This is one chapter in a series on Mathematical Quantum Field Theory.

The previous chapter is 14. Free quantum fields.

The next chapter is 16. Renormalization.

15. Interacting quantum fields

In this chapter we discuss the following topics:

In the previous chapter we have found the quantization of free Lagrangian field theories by first choosing a gauge fixed BV-BRST-resolution of the algebra of 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; 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).

In principle the perturbative quantization of such non-free field theory interacting field theories proceeds the same way: One picks a BV-BRST-gauge fixing, computes the Peierls-Poisson bracket on the resulting covariant phase space (Khavkine 14) 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, Hawkins-Rejzner 16, prop. 15.25 below) that the resulting interacting formal deformation quantization may equivalently be expressed in terms of scattering amplitudes (example 15.12 below): These are the probability amplitudes for plane waves of free fields to come in from the far past, then interact in a compact region of spacetime via the given interaction (adiabatically switched-off outside that region) and 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 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 adiabatically switched interaction action functional with itself (def. 15.3 below). The combinatorics of the terms in this exponential is captured by Feynman diagrams (prop. 15.51 below), which, with some care (remark 15.21 below), may be thought of as finite multigraphs (def. 15.50 below) whose edges are worldlines of virtual particles and whose vertices are the interactions that these particles undergo (def. 15.55 below).

The axiomatic definition of S-matrices for relativistic Lagrangian field theories and their rigorous construction via (“re”-)normalization of time-ordered products (def. 15.46 below) is called causal perturbation theory, due to (Epstein-Glaser 73). This makes precise and well-defined the would-be path integral quantization of interacting field theories (remark 15.16 below) and removes the errors (remark 15.19 below) and ensuing puzzlements (expressed in Feynman 85) that plagued the original informal conception of perturbative quantum field theory due to Schwinger-Tomonaga-Feynman-Dyson (remark 15.20 below).

The equivalent re-formulation of the formal deformation quantization of interacting field theories in terms of scattering amplitudes (prop. 15.25 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 powers of Planck’s constant and the coupling constant.

Moreover, the mathematical passage from scattering amplitudes to the actual interacting field algebra of quantum observables (def. 15.24 below) corresponding to the formal deformation quantization is well understood, given via “Bogoliubov’s formula” by the quantum Møller operators (def. 15.8 below).

Via Bogoliubov’s formula every perturbative S-matrix scheme (def. 15.3) induces for every choice of adiabatically switched interaction action functional a notion of perturbative interacting field observables (def. 15.8). These generate an algebra (def. 15.24 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. 15.27 below) when restricting attention to observables whose spacetime support is inside a compact causally closed subsets ##\mathcal{O}## of spacetime (def. 15.26 below). This is a sensible condition for an observable in physics, where any realistic experiment necessarily probes only a compact subset of spacetime, see also remark 15.18.

The resulting system (a “co-presheaf”) of well-defined perturbative interacting field algebras of observables (def. 15.29 below)

$$

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

$$

is in fact causally local (prop. 15.30 below). This fact was presupposed without proof already in Il’in-Slavnov 78; because this is one of two key properties that the Haag-Kastler axioms (Haag-Kastler 64) 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, Fredenhagen-Rejzner 12).

In this terminology the content of prop. 15.30 below is that while the input of causal perturbation theory is a 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})

\;:=\;

\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 15.18 below.

In conclusion so far, the algebraic adiabatic limit yields, starting with a BV-BRST gauge fixed free field vacuum, the perturbative construction of interacting field algebras of observables (def. 15.24) and their organization in increasing powers of ##\hbar## and ##g## (loop order, prop. 15.68) via the Feynman perturbation series (example 15.58, example 15.71).

But this interacting field algebra of observables still involves all the auxiliary fields of the BV-BRST gauge fixed free field vacuum (as in example 15.54 for QED), while the actual physical 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. 15.42 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. 15.72). The formula that characterizes it there is called the quantum master equation or equivalently the quantum master Ward identity (prop. 15.73 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 15.76 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 perturbative QFT (example 15.78 below).

As one extends the time-ordered products to coinciding interaction points in (“re”-)normalization of the perturbative QFT (def. 15.46 below), the quantum master equation/master Ward identity becomes a renormalization condition (prop. 15.49 below). If this condition fails one says that the 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 “(re)-“normalization we discuss in detail in the next chapter.

Free field vacua

In considering perturbative QFT, we are considering perturbation theory in formal deformation parameters around a fixed free 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:

Definition 15.1. (free relativistic Lagrangian quantum field vacuum)

Let

##\Sigma## be a spacetime (e.g. Minkowski spacetime); ##(E,\mathbf{L})## a free Lagrangian field theory (def. 5.25), with field bundle ##E \overset{fb}{\to} \Sigma##; ##\mathcal{G} \overset{fb}{\to} \Sigma## a gauge parameter bundle for ##(E,\mathbf{L})## (def. 10.5), with induced BRST-reduced Lagrangian field theory ##\left( E \times_\Sigma \mathcal{G}[1], \mathbf{L} – \mathbf{L}_{BRST}\right)## (example 10.28); ##(E_{\text{BV-BRST}}, \mathbf{L}’ – \mathbf{L}’_{BRST})## a gauge fixing (def. 12.1) with graded 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 12.7); ##\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 Euler-Lagrange equations of motion induced by the 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. 14.2). This is a star algebra with respect to (coefficient-wise) complex conjugation (prop. 14.5).

Write

$$

\label{HadamardVacuumStateForFreeFieldTheory}

\array{

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

&\overset{\langle – \rangle}{\longrightarrow}&

\mathbb{C}[ [\hbar] ]

\\

A &\mapsto& A(\Phi = 0)

}

$$ (224)

for the induced Hadamard vacuum state (prop. 14.15), hence the state whose 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:

}

$$ (225)

for the inclusion of local observables (def. 7.39) into microcausal polynomial observables (example 14.4), thought of as forming normal-ordered products in the Wick algebra (by def. 14.12).

We denote the Wick algebra-product (the star product ##\star_H## induced by the Wightman propagator ##\Delta_H## according to prop. 13.18) by juxtaposition (def. 14.12)

$$

A_1 A_2 \;:=\; 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. 7.31). For ##S_1, S_2 \subset \Sigma## two subsets of spacetime we write

$$

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

\phantom{AAA}

\left\{

\array{

“S_1 \, \text{does not intersect the past of} \, S_2”

\\

\Updownarrow

\\

“S_2 \, \text{does not intersect the future of} \, S_1”

}

\right.

$$

for the causal order-relation (def. 2.37) 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 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 15.14. The following collects our notational conventions for these matters:

Definition 15.2. (formal power series of observables for perturbative QFT)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1.

Write

$$

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

\;:=\;

\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

##\hbar## (“Planck’s constant”), ##g## (“coupling constant”) ##j## (“source field”)

with coefficients in the topological vector spaces of the off-shell polynomial local observables of the free field theory (def. 7.39); similarly for the off-shell microcausal polynomial observables (def. 14.2):

$$

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

\;:=\;

\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 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 (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. 15.52 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. 5.1), thought of as a local interaction Lagrangians, and if

$$

g_{sw}

\;\in\;

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

$$

is an adiabatically switched coupling as before, then the transgression (def. 7.32) 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 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 formal powers of ##\hbar##.

Perturbative S-Matrices

We introduce now the axioms for perturbative scattering matrices relative to a fixed relativistic free Lagrangian quantum field vacuum (def. 15.1 below) according to causal perturbation theory (def. 15.3 below). Since the first of these axioms requires the S-matrix to be a formal sum of multi-linear continuous functionals, it is convenient to impose axioms on these directly: this is the axiomatics for time-ordered products in def. 15.31 below. That these latter axioms already imply the former is the statement of prop. 15.39, prop. 15.40 below . Its proof requires a close look at the “reverse-time ordered products” for the inverse S-matrix (def. 15.35 below) and their induced reverse-causal factorization (prop. 15.38 below).

Definition 15.3. (S-matrix axioms — causal perturbation theory)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1.

Then a perturbative S-matrix scheme for perturbative QFT around this 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. 15.2), such that the following two conditions “perturbation” and “causal additivity (jointly: “causal perturbation theory”) hold:

(perturbation)There exist 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 ] ]

$$ (226) for all ##k \in \mathbb{N}##, such that: The nullary map is constant on the unit of the Wick algebra$$

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

$$ The unary map is the inclusion of local observables as normal-ordered products (225)$$

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

$$ 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)

\\

& :=

\underset{k = 0}{\overset{\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}

$$ (227) (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)

\,.

$$ (228)

(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 15.4 below.)

Def. 15.3 is due to (Epstein-Glaser 73 (1)), following (Stückelberg 49-53, Bogoliubov-Shirkov 59). 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. 15.31 below), in (Brunetti-Fredenhagen 99, section 3, Dütsch-Fredenhagen 04, appendix E, Hollands-Wald 04,around (20)). Review includes (Rejzner 16, around def. 6.7, Dütsch 18, section 3.3).

Remark 15.4. (invertibility of the S-matrix)

The mutliplicative inverse ##S(-)^{-1}## of the perturbative S-matrix in def. 15.3

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}

\\

& =

\underset{r = 0}{\overset{\infty}{\sum}}

\left(

{\, \atop \,}

-\mathcal{D}(g S_{int} + j A)

{\, \atop \,}

\right)^r

\end{aligned}

$$ (229)

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. 15.35 below), see prop. 15.36 below.

Notice that ##\mathcal{S}(-g S_{int} – j A )## is instead the inverse with respect to the time-ordered products (226) 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).

Remark 15.5. (adjoining further deformation parameters)

The definition of S-matrix schemes in def. 15.3 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. 15.3) it immediately induces a corresponding concept of observables:

Definition 15.6. (generating function scheme for interacting field observables)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3.

The corresponding generating function scheme (for interacting field observables, def. 15.8 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)

\;:=\;

\mathcal{S}(g S_{int})^{-1} \mathcal{S}( g S_{int} + j A )

\,.

$$ (230)

Proposition 15.7. (causal additivity in terms of generating functions)

In terms of the generating functions ##\mathcal{Z}## (def. 15.6) the axiom “causal additivity” on the S-matrix scheme ##\mathcal{S}## (def. 15.3) 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}

\,.

$$ (231)

(Whence “additivity”.)

Proof. This follows by elementary manipulations:

Multiplying both sides of (228) 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 (231).

Multiplying both sides of (228) 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 (231).

Definition 15.8. (interacting field observables — Bogoliubov’s formula)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, and let ##g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle## be a local observable regarded as an adiabatically switched interaction-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 (230):

$$

\label{BogoliubovsFormula}

\begin{aligned}

A_{int}

&:=

i \hbar \frac{d}{d j}

\left(

{\, \atop \,}

\mathcal{Z}_{ g S_{int} }( j A )

{\, \atop \,}

\right)_{\vert_{j = 0}}

\\

& :=

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}

$$ (232)

This expression is called Bogoliubov’s formula, due to (Bogoliubov-Shirkov 59).

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. 15.1) 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)); review includes (Dütsch-Fredenhagen 00, around (17), Dütsch 18, around (3.212)).

Remark 15.9. (interacting field observables are formal deformation quantization)

The interacting field observables in def. 15.8 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 powers of ##\hbar## (Dütsch-Fredenhagen 00, prop. 2 (ii), Hawkins-Rejzner 16, cor. 5.2). This is not immediate, since by def. 15.3 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. 15.25 below) and are thus the more fundamental concept.

As the name suggests, the S-matrices in def. 15.3 serve to express scattering amplitudes (example 15.12 below). But by remark 15.9

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:

Definition 15.10. (vacuum stability)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, and let ##g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]\langle g \rangle## be a local observable, regarded as an adiabatically switched interaction action functional.

We say that the given Hadamard vacuum state (prop. 14.15)

$$

\langle – \rangle

\;\colon\;

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

\longrightarrow

\mathbb{C}[ [ \hbar, g, j ] ]

$$

is 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

$$

Example 15.11. (time-ordered product of interacting field observables)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, and let ##g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle## be a local observable regarded as an adiabatically switched interaction-functional.

Consider two local observables

$$

A_1, A_2

\;\in\;

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

$$

with causally ordered spacetime support

$$

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

$$

Then causal additivity according to prop. 15.7 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. 15.8) 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. 15.3

from formal power series in ##\hbar, g, j## to formal power series in ##\hbar, g, j_1, j_2, \cdots## (remark 15.5). 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 15.11

is the following special case of scattering amplitudes, which is the example that gives the scattering matrix in def. 15.3 its name:

Example 15.12. (scattering amplitudes as vacuum expectation values of interacting field observables)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, and let ##g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle## be a local observable regarded as an adiabatically switched interaction-functional, such that the vacuum state is stable with respect to ##g S_{int}## (def. 15.10).

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. 15.8) 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

\\

& :=

\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 15.17 below).

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. 2.37)

$$

supp(A_{out})

{\vee\!\!\!\wedge}

supp(S_{int})

{\vee\!\!\!\wedge}

supp(A_{in})

$$

we compute with causal additivity via prop. 15.7 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. 15.10).

Remark 15.13. (computing S-matrices via Feynman perturbation series)

For practical computation of vacuum expectation values of interacting field observables (example 15.11) and hence in particular, via example 15.12, 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 15.58 below) and the effective action (def. 15.62), see example 15.71 below.

Conceptual remarks

The simple axioms for S-matrix schemes in causal perturbation theory (def. 15.3) and hence for interacting field observables (def. 15.8) 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:

Remark 15.14. (perturbative QFT and asymptotic expansion of probability amplitudes)

Given a perturbative S-matrix scheme (def. 15.3), then by remark 15.9 the expectation values of interacting field observables (def. 15.8) 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 converge for any positive value of ##\hbar## and/or ##g##. In terms of synthetic differential geometry this means that in perturbative QFT the deformation of the 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) suggests that in realistic field theories these series never converge for any positive value of ##\hbar## and/or ##g##. Namely convergence for ##g## would imply a positive radius of convergence around ##g = 0##, which would imply convergence also for ##-g## and even for 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 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 15.18).

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 quantization of Yang-Mills theory (as in QCD) has famously been advertized as one of the Millenium Problems of our age; and speculation about non-perturbative quantum gravity is the subject of much activity.

Now, as the name indicates, the axioms of causal perturbation theory (def. 15.3) do not address 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. 15.8) 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.

Remark 15.15. (Dyson series and Schrödinger equation in interaction picture)

The axiom “causal additivity” (228) on an S-matrix scheme (def. 15.3) implies immediately this seemingly weaker condition (which turns out to be equivalent, this is prop. 15.40 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” (227) we have ##\mathcal{S}(0) = 1##.)

If we now think of ##O_1 = g S_{1}## and ##O_2 = g S_2## themselves as 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 “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 at S-matrix; see also Scharf 95, second half of 0.3).

The relevance of manifest causal additivity of the S-matrix, over just causal factorization (even though both conditions happen to be equivalent, see prop. 15.40 below), is that it directly implies that the induced interacting field algebra of observables (def. 15.8) forms a causally local net (prop. 15.30 below).

Remark 15.16. (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) 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)). This is related to the intuitive picture of the Feynman perturbation series expressing a sum over all possible interactions of virtual particles (remark 15.21).

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. 15.8) 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!}}{:=}\;

\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. 15.3) as rigorously defining the path integral, not analytically as an actual integration, but 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 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. (“re”-)normalized) perturbative quantum field theory (see remark 15.20).

$$

\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}

}

}

$$

Remark 15.17. (scattering amplitudes)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, and let

$$

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

$$

be a local observable, regarded as an 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 15.12) 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

\;:=\;

\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

\\

& :=

\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

\\

& :=

\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}

\,.

$$ (233)

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:

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 15.17

describe scattering processes for interactions that vanish (are “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:

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 observation and reproduced in numerical lattice gauge theory simulation, but is invisible to 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), which is possibly a “theta-vacuum” exhibiting superposition of QCD instantons (Schäfer-Shuryak 98, section III.D). This remains open, closely related to the Millenium Problem 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:

Remark 15.19. (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. 15.31

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 15.33):

$$

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)).

This however is simply a mathematical error (as amplified in Scharf 95, below (3.2.4), below (3.2.44) and in fig. 3):

Both ##T## as well as ##\Theta## are distributions and their product of distributions is in general not defined, as Hörmander’s criterion (prop. 9.34), which is exactly what guarantees absence of UV-divergences (remark 9.27), may be violated. The notorious ultraviolet divergences which plagued (Feynman 85) the original conception of perturbative QFT due to Schwinger-Tomonaga-Feynman-Dyson are the signature of this ill-defined product (see remark 15.20).

On the other hand, when both distributions are 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 distributionsof 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 extensions. This freedom of choice is the freedom of “re-“normalization; or equivalently, by the main theorem of perturbative renormalization theory (theorem 16.19 below), this is the freedom of choosing “counterterms” (remark 16.24 below) for the local interactions. This we discuss below and in more detail in the next chapter.

Remark 15.20. (absence of ultraviolet divergences and re-normalization)

The simple axioms of causal perturbation theory (def. 15.3) do fully capture perturbative quantum field theory “in the ultraviolet”: A solution to these axioms induces, by definition, well-defined perturbative scattering amplitudes (remark 15.17) and well-defined perturbative probability amplitudes of interacting field observables (def. 15.8) induced by local action functionals (describing point-interactions such as the electron-photon interaction). By the main theorem of perturbative renormalization (theorem 16.19) such solutions exist. This means that, while these are necessarily formal power series in ##\hbar## and ##g## (remark 15.14), 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 diverging expressions for the coefficients (due to the mathematical error discussed in remark 15.19 below), which were then “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), 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. 15.3) is defined on local observables, which, by their very locality, encode point-interactions. See also remark 15.18 on infrared divergences.

It is suggestive to think of the edges in the Feynman diagrams (def. 15.55) 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. 15.56 may be read as saying that the scattering amplitude for given external source fields (remark 15.17) 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 15.16).

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

Remark 15.22. (renormalization scheme)

Beware the terminology in def. 15.3: 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 (adiabatically switched local) interaction ##S_{int}##, reflecting the scattering amplitudes (remark 15.17) 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. 15.3 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 (“re”-)normalization, the function ##\mathcal{S}## axiomatized by def. 15.3 may also be referred to as a (“re”-)normalization scheme.

This perspective on ##\mathcal{S}## as a renormalization scheme is amplified by the main theorem of perturbative renormalization (theorem 16.19) wich states that the space of choices for ##\mathcal{S}## is a torsor over the Stückelberg-Petermann renormalization group.

Remark 15.23. (quantum anomalies)

The axioms for the S-matrix in def. 15.3

(and similarly that for the time-ordered products below in def. 15.31) are sufficient to imply a causally local net of perturbative interacting field algebras of quantum observables (prop. 15.30 below), and thus its algebraic adiabatic limit (remark 15.18).

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

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.

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, and let ##g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]## be a local observable regarded as an adiabatically switched interaction-functional.

We write

$$

LocIntObs_{\mathcal{S}}(E_{\text{BV-BRST}}, g S_{int})

\;:=\;

\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. 15.8) corresponding to local observables ##A##, the local interacting field observables.

Furthermore we write

$$

\array{

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

&

\underset{\simeq}{\overset{\phantom{A}\mathcal{R}^{-1}\phantom{A}}{\longrightarrow}}

&

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

\\

A

&\mapsto&

A_{int}

:=

\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 15.4, is a linear isomorphism with inverse

$$

\array{

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

&

\underset{\simeq}{\overset{\phantom{A}\mathcal{R}\phantom{A}}{\longrightarrow}}

&

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

\\

A_{int}

&\mapsto&

A

:=

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)).

Finally we write

$$

\begin{aligned}

IntObs(E_{\text{BV-BRST}}, S_{int})

& :=

\left\langle

{\, \atop \,}

IntLocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]

{\, \atop \,}

\right\rangle

\\

&

\phantom{:=}

\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. 15.3) via Bogoliubov’s formula (def. 15.8) 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 15.25 says that this is nevertheless the case. (The special case of this statement for free field theory is discussed at Wick algebra, see remark 14.6).

Proposition 15.25. (interacting field algebra of observables is formal deformation quantization of interacting Lagrangian field theory)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, 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 adiabatically switched interaction Lagrangian density with corresponding action functional ##g S_{int} := \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. 15.24 is a formal deformation quantization of the interacting Lagrangian field theory ##(E_{\text{BV-BRST}}, \mathbf{L}’ + g_{sw} \mathbf{L}_{int})##.

(Hawkins-Rejzner 16, prop. 5.4, Collini 16)

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:

Definition 15.26. (system of spacetime-localized generating functions for interacting field observables)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, 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

$$

S_{int,sw}

\;:=\;

\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 adiabatically switched interaction action functional.

For ##\mathcal{O} \subset \Sigma## a causally closed subset of spacetime (def. 2.38) and for ##g_{sw} \in Cutoffs(\mathcal{O})## an adiabatic switching function (def. 2.39) which is constant on a neighbourhood of ##\mathcal{O}##, write

$$

Gen(E_{\text{BV-BRST}}, S_{int,sw} )(\mathcal{O})

\;:=\;

\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. 15.6) 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)

\;:=\;

\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))

The key technical fact is the following:

Proposition 15.27. (localized interacting field observables independent of adiabatic switching)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, 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

$$

g S_{int,sw} \;:=\; \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 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 (230) 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

\,.

$$ (234)

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 isomorphic to the abstract algebra ##Gen_{\mathbf{L}_{int}}(\mathcal{O})## (def. 15.26), 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}}

}

\,.

$$ (235)

(Brunetti-Fredenhagen 99, prop. 8.1)

Proof. By causal closure of ##\mathcal{O}##, lemma 2.40

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. 15.7:

$$

\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 (235).

(While the elements ##K## in (234) are far from being unique themselves, equation (234) says that the map on generators induced by conjugation with ##K## is independent of this choice.)

Proposition 15.28. (system of generating algebras is causally local net)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, 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. 15.26) is a causally local net in that it satisfies the following conditions:

(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)

$$ (causal locality) For ##\mathcal{O}_1, \mathcal{O}_2 \subset X## two causally closed subsets which are spacelike separated, in that their causal ordering (def. 2.37) satisfies$$

\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, following Brunetti-Fredenhagen 99, section 8, Il’in-Slavnov 78)

Proof. Isotony is immediate from the definition of the algebra homomorphisms in def. 15.26.

By the isomorphism (235) 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. 15.7):

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

Definition 15.29. (system of interacting field algebras of observables)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, 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

$$

g S_{int,sw} \;:=\; 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 adiabatically switched interaction action functional.

For ##\mathcal{O} \subset \Sigma## a causally closed subset of spacetime (def. 2.38) and for ##g_{sw} \in Cutoffs(\mathcal{O})## an compatible adiabatic switching function (def. 2.39) write

$$

IntObs(E_{\text{BV-BRST}}, S_{int,sw})(\mathcal{O})

:=

\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. 15.24) 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}

\;:=\;

\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. 15.28, we obtain the key result:

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1, let ##\mathcal{S}## be a corresponding S-matrix scheme according to def. 15.3, 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. 15.29) is a local net of observables in that

(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)

$$ (causal locality) For ##\mathcal{O}_1, \mathcal{O}_2 \subset X## two causally closed subsets which are spacelike separated, in that their causal ordering (def. 2.37) satisfies$$

\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), following Brunetti-Fredenhagen 99, section 8, Il’in-Slavnov 78)

Proof. The first point is again immediate from the definition (def. 15.29).

For the second point it is sufficient to check the commutativity relation on generators. For these the statement follows with prop. 15.28:

$$

\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

Definition 15.3 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. 15.31) and then prove that these axioms imply the axioms for the corresponding S-matrix (prop. 15.39 below).

Definition 15.31. (time-ordered products)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a free vacuum according to def. 15.1.

A time-ordered product is a sequence of 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. 15.2) such that the following conditions hold for all possible arguments:

(normalization)$$

T_0(O) = 1

$$ (perturbation)$$

T_1(O) = :O:

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

T_k(O_{\sigma(1)}, O_{\sigma(2)}, \cdots, O_{\sigma(k)})

\;=\;

T_k(O_1, O_2, \cdots, O_k)

$$ (causal factorization) If the spacetime support (def. 7.31) of local observables satisfies the causal ordering (def. 2.37)$$

\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 )

\,.

$$

Example 15.32. (S-matrix scheme implies time-ordered products)

Let ##(E_{\text{BV-BRST}}, \mathbf{L}’, \Delta_H )## be a relativistic free vacuum according to def. 15.1 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. 15.3.

Then the ##\{T_k\}_{k \in \mathbb{N}}## are time-ordered products in the sense of def. 15.31.

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 15.15) 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. 15.39 below).

Remark 15.33. (time-ordered products as generalized functions)

It is convenient (as in Epstein-Glaser 73) to think of time-ordered products (def. 15.31), 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_ \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)

\\

& :=

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. 15.31, 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} := \{x_1, \cdots, x_r\}## and ##\mathbf{Y} := \{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:

Example 15.34. (product of perturbation series in generalized function-notation)

Let

$$

U(g)

:=

\underset{n = 0}{\overset{\infty}{\sum}}

\frac{1}{n!}

\int U(x_1, \cdots, x_n)

\,

g(x_1) \cdots g(x_n) \, dvol

$$

and

$$

V(g)

:=

\underset{n = 0}{\overset{\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 15.33.

Then their product ##W(g) := U(g) V(g)## with generalized function-representation

$$

W(g)

:=

\underset{n = 0}{\overset{\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))

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. 15.39 below) we need to consider the corresponding reverse-time ordered products:

Given a time-ordered product ##T = \{T_k\}_{k \in \mathbb{N}}## (def. 15.31), 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 )

\;:=\;

\left\{

\array{

\underset{r = 1}{\overset{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 15.33, this reads

$$

\overline{T}( \mathbf{X} )

=

\underset{r = 1}{\overset{{\vert \mathbf{X} \vert}}{\sum}}

(-1)^r

\underset{ \array{

\mathbf{I}_1, \cdots, \mathbf{I}_r

eq \emptyset

\\

\underset{j

eq 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))

Proposition 15.36. (reverse-time ordered products express inverse S-matrix)

Given time-ordered products ##T(-)## (def. 15.31), then the corresponding reverse time-ordered product ##\overline{T}(-)## (def. 15.35) expresses the inverse ##S(-)^{-1}## (according to remark 15.4) of the corresponding perturbative S-matrix scheme ##\mathcal{S}(S_{int}) := \underset{k \in \mathbb{N}}{\sum} \tfrac{1}{k!} T(\underset{k\,\text{args}}{\underbrace{S_{int}, \cdots , S_{int}}})## (def. 15.3):

$$

\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. 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!}

\underset{r = 1}{\overset{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!}

\underset{r = 1}{\overset{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}

\underset{r = 1}{\overset{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 }} )

\\

& =

\underset{r = 0}{\overset{\infty}{\sum}}

\left(

– \underset{k = 0}{\overset{\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 (229).

In fact prop. 15.36 is a special case of the following more general statement:

Proposition 15.37. (inversion relation for reverse-time ordered products)

Let ##\{T_k\}_{k \in \mathbb{N}}## be time-ordered products according to def. 15.31. Then the reverse-time ordered products according to def. 15.35

satisfies the following inversion relation for all ##\mathbf{X}

eq \emptyset## (in the condensed notation of remark 15.33):

$$

\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. This is immediate from unwinding the definitions.

Proposition 15.38. (reverse causal factorization of reverse-time ordered products)

Let ##\{T_k\}_{k \in \mathbb{N}}## be time-ordered products according to def. 15.31. Then the reverse-time ordered products according to def. 15.35 satisfies reverse-causal factorization.

(Epstein-Glaser 73, around (15))

Proof. In the condensed notation of remark 15.33, 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 15.34.

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}

eq \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}’

eq \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}’

eq \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}

eq \emptyset }

}{\sum}

\overline{T}(\mathbf{K})

T (\mathbf{P} \setminus \mathbf{K} )

}}

\\

& =

\overline{T}(\mathbf{Q}) \overline{T}(\mathbf{P})

\end{aligned}

\,.

$$

Here

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. 15.31) and that of ##\overline{T}(-)## (which holds by the induction assumption, using that ##\mathbf{J}

eq \mathbf{X}## hence that ##{\vert \mathbf{J}\vert} \lt {\vert \mathbf{X}\vert}##). 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}##: 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. 15.37. 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. 15.37 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:

Proposition 15.39. (time-ordered products induce S-matrix)

Let ##\{T_k\}_{k \in \mathbb{N}}## be a system of time-ordered products according to def. 15.31. Then

$$

\begin{aligned}

\mathcal{S}(-)

& :=

T

\left(

\exp_\otimes

\left(

\tfrac{1}{i \hbar}(-)

\right)

\right)

\\

&

:=

\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. 15.3.

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 15.15) 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-ord