text{Dirac}
      \atop
      \text{field}
    }{
    \underbrace{
    (S_{odd} \times \Sigma)
    }}
    \times_\Sigma
   \underset{
     {\text{electromagnetic field &}}
     \atop
     {\text{Nakanishi-Lautrup field}}
    }{
    \underbrace{
      T^\ast\Sigma
       \times_\Sigma
      (\mathbb{R} \times \Sigma)
    }}
   \times_\Sigma
   \underset{
     \text{ghost field}
   }{
   \underbrace{
     (\mathbb{R}[1] \times \Sigma)
   }
   }
   \times_\Sigma
   \underset{
     \text{antighost field}
   }{
   \underbrace{
     (\mathbb{R}[-1] \times \Sigma)
   }
   }
$$

(by example \ref{LagrangianQED}) and example \ref{NLGaugeFixingOfElectromagnetism})).

The corresponding sum decomposition of the Feynman propagator, according to remark \ref{FeynmanPropagatorFieldSpecies},
is

$$
  \Delta_F
    \;=\;
  \underset{
    \text{Dirac}
    \atop
    \text{field}
  }{
  \underbrace{
    \Delta_F^{\text{electron}}
  }
  }
  +
  \underset{
    \text{electromagnetic field &}
    \atop
    \text{Nakanishi-Lautrup field}
  }{
  \underbrace{
    \left(
      \array{
        \Delta_F^{photon} & *
        \\
        * & *
      }
    \right)
  }
  }
  +
  \Delta_F^{ghost}
  +
  \Delta_F^{\text{antighost}}
  \,,
$$

where

1. $\Delta_F^{\text{electron}}$ is the [[electron propagator]] (def. \ref{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetim}));

1. $\Delta_F^{photon}$ is the [[photon propagator]] in [[Gaussian-averaged Lorenz gauge]] (prop. \ref{PhotonPropagatorInGaussianAveragedLorenzGauge});

1. the [[ghost field]] and [[antighost field]] [[Feynman propagators]] $\Delta_F^{ghost}$, and $\Delta_F^{antighost}$ are each one copy of the [[Feynman propagator]] of the [[real scalar field]] (prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue}), while the [[Nakanishi-Lautrup field]] contributes a mixing with the [[photon propagator]], notationally suppressed behind the star-symbols above.

=--

+-- {: .num_defn #FeynmanDiagram}
###### Definition
**([[Feynman diagrams]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Let moreover

$$
  E_{\text{BV-BRST}}
  \;\simeq\;
  \underset{sp \in Spec}{\times} E_{sp}
  \,,
$$

be a choice of field species, according to def \ref{VerticesAndFieldSpecies},

$$
  g S_{int} + j A
  \;=\;
  \underset{i \in Ext}{\sum} g S_{int,i}
  +
  \underset{j \in Int}{\sum} j A_j
$$

a choice of internal and external interaction vertices according to def. \ref{VerticesAndFieldSpecies}.

With these choices, we say that a _[[Feynman diagram]]_ $(\Gamma, vertlab, edgelab)$ is

1. a [[finite multigraph]] with [[linear order|linearly ordered]] vertices (def. \ref{Graphs})

   $$
     \Gamma \in \mathcal{G}
     \,,
   $$

1. a [[function]] from its [[vertices]]

   $$
     vertlab \;\colon\; V_{\Gamma} \longrightarrow Int \sqcup Ext
   $$

   to the [[disjoint union]] of the chosen sets of internal and external vertex labels;

1. a [[function]] from its [[edges]]

   $$
     edgelab \;\colon\; E_{\Gamma} \to Spec
   $$

   to the chosen set of field species.

We write

$$
  \array{
    \mathcal{G}^{Feyn}
      &\overset{\text{forget} \atop \text{labels}}{\longrightarrow}&
    \mathcal{G}
    \\
    (\Gamma,vertlab, edgelab) &\mapsto& \Gamma
  }
$$

for the set of [[isomorphism classes]] of Feynman diagrams with labels in $Sp$, refining the set of
isomorphisms of plain [[finite multigraphs]] with [[linear order|linearly ordered]] [[vertices]] from def. \ref{Graphs}.

=--

+-- {: .num_prop #FeynmanDiagramAmplitude}
###### Proposition
**([[Feynman amplitudes]] for [[Feynman diagrams]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Let moreover

$$
  E_{\text{BV-BRST}}
  \;\simeq\;
  \underset{sp \in Spec}{\times} E_{sp}
  \,,
$$

be a choice of field species, according to def \ref{VerticesAndFieldSpecies}, hence inducing, by remark \ref{FeynmanPropagatorFieldSpecies},
a sum decomposition of the [[Feynman propagator]]

$$
  \label{FeynmanPropagatorSumOverFieldSpecies}
  \Delta_F
  \;=\;
  \underset{sp \in Spec}{\sum}\Delta_{F,sp}
  \,,
$$

and let

$$
  \label{VertexDecompositionFeynmanAmplitude}
  g S_{int} + j A
  \;=\;
  \underset{i \in Ext}{\sum} g S_{int,i}
  +
  \underset{j \in Int}{\Sum} j A_j
$$

be a choice of internal and external interaction vertices according to def. \ref{VerticesAndFieldSpecies}.

Then by "multiplying out" the products of the sums (eq:FeynmanPropagatorSumOverFieldSpecies) and (eq:VertexDecompositionFeynmanAmplitude) in the
formula (eq:FeynmanAmplitude) for the [[Feynman amplitude]] $\Gamma\left( (g S_{int} + j A))_{i = 1}^\nu \right)$ (def. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) this decomposes as a sum of the form

$$
  \Gamma\left(
    (g S_{int} + j A)_{i = 1}^\nu
  \right)
  \;=\;
  \underset{
    { V_\Gamma \overset{vertlab}{\longrightarrow} Int \sqcup Ext}
    \atop
    { E_\Gamma \overset{edgelab}{\longrightarrow}  Spec }
  }{\sum}
    \left(
      \Gamma,
      edgelab, vertlab
    \right)
    (g S_{int} + j A)
$$

over all ways of labeling the [[vertices]] $v$ of $\Gamma$ by the internal or external vertex labels,
and  the [[edges]] $e$ of $\Gamma$ by field species. The corresponding summands

$$
   \left(
     \Gamma,
     edgelab,
     vertlab
   \right)
   (g S_{int} + j A)
   \;\in\;
   PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$

or rather their [[vacuum expectation value]]

$$
  \left\langle
   \left(
     \Gamma,
     edgelab, vertlab
   \right)
   (g S_{int} + j A)
  \right\rangle
  \;\in\;
  \mathbb{C}[ [ \hbar, g, j ] ]
$$

are called the _[[Feynman amplitude]] associated with these [[Feynman diagrams]].

=--

+-- {: .num_example #FeynmanAmplitudesInCausalPerturbationTheoryExampleOfQED}
###### Example
**([[Feynman amplitudes]] in [[causal perturbation theory]] -- example of [[QED]])**

To recall, in [[perturbative quantum field theory]], [[Feynman diagrams]] (def. \ref{FeynmanDiagram}) are labeled [[finite multigraphs]] (def. \ref{Graphs}) that encode [[product of distributions|products of]] [[Feynman propagators]], called _[[Feynman amplitudes]]_ (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) which in turn contribute to [[probability amplitudes]] for physical [[scattering]] processes -- _[[scattering amplitudes]]_ (example \ref{ScatteringAmplitudeFromInteractingFieldObservables}):

The [[Feynman amplitudes]] are the summands in the [[Feynman perturbation series]]-expansion (example \ref{FeynmanPerturbationSeries}) of the _[[scattering matrix]]_ (def. \ref{LagrangianFieldTheoryPerturbativeScattering})

$$
  \mathcal{S}
  \left(
    S_{int}
  \right)
  =
  \underset{k \in \mathbb{N}}{\sum}
  \frac{1}{k!}
  \frac{1}{(i \hbar)^k}
  T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots , S_{int}}} )
$$

of a given [[interaction]] [[Lagrangian density]] $L_{int}$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).

The [[Feynman amplitudes]] are the summands in an expansion of the _[[time-ordered products]]_ $T(\cdots)$ (def. \ref{TimeOrderedProduct}) of the [[interaction]] with itself, which, away from coincident vertices, is given by the [[star product]] of the [[Feynman propagator]] $\Delta_F$ (prop. \ref{TimeOrderedProductAwayFromDiagonal}), via the [[exponential]] contraction

$$
  T(S_{int}, S_{int})
  \;=\;
  prod \circ \exp
  \left(
     \hbar
      \int \Delta_{F}^{a b}(x,y)
        \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
          \otimes
        \frac{\delta}{\delta \mathbf{\Phi}(y)}
    \right)
    ( S_{int} \otimes S_{int} )
   \,.
$$

Each [[edge]] in a [[Feynman diagram]] corresponds to a factor of a [[Feynman propagator]]  in $T( \underset{k \, \text{factors}}{\underbrace{S_{int} \cdots S_{int}}} )$, being a [[distribution of two variables]]; and each [[vertex]] corresponds to a factor of the [[interaction]] [[Lagrangian density]] at $x_i$.

For example [[quantum electrodynamics]] (example \ref{LagrangianQED}) in [[Gaussian-averaged Lorenz gauge]] (example \ref{NLGaugeFixingOfElectromagnetism}) involves (via example \ref{FieldSpeciesQED}):

1. the [[Dirac field]] modelling the [[electron]], with [[Feynman propagator]] called the _[[electron propagator]]_ (def. \ref{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime}), here to be denoted

   $$
     \Delta \phantom{AAAA} \text{electron propagator}
   $$

1. the [[electromagnetic field]] modelling the [[photon]], with [[Feynman propagator]] called the _[[photon propagator]]_ (prop. \ref{PhotonPropagatorInGaussianAveragedLorenzGauge}), here to be denoted

   $$
     G \phantom{AAAA} \text{photon propagator}
   $$

1. the [[electron-photon interaction]] (eq:ElectronPhotonInteractionLocalLagrangian)

   $$
     L_{int}
     \;=\;
     \underset{
       \text{interaction}
     }{
     \underbrace{
       i g
       (\gamma^\mu)^\alpha{}_\beta
     }
     }
     \,
     \underset{
       { \text{incoming} \atop \text{electron}  }
       \atop
       \text{field}
     }{\underbrace{\overline{\psi_\alpha}}}
     \;
     \underset{
       {
         \,
         \atop
         \text{photon}
       }
       \atop
       \text{field}
     }{\underbrace{a_\mu}}
     \;
     \underset{
       {\text{outgoing} \atop \text{electron} }
       \atop
       \text{field}
     }{\underbrace{\psi^\beta}}
   $$

The [[Feynman diagram]] for the [[electron-photon interaction]] alone is

<center>
<img src="https://ncatlab.org/nlab/files/InteractionVertexOfQED.jpg" width="150">
</center>

where the solid lines correspond to the [[electron]], and the wiggly line to the [[photon]]. The corresponding [[product of distributions]] (prop. \ref{HoermanderCriterionForProductOfDistributions}) is (written in [[generalized function]]-notation, example \ref{SomeNonSingularTemperedDistributions})

$$
  \underset{
    \text{loop order}
  }{
  \underbrace{
    \hbar^{3/2-1}
  }
  }
  \underset{
    \text{electron-photon}
    \atop
    \text{interaction}
  }{
  \underbrace{
    i g (\gamma^\mu)^\alpha{}_\beta
  }
  }
  \,.
  \,
  \underset{
    {\text{incoming} \atop \text{electron}}
    \atop
    \text{propagator}
  }{
  \underbrace{
    \overline{\Delta(-,x)}_{-, \alpha}
  }
  }
  \underset{
    {
      \,
      \atop
      \text{photon}
    }
    \atop
    \text{propagator}
  }{
  \underbrace{
     G(x,-)_{\mu,-}
  }
  }
  \underset{
    { \text{outgoing} \atop \text{electron} }
    \atop
    \text{propagator}
  }{
  \underbrace{
    \Delta(x,-)^{\beta, -}
  }
  }
$$

Hence a typical [[Feynman diagram]] in the [[QED]] [[Feynman perturbation series]] induced by this [[electron-photon interaction]] looks as follows:

<center>
<img src="https://ncatlab.org/nlab/files/FeynmanDiagramGlobal.jpg" width="560"/>
</center>

where on the bottom the corresponding [[Feynman amplitude]] [[product of distributions]] is shown;
now notationally suppressing the contraction of the internal indices and all prefactors.

For instance the two solid [[edges]] between the [[vertices]] $x_2$ and $x_3$ correspond to the two factors of $\Delta(x_2,x_2)$:

<center>
<img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponent1.jpg" width="560"/>
</center>

This way each sub-graph encodes its corresponding subset of factors in the [[Feynman amplitude]]:

<center>
<img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponentTwo.jpg" width="560"/>
</center>

<center>
<img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponentThree.jpg" width="560"/>
</center>

> graphics grabbed from  [Brouder 10](Feynman+diagram#Brouder10)

A priori this [[product of distributions]] is defined away from coincident vertices: $x_i \neq x_j$ (prop. \ref{TimeOrderedProductAwayFromDiagonal} below). The definition at coincident vertices $x_i = x_j$ requires a choice of _[[extension of distributions]]_ (def. \ref{ExtensionOfDistributions} below) to the [[diagonal]] locus of coincident interaction points. This choice is the _[[renormalization|("re-")normalization]]_ (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization} below) of the [[Feynman amplitude]].

=--

+-- {: .num_example #FeynmanPerturbationSeries}
###### Example
**([[Feynman perturbation series]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let

$$
  g S_{int} + j A
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, h ] ]\langle g , j\rangle
$$

be a [[local observable]], regarded as a [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

By prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} every choice of perturbative [[S-matrix]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering})

$$
  \mathcal{S}(g S_{int} + j A)
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ] +
$$

has an expansion as a [[formal power series]] of the form

$$
  \mathcal{S}(g S_{int} + j A)
  \;=\;
  \underset{\Gamma \in \mathcal{G}}{\sum}
  \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)}\right)
  \,,
$$

where the series is over all [[finite multigraphs]] with [[linear order|linearly ordered]] [[vertices]] $\Gamma$ (def. \ref{Graphs}), and the summands are the corresponding [[renormalization|("re"-)normalized]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}) [[Feynman amplitudes]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}).

If moreover a choice of field species and of internal and external interaction vertices is made, according to def. \ref{VerticesAndFieldSpecies},
then this series expansion refines to an expansion over all [[Feynman diagrams]] $(\Gamma,edgelab, vertlab)$ (def. \ref{FeynmanDiagram})
of [[Feynman amplitudes]] $(\Gamma, edgelab,vertlab)(g S_{int} + j A)$ (def. \ref{FeynmanDiagramAmplitude}):

$$
  \mathcal{S}(g S_{int} + j A)
  \;=\;
  \underset{(\Gamma,edgelab, vertlab) \in \mathcal{G}^{Feyn}}{\sum}
    (\Gamma, edgelab,vertlab)(g S_{int} + j A)
  \,,
$$

Expressed in this form the [[S-matrix]] is known as the _[[Feynman perturbation series]]_.

=--

+-- {: .num_remark #Tadpoles}
###### Remark
**(no [[tadpole]] [[Feynman diagrams]])**

In the definition of [[finite multigraphs]] in def. \ref{Graphs} there are _no_ edges considered that go from any [[vertex]] to _itself_. Accordingly, there are _no_ such labeled edges in [[Feynman diagrams]] (def. \ref{FeynmanDiagram}):

<center>
  <img src="https://ncatlab.org/nlab/files/tadpole.png" width="70">
</center>

In [[pQFT]] these diagrams are called _[[tadpoles]]_, and their non-appearance is considered part of the _[[Feynman rules]]_ (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}).
Via prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints} this condition reflects the nature of the [[star product]] (def. \ref{PropagatorStarProduct}) which always contracts _different_ [[tensor product]] factors with the [[Feynman propagator]] before taking their pointwise product.

Beware that in [[graph theory]] these [[tadpoles]] are called "[[loops]]", while here in [[pQFT]] a "loop" in a [[planar graph]] refers instead to what in [[graph theory]] is called a _[[face]]_ of the graph, see the discussion of _[[loop order]]_ in prop. \ref{FeynmanDiagramLoopOrder} below.

=--

([Keller 10, remark II.8 and proof of prop. II.7](S-matrix#Keller10))

$\,$

**Effective action**
 {#EffectiveAction}

We have seen that the [[Feynman perturbation series]] expresses the [[S-matrix]] as a [[formal power series]] of _[[Feynman amplitudes]]_ labeled by _[[Feynman diagrams]]_. Now the [[Feynman amplitude]] associated with a [[disjoint union]] of [[connected graph|connected]] [[Feynman diagrams]] (def. \ref{ConnectedGraphs} below) is just the product of the amplitudes of the [[connected components]] (prop. \ref{LogarithmEffectiveAction} below). This allows to re-organize the [[Feynman perturbation series]] as the ordinary [[exponential]] of the Feynman perturbation series restricted to just [[connected graph|connected]] Feynman diagrams. The latter is called the _[[effective action]]_ (def. \ref{InPerturbationTheoryActionEffective} below) because it allows to express [[vacuum expectation values]] of the [[S-matrix]] as an ordinary exponential (equation (eq:ExponentialSeffVEVOfSMatrix) below).

+-- {: .num_defn #ConnectedGraphs}
###### Definition
**([[connected graphs]])**

Given two [[finite multigraphs]] $\Gamma_1, \Gamma_2 \in \mathcal{G}$ (def. \ref{Graphs}), their [[disjoint union]]

$$
  \Gamma_1 \sqcup \Gamma_2
  \;\in\;
  \mathcal{G}
$$

is the finite multigraph whose set of [[vertices]] and set of [[edges]] are the [[disjoint unions]] of the corresponding sets of $\Gamma_1$ and $\Gamma_2$

$$
  V_{\Gamma_1 \sqcup \Gamma_2}
    \;\coloneqq\;
  V_{\Gamma_1} \sqcup V_{\Gamma_2}
$$

$$
  E_{\Gamma_1 \sqcup \Gamma_2}
    \;\coloneqq\;
  E_{\Gamma_1} \sqcup E_{\Gamma_2}
$$

and whose vertex-assigning function $p$ is the corresponding function on disjoint unions

$$
  p_{\Gamma_1 \sqcup \Gamma_2}
  \;\coloneqq\;
  p_{\Gamma_1} \sqcup p_{\Gamma_2}
  \,.
$$

The operation induces a pairing on the set $\mathcal{G}$ of [[isomorphism classes]] of [[finite multigraphs]]

$$
  (-) \sqcup (-)
  \;\colon\;
  \mathcal{G} \times \mathcal{G}
    \longrightarrow
  \mathcal{G}
  \,.
$$

A [[finite multigraph]] $\Gamma \in \mathcal{G}$ (def. \ref{Graphs}) is called _[[connected graph|connected]]_ if it is not the [[disjoint union]] of two [[inhabited|non-empty]] finite multigraphs.

We write

$$
  \mathcal{G}_{conn}
  \subset
  \mathcal{G}
$$

for the subset of [[isomorphism classes]] of [[connected graph|connected]] [[finite multigraphs]].

=--

+-- {: .num_lemma #MultiplicativeFeynmanAmplitudes}
###### Lemma
**([[Feynman amplitudes]] multiply under [[disjoint union]] of [[graphs]])**

Let

$$
  \Gamma
    \;=\;
  \Gamma_1
    \sqcup
  \Gamma_2
    \sqcup
    \cdots
    \sqcup
  \Gamma_n
  \;\in\; \mathcal{G}
$$

be [[disjoint union]] of graphs (def. \ref{ConnectedGraphs}). then then corresponding [[Feynman amplitudes]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints})
multiply by the pointwise product (def. \ref{Observable}):

$$
  \Gamma\left( g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
  \;=\;
  \Gamma_1\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma_1)}\right)
   \cdot
  \Gamma_2\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma_2)} \right)
    \cdot
    \cdots
    \cdot
  \Gamma_n\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma_n)} \right)
  \,.
$$

=--

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

By prop. \ref{TimeOrderedProductAwayFromDiagonal} the contributions to the S-matrix away from coinciding interaction points are given by the [[star product]] induced by the [[Feynman propagator]], and specifically, by prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}, the [[Feynman amplitudes]] are given this way.
Moreover the [[star product]] (def. \ref{PropagatorStarProduct}) is given by first contracting with powers of the [[Feynman propagator]] and then multiplying all resulting terms with the pointwise product of observables.
This implies the claim by the nature of the combinatorial factor in the definition of the [[Feynman amplitudes]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}).

=--

+-- {: .num_defn #InPerturbationTheoryActionEffective}
###### Definition
**([[effective action]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be an [[S-matrix]] scheme for [[perturbative QFT]] around this vacuum (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) and let

$$
  g S_{int} + j A
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, h ] ]
$$

be a [[local observable]].

Recall that for each [[finite multigraph]] $\Gamma \in \mathcal{G}$ (def. \ref{Graphs}) the [[Feynman perturbation series]] for $\mathcal{S}(g S_{int} + j A)$ (example \ref{FeynmanPerturbationSeries})

$$
  \mathcal{S}(g S_{int} + j A)
  \;=\;
  \underset{\Gamma \in \mathcal{G}}{\sum}
    \Gamma\left( (g S_{int} + j A)_{i = 1}^{v(\Gamma)} \right)
$$

contributes with a [[renormalization|("re"-)nromalized]] [[Feynman amplitude]] $\Gamma\left( (g S_{int} + j A)_{i = 1}^v\right) \in PolyObs(E_{\text{BV-BRST}})((\hbar))[ [ g, j ] ]$.

We say that the corresponding _[[effective action]]_ is $i \hbar$ times the sub-series

$$
  \label{ExpansionEffectiveAction}
  S_{eff}(g,j)
  \;\coloneqq\;
  i \hbar
  \underset{\Gamma \in \mathcal{G}_{conn}}{\sum}
  \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})((\hbar))[ [ g, j ] ]
$$

of [[Feynman amplitudes]] that are labeled only by the _[[connected graphs]]_  $\Gamma \in \mathcal{G}_{conn} \subset \mathcal{G}$ (def. \ref{ConnectedGraphs}).

(A priori $S_{eff}(g,j)$ could contain negative powers of $\hbar$, but it turns out that it does not; this is prop. \ref{FeynmanDiagramLoopOrder} below.)

=--

+-- {: .num_remark #TerminologyForEffectiveAction}
###### Remark
**(terminology for "effective action")**

Beware differing conventions of terminology:

1. In the perspective of [[effective quantum field theory]] (remark \ref{pQFTEffective} below), the [[effective action]] in def. \ref{InPerturbationTheoryActionEffective} is sometimes called the _effective potential_ at scale $\Lambda = 0$ (see prop. \ref{InPerturbationTheoryActionEffective} below).

   This terminology originates in restriction to the special example of the [[scalar field]] (example \ref{RealScalarFieldBundle}), where the non-derivative [[Phi^n interactions]] $g S_{int} = \underset{n}{\sum} \underset{\Sigma}{\int} g_{sw}^{(n)}(x) (\mathbf{\Phi}(x))^n \, dvol_\Sigma(x)$ (example \ref{phintheoryLagrangian}) are naturally thought of as [[potential energy]]-terms.

   From this perspective the [[effective action]] in def. \ref{InPerturbationTheoryActionEffective} is a special case of _[[relative effective actions]]_ $S_{eff,\Lambda}$ ("relative effective potentials", in the case of [[Phi^n interactions]]) relative to an arbitrary [[UV cutoff]]-scales $\Lambda$ (def. \ref{EffectiveActionRelative} below).

1. For the special case that

   $$
     j A
       \coloneqq
     \underset{\Sigma}{\int} j_{sw,a}(x) \mathbf{\Phi}^a(x)\, dvol_{\Sigma}(x)
   $$

   is a [[regular polynomial observable|regular]] [[linear observable]] (def. \ref{RegularLinearFieldObservables}) the [[effective action]] according to def. \ref{InPerturbationTheoryActionEffective} is often denoted $W(j)$ or $E(j)$, and then its _functional [[Legendre transform]]_ (if that makes sense) is instead called the effective action, instead.

   This is because the latter encodes the [[equations of motion]] for the [[vacuum expectation values]] $\langle \mathbf{\Phi}(x)_int\rangle$ of the [[interacting field observables|interacting]] [[field observables]]; see example \ref{EquationsOfMotionForVacuumExpectationValues} below.

Notice the different meaning of "effective" in both cases: In the first case it refers to what is effectively seen of the full [[pQFT]] _at some [[UV-cutoff scale]]_, while in the second case it refers to what is effectively seen when restricting attention only to the [[vacuum expectation values]] of [[regular polynomial observable|regular]] [[linear observables]].

=--

+-- {: .num_prop #LogarithmEffectiveAction}
###### Proposition
**([[effective action]] is [[logarithm]] of [[S-matrix]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, let $\mathcal{S}$ be an [[S-matrix]] scheme for [[perturbative QFT]] around this vacuum (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) and let

$$
  g S_{int} + j A
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, h ] ]
$$

be a [[local observable]] and let

$$
  S_{eff}(g,j) \;\in\; PolyObs(E_{\text{BV-BRST}})((\hbar))[ [ g, j] ]
$$

be the corresponding [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}).

Then then [[S-matrix]] for $g S_{int} + j A$ is the [[exponential]] of the [[effective action]] with respect to the pointwise product $(-)\cdot (-)$ of observables (def. \ref{Observable}):

$$
  \begin{aligned}
    \mathcal{S}(g S_{int} + j A)
    & =
    \exp_\cdot\left( \tfrac{1}{i \hbar} S_{eff}(g,j)  \right)
    \\
    & \coloneqq
    1
      +
    \frac{1}{i \hbar} S_{eff}(g,j)
      +
    \frac{1}{(i \hbar)^2} S_{eff}(g,j) \cdot S_{eff}(g,j)
      +
    \frac{1}{(i \hbar)^3} S_{eff}(g,j) \cdot S_{eff}(g,j) \cdot S_{eff}(g,j)
      +
    \cdots
  \end{aligned}
$$

Moreover, this relation passes to the [[vacuum expectation values]]:

$$
  \label{ExponentialSeffVEVOfSMatrix}
  \begin{aligned}
    \left\langle
      {\, \atop \,}
      \mathcal{S}(g S_{int} + j A)
      {\, \atop \,}
    \right\rangle
    & =
    \left\langle
      {\, \atop \,}
      \exp\left(
        \tfrac{1}{i \hbar} S_{eff}(g,j)
      \right)
      {\, \atop \,}
    \right\rangle
    \\
    & =
    e^{\tfrac{1}{i \hbar} \langle S_{eff}(g,j) \rangle}
  \end{aligned}
  \,.
$$

Conversely the [[vacuum expectation value]] of the [[effective action]] is to the [[logarithm]] of that of the S-matrix:

$$
  \left\langle
    S_{eff}(g,j)
  \right\rangle
  \;=\;
  i \hbar
  \,
  \ln
  \left\langle
    \mathcal{S}(g S_{int} + j A)
  \right\rangle
  \,.
$$

=--

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

By lemma \ref{MultiplicativeFeynmanAmplitudes} the summands in the $n$th pointwise power of $\frac{1}{i \hbar}$ times the effective action are precisely the Feynman amplitudes $\Gamma\left((g S_{int} + j A)_{i = 1}^{\nu(\Gamma)}\right)$ of [[finite multigraphs]] $\Gamma$ with $n$ [[connected components]], where each such appears with multiplicity given by the [[factorial]] of $n$:

$$
  \frac{1}{n!}
  \left(
    \frac{1}{i \hbar}
    S_{eff}(g,j)
  \right)^n
  \;=\;
  \underset{
     { \Gamma = \underoverset{j = 1}{n}{\sqcup} \Gamma_j }
     \atop
     { \Gamma_j \in \mathcal{G}_{conn} }
  }{\sum}
  \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
  \,.
$$

It follows that

$$
  \begin{aligned}
    \exp_\cdot\left(
      \frac{1}{i \hbar} S_{int}
    \right)
    & =
    \underset{n \in \mathbb{N}}{\sum}
    \underset{
       { \Gamma = \underoverset{j = 1}{n}{\sqcup} \Gamma_j }
       \atop
       { \Gamma_j \in \mathcal{G}_{conn} }
    }{\sum}
    \Gamma\left( (g S_{int} + j A)_{i = 1}^{v(\Gamma)} \right)
    \\
    & =
    \underset{\Gamma \in \mathcal{G}}{\sum}
    \Gamma\left( (g S_{int} + j A)_{i = 1}^{v(\Gamma)} \right)
  \end{aligned}
$$

yields the [[Feynman perturbation series]] by expressing it as a series (re-)organized by number of [[connected components]] of the [[Feynman diagrams]].

To conclude the proof it is now sufficient to observe that taking [[vacuum expectation values]] of [[polynomial observables]] respects the pointwise product of observables

$$
  \left\langle A_1 \cdot A_2 \right\rangle
  \;=\;
  \left\langle
    A_1
  \right\rangle
  \,
  \left\langle
    A_2
  \right\rangle
  \,.
$$

This is because the [[Hadamard vacuum state]] $\langle -\rangle \colon PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]  \to \mathbb{C}[ [\hbar, g, j ] ]$ simply picks the zero-order monomial term, by prop. \ref{WickAlgebraCanonicalState}),
and under multiplication of polynomials the zero-order terms are multiplied.

=--

This immediately implies the following important fact:

+-- {: .num_prop #EffectiveActionIsGeneratingFunction}
###### Proposition
**(in [[vacuum stability|stable vacuum]] the [[effective action]] is [[generating function]] for [[vacuum expectation values]] of [[interacting field observables]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

If the given [[vacuum state]] is [[vacuum stability|stable]] (def. \ref{VacuumStability}) then the [[vacuum expectation value]] $\langle S_{eff}(g,j)\rangle$ of the [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) is the generating function for the [[vacuum expectation value]] of the [[interacting field observable]] $A_{int}$ (def. \ref{InteractingFieldObservables}) in that

$$
  \left\langle
    A_{int}
  \right\rangle
  \;=\;
  \frac{d}{d j}
  S_{eff}(g,j)\vert_{j = 0}
  \,.
$$

=--

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

We compute as follows:

$$
  \begin{aligned}
    \frac{d}{d j}
    S_{eff}(g,j)
    & =
    i \hbar
    \frac{d}{d j}
    \ln
    \left\langle
      \mathcal{S}(g S_{int} + j A)
    \right\rangle
    \vert_{j = 0}
    \\
    & =
    i \hbar
    \left\langle
      \mathcal{S}(g S_{int})
    \right\rangle^{-1}
    \frac{d}{d j}
    \left\langle
      \mathcal{S}(g S_{int} + j A)
    \right\rangle
    \vert_{j = 0}
    \\
    & =
    \left\langle
      \frac{d}{d j}
      \underset{ \mathcal{Z}(j A) }{
      \underbrace{\mathcal{S}(g S_{int})^{-1}
      \mathcal{S}(g S_{int} + j A)
      }}
      \vert_{j = 0}
    \right\rangle
    \\
    & =
    \left\langle
      A_{int}
    \right\rangle
    \,.
  \end{aligned}
$$

Here in the first step we used prop \ref{LogarithmEffectiveAction}, in the second step we applied the [[chain rule]] of [[differentiation]], in the third step we used the definition of [[vacuum stability]] (def. \ref{VacuumStability}) and in the fourth step we recognized the definition of the [[interacting field observables]] (def. \ref{InteractingFieldObservables}).

=--

+-- {: .num_example #EquationsOfMotionForVacuumExpectationValues}
###### Example
**([[equations of motion]] for [[vacuum expectation values]] of [[interacting field observables]])**

Consider the [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) for the case that

$$
  \begin{aligned}
    j A
    & =
    \tau{\Sigma}( j_{sw} \phi)
    \\
    & =
    \underset{\Sigma}{\int}
      j_{sw}(x) \mathbf{\Phi}(x)
    \, dvol_\Sigma(x)
  \end{aligned}
$$

is a [[regular polynomial observable|regular]] [[linear observable]] ([this def.](A+first+idea+of+quantum+field+theory#RegularLinearFieldObservables)), hence the smearing of a [[field observable]] ([this def.](A+first+idea+of+quantum+field+theory#PointEvaluationObservables)) by an [[adiabatic switching]] of the [[source field]]

$$
  j_{sw} \;\in\; C^\infty_{cp}(\Sigma) \langle j\rangle
  \,.
$$

(Here we are notationally suppressing internal field indices, for convenience.)

In this case the [[vacuum expectation value]] of the corresponding [[effective action]] is often denoted

$$
  W(j_{sw})
$$

and regarded as a functional of the [[adiabatic switching]] $j_{sw}$ of the [[source field]].

In this case prop. \ref{EffectiveActionIsGeneratingFunction} says that if the [[vacuum state]] is [[vacuum stability|stable]], then  $W$ is the [[generating functional]] for [[interacting field observables|interacting]] (def. \ref{InteractingFieldObservables}) [[field observables]] (def. \ref{PointEvaluationObservables}) in that

$$
  \label{WFunctionalDerivative}
  \left\langle
    \mathbf{\Phi}(x)_{int}
  \right\rangle
  \;=\;
  \frac{\delta}{\delta j_{sw}(x)}
  W(j_{sw} = 0)
  \,.
$$

Assume then that there exists a corresponding functional  $\Gamma(\Phi)$ of the [[field histories]] $\Phi \in \Gamma_{\Sigma}(E_{\text{BV-BRST}})$ (def. \ref{FieldsAndFieldBundles}), which behaves like a functional [[Legendre transform]] of $W$ in that it satisfies the functional version of the defining equation of Legendre transforms (first derivatives are [[inverse functions]] of each other, see [this equation](Legendre+transformation#eq:DerivativesOfLegendreTransformsAreInverseFunctions)):

$$
 \frac{\delta }{\delta \Phi(x)}
 \Gamma
  \left(
      \frac{\delta}{\delta j_{sw}(y)} W
  \right)
  \;=\;
  \delta(x,y) j_{sw}(x)
  \,.
$$

By (eq:WFunctionalDerivative) this implies that

$$
 \frac{\delta }{\delta \Phi(x)}
 \Gamma
  \left(
    \left\langle \mathbf{\Phi}(x)_{int} \right\rangle  \right)
  \;=\;
  0
  \,.
$$

This may be read as a quantum version of the [[principle of extremal action]] (prop. \ref{PrincipleOfExtremalAction})
formulated now not for the [[field histories]] $\Phi(x)$, but for the [[vacuum expectation values]] $\langle \mathbf{\Phi}(x)_{int}\rangle$ of their corresponding [[interacting quantum field observables]].

Beware, (as in remark \ref{TerminologyForEffectiveAction}) that many texts refer to $\Gamma(\Phi)$ as the _effective action_, instead of its [[Legendre transform]], the generating functional $W(j_{sw})$.

=--

The perspective of the [[effective action]] gives a transparent picture of the order of quantum effects involved in the
[[S-matrix]], this is prop. \ref{FeynmanDiagramLoopOrder} below. In order to state this conveniently, we invoke
two basic concepts from [[graph theory]]:

+-- {: .num_defn #GraphPlanar}
###### Definition
**([[planar graphs]] and [[trees]])**

A [[finite multigraph]] (def. \ref{Graphs}) is called a _[[planar graph]]_ if it
admits an [[embedding]] into the [[plane]], hence if it may be "drawn into the plane" without intersections, in the evident way.

A [[finite multigraph]] is called a _[[tree]]_ if for any two of its [[vertices]]
there is at most one [[path]] of [[edges]] connecting them, these are examples of planar graphs. We write

$$
  \mathcal{G}_{tree}
    \subset
  \mathcal{G}
$$

for the [[subset]] of [[isomorphism classes]] of [[finite multigraphs]] with [[linear order|linearly orrdered]] [[vertices]] (def. \ref{Graphs})
on those which are [[trees]].

=--

+-- {: .num_prop #FeynmanDiagramLoopOrder}
###### Proposition
**([[loop order]] and [[tree level]] of [[Feynman perturbation series]])**

The [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) contains no negative powers of $\hbar$, hence is indeed
a [[formal power series]] also in $\hbar$:

$$
  S_{eff}(g,j)
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
  \,.
$$

and in particular

$$
  \left\langle
    S_{eff}(g,j)
  \right\rangle
  \;\in\;
  \mathbb{C}[ [ \hbar, g, j] ]
  \,.
$$

Moreover, the contribution to the effective action in the [[classical limit]] $\hbar \to 0$
is precisely that of [[Feynman amplitudes]] of those [[finite multigraphs]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) which are [[trees]] (def. \ref{GraphPlanar}); thus called the _[[tree level]]_-contribution:

$$
  S_{eff}(g,j)\vert_{\hbar = 0}
  \;=\;
  i \hbar
  \underset{\Gamma \in \mathcal{G}_{conn} \cap \mathcal{G}_{tree}}{\sum}
  \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
  \,.
$$

Finally, a [[finite multigraph]] $\Gamma$ (def. \ref{Graphs}) which is [[planar graph|planar]] (def. \ref{GraphPlanar}) and [[connected graph|connected]] (def. \ref{ConnectedGraphs}) contributes to the effective action
precisely at order

$$
  \hbar^{L(\Gamma)}
  \,,
$$

where $L(\Gamma) \in \mathbb{N}$ is the number of _[[faces]]_ of $\Gamma$, here called the _number of loops_ of the diagram;
here usually called the _[[loop order]]_ of $\Gamma$.

(Beware the terminology clash with [[graph theory]], see the discussion of [[tadpoles]] in remark \ref{Tadpoles}.)

=--

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

By def. \ref{LagrangianFieldTheoryPerturbativeScattering} the explicit $\hbar$-dependence of the [[S-matrix]] is

$$
  \mathcal{S}
  \left(
    S_{int}
  \right)
  \;=\;
  \underset{k \in \mathbb{N}}{\sum}
    \frac{1}{k!}
    \frac{1}{(i \hbar)^k}
    T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots,  S_{int}}} )
$$

and by prop. \ref{TimeOrderedProductAwayFromDiagonal} the further $\hbar$-dependence of the [[time-ordered product]] $T(\cdots)$ is

$$
  T(S_{int}, S_{int})
  \;=\;
  prod \circ
  \exp\left(
    \hbar
    \left\langle
      \Delta_F,
      \frac{\delta}{\delta \mathbf{\Phi}}
        \otimes
      \frac{\delta}{\delta \mathbf{\Phi}}
    \right\rangle
  \right)
 ( S_{int} \otimes S_{int} )
  \,,
$$

By the [[Feynman rules]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) this means that

1. each [[vertex]] of a Feynman diagram contributes a power $\hbar^{-1}$ to its Feynman amplitude;

1. each [[edge]] of a Feynman diagram contributes a power $\hbar^{+1}$ to its Feynman amplitude.

If we write

$$
  E(\Gamma), V(\Gamma) \;\in\; \mathbb{N}
$$

for the total number of [[vertices]] and [[edges]], respectively, in $\Gamma$, this means that a Feynman amplitude
corresponding to some $\Gamma \in \mathcal{G}$ contributes precisely at order

$$
  \label{GeneralFeynmanDiagramhbarContribution}
  \hbar^{E(\Gamma) - V(\Gamma)}
  \,.
$$

So far this holds for arbitrary $\Gamma$. If however $\Gamma$ is [[connected graph|connected]] (def. \ref{ConnectedGraphs}) and [[planar graph|planar]] (def. \ref{GraphPlanar}), then _[[Euler's formula]]_ asserts that

$$
  \label{ConnectedPlanarGraphEulerCharacteristic}
  E(\Gamma) - V(\Gamma)
  \;=\;
  L(\Gamma) - 1
  \,.
$$

Hence $\hbar^{L(\Gamma)- 1}$ is the order of $\hbar$ at which $\Gamma$ contributes to the [[scattering matrix]] expressed as the [[Feynman perturbation series]].

But the [[effective action]], by definition (eq:ExpansionEffectiveAction), has the same contributions
of Feynman amplitudes, but multiplied by another power of $\hbar^1$, hence it contributes at order

$$
  \hbar^{E(\Gamma) - V(\Gamma) + 1} = \hbar^{L(\Gamma)}
  \,.
$$

This proves the second claim on [[loop order]].

The first claim, due to the extra factor of $\hbar$ in the definition of the effective action, is equivalent to saying
that the Feynman amplitude of every [[connected graph|connected]] [[finite multigraph]]
contributes powers in $\hbar$ of order $\geq -1$ and contributes at order $\hbar^{-1}$ precisely if the graph is a tree.

Observe that a [[connected graph|connected]] [[finite multigraph]] $\Gamma$ with $\nu \in \mathbb{N}$ vertices (necessarily $\nu \geq 1$) has at least $\nu-1$ edges and precisely $\nu - 1$ edges if it is a tree.

To see this, consecutively remove edges from $\Gamma$ as long as possible while retaining connectivity. When this process stops, the result must be a connected tree $\Gamma'$, hence a [[connected graph|connected]] [[planar graph]] with $L(\Gamma') = 0$. Therefore [[Euler's formula]] (eq:ConnectedPlanarGraphEulerCharacteristic) implies that that $E(\Gamma') = V(\Gamma') -1$.

This means that the connected multigraph $\Gamma$ in general has a Feynman amplitude of order

$$
  \hbar^{E(\Gamma) - V(\Gamma)}
  =
  \hbar^{ \overset{\geq 0}{\overbrace{E(\Gamma) - E(\Gamma')}} + \overset{= -1}{\overbrace{E(\Gamma') - V(\Gamma)}}  }
$$

and precisely if it is a tree its Feynman amplitude is of order $\hbar^{-1}$.

=--

$\,$

**Vacuum diagrams**
 {#VacuumDiagrams}

With the [[Feynman perturbation series]] and the [[effective action]] in hand, it is now immediate to see that there
is a general contribution by [[vacuum diagrams]] (def. \ref{VacuumDiagram} below) in the [[scattering matrix]] which,
in a [[vacuum stability|stable vacuum state]],
cancels out against the prefactor $\mathcal{S}(g S_{int})$ in [[Bogoliubov's formula]] for [[interacting field observables]].

+-- {: .num_defn #VacuumDiagram}
###### Definition
**([[vacuum diagrams]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]],
and consider a choice of decomposition for field species and interaction vertices according to def. \ref{VerticesAndFieldSpecies}.

Then a [[Feynman diagram]] all whose vertices are internal vertices (def. \ref{FeynmanDiagram})
is called a _[[vacuum diagram]]_.

Write

$$
  \mathcal{G}^{Feyn}_{vac}
   \subset
  \mathcal{G}^{Feyn}
$$

for the subset of [[isomorphism classes]] of vacuum diagrams among the set of isomorphism classes of all Feynman diagrams, def. \ref{FeynmanDiagram}. Similarly write

$$
  \mathcal{G}^{Feyn}_{conn,vac}
  \;\coloneqq\;
  \mathcal{G}^{Feyn}_{conn}
    \cap
  \mathcal{G}^{Feyn}_{vac}
  \;\subset\;
  \mathcal{G}^{Feyn}
$$

for the subset of [[isomorphism classes]] of Feynman diagrams which are both vacuum diagrams as well as [[connected graphs]] (def. \ref{ConnectedGraphs}).

Finally write

$$
  S_{eff,vac}(g)
  \;\coloneqq\;
  \underset{ { (\Gamma,vertlab,edgelab) } \atop { \in \mathcal{G}_{conn,vac} } }{\sum}
  (\Gamma,vertlab, edgelab)(g S_{int})
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar , g ] ]
$$

for the sub-series of that for the [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) given
only by those connected diagrams which are also vacuum diagrams.

=--

+-- {: .num_example}
###### Example
**(2-vertex [[vacuum diagram]] in [[QED]])**

The [[vacuum diagram]] (def. \ref{VacuumDiagram}) with two [[electron-photon interaction]]-vertices in [[quantum electrodynamics]] (example \ref{LagrangianQED}) is:

<center>
<img src="https://ncatlab.org/nlab/files/QEDVacuumDiagram.png" width="200">
</center>

=--

+-- {: .num_example #SMatrixVacuumContribution}
###### Example
**([[vacuum diagram]]-contribution to [[S-matrices]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and let $g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g , j] ]\langle g,j \rangle$
be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]],
and consider a choice of decomposition for field species and interaction vertices according to def. \ref{VerticesAndFieldSpecies}.

Then the [[Feynman perturbation series]]-expansion of the [[S-matrix]] (example \ref{FeynmanPerturbationSeries})
of the [[interaction]]-term $g S_{int}$ alone (no [[source field]]-contribution) is the series of [[Feynman amplitudes]] that are labeled by [[vacuum diagrams]] (def. \ref{VacuumDiagram}), hence (by prop. \ref{LogarithmEffectiveAction}) the exponential of the
vacuum [[effective action]] $S_{eff,vac}$ (def. \ref{VacuumDiagram}):

$$
  \begin{aligned}
    \mathcal{S}(g S_{int})
    & =
    \exp_\cdot\left( \tfrac{1}{i \hbar} S_{eff,vac}(g,j) \right)
    \\
    & =
    \underset{\Gamma \in \mathcal{G}_{vac}}{\sum}
      \Gamma\left(g S_{int}\right)
  \end{aligned}
  \,.
$$

More generally, the S-matrix with [[source field]]-contribution $j A$ included always splits as a
_pointwise_ product of the vacuum S_matrix with the [[Feynman perturbation series]] over all [[Feynman graphs]] with at least
one external vertex:

$$
  \begin{aligned}
  \mathcal{S}(g S_{int} + j A)
  \;=\;
  \mathcal{S}(g S_{int})
  \cdot
  \underset{
    \text{Feynman perturbation series}
    \atop
    \text{over diagrams with at least one external vertex}
  }{
   \underbrace{
     \exp_\cdot
     \left(
       \tfrac{1}{i \hbar} \left( S_{eff}(g,j) - S_{eff,vac}(g) \right)
     \right)
   }
  }
  \,,
  \end{aligned}
$$

Hence if the [[free field]] [[vacuum state]] is stable with respect to the interaction $g S_{int}$, according to def. \ref{VacuumStability}, then the [[vacuum expectation value]] of a [[time-ordered product]] of [[interacting field observables]] $j (A_i)_{int}$ (example \ref{InteractinFieldTimeOrderedProduct}) and hence in particular of [[scattering amplitudes]] (example \ref{ScatteringAmplitudeFromInteractingFieldObservables}) is
given by the [[Feynman perturbation series]] (example \ref{FeynmanPerturbationSeries}) over
just the non-vacuum [[Feynman diagrams]], hence over all those diagram that have at least one one external vertex

$$
  \begin{aligned}
  &
  \left(
    {\, \atop \,}
    supp(A_1)
      {\vee\!\!\!\wedge}
    supp(A_2)
      {\vee\!\!\!\wedge}
      \cdots
      {\vee\!\!\!\wedge}
    supp(A_n)
    {\, \atop \,}
  \right)
  \\
  & \Rightarrow
  \left\langle
    {\, \atop \,}
    (A_1)_int
    (A_2)_{int}
      \cdots
    (A_n)_{int}
    {\, \atop \,}
  \right\rangle
  \;=\;
  \frac{d^n}{ d j_1 \cdots d j_n}
  \left(
    \underset{\Gamma \in \mathcal{G} \setminus \mathcal{G}_{vac}   }{\sum}
    \Gamma(g S_{int} + \sum_i j_i A_i)
  \right)_{
    \vert j_1, \cdots, j_n = 0
  }
  \,.
  \end{aligned}
$$

This is the way in which the [[Feynman perturbation series]] is used in practice for computing [[scattering amplitudes]].

=--

$\,$

**Interacting quantum BV-Differential**
 {#InteractingQantumBVDifferential}

So far we have discussed, starting with a [[BV-BRST formalism|BV-BRST]] [[gauge fixing|gauge fixed]] [[free field]] [[vacuum]],
the perturbative construction of [[interacting field algebras of observables]] (def. \ref{QuntumMollerOperator}) and their organization
in increasing powers of $\hbar$ and $g$ ([[loop order]], prop. \ref{FeynmanDiagramLoopOrder}) via the [[Feynman perturbation series]] (example \ref{FeynmanPerturbationSeries}, example \ref{SMatrixVacuumContribution}).

But this [[interacting field algebra of observables]] still involves all the [[auxiliary fields]] of the [[BV-BRST formalism|BV-BRST]] [[gauge fixing|gauge fixed]] [[free field]] [[vacuum]] (example \ref{FieldSpeciesQED}), while the
actual physical [[gauge invariance|gauge invariant]] [[on-shell]] observables should be (just) the [[cochain cohomology]] of the
[[BV-BRST differential]] on this enlarged space of observables. Hence for the construction of [[perturbative QFT]] to conclude,
it remains to pass the [[BV-BRST differential]] of the [[free field]] [[Wick algebra]] of observables to
a [[differential]] on the [[interacting field algebra]], such that its [[cochain cohomology]] is well defined.

Since the [[time-ordered products]] away from coinciding interaction points and as well as on [[regular polynomial observables]] are uniquely fixed
(prop. \ref{TimeOrderedProductAwayFromDiagonal}), one finds that also this _interacting quantum BV-differential_
is uniquely fixed, on [[regular polynomial observables]], by [[conjugation]] with the [[quantum Møller operators]] (def. \ref{BVDifferentialInteractingQuantum}).
The formula that characterizes it there is called the _[[quantum master equation]]_ or equivalently the _[[quantum master Ward identity]]_ (prop. \ref{QuantumMasterEquation} below).

When [[extension of distributions|extending]] to coinciding interaction points via [[renormalization|("re"-)normalization]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}) these identities are not guaranteed to hold anymore, but may
be imposed as [[renormalization conditions]] (def. \ref{RenormalizationConditions}, prop. \ref{BasicConditionsRenormalization}).
Quantum correction to the [[master Ward identity]] then imply corrections to [[Noether's theorem|Noether current]] [[conserved current|conservation laws]]; this we discuss [below](#WardIdentities).

$\,$

For the following discussion, recall from the [previous chapter](#FreeQuantumFields) how the global BV-differential

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

on [[regular polynomial observables]] (def. \ref{BVDifferentialGlobal}) as well as the global [[antibracket]] $\{-,-\}$ (def. \ref{ComplexBVBRSTGlobal}) are [[conjugation|conjugated]] into the [[time-ordered product]] via the time ordering operator $\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-}$ (def. \ref{AntibracketTimeOrdered}, prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket}), which makes

In the same way we may use the [[quantum Møller operators]] to conjugate the BV-differential into the regular part of the [[interacting field algebra of observables]]:

+-- {: .num_defn #BVDifferentialInteractingQuantum}
###### Definition
**(interacting quantum [[BV-differential]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree} and let

$$
  S_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar, g, j] ]
$$

be a [[regular polynomial observables]], regarded as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]].

Then the _interacting quantum [[BV-differential]]_ on the [[interacting field algebra]] on [[regular polynomial observables]]
(def. \ref{FieldAlgebraObservablesInteracting}) is the [[conjugation]] of the plain global [[BV-differential]] $\{-S',-\}$ (def. \ref{ComplexBVBRSTGlobal}) by the [[quantum Møller operator]] induced by $S_{int}$ (def. \ref{MollerOperatorOnRegularPolynomialObservables}):

$$
  \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  \,.
$$

=--

([Rejzner 11, (5.38)](quantum+master+equation#Rejzner11))

+-- {: .num_prop #QuantumMasterEquation}
###### Proposition
**([[quantum master equation]] and [[quantum master Ward identity]] on [[regular polynomial observables]])**

Consider an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]] in the form of a [[regular polynomial observable]] in degree 0

$$
  S_{int}
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ]
  \,,
$$

Then the following are equivalent:

1. The _[[quantum master equation]]_ (QME)

   $$
     \label{OnRegularObservablesQuantumMasterEquation}
      \tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}}
      +
      i \hbar  \Delta_{BV}( S' + S_{int} )
     \;=\;
     0
     \,.
   $$

1. The  [[perturbative S-matrix]] (def. \ref{OnRegularObservablesPerturbativeSMatrix}) is $BV$-closed

   $$
     \{-S', \mathcal{S}(S_{int})\} = 0
     \,.
   $$

1. The quantum _[[master Ward identity]]_ (MWI) on [[regular polynomial observables]] _in terms of [[retarded products]]_:

   $$
     \label{OnRegularObservablesQuantumMasterWardIdentity}
     \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1}
     \;=\;
     -
     \left(
       \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}}
       + i \hbar   \Delta_{BV}
     \right)
   $$

   ([Dütsch 18, (4.2)](Ward+identity#Duetsch18))

   expressing the interacting quantum [[BV-differential]] (def. \ref{BVDifferentialInteractingQuantum}) as the sum of the [[time-ordered product|time-ordered]] [[antibracket]] (def. \ref{AntibracketTimeOrdered}) with the _total_ [[action functional]] $S' + S_{int}$ and $i \hbar$ times the [[BV-operator]] ([BV-operator](BV-operator#ForGaugeFixedFreeLagrangianFieldTheoryBVOperator)).

1. The quantum _[[master Ward identity]]_ (MWI) on [[regular polynomial observables]] _in terms of [[time-ordered products]]_:

   $$
     \label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}
     \mathcal{S}(-S_{int})
       \star_F
     \{-S', \mathcal{S}(S_{int}) \star_F  (-)\}
     \;=\;
     -
     \left(
       \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}}
       + i \hbar   \Delta_{BV}
     \right)
   $$

   ([Dütsch 18, (4.8)](Ward+identity#Duetsch18))

=--

([Rejzner 11, (5.35) - (5.38)](quantum+master+equation#Rejzner11), following [Hollands 07, (342)-(345)](Ward+identity#Hollands07))

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

To see that the first two conditions are equivalent, we compute as follows

$$
  \label{QuantumMasterOnRegularObservablesBVDifferentialOfSMatrixInTerms}
  \begin{aligned}
    \left\{
      -S',  \mathcal{S}(S_{int})
    \right\}
    & =
    \left\{
      -S'
      ,
      \exp_{\mathcal{T}}
      \left(
        \tfrac{1}{i \hbar} S_{int}
      \right)
    \right\}
    \\
    & =
    \underset{
      {
        \tfrac{-1}{i \hbar} \{S',S\}_{\mathcal{T}}
      }
      \atop
      {
        \star_F
        \exp_{\mathcal{T}}
        \left(
          \tfrac{1}{i \hbar} S_{int}
        \right)
      }
    }{
    \underbrace{
    \left\{
      -S'
      ,
      \exp_{\mathcal{T}}
      \left(
        \tfrac{1}{i \hbar} S_{int}
      \right)
    \right\}_{\mathcal{T}}
    }
    }
    -
    i \hbar
    \underset{
      {
        \left(
          \tfrac{1}{i \hbar}
          \Delta_{BV}(S_{int})
          +
          \tfrac{1}{2 (i \hbar)^2}
          \left\{
            S_{int}, S_{int}
          \right\}_{\mathcal{T}}
        \right)
      }
      \atop
      {
        \star_{F}
        \exp_{\mathcal{T}}
        \left(
          \tfrac{1}{i \hbar}
          S_{int}
        \right)
      }
    }{
    \underbrace{
    \Delta_{BV}
    \left(
      \exp_{\mathcal{T}}
      \left(
        \tfrac{1}{i \hbar} S_{int}
      \right)
    \right)
    }
    }
    \\
    & =
    \tfrac{-1}{i \hbar}
    \underset{ \text{QME} }{
    \underbrace{
    \left(
      \{S',S_{int}\}
      +
      \tfrac{1}{2}\{S_{int}, S_{int}\}
      +
      i \hbar \Delta_{BV}(S_{int})
    \right)
    }
    }
      \star_F
    \exp_{\mathcal{T}}
    \left(
      \tfrac{1}{i \hbar}
      S_{int}
    \right)
  \end{aligned}
$$

Here in the first step we used the definition of the [[BV-operator]] (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) to rewrite the plain antibracket in terms of the time-ordered antibracket (def. \ref{AntibracketTimeOrdered}), then under the second brace we used that the time-ordered antibracket is the failure of the BV-operator to be a derivation (prop. \ref{AntibracketBVOperatorRelation}) and under the first brace the consequence of this statement for application to exponentials (example \ref{TimeOrderedExponentialBVOperator}). Finally we collected terms, and to "complete the square" we added the terms on the left of

$$
  \frac{1}{2} \underset{= 0}{\underbrace{\{S', S'\}_{\mathcal{T}}}}
  -
  i \hbar \underset{ = 0}{\underbrace{ \Delta_{BV}(S')}} = 0
$$

which vanish because, by definition of [[gauge fixing]] (def. \ref{GaugeFixingLagrangianDensity}), the free gauge-fixed action functional $S'$ is independent of [[antifields]].

But since the operation $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)$ has the [[inverse]] $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar}  S_{int} \right)$, this implies the claim.

Next we show that the [[quantum master equation]] implies the [[quantum master Ward identities]].

We use that the BV-differential $\{-S',-\}$ is a [[derivation]] of the [[Wick algebra]] product $\star_H$ (lemma \ref{DerivationBVDifferentialForWickAlgebra}).

First of all this implies that with $\{-S', \mathcal{S}(S_{int})\} = 0$ also
$\{-S', \mathcal{S}(S_{int})^{-1}\} = 0$.

Thus we compute as follows:

$$
  \begin{aligned}
    \{-S', -\} \circ \mathcal{R}^{-1}(A)
    & =
    \{-S', \mathcal{R}^{-1}(A)\}
    \\
    & =
    \left\{
      { \, \atop \, }
      -S',
      \mathcal{S}(S_{int})^{-1}
        \star_H
      \left( \mathcal{S}(S_{int}) \star_F a \right)
      {\, \atop \,}
    \right\}
    \\
    & = \phantom{+}
    \underset{
      = 0
    }{
    \underbrace{
    \left\{
      -S', \mathcal{S}(S_{int})^{-1}
    \right\}
    }
    }
      \star_H
    \left(
      \mathcal{S}(S_{int}) \star_F A
    \right)
    \\
    & \phantom{=}
    +
    \mathcal{S}(S_{int})^{-1}
      \star_H
    \left\{
      -S',
      \mathcal{S}(S_{int}) \star_F A
    \right\}
    \\
    & =
    \mathcal{S}(S_{int})^{-1}
      \star_H
    \left(
      \underset{
        = 1
      }{
      \underbrace{
        \mathcal{S}(+ S_{int})
          \star_F
        \mathcal{S}(- S_{int})
      }
      }
        \star_F
      \left\{
        -S',
        \mathcal{S}(S_{int}) \star_F A
      \right\}
   \right)
   \\
   & =
    \mathcal{S}(S_{int})^{-1}
      \star_H
    \left(
      \mathcal{S}(+ S_{int})
        \star_F
      \underset{ (\ast) }{
      \underbrace{
        \mathcal{S}(- S_{int})
          \star_F
        \left\{
          -S',
          \mathcal{S}(S_{int}) \star_F A
        \right\}
     }
     }
   \right)
   \\
   & =
   \mathcal{R}^{-1}
   \left(
     \underset{ (\ast) }{
     \underbrace{
       \phantom{\, \atop \,}
       \mathcal{S}(-S_{int})
         \star_F
       \left\{
         -S',
         \mathcal{S}(S_{int}) \star_F A
       \right\}
       }
       }
   \right)
  \end{aligned}
$$

By applying $\mathcal{R}$ to both sides of this equation, this means
first of all that the interacting quantum BV-differential
is equivalently given by

$$
  \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}
  \;=\;
  \mathcal{S}(-S_{int}) \star_F \{-S', \mathcal{S}(S_{int}) \star_F  (-)\}
  \,,
$$

hence that if either version (eq:OnRegularObservablesQuantumMasterWardIdentity) or (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered)  of the [[master Ward identity]] holds, it implies the other.

Now expanding out the definition of $\mathcal{S}$ (def. \ref{OnRegularObservablesPerturbativeSMatrix}) and expressing $\{-S',-\}$ via the [[time-ordered product|time-ordered]] [[antibracket]] (def. \ref{AntibracketTimeOrdered}) and the [[BV-operator]] $\Delta_{BV}$ (prop. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) as

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

(on [[regular polynomial observables]]), we continue computing as follows:

$$
  \label{QMESecondStep}
  \begin{aligned}
    & \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}( A )
    \\
    & =
   \exp_{\mathcal{T}}
    \left(
      \tfrac{-1}{i \hbar} S_{int}
    \right)
      \star_F
    \left\{
      -S',
      \exp_{\mathcal{T}}
      \left(
         \tfrac{1}{i \hbar} S_{int}
      \right)
        \star_F
      A
    \right\}
    \\
    & =
    \exp_{\mathcal{T}}
    \left(
      \tfrac{-1}{i \hbar}
        S_{int}
    \right)
      \star_F
    \left(
      \left\{
        -S',
        \exp_{\mathcal{T}}
        \left(
          \tfrac{ 1 }{i \hbar} S_{int}
        \right)
          \star_F
        A
      \right\}_{\mathcal{T}}
      -
      i \hbar
      \Delta_{BV}
      \left(
        \exp_{\mathcal{T}}
        \left(
           \tfrac{ 1 }{i \hbar} S_{int}
        \right)
          \star_F
        A
      \right)
    \right)
    \\
    &
    \phantom{+}
    =
    \tfrac{1}{i \hbar} \{ -S', S_{int}  \}_{\mathcal{T}} \star_F   A
    +
    \{-S', A\}_{\mathcal{T}}
    \\
    &
    \phantom{=}
    -
    i \hbar
    \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int}\right)
    \star_F
    \left(
      \underset{
        {
          \left(
          \tfrac{1}{i \hbar}\Delta_{BV}(S_{int})
          +
          \tfrac{1}{2 (i \hbar)^2} \left\{ S_{int}, S_{int} \right\}
          \right)
        }
        \atop
        {
          \star_F \exp_{\mathcal{T}}\left( \tfrac{ 1 }{i \hbar} S_{int} \right)
        }
      }{
      \underbrace{
        \Delta_{BV}
        \left(
          \exp_{\mathcal{T}}
          \left(
            \tfrac{ 1}{i \hbar} S_{int}
          \right)
        \right)
      }
      }
      \star_F A
      \,+\,
       \exp_{\mathcal{T}}
       \left(
         \tfrac{ 1}{i \hbar} S_{int}
       \right)
       \star_F
       \Delta_{BV}(A)
       \,+\,
       \underset{
         {
           \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)
         }
         \atop
         {
           \star_F
           \tfrac{ 1}{i \hbar}
           \{S_{int}, A\}
         }
       }{
       \underbrace{
         \left\{
           \exp_{\mathcal{T}}
           \left(
             \tfrac{ 1}{i \hbar} S_{int}
           \right)
           \,,\,
           A
         \right\}_{\mathcal{T}}
      }
      }
    \right)
    \\
    & =
    -
    \left(
      \{ S' + S_{int}\,,\, A\}_{\mathcal{T}}
        +
      i \hbar \Delta_{BV}(A)
    \right)
    \\
    & \phantom{=}
    -
    \tfrac{1}{i \hbar}
    \underset{ \text{QME} }{
    \underbrace{
    \left(
      \tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}}
      +
      i \hbar  \Delta_{BV}( S' + S_{int} )
    \right)
    }}
      \star_F A
    \\
    & =
    -
    \left(
      \{ S' + S_{int}\,,\, A\}_{\mathcal{T}}
        +
      i \hbar \Delta_{BV}(A)
    \right)
  \end{aligned}
$$

Here in the line with the braces we used that the [[BV-operator]] is a [[derivation]] of the [[time-ordered product]] up to correction by the time-ordered [[antibracket]] (prop. \ref{AntibracketBVOperatorRelation}), and under the first brace we used the effect of that property on time-ordered exponentials (example \ref{TimeOrderedExponentialBVOperator}), while under the second brace we used that $\{(-),A\}_{\mathcal{T}}$ is a derivation of the time-ordered product. Finally we have collected terms, added $0 = \{S',S'\} + i \hbar \Delta_{BV}(S')$ as before, and then used the QME.

This shows that the quantum [[master Ward identities]] follow from the [[quantum master equation]]. To conclude, it is now sufficient to show that, conversely, the MWI in terms of, say, retarded products implies the QME.

To see this, observe that with the BV-differential being nilpotent, also its conjugation by $\mathcal{R}$ is, so that with the above we have:

$$
  \begin{aligned}
  & \left( \{-S',-\}\right)^2 = 0
  \\
  \Leftrightarrow
  \;
  &
  \left( \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \right)^2 = 0
  \\
  \Leftrightarrow
  \;
  &
  \underset{
    \left\{
      {\, \atop \,}
      \tfrac{1}{2}\{S' + S_{int}, S' + S_{int}\}_{\mathcal{T}}
      +
      i \hbar \Delta_{BV}(S' + S_{int})
      \,,\,
      (-)
    \right\}
  }{
  \underbrace{
    \left(
      \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV}
    \right)^2
  }
  } = 0
  \end{aligned}
$$

Here under the brace we computed as follows:

$$
  \begin{aligned}
  \left(
    \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV}
  \right)^2
  & =
  \phantom{+}
  \underset{
    \tfrac{1}{2} \{ \{S' + S, S'+ S\}_{\mathcal{T}}, (-) \}_{\mathcal{T}}
  }{
  \underbrace{
    \{S' + S_{int}, \{S' + S_{int}\}_{\mathcal{T}}, (-) \}_{\mathcal{T}}
  }}
  \\
  &
  \phantom{=}
  +
  i \hbar
  \underset{
    \{  \Delta_{BV}(S'+ S)\,,\, (-)  \}_{\mathcal{T}}
  }{
  \underbrace{
  \left(
    \{S' + S_{int}, (-)\}_{\mathcal{T}} \circ \Delta_{BV}
    +
    \Delta_{BV} \circ \{S' + S_{int}, (-)\}_{\mathcal{T}}
  \right)
  }}
  \\
  &
  \phantom{=}
  +
  (i \hbar)^2
   \underset{= 0}
   {
     \underbrace{
       \Delta_{BV} \circ \Delta_{BV}
      }
    }
  \end{aligned}
  \,.
$$

where, in turn, the term under the first brace follows by the graded [[Jacobi identity]], the one under the second brace by Henneaux-Teitelboim (15.105c) and the one under the third brace by Henneaux-Teitelboim (15.105b).

=--

$\,$

**[[Ward identities]]**
  {#WardIdentities}

The _[[quantum master Ward identity]]_ (prop. \ref{QuantumMasterEquation}) expresses the relation between the [[quantum field theory|quantum]] (measured by [[Planck's constant]] $\hbar$) [[interacting field theory|interacting]] (measured by the [[coupling constant]] $g$) [[equations of motion]] to the [[classical field theory|classical]] [[free field]] [[equations of motion]] at $\hbar, g\to 0$ (remark \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below). As such it generalizes the [[Schwinger-Dyson equation]] (prop. \ref{DysonSchwinger}), to which it reduces for $g = 0$ (example \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below) as well as the _classical master Ward identity_, which is the case for $\hbar = 0$ (example \ref{MasterWardIdentityClassical} below).

Applied to products of the [[equations of motion]] with any given [[observable]], the master Ward identity becomes a particular _Ward identity_.

This is of interest notably in view of [[Noether's theorem]] (prop. \ref{NoethersFirstTheorem}), which says that every [[infinitesimal symmetry of the Lagrangian]] of, in particular, the given [[free field theory]], corresponds to a [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}), hence a [[horizontal differential form]] whose [[total spacetime derivative]] vanishes up to a term proportional to the [[equations of motion]]. Under [[transgression of variational differential forms|transgression]] to [[local observables]] this is a relation of the form

$$
  div \mathbf{J} = 0 \phantom{AAA} \text{on-shell}
  \,,
$$

where "on shell" means up to the ideal generated by the [[classical field theory|classical]] [[free field theory|free]] [[equations of motion]]. Hence for the case of [[local observables]] of the form $div \mathbf{J}$, the quantum Ward identity expresses the possible failure of the original [[conserved current]] to actually be conserved, due to both quantum effects ($\hbar$) and interactions ($g$). This is the form in which Ward identities are usually understood (example \ref{NoetherCurrentConservationQuantumCorrection} below).

As one [[extension of distributions|extends]] the [[time-ordered products]] to coinciding interaction points in [[renormalization|("re"-)normalization]]
of the [[perturbative QFT]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}), the [[quantum master equation]]/[[master Ward identity]] becomes a _[[renormalization condition]]_ (def. \ref{RenormalizationConditions}, prop. \ref{BasicConditionsRenormalization}). If this condition fails, one speaks of a _[[quantum anomaly]]_. Specifically if the Ward identity for an [[infinitesimal gauge symmetry]] is violated, one speaks of a _[[gauge anomaly]]_.

$\,$

+-- {: .num_defn #OnRegularPolynomialObservablesMasterWardIdentity}
###### Definition

Consider a [[free field theory|free]] [[gauge fixing|gauge fixed]] [[Lagrangian field theory]] $(E_{\text{BV-BRST}}, \mathbf{L}')$ (def. \ref{GaugeFixingLagrangianDensity}) with global [[BV-differential]] on [[regular polynomial observables]]

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

(def. \ref{ComplexBVBRSTGlobal}).

Let moreover

$$
  g S_{int}
   \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar , g  ] ]
$$

be a [[regular polynomial observable]] (regarded as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]]) such that the total action $S' + g S_{int}$ satisfies the [[quantum master equation]] (prop. \ref{QuantumMasterEquation}); and write

$$
  \mathcal{R}^{-1}(-)
  \;\coloneqq\;
  \mathcal{S}(g S_{int})^{-1}
    \star_H
  (\mathcal{S}(g S_{int}) \star_F (-))
$$

for the corresponding [[quantum Møller operator]] (def. \ref{MollerOperatorOnRegularPolynomialObservables}).

Then by prop. \ref{QuantumMasterEquation} we have

$$
  \label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}
  \{-S',(-)\} \circ \mathcal{R}^{-1} \;=\;
  \mathcal{R}^{-1}
  \left(\left\{ -(S' + g S_{int}) \,,\, (-) \right\}_{\mathcal{T}} -i \hbar   \Delta_{BV}\right)
$$

This is the _quantum master Ward identity_ on [[regular polynomial observables]], i.e. before [[renormalization]].

=--

([Rejzner 13, (37)](Ward+identity#Rejzner13))

+-- {: .num_remark #QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations}
###### Remark
**([[quantum master Ward identity]] relates [[quantum field theory|quantum]] [[interacting field theory|interacting field]] [[equation of motion|EOMs]] to [[classical field theory|classical]] [[free field]] [[equation of motion|EOMs]])**

For $A \in PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar, g] ] $ the [[quantum master Ward identity]] on [[regular polynomial observables]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reads

$$
  \label{RearrangedMasterQuantumWard}
  \mathcal{R}^{-1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}}
    -
    i \hbar   \Delta_{BV}(A)
  \right)
  \;=\;
  \{-S', \mathcal{R}^{-1}(A) \}
$$

The term on the right is manifestly in the [[image]] of the global [[BV-differential]] $\{-S',-\}$ of the [[free field theory]] (def. \ref{ComplexBVBRSTGlobal}) 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}).

Hence

$$
  \mathcal{R}^{-1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}}
    -
    i \hbar   \Delta_{BV}(A)
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
$$

In contrast, the left hand side is the [[interacting field observable]] (via def. \ref{MollerOperatorOnRegularPolynomialObservables}) of the sum of the [[time-ordered product|time-ordered]] [[antibracket]] with the [[action functional]] of the [[interacting field theory]] and a quantum correction given by the [[BV-operator]]. If we use the definition of the [[BV-operator]] $\Delta_{BV}$ (def. \ref{RearrangedMasterWardWithOnShell}) we may equivalently re-write this as

$$
  \label{RearrangedMasterWardWithOnShell}
  \mathcal{R}^{-1}
  \left(
    \left\{ -S'  \,,\, A \right\}
    +
    \left\{ -g S_{int} \,,\, A  \right\}_{\mathcal{T}}
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
$$

Hence the [[quantum master Ward identity]] expresses a relation between the ideal spanned by the [[classical field theory|classical]] [[free field theory|free field]] [[equations of motion]] and the [[quantum field theory|quantum]] [[interacting field theory|interacting field]] equations of motion.

=--

+-- {: .num_example #SchwingerDysonReductionOfQuantumMasterWardIdentity}
###### Example
**([[free field]]-limit of [[master Ward identity]] is [[Schwinger-Dyson equation]])**

In the [[free field]]-limit $g \to 0$ (noticing that in this limit $\mathcal{R}^{-1} = id$) the [[quantum master Ward identity]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reduces to

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

which is the defining equation for the [[BV-operator]] (eq:BVOperatorDefiningRelation), hence is isomorphic (under $\mathcal{T}$) to the [[Schwinger-Dyson equation]] (prop. \ref{DysonSchwinger})

=--

+-- {: .num_example #MasterWardIdentityClassical}
###### Example
**([[classical limit]] of [[quantum master Ward identity]])**

In the [[classical limit]] $\hbar \to 0$ (noticing that the classical limit of $\{-,-\}_{\mathcal{T}}$ is $\{-,-\}$) the [[quantum master Ward identity]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reduces to

$$
  \mathcal{R}^{1}
  \left(
    \left\{ -(S' + g S_{int}) \,,\, A \right\}
  \right)
  \;=\;
  \{-S', \mathcal{R}^{-1}(A) \}
$$

This says that the [[interacting field observable]] corresponding to the global [[antibracket]] with the action functional of the [[interacting field theory]] vanishes on-shell, classically.

Applied to an observable which is [[linear map|linear]] in the [[antifields]]

$$
  A
    \;=\;
  \underset{\Sigma}{\int}
    A^a(x)
    \mathbf{\Phi}^\ddagger_a(x)
  \,
  dvol_\Sigma(x)
$$

this yields

$$
  \begin{aligned}
    0
    & =
    \{-S', \mathcal{R}^{-1}(A)\}
    +
    \mathcal{R}^{-1}
    \left(
      \left\{ -(S' + S_{int}) \,,\, A \right\}_{\mathcal{T}}
    \right)
    \\
    & =
    \underset{\Sigma}{\int}
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \mathcal{R}^{-1}(A^a(x))
    \, dvol_\Sigma(x)
    +
    \mathcal{R}^{-1}
    \left(
      \underset{\Sigma}{\int}
          A^a(x) \frac{\delta (S' + S_{int})}{\delta \mathbf{\Phi}^a(x)}
        \, dvol_\Sigma(x)
    \right)
  \end{aligned}
$$

This is the _classical master Ward identity_ according to ([Dütsch-Fredenhagen 02](Ward+identity#DuetschFredenhagen02), [Brennecke-Dütsch 07, (5.5)](Ward+identity#BrennecketDuetsch07)), following ([Dütsch-Boas 02](Ward+identity#DuetschBoas02)).

=--

+-- {: .num_example #NoetherCurrentConservationQuantumCorrection}
###### Example
**(quantum correction to [[Noether's theorem|Noether current]] [[conserved current|conservation]])**

Let $v \in \Gamma^{ev}_\Sigma(T_\Sigma(E_{\text{BRST}}))$ be an [[evolutionary vector field]], which is an [[infinitesimal symmetry of the Lagrangian]] $\mathbf{L}'$, and let $J_{\hat v} \in \Omega^{p,0}_\Sigma(E_{\text{BV-BRST}})$ the corresponding [[conserved current]], by [[Noether's theorem|Noether's theorem I]] (prop. \ref{NoethersFirstTheorem}), so that

$$
  \begin{aligned}
    d J_{\hat v}
    & =
    \iota_{\hat v}  \delta \mathbf{L}'
    \\
    & =
    (v^a dvol_\Sigma) \frac{\delta_{EL} L'}{\delta \phi^a}
    \phantom{AAA}
    \in
    \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})
  \end{aligned}
$$

by (eq:CurrentNoetherConservation), where in the second line we just rewrote the expression in components (eq:EulerLagrangeEquationGeneral)

$$
  v^a
  \,,
  \frac{\delta_{EL} L'}{\delta \phi^a}
  \;\in \Omega^{0,0}_\Sigma(E_{\text{BV-BRST}})
$$

and re-arranged suggestively.

Then for $a_{sw} \in C^\infty_{cp}(\Sigma)$ any choice of [[bump function]], we obtain the [[local observables]]

$$
  \begin{aligned}
    A_{sw}
    & \coloneqq
    \underset{\Sigma}{\int}
      \underset{
        A^a(x)
      }{
      \underbrace{
        a_{sw}(x)
        v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
      }
      }
      \mathbf{\Phi}^\ddagger_a(x)
      \,
      dvol_\Sigma(x)
    \\
    & \coloneqq
    \tau_\Sigma( a_{sw}  v^a \phi^{\ddagger}_a \, dvol_\Sigma)
  \end{aligned}
$$

and

$$
  \begin{aligned}
    (div \mathbf{J})_{sw}
    & \coloneqq
    \underset{\Sigma}{\int}
      \underset{
        A^a(x)
      }{
      \underbrace{
        a_{sw}(x)
        v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
      }
      }
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
      \,
      dvol_\Sigma(x)
    \\
    & \coloneqq
    \tau_\Sigma
    \left(
      a_{sw}  v^a \frac{\delta_{EL} \mathbf{L}'}{\delta \phi^a}
      \, dvol_\Sigma
    \right)
  \end{aligned}
$$

by [[transgression of variational differential forms]].

This is such that

$$
  \left\{
    -S' , A_{sw}
  \right\}
  =
  (div \mathbf{J})_{sw}
  \,.
$$

Hence applied to this choice of local observable $A$, the quantum master Ward identity (eq:RearrangedMasterWardWithOnShell) now says that

$$
  \mathcal{R}^{-1}
  \left(
    {\, \atop \,}
    (div \mathbf{J})_{sw}
  \right)
  \;=\;
  \mathcal{R}^{-1}
  \left(
    {\, \atop \,}
      \{g S_{int}, A_{sw} \}_{\mathcal{T}}
    {\, \atop \,}
  \right)
    \phantom{AAA}
  \text{on-shell}
$$

Hence the [[interacting field observable]]-version $\mathcal{R}^{-1}(div\mathbf{J})$ of $div \mathbf{J}$ need not vanish itself on-shell, instead there may be a correction as shown on the right.

=--

$\,$

This concludes our discussion of perturbative [[quantum observables]] of [[interacting field theories]]. In the _[next chapter](#Renormalization)_ wé discuss explicitly the [[induction|inductive]] construction via _[[renormalization|("re"-)normalization]]_
of [[time-ordered products]]/[[Feynman amplitudes]] as well as the various incarnations of the [[renormalization group|re-normalization group]]
passing between different choices of such [[renormalization|("re"-)normalizations]].