Contents

# Contents

## Idea

In quantum field theory a scattering amplitude or scattering matrix, usually just S-matrix for short, encodes the probability amplitudes for scattering processes of particles off each other.

### General idea

Every Lagrangian perturbative quantum field theory has an S-matrix associated with its (after renormalization), usually thought of as a perturbation series over Feynman diagrams extracted from the Lagrangian density. The rigorous construction of this as an operator-valued distribution is the content of causal perturbation theory (Epstein-Glaser 73).

But there are also S-matrices not arising from a local field theory, for instance the string scattering amplitudes.

There have been attempts to define perturbative quantum field theory by directly axiomatizing properties of the S-matrix, without requiring concepts of fields in spacetime. This perspective goes back to (Heisenberg 43) and was vocally promoted in Geoffrey Chew‘s “bootstrap program” (a textbook account is in Eden-Ladshoff-Olive-Polkinhorne 66).

In the field theory-picture the crucial condition on the S-matrix is its causal additivity (def. below) which reflects the microcausality of quantum observables (prop. below), whence the name “causal perturbation theory”.

This causality of the S-matrix, when understood in terms of underlying spacetime and fields, is supposed to be detected more abstractly by the S-matrix being a suitable analytic function of the wave vectors of the scattering asymptotic states (Newton 82, 10.3.3, Arkani-Hamed et al. 06), often refereed to via “dispersion relations” (e.g. Eden-Ladshoff-Olive-Polkinhorne 66 (1.1.1)-(1.1.5), Gribov 69, 1.1.2). Since thereby analyticity is recognized as the crucial property of the S-matrix in the spacetime/field-independent axiomatization, this is often referred to as “the analytic S-matrix” (e.g. Eden-Ladshoff-Olive-Polkinhorne 66). More specifically microcausality is what induces “crossing symmetry” of the S-matrix (Weinberg 95, section 10.8).

The perception of the nature of the S-matrix as a primary or derived concept in the foundations of quantum field theory has a convoluted (and ongoing) history, see below.

### The S-matrix bootstrap

The idea of the S-matrix “bootstrap” is that one may compute the S-matrix directly from suitable axioms without using a local quantum field theory involving fields on spacetime. In order for the theory to be interesting, the S-matrix should obey certain properties abstracted away from field theory

• It should be unitary
• It should be Lorentz invariant
• It should be crossing invariant: this means that the antiparticle scattering should be described by the analytic continuation of the particle scattering
• It should obey the Landau property— that all singularities of scattering are poles and cuts corresponding to exchange of collections of real particles on shell.
• It should obey (Mandelstam) analyticity: the amplitude should be writable as an integral over the imaginary part of the cut discontinuity from production of physical particles. Further, this cut discontinuity itself can be expanded in terms of another cut discontinuity (these are the mysterious then and still mysterious now double dispersion relations of Mandelstam).

This is a sketchy summary, because each of these conditions is involved. The unitarity condition in particular, is very difficult, because it is so nonlinear. The only practical way to solve it is in a perturbation series which starts with weakly interacting nearly stable particles (described by poles of the S-matrix) which exchange each other (the exchange picture is required by crossing, and the form of the scattering is fixed by the Landau and Mandelstam analyticity, once you know the spectrum).

The “Bootstrap property” is then the following heuristic idea, which is included in the above formal relations:

• The particles and interactions which emerge as the spectrum of the S-matrix from the scattering of states, including their binding together into bound states, should be the same spectrum of particles that come in ias in-states.

This is a heuristic idea, because it is only saying that the S-matrix is consistent, and the formal consistency relations are those above. But the bootstrap was a slogan that implied that all the consistency conditions were not yet discovered, and there might be more.

This idea was very inspirational to many great people in the 1960s, because it was an approach to strong interactions that could accommodate non-field theories of infinitely many particle types of high spin, without postulating constituent particles (like quarks and gluons).

### Regge theory

Continuing with this Physics.SE comment by Ron Maimon:

The theory above doesn’t get you anywhere without the following additional stuff. If you don’t do this, you end up starting with a finite number of particles and interactions, and then you end up in effective field theory land. The finite-number-of-particles version of S-matrix theory is a dead end, or at least, it is equivalent to effective field theory, and this was understood in the late 1960s by Weinberg, and others, and this led S-matrix theory to die. This was the road the Chew travelled on, and the end of this road must be very personally painful to him.

But there is another road for S-matrix theory which is much more interesting, so that Chew should not be disheartened. You need to know that the scattering amplitude is analytic in the angular momentum of the exchanged particles, so that the particles lie on Regge trajectories, which give their angular momentum as a function of their mass squared, s.

Where the Regge trajectories hit an integer angular momentum, you see a particle. The trajectory interpolates the particle mass-squared vs. angular momentum graph, and it gives the asymptotic scattering caused by exchanging all these particles together. This scattering can be softer than the exchange of any one of these particles, because exchanging a particle of high spin necessarily has very singular scattering amplitudes at high energy. The Regge trajectory cancels out this growth with an infinite series of higher particles which soften the blowup, and lead to a power-law near-beam scattering at an angle which shrinks to zero as the energy goes to infinity in a way determined by the shape of the trajectory.

So the Regge bootstrap adds the following conditions

• All the particles in the theory lie on Regge trajectories, and the scattering of these particles is by Regge theory.

This condition is the most stringent, because you can’t deform a pure Regge trajectory by adding a single particle— you have to add new trajectories. The following restriction was suggested by experiment

• The Regge trajectories are linear in s

This was suggested by Chew and Frautschi from the resonances known in 1960! The straight lines mostly had two points. The next condition is also ad-hoc and experimental

• The Regge slope is universal (for mesons), it’s the same for all the trajectories.

There are also “pomerons” in this approach which are not mesons, which have a different Regge slopem but ignore this for now.

Finally, there is the following condition, which was experimentally motivated, but has derivations by Mandelstam and others from more theoretical foundations (although this is S-matrix theory, it doesn’t have axioms, so derivation is a loose word).

• The exchange of trajectories is via the s-channel or the t-channel, but not both. It is double counting to exchange the same trajectories in both channels.

These conditions essentially uniquely determine Veneziano’s amplitude and bosonic string theory. Adding Fermion trajectories requires Ramond style supersymmetry, and then the road to string theory is to reinterpret all these conditions in the string picture which emerges.

String theory incorporates and gives concrete form to all the boostrap ideas, so much so that anyone doing bootstrap today is doing string theory, especially since AdS/CFT showed why the bootstrap is relevant to gauge theories like QCD in the first place.

The highlight of Regge theory is the Reggeon calculus, a full diagrammatic formalism, due to Gribov, for calculating the exchange of pomerons in a perturbation framework. This approach inspired a 2d parton picture of QCD which is studied heavily by several people, notably, Gribov, Lipatov, Feynman (as part of his parton program), and more recently Rajeev. Nearly every problem here is open and interesting.

For an example of a reasearch field which (partly) emerged from this, one of the major motivations for taking PT quantum mechanics seriously was the strange non-Hermitian form of the Reggeon field theory Hamiltonian.

### Pomerons and Reggeon Field theory

Further from this Phyics.SE comment by Ron Maimon:

The main success of this picture is describing near-beam scattering, or diffractive scattering, at high energies. The idea here is that there is a Regge trajectory which is called the pomeron, which dominates high energy scattering, and which has no quantum numbers. This means that any particle will exchange the pomeron at high energies, so that p-pbar and p-p total cross sections will become equal.

This idea is spectacularly confirmed by mid 90’s measurements of total p-p and p-pbar cross sections, and in a better political climate, this would have won some boostrap theorists a Nobel prize. Instead, it is never mentioned.

The pomeron in string theory becomes the closed string, which includes the graviton, which couples universally to stress energy. The relation between the closed string and the QCD pomeron is the subject of active research, associated with the names of Lipatov, Polchinski, Tan, and collaborators.

Regge scattering also predicts near beam scattering amplitudes from the sum of the appropriate trajectory function you can exchange. These predictions have been known to roughly work since the late 1960s.

## Details

We first discuss the simple situation of S-matrices in quantum mechanics:

Then we give a detailed account of S-matrix theory for perturbative quantum field theory induced from interaction action functionals on spacetime:

This is essentially chapter 15. in A first idea of quantum field theory.

(We should eventually also discuss the abstract S-matrix bootstrap here in detail.)

### In quantum mechanics

In quantum mechanics, let $\mathcal{H}$ be some Hilbert space and let

$H = H_{free} + V$

be an Hermitian operator, thought of as a Hamiltonian, decomposed as the sum of a free part (kinetic energy) and an interaction part (potential energy).

For example for a non-relativistic particle of mass $m$ propagating on the line subject to a potential energy $V_{pot} \colon \mathbb{R} \to \mathbb{R}$, then $\mathcal{H} = L^2(\mathbb{R})$ is the Hilbert space space of square integrable functions and

$H = \underset{H_{free}}{\underbrace{\tfrac{- \hbar^2}{2m} \frac{\partial^2}{\partial^2 x}}} + V \,,$

where $V = V_{pot}(x)$ is the operator of multiplying square integrable functions with the given potential energy function.

Now for

$\array{ \mathbb{R} &\overset{\vert \psi (-)\rangle }{\longrightarrow}& \mathcal{H} \\ t &\mapsto& \vert \psi(t) \rangle }$

a one-parameter family of quantum states, the Schrödinger equation for this state reads

$\frac{d}{d t} \vert \psi(t) \rangle \;=\; \tfrac{1}{i \hbar} H \vert \psi\rangle \,.$

It is easy to solve this differential equation formally via its Green function: for $\vert \psi \rangle \in \mathcal{H}$ any state, then the unique solution $\vert \psi(-) \rangle$ to the Schrödinger equation subject to $\vert \psi(0) \rangle = \vert \psi \rangle$ is

$\vert \psi(t)\rangle_S \coloneqq \exp( \tfrac{t}{i \hbar} H ) \vert \psi \rangle \,.$

(One says that this is the solution “in the Schrödinger picture”, whence the subscript.)

However, if $H$ is sufficiently complicated, it may still be very hard to extract from this expression a more explicit formula for $\vert \psi(t) \rangle$, such as, in the example of the free particle on the line, its expression as a function (“wave function”) of $x$ and $t$.

But assume that the analogous expression for $H_{free}$ alone is well understood, hence that the operator

$U_{S,free}(t_1, t_2) \coloneqq \exp\left({\tfrac{t_2 - t_1}{i \hbar} H_{free}}\right)$

is sufficiently well understood. The “interaction picture” is a way to decompose the Schrödinger equation such that its dependence on $V$ gets separated from its dependence on $H_{free}$ in a way that admits to treat $H_{int}$ in perturbation theory.

Namely define analogously

(1)\begin{aligned} \vert \psi(t)\rangle_I &\coloneqq \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \vert \psi(t)\rangle_S \\ & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \exp\left({ \tfrac{+ t}{i \hbar} H} \right)\vert \psi \rangle \\ & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right) \vert \psi \rangle \end{aligned} \,.

This is called the solution of the Schrödinger equation “in the interaction picture”, whence the subscript. Its definition may be read as the result of propagating the actual solution $\vert \psi(-)\rangle_S$ at time $t$ back to time $t = 0$, but using just the free Hamiltonian, hence with “the interaction switched off”.

Notice that if the operator $V$ were to commute with $H_{free}$ (which it does not in all relevant examples) then we would simply have $\vert \psi(t)\rangle_I = \exp( \tfrac{t}{i \hbar } V ) \vert \psi\rangle$, hence then the solution (1) in the interaction picture would be the result of “propagating” the initial conditions using only the interaction. Now since $V$ may not be assumed to commute with $H_{free}$, the actual form of $\vert \psi(-) \rangle_{I}$ is more complicated. But infinitesimally it remains true that $\vert \psi(-)\rangle_I$ is propagated this way, not by the plain operator $V$, though, but by $V$ viewed in the Heisenberg picture of the free theory. This is the content of the differential equation (2) below.

But first notice that this will indeed be useful: If an explicit expression for the “state in the interaction picture(1) is known, then the assumption that also the operator $\exp\left({\tfrac{t}{i \hbar} H_{free}}\right)$ is sufficiently well understood implies that the actual solution

$\vert \psi(t) \rangle_S \;=\; \exp\left({\tfrac{t}{i \hbar} H_{free}}\right) \vert \psi(t) \rangle_I$

is under control. Hence the question now is how to find $\vert \psi(-)\rangle_I$ given its value at some time $t$. (It is conventional to consider this for $t \to \pm \infty$, see (4) below.)

Now observe that $\vert \psi(-)\rangle_i$ satisfies the following differential equation (“Schrödinger equation in interaction picture”):

(2)$\frac{d}{d t} \vert \psi(t) \rangle_I \;=\; V_I(t) \vert \psi(t)\rangle_I \,,$

where

$V_I(t) \coloneqq \exp\left( -\tfrac{t}{i \hbar} H_{free} \right) V \exp\left( +\tfrac{t}{i \hbar} H_{free} \right)$

is known as the interaction term $V$ “viewed in the interaction picture”.

Here is the derivation of (2), where we use the product law for differentiation:

\begin{aligned} \frac{d}{d t} \vert \psi(r) \rangle_I & = \frac{d}{d t} \left( \exp\left( \tfrac{- t}{i \hbar} H_{free}\right) \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right) \right) \vert \psi \rangle \\ & = \left( \left( \frac{d}{d t} \exp\left( \tfrac{- t}{i \hbar} H_{free} \right) \right) \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right) + \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \left( \frac{d}{d t} \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right) \right) \right) \vert \psi \rangle \\ & = \left( \exp\left( \tfrac{- t}{i \hbar} H_{free} \right) \left( \tfrac{- 1}{i \hbar} H_{free} \right) \exp\left( \tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V \right) + \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \left( \tfrac{1}{i \hbar} H_{free} + \tfrac{1}{i \hbar} V \right) \exp\left( \tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V \right) \right) \vert \psi \rangle \\ & = \tfrac{1}{i \hbar} \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) V \exp\left( \tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V \right) \vert \psi \rangle \\ & = \tfrac{1}{i \hbar} \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) V \exp\left({\tfrac{+ t}{i \hbar} H_{free}}\right) \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \exp\left( \tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V \right) \vert \psi \rangle \\ & = \tfrac{1}{i \hbar} \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) V \exp\left({\tfrac{+ t}{i \hbar} H_{free}}\right) \vert \psi(t) \rangle_I \\ & = \tfrac{1}{i \hbar} V_I(t) \vert \psi(t) \rangle_I \end{aligned}

Now in fact $V_I$ is just $V$ “viewed in the Heisenberg picture”, but for the free theory. By our running assumption that the free theory is well understood, also $V_I(t)$ is well understood, and hence all that remains now is to find a sufficiently concrete solution to equation (2). This is the heart of working in the interaction picture.

Solutions to equations of the “parallel transport”-type such as (2) are given by time-ordering of Heisenberg picture operators, denoted $T$, applied to the naive exponential solution as above. This is known as the Dyson formula:

(3)$\vert \psi(t)\rangle_I \;=\; T\left( \exp\left( \int_{t_0}^t V_I(t) \tfrac{d t}{i \hbar} \right) \right) \vert \psi(t_0)\rangle \,.$

Here time-ordering means

$T( V_I(t_1) V_I(t_2) ) \;\coloneqq\; \left\{ \array{ V_I(t_1) V_I(t_2) &\vert& t_1 \geq t_2 \\ V_I(t_2) V_I(t_1) &\vert& t_2 \geq t_1 } \right. \,.$

Beware the conventional abuse of notation here: Strictly speaking time ordering acts on the tensor algebra spanned by the $\{V_I(t)\}_{t \in \mathbb{R}}$ and has to be followed by taking tensor products to actual products.

In applications to scattering processes one is interest in prescribing the quantum state/wave function far in the past, hence for $t \to - \infty$, and computing its form far in the future, hence for $t \to \infty$.

The operator that sends such “asymptotic ingoing-states” $\vert \psi(-\infty) \rangle_I$ to “asymptic outgoing states” $\vert \psi(+ \infty) \rangle_I$ is hence the limit

(4)$\mathcal{S}(V_I) \;\coloneqq\; \underset{t \to \infty}{\lim} T\left( \exp\left( \int_{-t}^t V_I(t) \tfrac{d t}{i \hbar} \right) \right) \,.$

This limit (if it exists) is called the scattering matrix or S-matrix, for short.

For example if $V_{I,1}$ and $V_{I,2}$ are two interactions such that the support in time of $V_{I,1}$ is after the support of $V_{I,2}$:

$\left\{ \array{ \left(t \gt t_0 \right) &\Rightarrow& V_{I,2} = 0 \\ \left(t \lt t_0 \right) &\Rightarrow& V_{I,1} = 0 } \right.$

then, assuming the S-matrix for $V_{I,1}$, $V_{I,2}$ and $V_I \coloneqq V_{I,1} + V_{I,2}$ exists, the Dyson formula (3) implies the “causal factorization

(5)$\mathcal{S}(V_{I,1} + V_{I,2}) \;=\; \mathcal{S}(V_{I,1}) \, \mathcal{S}(V_{I,2})$

Conversely, decomposing any $V_I(t)$ with the step function $\Theta$ as

$V_I(t) \;=\; \underset{ \coloneqq V_{I,1}(t) }{\underbrace{V_I(t) \Theta(t-t_0)}} \;+\; \underset{ \coloneqq V_{I,2}(t) }{\underbrace{V_I(t)\Theta(t_0 - t)}}$

then this causal factorization-relation may be understood as the integral version of the “Schrödinger equation in the interaction picture(2).

It is this “integral-version of the Schrödinger equation in the interaction picture(5) that has a fairly evident generalization from quantum mechanics to relativistic perturbative quantum field theory in the form of causal perturbation theory, def. below, see remark below that.

### In perturbative relativistic Lagrangian QFT – Causal perturbation theory

In perturbative algebraic quantum field theory the broad structure of the interaction picture in quantum mechanics (above) remains a very good guide, but various technical details have to be generalized with due care:

1. The algebra of operators in the Heisenberg picture of the free theory becomes the Wick algebra of the free field theory (taking into account “normal ordering” of field operators) defined on microcausal functionals built from operator-valued distributions with constraints on their wave front set.

2. The time-ordered products in the Dyson formula have to be refined to causally ordered products and the resulting product at coincident points has to be defined by point-extension of distributions – the freedom in making this choice is the renormalization freedom (“conter-terms”).

3. The sharp interaction cutoff in the Dyson formula that is hidden in the integration over $[t_0,t]$ has to be smoothed out by adiabatic switching of the interaction (making the whole S-matrix an operator-valued distribution).

Together these three points are taken care of by the axiomatization of the “adiabatically switched S-matrix” according to causal perturbation theory (def. below)

$\,$

#### 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

(free relativistic Lagrangian quantum field vacuum)

Let

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

2. $(E,\mathbf{L})$ a free Lagrangian field theory (this def.), with field bundle $E \overset{fb}{\to} \Sigma$;

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

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

5. $\Delta_H \in \Gamma'( E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}} )$ 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. This is a star algebra with respect to (coefficient-wise) complex conjugation.

Write

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

for the induced Hadamard vacuum state (this prop.), 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

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

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

We denote the Wick algebra-product (the star product $\star_H$ induced by the Wightman propagator $\Delta_H$) by juxtaposition (this def.)

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

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

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

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

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

for the causal ordering-relation 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.

###### Remark

For the purposes of constructing or defining the Wick algebra, the conditions on $\Delta_H$ or $H$ could be relaxed. Requiring $\Delta_H$ to be an honest Wightman propagator means that it is a distribution satisfying the Hadamard wavefront condition, as well as addition positivity and normalization requirements. Dropping the positivity and some of the normalization requirements, $\Delta_H$ is then only a Hadamard parametrix for the Wightman propagator. The construction of the Wick algebra with respect to $\Delta_H$ still makes sense, but $:(-):$ can no longer be interpreted as normal ordering with respect to a fixed vacuum state. In fact, in locally covariant pAQFT, the property for $\Delta_H$ to be the Wightman propagator for a state is in conflict with local covariance. On the other hand, there is no problem with selecting a locally covariant Hadamard parametrix $\Delta_H$, which allows the construction or definition of the Wick algebra to be locally covariant.

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 . The following collects our notational conventions for these matters:

###### Definition

(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. .

Write

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

for the space of formal power series in three formal variables

1. $\hbar$ (“Planck's constant”),

2. $g$ (“coupling constant”)

3. $j$ (“source field”)

with coefficients in the topological vector spaces of the off-shell polynomial local observables of the free field theory; similarly for the off-shell microcausal polynomial observables:

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

Similary

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

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

Accordingly

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

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

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

may be regarded as spacetime-dependent “coupling constants” with compact support, called 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.

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, thought of as a local interaction Lagrangian, and if

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

is an adiabatically switched coupling as before, then the transgression 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. below) according to causal perturbation theory (def. 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. below. That these latter axioms already imply the former is the statement of prop. below. Its proof requires a close look at the “reverse-time ordered products” for the inverse S-matrix (def. below) and their induced reverse-causal factorization (prop. below).

###### Definition

(S-matrix axiomscausal perturbation theory)

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

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. ), such that the following two conditions “perturbation” and “causal additivity (jointly: ”causal perturbation theory“) hold:

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

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

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

1. The nullary map is constant on the unit of the Wick algebra

$T_0( g S_{int} + j A) = 1$
2. The unary map is the inclusion of local observables as normal-ordered products (7)

$T_1(g S_{int} + j A) = g :S_{int}: + j :A:$
3. The perturbative S-matrix is the exponential series of these maps in that for all $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [\hbar , g, j] ]\langle g,j\rangle$

(9)\begin{aligned} \mathcal{S}( g S_{int} + j A) & = T \left( \exp_{\otimes} \left( \tfrac{ 1 }{i \hbar} \left( g S_{int} + j A \right) \right) \right) \\ & \coloneqq \underoverset{k = 0}{\infty}{\sum} \frac{1}{k!} \left( \frac{1}{i \hbar} \right)^k T_k \left( {\, \atop \,} \underset{k\,\text{arguments}}{\underbrace{ (g S_{int} + jA) , \cdots, (g S_{int} + j A) }} {\, \atop \,} \right) \end{aligned}
2. For all perturbative local observables $O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]$ we have

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

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

Def. 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. 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

(invertibility of the S-matrix)

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

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

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

This expression for the inverse of S-matrix may usefully be re-organized in terms of “rever-time ordered products” (def. below), see prop. below.

Notice that $\mathcal{S}(-g S_{int} - j A )$ is instead the inverse with respect to the time-ordered products (8) 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

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

###### Definition

(generating function scheme for interacting field observables)

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

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

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

(causal additivity in terms of generating functions)

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

(13)\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} \,.

###### Proof

This follows by elementary manipulations:

Multiplying both sides of (10) 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 (13).

Multiplying both sides of (10) 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 (13).

###### Definition

(interacting field observablesBogoliubov's formula)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a relativistic free vacuum according to def. , let $\mathcal{S}$ be a corresponding S-matrix scheme according to def. , 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 (12):

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

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

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. ) 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

(interacting field observables are formal deformation quantization)

The interacting field observables in def. 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. 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. below) and are thus the more fundamental concept.

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

(vacuum stability)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a relativistic free vacuum according to def. , let $\mathcal{S}$ be a corresponding S-matrix scheme according to def. , 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 (this prop.)

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

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

(time-ordered product of interacting field observables)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a relativistic free vacuum according to def. , let $\mathcal{S}$ be a corresponding S-matrix scheme according to def. , 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. 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. ) 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. from formal power series in $\hbar, g, j$ to formal power series in $\hbar, g, j_1, j_2, \cdots$ (remark ). 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 is the following special case of scattering amplitudes, which is the example that gives the scattering matrix in def. its name:

###### Example

(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. , let $\mathcal{S}$ be a corresponding S-matrix scheme according to def. , 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. ).

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. ) is

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

These vacuum expectation values are interpreted, in the adiabatic limit where $g_{sw} \to 1$, as scattering amplitudes (remark 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

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

we compute with causal additivity via prop. 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. ).

###### Remark

(computing S-matrices via Feynman perturbation series)

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

$\,$

#### Conceptual remarks

The simple axioms for S-matrices in causal perturbation theory (def. ) and hence for interacting field observables (def. ) 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

(perturbative QFT and asymptotic expansion of probability amplitudes)

Given a perturbative S-matrix scheme (def. ), then by remark the expectation values of interacting field observables (def. ) 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 ).

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. ) 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. ) 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

(Dyson series and Schrödinger equation in interaction picture)

The axiom “causal additivity(10) on an S-matrix scheme (def. ) implies immediately this seemingly weaker condition (which turns out to be equivalent, this is prop. below):

• 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” (9) 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(5). (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. below), is that it directly implies that the induced interacting field algebra of observables (def. ) forms a causally local net (prop. below).

###### Remark

(path integral-intuition)

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

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

Here the would-be integration is thought to be over the space of field histories $\Gamma_\Sigma(E_{\text{BV-BRST}})_{asm}$ (the space of sections of the given field bundle) 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 ).

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. ) symbolically reads

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

If here we were to regard the expression

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

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

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

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

But we may think of the axioms for the S-matrix in causal perturbation theory (def. ) 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 ).

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

(scattering amplitudes)

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

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

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

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

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

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

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

or similarly, under Fourier transform of distributions,

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

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

Or rather:

###### Remark

(adiabatic limit, infrared divergences and interacting vacuum)

Since a local observable $S_{int} \in LocObs(E_{\text{BV-BRST}})[ [\hbar, g, j] ]$ by definition has compact spacetime support, the scattering amplitudes in remark 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

Naively it might seem that (say over Minkowski spacetime, for simplicity) examples of time-ordered products according to def. 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 ):

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

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 (Hörmander's criterion 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 ).

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 distributions of the above product expression back from the complement of the diagonal to the whole space of tuples of points. Any such extension will produce time-ordered products.

There are in general several different such extensions. This freedom of choice is the freedom of "re-"normalization; or equivalently, by the main theorem of perturbative renormalization theory (theorem ), this is the freedom of choosing “counter terms” for the local interactions. This we discuss below in Feynman diagrams and (re-)normalization.

###### Remark

(absence of ultraviolet divergences and re-normalization)

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

###### Remark

(virtual particles, worldline formalism and perturbative string theory)

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

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

(renormalization scheme)

Beware the terminology in def. : 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 ) 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. 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. 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 ) wich states that the space of choices for $\mathcal{S}$ is a torsor over the Stückelberg-Petermann renormalization group.

###### Remark

(quantum anomalies)

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

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. This extra axiom is needed to imply the main theorem of perturbative renormalization (theorem ).

$\,$

#### Interacting field observables

We have seen that via Bogoliubov's formula (def. ) every perturbative S-matrix scheme (def. ) induces for every choice of adiabatically switched interaction action functional $S_{int,sw} \coloneqq \tau_\Sigma( g_{sw} \mathbf{L}_{int} )$ a notion of perturbative interacting field observables (def. ). These generate an algebra (def. 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. below) when restricting attention to observables whose spacetime support is inside a compact causally closed subsets $\mathcal{O}$ of spacetime (def. below). This is a sensible condition for an observable in physics, where any realistic experiment nessecarily probes only a compact subset of spacetime, see also remark .

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

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

is in fact causally local (prop. 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. 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

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

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

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

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

$\,$

###### Definition

(interacting field algebra of observablesquantum Møller operator)

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

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

for the subspace of interacting field observables $A_{int}$ (def. ) corresponding to local observables $A$, the local interacting field observables.

Furthermore we write

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

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

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

This may be called the quantum Møller operator (Hawkins-Rejzner 16, (33)).

Finally we write

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

for the smallest subalgebra of the Wick algebra containing the interacting local observables. This is the perturbative interacting field algebra of observables.

The definition of the interacting field algebra of observables from the data of a scattering matrix (def. ) via Bogoliubov's formula (def. ) 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 says that this is nevertheless the case. (The special case of this statement for free field theory is discussed at Wick algebra, see this remark).

###### Proposition

(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. , 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 $g S_{int} \coloneqq \tau_\Sigma( g_{sw} \mathbf{L}_{int} )$.

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

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

(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. , let $\mathcal{S}$ be a corresponding S-matrix scheme according to def. , 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

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

is a local observable, to be thought of as an adiabatically switched interaction action functional.

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

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

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

Moreover, write

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

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

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

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

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

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

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

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

This construction defines a functor

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

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

(extends to star algebras if scattering matrices are chosen unitary…)

The key technical fact is the following:

###### Proposition

(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. , let $\mathcal{S}$ be a corresponding S-matrix scheme according to def. , 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

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

is a local observable, to be thought of as an 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 (12) are related via conjugation by $K$:

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

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

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

By causal closure of $\mathcal{O}$, this lemma 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. :

\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 (17).

(While the elements $K$ in (16) are far from being unique themselves, equation (16) says that the map on generators induced by conjugation with $K$ is independent of this choice.)

###### Proposition

(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. , let $\mathcal{S}$ be a corresponding S-matrix scheme according to def. , 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. ) is a causally local net in that it satisfies the following conditions:

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

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

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

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

By the isomorphism (17) 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. ):

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

(system of interacting field algebras of observables)

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

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

is a local observable, to be thought of as an adiabatically switched interaction action functional.

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

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

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

Let then

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

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

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

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

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

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

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

This construction defines a functor

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

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

Finally, as a direct corollary of prop. , we obtain the key result:

###### Proposition

(system of interacting field algebras of observables is causally local)

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

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

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

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

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

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

\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 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. ) and then prove that these axioms imply the axioms for the corresponding S-matrix (prop. below).

###### Definition

(time-ordered products)

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

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. ) such that the following conditions hold for all possible arguments:

1. (normalization)

$T_0(O) = 1$
2. (perturbation)

$T_1(O) = :O:$
3. (symmetry) each $T_k$ is symmetric in its arguments, in that for every permutation $\sigma \in \Sigma(k)$ of $k$ elements

$T_k(O_{\sigma(1)}, O_{\sigma(2)}, \cdots, O_{\sigma(k)}) \;=\; T_k(O_1, O_2, \cdots, O_k)$
4. (causal factorization) If the spacetime support (this def.) of local observables satisfies the causal ordering

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

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

(S-matrix scheme implies time-ordered products)

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

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

###### 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 $\kappa_j = 0$:

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

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

Now the causal additivity of the S-matrix $\mathcal{S}$ implies its causal factorization (remark ) 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. below).

###### Remark

(time-ordered products as generalized functions)

It is convenient (as in Epstein-Glaser 73) to think of time-ordered products (def. ), 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) \\ & \coloneqq T( S_1, \cdots, S_r, A_{r+1}, \cdots, A_n ) \end{aligned} \,.

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

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

In this condensed notation the above reads

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

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

###### Example

(product of perturbation series in generalized function-notation)

Let

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

and

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

be power series of Wick algebra-valued distributions in the generalized function-notation of remark .

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

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

is given simply by

$W(\mathbf{X}) \;=\; \underset{\mathbf{I} \subset \mathbf{X}}{\sum} U(\mathbf{I}) V(\mathbf{X} \setminus \mathbf{I}) \,.$
###### 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. below) we need to consider the corresponding reverse-time ordered products:

###### Definition

(reverse-time ordered products)

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

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

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

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

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

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

(reverse-time ordered products express inverse S-matrix)

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

$\left( {\, \atop \,} \mathcal{S}(g S_{int} + j A ) {\, \atop \,} \right)^{-1} \;=\; \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} \left( \frac{1}{i \hbar} \right)^k \overline{T}( \underset{k \, \text{arguments}}{\underbrace{ (g S_{int} + j A), \cdots, (g S_{int} + j A)}} ) \,.$
###### 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!} \underoverset{r = 1}{k}{\sum} (-1)^r \!\!\!\underset{\sigma \in Unshuffl(k,r)}{\sum}\!\!\! T( A_{\sigma(1)} \cdots A_{\sigma(k_1)} ) T( A_{\sigma(k_1 + 1)} \cdots A_{\sigma(k_2)} ) \cdots T( A_{\sigma(k_{r-1}+1)} \cdots A_{\sigma_{k_r}} ) \end{aligned}

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

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

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

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

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

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

where in the last line we used (11).

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

###### Proposition

(inversion relation for reverse-time ordered products)

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

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

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

(reverse causal factorization of reverse-time ordered products)

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

###### Proof

In the condensed notation of remark , 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 .

From the formal inversion

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

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

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

Here

1. in the second line we used that $\mathbf{X} = \mathbf{Q} \sqcup \mathbf{P}$, together with the

causal factorization property of $T(-)$ (which holds by def. ) and that of $\overline{T}(-)$

(which holds by the induction assumption, using that $\mathbf{J} \neq \mathbf{X}$ hence that ${\vert \mathbf{J}\vert} \lt {\vert \mathbf{X}\vert}$).

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

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

2. 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. 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

(time-ordered products induce S-matrix)

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

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

is indeed a perturbative S-matrix according to def. .

###### 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 ) is immediate from the causal factorization condition on the time-ordered products.

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

To see this, first expand the generating function $\mathcal{Z}$ (12) into powers of $g$ and $j$

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

and then compare order-by-order with the given time-ordered product $T$ and its induced reverse-time ordered product (def. ) via prop. . (These $R(-,-)$ are also called the “generating retarded products, discussed in their own right around def. below.)

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

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

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

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

and by prop. this is equivalent to causal additivity of the S-matrix.

It remains to prove the claim:

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

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

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

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

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

With this decomposition of $\mathbf{Y}$, the sum in (18) over subsets $\mathbf{I}$ of $\mathbf{Y}$ may be decomposed into a sum over subsets $\mathbf{J}$ of $\mathbf{P}$ and $\mathbf{K}$ of $\mathbf{Q}$, respectively. These subsets inherit the above causal ordering, so that by the causal factorization property of $T(-)$ (def. ) and $\overline{T}(-)$ (prop. ) the time-ordered and reverse time-ordered products factor on these arguments:

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

Here the sub-sum in brackets vanishes by the inversion formula, prop. .

In conclusion:

###### Proposition

(S-matrix scheme via causal factorization)

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

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

from local observables to microcausal polynomial observables which satisfies the condition “perturbation” from def. . Then the following two conditions on $\mathcal{S}$ are equivalent

2. causal factorization (remark )

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

###### Proof

That causal factorization follows from causal additivity is immediate (remark ).

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

#### (“Re”-)Normalization

We discuss now that time-ordered products as in def. , hence, by prop. , perturbative S-matrix schemes (def. ) exist in fact uniquely away from coinciding interaction points (prop. below).

This means that the construction of full time-ordered products/S-matrix schemes may be phrased as an extension of distributions of time-ordered products to the diagonal locus of coinciding spacetime arguments (prop. below). This choice in their definition is called the choice of ("re"-)normalization of the time-ordered products (remark ), and hence of the interacting pQFT that these define (def. below).

The space of these choices may be accurately characterized, it is a torsor over a group of re-definitions of the interaction-terms, called the “Stückelberg-Petermann renormalization group”. This is called the main theorem of perturbative renormalization, theorem below.

Here we discuss just enough of the ingredients needed to state this theorem. For proof of theorem and discussion of the various methods of picking ("re"-)normalizations see there.

$\,$

###### Definition

(tuples of local observables with pairwise disjoint spacetime support)

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

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

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

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

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

(time-ordered product unique away from coinciding spacetime arguments)

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

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

Then their restriction to the subspace of tuples of local observables of pairwise disjoint spacetime support (def. ) is unique (independent of the "re-"normalization freedom in choosing $T$) and is given by the star product

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

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

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

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

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

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

The idea of this statement goes back at least to Epstein-Glaser 73, as in remark . One formulation appears as (Brunetti-Fredenhagen 00, theorem 4.3). The above formulation in terms of the star product is stated in (Fredenhagen-Rejzner 12, p. 27, Dütsch 18, lemma 3.63 (b)).

###### Proof

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

Consider two partitions of unity

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

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

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

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

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

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

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

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

But now it follows as in the proof of this prop. (via this equation) that

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

Finally the associativity-statement follows as in this prop..

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

By prop. , on regular polynomial observables the S-matrix is given as follows:

###### Definition

(perturbative S-matrix on regular polynomial observables)

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

Recall that the time-ordered product on regular polynomial observables is the star product $\star_F$ induced by the Feynman propagator (this. def.) and that, due to the non-singular nature of regular polynomial observables, this is given by conjugation of the pointwise product (this equation) with $\mathcal{T}$ (this equation) as

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

(this prop.).

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

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

given by

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

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

We write $\mathcal{S}(S_{int})^{-1}$ for the inverse with respect to the Wick product (which exists by this remark)

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

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

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

Similarly, by def. , on regular polynomial observables the quantum Møller operator is given as follows:

###### Definition

(quantum Møller operator on regular polynomial observables)

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

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

then the corresponding quantum Møller operator on regular polynomial observables

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

is given by the derivative of Bogoliubov's formula

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

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

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

Hence the inverse map is

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

(Bogoliubov-Shirkov 59; the above terminology follows Hawkins-Rejzner 16, below def. 5.1)

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

Still by def. , on regular polynomial observables the interacting field algebra of observables is given as follows:

###### Definition

(interacting field algebra structure on regular polynomial observables)

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

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

is the conjugation of the Wick algebra-structure by the quantum Møller operator (def. ):

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

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

endomorphism of
regular polynomial observables
meaningdepends on choice of
$\phantom{AA}\mathcal{T}$time-orderingfree Lagrangian density and Wightman propagator
$\phantom{AA}\mathcal{S}$S-matrixfree Lagrangian density and Wightman propagator
$\phantom{AA}\mathcal{R}$quantum Møller operatorfree Lagrangian density and Wightman propagator and interaction

$\,$

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

###### Definition

(Epstein-Glaser ("re"-)normalization of perturbative QFT)

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

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

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

Via the associativity (19) of the restricted time-ordered product thesese choices are naturally made by induction over $k$, choosing the $(k+1)$-ary time-ordered product $T_{k+1}$ as an extension of distributions of $T_k(\underset{k \, \text{args}}{\underbrace{-, \cdots, -}}) \star_F (-)$.

This inductive choice of extension of distributions of the time-ordered product to coinciding interaction points deserves to be called a choice of normalization of the time-ordered product (e.g. Scharf 94, section 4.3), but for historical reasons (see remark and remark ) it is known as re-normalization. Specifically the inductive construction by extension to coinciding interaction points is known as Epstein-Glaser renormalization.

In (Epstein-Glaser 73) this is phrased in terms of splitting of distributions. In (Brunetti-Fredenhagen 00, sections 4 and 7) the perspective via extension of distributions is introduced, following (Stora 93). Review is in (Dütsch 18, section 3.3.2).

Proposition already shows that the freedom in choosing the ("re"-)normalization of time-ordered products is at most that of extending them to the “fat diagonal”, where at least one pair of interaction points coincides. The following proposition says that when making these choices inductively in the arity of the time-ordered products as in def. then the available choice of ("re"-)normalization) at each stage is in fact only that of extension to the actual diagonal, where all interaction points coincide:

###### Proposition

(("re"-)normalization is inductive extension of time-ordered products to diagonal)

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

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

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

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

This statement appears in (Popineau-Stora 82), with (unpublished) details in (Stora 93), following personal communication by Henri Epstein (according to Dütsch 18, footnote 57). Following this, statement and detailed proof appeared in (Brunetti-Fredenhagen 99).

###### Proof

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

We now say this in detail:

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

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

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

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

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

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

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

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

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

By definition this implies that for

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

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

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

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

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

Using this we find that the identifications of $T_{n+1}$ on $\mathcal{C}_{I_1}$ and on $\mathcal{C}_{I_2}$, accrding to (20), agree on the intersection: in that for $\mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}$ we have

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

Here in the first step we expanded out the two factors using (20) for $I_2$, then under the brace we used (21) and in the last step we used again (20), but now for $I_1$.

To conclude, let

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

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

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

is well defined and satisfies causal factorization.

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

###### Definition

(renormalization conditions, protection from quantum corrections and quantum anomalies)

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

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

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

is called a renormalization condition if

1. it holds for the unique time-ordered products away from coinciding spacetime arguments (according to prop. );

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

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

This means that a renormalization condition is a condition that may consistently be imposed degreewise in an inductive construction of time-ordered products by degreewise extension to the diagonal, according to prop. .

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

If for specified renormalization conditions $\{P_i\}$ there is no choice of time-ordered products $\{T_k\}_{k \in \mathbb{N}}$ (def. ) that satisfies all these conditions, then one says that an interacting perturbative QFT satisfying $\{P_i\}$ fails to exist due to a quantum anomaly.

###### Proposition

(basic renormalization conditions)

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

Then the following conditions are renormalization conditions (def. ):

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

(22)$\frac{\delta}{\delta \mathbf{\Phi}^a(x)} T(A_1, \cdots, A_n) \;=\; \underset{1 \leq k \leq n}{\sum} T\left( A_1, \cdots, A_{k-1}, \frac{\delta}{\delta \mathbf{\Phi}^a(x)}A_k, A_{k+1}, \cdots, A_n \right)$
2. (translation equivariance) If the underlying spacetime is Minkowski spacetime, $\Sigma = \mathbb{R}^{p,1}$, with the induced action of the translation group on polynomial observables

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

then

$\rho_v \left( {\, \atop \,} T(A_1, \cdots, A_n) {\, \atop \,}\right) \;=\; T(\rho_{v}(A_1), \cdots, \rho_v(A_n))$
3. (quantum master equation, master Ward identity) see prop.

(if this condition fails, the corresponding quantum anomaly (def. ) is called a gauge anomaly)

###### Proof

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

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

###### Theorem

(main theorem of perturbative renormalization)

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

1. An S-matrix renormalization scheme (def. ) around this free vacuum, satisfying the renormalization conditions (def. ) “field independence” (prop. ), exists, and its construction by choices of ("re"-)normalization of time-ordered products $\{T_k\}_{k \in \mathbb{N}}$ according to def. involves precisely a finite-dimensional vector space of choices (“renormalization constants”) at each order $k \in \mathbb{N}$.

2. Every pair $\mathcal{S}$, $\widetilde{\mathcal{S}}$ of such choices is related by a unique interaction vertex redefinition

$\mathcal{Z} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ] \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, h ] ]$
$\widetilde{\mathcal{S}} \;=\; \mathcal{S} \circ Z \,.$
3. The group of transformations $\mathcal{Z}$ arising this way is the Stückelberg-Petermann renormalization group.

In summary this says that for each free field vacuum, the space of renormalization schemes for perturbative QFT around this vacuum is non-empty and is canonically a torsor over the Stückelberg-Petermann renormalization group.

Notice that the Stückelberg-Petermann renormalization group involves neither scaling transformations as in Gell-Mann-Low renormalization cocycles, nor cutoffs as in Wilsonian effective field theory. But these alternative perspectives may be extracted as specia cases (Brunetti-Dütsch-Fredenhagen 09).

$\,$

#### Feynman perturbation series

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

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

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

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

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

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

free field algebra of quantum observablesphysics terminologymaths terminology
1)supercommutative product$\phantom{AA} :A_1 A_2:$
normal ordered product
$\phantom{AA} A_1 \cdot A_2$
pointwise product of functionals
2)non-commutative product
(deformation induced by Poisson bracket)
$\phantom{AA} A_1 A_2$
operator product
$\phantom{AA} A_1 \star_H A_2$
star product for Wightman propagator
3)$\phantom{AA} T(A_1 A_2)$
time-ordered product
$\phantom{AA} A_1 \star_F A_2$
star product for Feynman propagator
perturbative expansion
of 2) via 1)
Wick's lemma
Moyal product for Wightman propagator $\Delta_H$
\begin{aligned} & A_1 \star_H A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_H)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}
perturbative expansion
of 3) via 1)
Feynman diagrams
Moyal product for Feynman propagator $\Delta_F$
\begin{aligned} & A_1 \star_F A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_F)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}

$\,$

We now discuss Feynman diagrams and their Feynman amplitudes in two stages: First we consider plain finite multigraphs with linearly ordered vertices but no other labels (def. below) and discuss how these generally organize an expansion of the time-ordered products as a sum of distributional products of the given Feynman propagator (prop. below). These summands (or their vacuum expectation values) are called the Feynman amplitudes if one thinks of the underlying free field vacuum as having a single “field species” and of the chosen interaction to be a single “interaction vertex”.

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

###### Definition

(finite multigraphs)

A finite multigraph is

1. a finite set $V$ (“of vertices”);

2. a finite set $E$ (“of edges”);

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

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

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

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

Hence the isomorphism classes of a finite multigraphs with linearly ordered vertices are characterized by

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

(the number of vertices);

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

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

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

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

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

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

###### Proposition

(Feynman amplitudes of finite multigraphs)

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

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

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

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

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

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

Here the summands of the expansion (23)

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

and/or their vacuum expectation values

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

are called the Feynman amplitudes for scattering processes in the given free field vacuum of shape $\Gamma$ with interaction vertices $O_i$. Their expression as products of distributions via algebraic expression on the right hand side of (24) is also called the Feynman rules.

###### Proof

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

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

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

Here in the first step we used the associativity (19) of the restricted time-ordered product, in the second step we used the induction assumption, in the third we passed the outer functional derivatives through the pointwise product using the product rule, and in the fourth step we recognized that this amounts to summing in addition over all possible choices of sets of edges from the first $v$ vertices to the new $\nu+1$st vertex, which yield in total the sum over all diagrams with $\nu+1$ vertices.

If the free field theory is decomposed as a direct sum of free field theories, we obtain a more fine-grained concept of Feynman amplitudes:

###### Definition

(field species and interaction vertices)

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

Then

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

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

such that the gauge fixed free Lagrangian density $\mathbf{L}'$ is the sum

$\mathbf{L}' \;=\; \mathbf{L}'_{sp_1} + \cdots + \mathbf{L}'_{sp_n}$
$\mathbf{L}'_{sp_i} \in \Omega^{p+1,0}_\Sigma(E_i)$

on these separate field bundles.

1. a choice of interaction vertices and external vertices is a choice of sum decomposition

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

parameterized by finite sets $Int$ and $Ext$, to be called the sets of internal vertex labels and external vertex labels, respectively.

###### Remark

(Feynman propagator for separate field species)

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

Then a choice of field species as in def. induces a corresponding decomposition of the Feynman propagator of the gauge fixed free field theory

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

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

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

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

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

(field species in quantum electrodynamics)

The field bundle for Lorenz gauge fixed quantum electrodynamics on Minkowski spacetime $\Sigma$ admits a decomposition into field species, according to def. , as

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

(by this example and this example).

The corresponding sum decomposition of the Feynman propagator, according to remark , is

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

where

1. $\Delta_F^{\text{electron}}$ is the electron propagator (this def.)

2. $\Delta_F^{photon}$ is the photon propagator in Gaussian-averaged Lorenz gauge (this prop.)

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

###### Definition

(Feynman diagrams)

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

Let moreover

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

be a choice of field species, according to def ,

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

a choice of internal and external interaction vertices according to def. .

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

1. a finite multigraph with linearly ordered vertices (def. )

$\Gamma \in \mathcal{G} \,,$
2. a function from its vertices

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

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

3. a function from its edges

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

to the chosen set of field species.

We write

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

for the set of isomorphism classes of Feynman diagrams with labels in $Sp$, refining the set of isomorphisms of plain finite multigraphs with linearly ordered vertices from def. .

###### Proposition

(Feynman amplitudes for Feynman diagrams)

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

Let moreover

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

be a choice of field species, according to def , hence inducing, by remark , a sum decomposition of the Feynman propagator

(25)$\Delta_F \;=\; \underset{sp \in Spec}{\sum}\Delta_{F,sp} \,,$

and let

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

be a choice of internal and external interaction vertices according to def. .

Then by “multiplying out” the products of the sums (25) and (26) in the formula (24) for the Feynman amplitude $\Gamma\left( (g S_{int} + j A))_{i = 1}^\nu \right)$ (def. ) this decomposes as a sum of the form

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

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

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

or rather their vacuum expectation value

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

are called the Feynman amplitude associated with these Feynman diagrams.

###### Example

(Feynman amplitudes in causal perturbation theory – example of QED)

In perturbative quantum field theory, Feynman diagrams are labeled multigraphs that encode products of Feynman propagators, called Feynman amplitudes (this prop.) which in turn contribute to probability amplitudes for physical scattering processes – scattering amplitudes:

The Feynman amplitudes are the summands in the Feynman perturbation series-expansion of the scattering matrix

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

of a given interaction Lagrangian density $L_{int}$.

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

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

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

For example quantum electrodynamics in Gaussian-averaged Lorenz gauge involves (via this example):

1. the Dirac field modelling the electron, with Feynman propagator called the electron propagator (this def.), here to be denoted

$\Delta \phantom{AAAA} \text{electron propagator}$
2. the electromagnetic field modelling the photon, with Feynman propagator called the photon propagator (this prop.), here to be denoted

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

The Feynman diagram for the electron-photon interaction alone is

where the solid lines correspond to the electron, and the wiggly line to the photon. The corresponding product of distributions is (written in generalized function-notation)

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

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

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

For instance the two solid edges between the vertices $x_2$ and $x_3$ correspond to the two factors of $\Delta(x_2,x_2)$:

This way each sub-graph encodes its corresponding subset of factors in the Feynman amplitude:

graphics grabbed from Brouder 10

A priori this product of distributions is defined away from coincident vertices: $x_i \neq x_j$. The definition at coincident vertices $x_i = x_j$ requires a choice of extension of distributions to the diagonal locus. This choice is the ("re-")normalization of the Feynman amplitude.

###### Example

(Feynman perturbation series)

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

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

be a local observable, regarded as a adiabatically switched interaction action functional.

By prop. every choice of perturbative S-matrix (def. )

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

has an expansion as a formal power series of the form

$\mathcal{S}(g S_{int} + j A) \;=\; \underset{\Gamma \in \mathcal{G}}{\sum} \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)}\right) \,,$

where the series is over all finite multigraphs with linearly ordered vertices $\Gamma$ (def. ), and the summands are the corresponding ("re"-)normalized (def. ) Feynman amplitudes (prop. ).

If moreover a choice of field species and of internal and external interaction vertices is made, according to def. , then this series expansion refines to an expansion over all Feynman diagrams $(\Gamma,edgelab, vertlab)$ (def. ) of Feynman amplitudes $(\Gamma, edgelab,vertlab)(g S_{int} + j A)$ (def. ):

$\mathcal{S}(g S_{int} + j A) \;=\; \underset{(\Gamma,edgelab, vertlab) \in \mathcal{G}^{Feyn}}{\sum} (\Gamma, edgelab,vertlab)(g S_{int} + j A) \,,$

Expressed in this form the S-matrix is known as the Feynman perturbation series.

###### Remark

In the definition of finite multigraphs in def. there are no edges considered that go from any vertex to itself. Accordingly, there are no such labeled edges in Feynman diagrams (def. ):

In pQFT these diagrams are called tadpoles, and their non-appearance is considered part of the Feynman rules (prop. ). Via prop. this condition reflects the nature of the star product (this def.) which always contracts different tensor product factors with the Feynman propagator before taking their pointwise product.

Beware that in graph theory these tadpoles are called “loops”, while here in pQFT a “loop” in a planar graph refers instead to what in graph theory is called a face of the graph, see the discussion of loop order in prop. below.

$\,$

#### Effective action

We have seen that the Feynman perturbation series expresses the S-matrix as a formal power series of Feynman amplitudes labeled by Feynman diagrams. Now the Feynman amplitude associated with a disjoint union of connected Feynman diagrams (def. below) is just the product of the amplitudes of the connected components (prop. below). This allows to re-organize the Feynman perturbation series as the ordinary exponential of the Feynman perturbation series restricted to just connected Feynman diagrams. The latter is called the effective action (def. below) because it allows to express vacuum expectation values of the S-matrix as an ordinary exponential (equation (28) below).

###### Definition

(connected graphs)

Given two finite multigraphs $\Gamma_1, \Gamma_2 \in \mathcal{G}$ (def. ), their disjoint union

$\Gamma_1 \sqcup \Gamma_2 \;\in\; \mathcal{G}$

is the finite multigraph whose set of vertices and set of edges are the disjoint unions of the corresponding sets of $\Gamma_1$ and $\Gamma_2$

$V_{\Gamma_1 \sqcup \Gamma_2} \;\coloneqq\; V_{\Gamma_1} \sqcup V_{\Gamma_2}$
$E_{\Gamma_1 \sqcup \Gamma_2} \;\coloneqq\; E_{\Gamma_1} \sqcup E_{\Gamma_2}$

and whose vertex-assigning function $p$ is the corresponding function on disjoint unions

$p_{\Gamma_1 \sqcup \Gamma_2} \;\coloneqq\; p_{\Gamma_1} \sqcup p_{\Gamma_2} \,.$

The operation induces a pairing on the set $\mathcal{G}$ of isomorphism classes of finite multigraphs

$(-) \sqcup (-) \;\colon\; \mathcal{G} \times \mathcal{G} \longrightarrow \mathcal{G} \,.$

A finite multigraph $\Gamma \in \mathcal{G}$ (def. ) is called connected if it is not the disjoint union of two non-empty finite multigraphs.

We write

$\mathcal{G}_{conn} \subset \mathcal{G}$

for the subset of isomorphism classes of connected finite multigraphs.

###### Lemma

(Feynman amplitudes multiply under disjoint union of graphs)

Let

$\Gamma \;=\; \Gamma_1 \sqcup \Gamma_2 \sqcup \cdots \sqcup \Gamma_n \;\in\; \mathcal{G}$

be disjoint union of graphs (def. ). then then corresponding Feynman amplitudes (prop. ) multiply by the pointwise product (this def.):

$\Gamma\left( g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right) \;=\; \Gamma_1\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma_1)}\right) \cdot \Gamma_2\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma_2)} \right) \cdot \cdots \cdot \Gamma_n\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma_n)} \right) \,.$
###### Proof

By prop. the contributions to the S-matrix away from coinciding interaction points are given by the star product induced by the Feynman propagator, and specifically, by prop. , the Feynman amplitudes are given this way. Moreover the star product (this def.) is given by first contracting with powers of the Feynman propagator and then multiplying all resulting terms with the pointwise product of observables. This implies the claim by the nature of the combinatorial factor in the definition of the Feynman amplitudes (prop. ).

###### Definition

(effective action)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a relativistic free vacuum according to def. , let $\mathcal{S}$ be an S-matrix scheme for perturbative QFT around this vacuum (def. ) and let

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

be a local observable.

Recall that for each finite multigraph $\Gamma \in \mathcal{G}$ (def. ) the Feynman perturbation series for $\mathcal{S}(g S_{int} + j A)$ (example )

$\mathcal{S}(g S_{int} + j A) \;=\; \underset{\Gamma \in \mathcal{G}}{\sum} \Gamma\left( (g S_{int} + j A)_{i = 1}^{v(\Gamma)} \right)$

contributes with a ("re"-)nromalized Feynman amplitude $\Gamma\left( (g S_{int} + j A)_{i = 1}^v\right) \in PolyObs(E_{\text{BV-BRST}})((\hbar))[ [ g, j ] ]$.

We say that the corresponding effective action is $i \hbar$ times the sub-series

(27)$S_{eff}(g,j) \;\coloneqq\; i \hbar \underset{\Gamma \in \mathcal{G}_{conn}}{\sum} \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right) \;\in\; PolyObs(E_{\text{BV-BRST}})[ [\hbar, g, j ] ]$

of Feynman amplitudes that are labeled only by the connected graphs $\Gamma \in \mathcal{G}_{conn} \subset \mathcal{G}$ (def. ).

(A priori $S_{eff}(g,j)$ could contain negative powers of $\hbar$, but it turns out that it does not; this is prop. below.)

###### Remark

(terminology for “effective action”)

Beware differing conventions of terminology:

1. In effective quantum field theory, the effective action in def. is sometimes called the effective potential at scale $\Lambda = 0$ (see this prop.).

This terminology originates in restriction to the special example of the scalar field, where the typical non-derivative Phi^n interactions $g S_{int} = \underset{n}{\sum} \underset{\Sigma}{\int} g_{sw}^{(n)}(x) (\mathbf{\Phi}(x))^n \, dvol_\Sigma(x)$ are naturally thought of as potential energy-terms.

From this perspective the effective action in def. is a special case of relative effective actions $S_{eff,\Lambda}$ (“relative effective potentials”, in the case of Phi^n interactions) relative to an arbitrary UV cutoff-scales $\Lambda$ (this def.).

2. For the special case that

$j A \coloneqq \underset{\Sigma}{\int} j_{sw,a}(x) \mathbf{\Phi}^a(x)\, dvol_{\Sigma}(x)$

is a regular linear observable (this def.) the effective action according to def. is often denoted $W(j)$ or $E(j)$, and then its functional Legendre transform (if that makes sense) is instead called the effective action, instead.

This is because the latter encodes the equations of motion for the vacuum expectation values $\langle \mathbf{\Phi}(x)_int\rangle$ of the interacting field observables; we discuss this as example below.

Notice the different meaning of “effective” in both cases: In the first case it refers to what is effectively seen of the full pQFT at some UV-cutoff scale, while in the second case it refers to what is effectively seen when restricting attention only to the vacuum expectation values of regular linear observables.

###### Proposition

(effective action is logarithm of S-matrix)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a relativistic free vacuum according to def. , let $\mathcal{S}$ be an S-matrix scheme for perturbative QFT around this vacuum (def. ) and let

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

be a local observable and let

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

be the corresponding effective action (def. ).

Then then S-matrix for $g S_{int} + j A$ is the exponential of the effective action with respect to the pointwise product $(-)\cdot (-)$ of observables (this def.):

\begin{aligned} \mathcal{S}(g S_{int} + j A) & = \exp_\cdot\left( \tfrac{1}{i \hbar} S_{eff}(g,j) \right) \\ & \coloneqq 1 + \frac{1}{i \hbar} S_{eff}(g,j) + \frac{1}{(i \hbar)^2} S_{eff}(g,j) \cdot S_{eff}(g,j) + \frac{1}{(i \hbar)^3} S_{eff}(g,j) \cdot S_{eff}(g,j) \cdot S_{eff}(g,j) + \cdots \end{aligned}

Moreover, this relation passes to the vacuum expectation values:

(28)\begin{aligned} \left\langle {\, \atop \,} \mathcal{S}(g S_{int} + j A) {\, \atop \,} \right\rangle & = \left\langle {\, \atop \,} \exp\left( \tfrac{1}{i \hbar} S_{eff}(g,j) \right) {\, \atop \,} \right\rangle \\ & = e^{\tfrac{1}{i \hbar} \langle S_{eff}(g,j) \rangle} \end{aligned} \,.

Conversely the vacuum expectation value of the effective action is to the logarithm of that of the S-matrix:

$\left\langle S_{eff}(g,j) \right\rangle \;=\; i \hbar \, \ln \left\langle \mathcal{S}(g S_{int} + j A) \right\rangle \,.$
###### Proof

By lemma the summands in the $n$th pointwise power of $\frac{1}{i \hbar}$ times the effective action are precisely the Feynman amplitudes $\Gamma\left((g S_{int} + j A)_{i = 1}^{\nu(\Gamma)}\right)$ of finite multigraphs $\Gamma$ with $n$ connected components, where each such appears with multiplicity given by the factorial of $n$:

$\frac{1}{n!} \left( \frac{1}{i \hbar} S_{eff}(g,j) \right)^n \;=\; \underset{ { \Gamma = \underoverset{j = 1}{n}{\sqcup} \Gamma_j } \atop { \Gamma_j \in \mathcal{G}_{conn} } }{\sum} \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right) \,.$

It follows that

\begin{aligned} \exp_\cdot\left( \frac{1}{i \hbar} S_{int} \right) & = \underset{n \in \mathbb{N}}{\sum} \underset{ { \Gamma = \underoverset{j = 1}{n}{\sqcup} \Gamma_j } \atop { \Gamma_j \in \mathcal{G}_{conn} } }{\sum} \Gamma\left( (g S_{int} + j A)_{i = 1}^{v(\Gamma)} \right) \\ & = \underset{\Gamma \in \mathcal{G}}{\sum} \Gamma\left( (g S_{int} + j A)_{i = 1}^{v(\Gamma)} \right) \end{aligned}

yields the Feynman perturbation series by expressing it as a series (re-)organized by number of connected components of the Feynman diagrams.

To conclude the proof it is now sufficient to observe that taking vacuum expectation values of polynomial observables respects the pointwise product of observables

$\left\langle A_1 \cdot A_2 \right\rangle \;=\; \left\langle A_1 \right\rangle \, \left\langle A_2 \right\rangle \,.$

This is because the Hadamard vacuum state $\langle -\rangle \colon PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ] \to \mathbb{C}[ [\hbar, g, j ] ]$ simply picks the zero-order monomial term, by this prop., and under multiplication of polynomials the zero-order terms are multiplied.

This immediately implies the following important fact:

###### Proposition

(in stable vacuum the effective action is generating function for vacuum expectation values of interacting field observables)

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

If the given vacuum state is stable (def. ) then the vacuum expectation value $\langle S_{eff}(g,j)\rangle$ of the effective action (def. ) is the generating function for the vacuum expectation value of the interacting field observable $A_{int}$ (def. ) in that

$\left\langle A_{int} \right\rangle \;=\; \frac{d}{d j} S_{eff}(g,j)\vert_{j = 0} \,.$
###### Proof

We compute as follows:

\begin{aligned} \frac{d}{d j} S_{eff}(g,j) & = i \hbar \frac{d}{d j} \ln \left\langle \mathcal{S}(g S_{int} + j A) \right\rangle \vert_{j = 0} \\ & = i \hbar \left\langle \mathcal{S}(g S_{int}) \right\rangle^{-1} \frac{d}{d j} \left\langle \mathcal{S}(g S_{int} + j A) \right\rangle \vert_{j = 0} \\ & = \left\langle \frac{d}{d j} \underset{ \mathcal{Z}(j A) }{ \underbrace{\mathcal{S}(g S_{int})^{-1} \mathcal{S}(g S_{int} + j A) }} \vert_{j = 0} \right\rangle \\ & = \left\langle A_{int} \right\rangle \,. \end{aligned}

Here in the first step we used prop , in the second step we applied the chain rule of differentiation, in the third step we used the definition of vacuum stability (def. ) and in the fourth step we recognized the definition of the interacting field observables (def. ).

###### Example

(equations of motion for vacuum expectation values of interacting field observables)

Consider the effective action (def. ) for the case that

\begin{aligned} j A & = \tau{\Sigma}( j_{sw} \phi) \\ & = \underset{\Sigma}{\int} j_{sw}(x) \mathbf{\Phi}(x) \, dvol_\Sigma(x) \end{aligned}

is a regular linear observable (this def.), hence the smearing of a field observable (this def.) by an adiabatic switching of the source field

$j_{sw} \;\in\; C^\infty_{cp}(\Sigma) \langle j\rangle \,.$

(Here we are notationally suppressing internal field indices, for convenience.)

In this case the vacuum expectation value of the corresponding effective action is often denoted

$W(j_{sw})$

and regarded as a functional of the adiabatic switching $j_{sw}$ of the source field.

In this case prop. says that if the vacuum state is stable, then $W$ is the generating functional for interacting (def. ) field observables (this def.) in that

(29)$\left\langle \mathbf{\Phi}(x)_{int} \right\rangle \;=\; \frac{\delta}{\delta j_{sw}(x)} W(j_{sw} = 0) \,.$

Assume then that there exists a corresponding functional of the field histories $\Gamma(\Phi)$, which behaves like a functional Legendre transform of $W$ in that it satisfies the functional version of the defining equation of Legendre transforms (first derivatives are inverse functions of each other, see this equation), in that

$\frac{\delta }{\delta \Phi(x)} \Gamma \left( \frac{\delta}{\delta j_{sw}(y)} W \right) \;=\; \delta(x,y) j_{sw}(x) \,.$

By (29) this implies that

$\frac{\delta }{\delta \Phi(x)} \Gamma \left( \left\langle \mathbf{\Phi}(x)_{int} \right\rangle \right) \;=\; 0 \,.$

This may be read as a quantum version of the principle of extremal action (this prop.) formulated now not for the field histories $\Phi(x)$, but for the vacuum expectation values $\langle \mathbf{\Phi}(x)_{int}\rangle$ of their corresponding interacting quantum field observables.

Beware, (as in remark ) that many texts refer to $\Gamma(\Phi)$ as the effective action, instead of its Legendre transform, the generating functional $W(j_{sw})$.

The perspective of the effective action also gives a transparent picture of the order of quantum effects involved in the S-matrix, this is prop. below. In order to state this conveniently, we invoke two basic concepts from graph theory:

###### Definition

(planar graphs and trees)

A finite multigraph (def. ) is called a planar graph if it admits an embedding into the plane, hence if it may be “drawn into the plane” without intersections, in the evident way.

A finite multigraph is called a tree if for any two of its vertices there is at most one path of edges connecting them, these are examples of planar graphs. We write

$\mathcal{G}_{tree} \subset \mathcal{G}$

for the subset of isomorphism classes of finite multigraphs with linearly orrdered vertices (def. ) on those which are trees.

###### Proposition

(loop order and tree level of Feynman perturbation series)

The effective action (def. ) contains no negative powers of $\hbar$, hence is indeed a formal power series also in $\hbar$:

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

and in particular

$\left\langle S_{eff}(g,j) \right\rangle \;\in\; \mathbb{C}[ [ \hbar, g, j] ] \,.$

Moreover, the contribution to the effective action in the classical limit $\hbar \to 0$ is precisely that of Feynman amplitudes of those finite multigraphs (prop. ) which are trees (def. ); thus called the tree level-contribution:

$S_{eff}(g,j)\vert_{\hbar = 0} \;=\; i \hbar \underset{\Gamma \in \mathcal{G}_{conn} \cap \mathcal{G}_{tree}}{\sum} \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right) \,.$

Finally, a finite multigraph $\Gamma$ (def. ) which is planar (def. ) and connected (def. ) contributes to the effective action precisely at order

$\hbar^{L(\Gamma)} \,,$

where $L(\Gamma) \in \mathbb{N}$ is the number of faces of $\Gamma$, here called the number of loops of the diagram; here usually called the loop order of $\Gamma$.

(Beware the terminology clash with graph theory, see the discussion of tadpoles in remark .)

###### Proof

By def. the explicit $\hbar$-dependence of the S-matrix is

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

and by prop. the further $\hbar$-dependence of the time-ordered product $T(\cdots)$ is

$T(S_{int}, S_{int}) \;=\; prod \circ \exp\left( \hbar \left\langle \Delta_F, \frac{\delta}{\delta \mathbf{\Phi}} \otimes \frac{\delta}{\delta \mathbf{\Phi}} \right\rangle \right) ( S_{int} \otimes S_{int} ) \,,$

By the Feynman rules (prop. ) this means that

1. each vertex of a Feynman diagram contributes a power $\hbar^{-1}$ to its Feynman amplitude;

2. each edge of a Feynman diagram contributes a power $\hbar^{+1}$ to its Feynman amplitude.

If we write

$E(\Gamma), V(\Gamma) \;\in\; \mathbb{N}$

for the total number of vertices and edges, respectively, in $\Gamma$, this means that a Feynman amplitude corresponding to some $\Gamma \in \mathcal{G}$ contributes precisely at order

(30)$\hbar^{E(\Gamma) - V(\Gamma)} \,.$

So far this holds for arbitrary $\Gamma$. If however $\Gamma$ is connected (def. ) and planar (def. ), then Euler's formula asserts that

(31)$E(\Gamma) - V(\Gamma) \;=\; L(\Gamma) - 1 \,.$

Hence $\hbar^{L(\Gamma)- 1}$ is the order of $\hbar$ at which $\Gamma$ contributes to the scattering matrix expressed as the Feynman perturbation series.

But the effective action, by definition (27), has the same contributions of Feynman amplitudes, but multiplied by another power of $\hbar^1$, hence it contributes at order

$\hbar^{E(\Gamma) - V(\Gamma) + 1} = \hbar^{L(\Gamma)} \,.$

This proves the second claim on loop order.

The first claim, due to the extra factor of $\hbar$ in the definition of the effective action, is equivalent to saying that the Feynman amplitude of every connected finite multigraph contributes powers in $\hbar$ of order $\geq -1$ and contributes at order $\hbar^{-1}$ precisely if the graph is a tree.

Observe that a connected finite multigraph $\Gamma$ with $\nu \in \mathbb{N}$ vertices (necessarily $\nu \geq 1$) has at least $\nu-1$ edges and precisely $\nu - 1$ edges if it is a tree.

To see this, consecutively remove edges from $\Gamma$ as long as possible while retaining connectivity. When this process stops, the result must be a connected tree $\Gamma'$, hence a connected planar graph with $L(\Gamma') = 0$. Therefore Euler's formula (31) implies that that $E(\Gamma') = V(\Gamma') -1$.

This means that the connected multigraph $\Gamma$ in general has a Feynman amplitude of order

$\hbar^{E(\Gamma) - V(\Gamma)} = \hbar^{ \overset{\geq 0}{\overbrace{E(\Gamma) - E(\Gamma')}} + \overset{= -1}{\overbrace{E(\Gamma') - V(\Gamma)}} }$

and precisely if it is a tree its Feynman amplitude is of order $\hbar^{-1}$.

$\,$

#### Vacuum diagrams

With the Feynman perturbation series and the effective action in hand, it is now immediate to see that there is a general contribution by vacuum diagrams (def. below) in the scattering matrix which, in a stable vacuum state, cancels out against the prefactor $\mathcal{S}(g S_{int})$ in Bogoliubov's formula for interacting field observables.

###### Definition

(vacuum diagrams)

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

Then a Feynman diagram all whose vertices are internal vertices (def. ) is called a vacuum diagram.

Write

$\mathcal{G}^{Feyn}_{vac} \subset \mathcal{G}^{Feyn}$

for the subset of isomorphism classes of vacuum diagrams among the set of isomorphism classes of all Feynman diagrams, def. . Similarly write

$\mathcal{G}^{Feyn}_{conn,vac} \;\coloneqq\; \mathcal{G}^{Feyn}_{conn} \cap \mathcal{G}^{Feyn}_{vac} \;\subset\; \mathcal{G}^{Feyn}$

for the subset of isomorphism classes of Feynman diagrams which are both vacuum diagrams as well as connected graphs (def. ).

Finally write

$S_{eff,vac}(g) \;\coloneqq\; \underset{ { (\Gamma,vertlab,edgelab) } \atop { \in \mathcal{G}_{conn,vac} } }{\sum} (\Gamma,vertlab, edgelab)(g S_{int}) \;\in\; PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar , g ] ]$

for the sub-series of that for the effective action (def. ) given only by those connected diagrams which are also vacuum diagrams.

###### Example

(2-vertex vacuum diagram in QED)

The vacuum diagram (def. ) with two electron-photon interaction-vertices in quantum electrodynamics (this example.) is:

###### Example

(vacuum diagram-contribution to S-matrices)

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

Then the Feynman perturbation series-expansion of the S-matrix (example ) of the interaction-term $g S_{int}$ alone (no source field-contribution) is the series of Feynman amplitudes that are labeled by vacuum diagrams (def. ), hence (by prop. ) the exponential of the vacuum effective action $S_{eff,vac}$ (def. ):

\begin{aligned} \mathcal{S}(g S_{int}) & = \exp_\cdot\left( \tfrac{1}{i \hbar} S_{eff,vac}(g,j) \right) \\ & = \underset{\Gamma \in \mathcal{G}_{vac}}{\sum} \Gamma\left(g S_{int}\right) \end{aligned} \,.

More generally, the S-matrix with source field-contribution $j A$ included always splits as a pointwise product of the vacuum S_matrix with the Feynman perturbation series over all Feynman graphs with at least one external vertex:

\begin{aligned} \mathcal{S}(g S_{int} + j A) \;=\; \mathcal{S}(g S_{int}) \cdot \underset{ \text{Feynman perturbation series} \atop \text{over diagrams with at least one external vertex} }{ \underbrace{ \exp_\cdot \left( \tfrac{1}{i \hbar} \left( S_{eff}(g,j) - S_{eff,vac}(g) \right) \right) } } \,, \end{aligned}

Hence if the free field vacuum state is stable with respect to the interaction $g S_{int}$, according to def. , then the vacuum expectation value of a time-ordered product of interacting field observables $j (A_i)_{int}$ (example ) and hence in particular of scattering amplitudes (example ) is given by the Feynman perturbation series (example ) over just the non-vacuum Feynman diagrams, hence over all those diagram that have at least one one external vertex

\begin{aligned} & \left( {\, \atop \,} supp(A_1) {\vee\!\!\!\wedge} supp(A_2) {\vee\!\!\!\wedge} \cdots {\vee\!\!\!\wedge} supp(A_n) {\, \atop \,} \right) \\ & \Rightarrow \left\langle {\, \atop \,} (A_1)_int (A_2)_{int} \cdots (A_n)_{int} {\, \atop \,} \right\rangle \;=\; \frac{d^n}{ d j_1 \cdots d j_n} \left( \underset{\Gamma \in \mathcal{G} \setminus \mathcal{G}_{vac} }{\sum} \Gamma(g S_{int} + \sum_i j_i A_i) \right)_{ \vert j_1, \cdots, j_n = 0 } \,. \end{aligned}

This is the way in which the Feynman perturbation series is used in practice for computing scattering amplitudes.

$\,$

#### Interacting quantum BV-Differential

So far we have discussed, starting with a BV-BRST gauge fixed free field vacuum, the perturbative construction of interacting field algebras of observables (def. ) and their organization in increasing powers of $\hbar$ and $g$ (loop order, prop. ) via the Feynman perturbation series (example , example ).

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 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 and as well as on regular polynomial observables are uniquely fixed (prop. ), one finds that also this interacting quantum BV-differential is uniquely fixed, on regular polynomial observables, by conjugation with the quantum Møller operators (def. ). The formula that characterizes it there is called the quantum master equation or equivalently the quantum master Ward identity (prop. below).

When extending to coinciding interaction points via ("re"-)normalization (def. ) these identities are not guaranteed to hold anymore, but may be imposed as renormalization conditions (def. , prop. ). Quantum correction to the master Ward identity then imply corrections to Noether current conservation laws; this we discuss below.

$\,$

Recall how the global BV-differential

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

on regular polynomial observables (this def.) is conjugated into the time-ordered product via the time ordering operator $\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-}$ (this prop.).

In the same way we may use the quantum Møller operators to conjugate the BV-differential into the regular part of the interacting field algebra of observables:

###### Definition

(interacting quantum BV-differential)

Given an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable $S_{int}$, then the interacting quantum BV-differential on the interacting field algebra (def. ) on regular polynomial observables is the conjugation of the plain BV-differential $\{-S',-\}$ by the quantum Møller operator induced by $S_{int}$ (def. ):

$\mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \,.$
###### Proposition

(quantum master equation and quantum master Ward identity on regular polynomial observables)

Consider an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable in degree 0

$S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ] \,,$

Then the following are equivalent:

1. The quantum master equation (QME)

(32)$\tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}} + i \hbar \Delta_{BV}( S' + S_{int} ) \;=\; 0 \,.$
2. The perturbative S-matrix (def. ) is $BV$-closed

$\{-S', \mathcal{S}(S_{int})\} = 0 \,.$
3. The quantum master Ward identity (MWI) on regular polynomial observables in terms of retarded products:

(33)$\mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \;=\; - \left( \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right)$

expressing the interacting quantum BV-differential (def. ) as the sum of the time-ordered antibracket (this def.) with the total action functional $S' + S_{int}$ and $i \hbar$ times the BV-operator (BV-operator).

4. The quantum master Ward identity (MWI) on regular polynomial observables in terms of time-ordered products:

(34)$\mathcal{S}(-S_{int}) \star_F \{-S', \mathcal{S}(S_{int}) \star_F (-)\} \;=\; - \left( \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right)$

(Rejzner 11, (5.35) - (5.38), following Hollands 07, (342)-(345))

###### Proof

To see that the first two conditions are equivalent, we compute as follows

(35)\begin{aligned} \left\{ -S', \mathcal{S}(S_{int}) \right\} & = \left\{ -S' , \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \right\} \\ & = \underset{ { \tfrac{-1}{i \hbar} \{S',S\}_{\mathcal{T}} } \atop { \star_F \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) } }{ \underbrace{ \left\{ -S' , \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \right\}_{\mathcal{T}} } } - i \hbar \underset{ { \left( \tfrac{1}{i \hbar} \Delta_{BV}(S_{int}) + \tfrac{1}{2 (i \hbar)^2} \left\{ S_{int}, S_{int} \right\}_{\mathcal{T}} \right) } \atop { \star_{F} \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) } }{ \underbrace{ \Delta_{BV} \left( \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \right) } } \\ & = \tfrac{-1}{i \hbar} \underset{ \text{QME} }{ \underbrace{ \left( \{S',S_{int}\} + \tfrac{1}{2}\{S_{int}, S_{int}\} + i \hbar \Delta_{BV}(S_{int}) \right) } } \star_F \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \end{aligned}

Here in the first step we used the definition of the BV-operator (this def.) to rewrite the plain antibracket in terms of the time-ordered antibracket (this def.), then under the second brace we used that the time-ordered antibracket is the failure of the BV-operator to be a derivation (this prop) and under the first brace the consequence of this statement for application to exponentials (this example). Finally we collected terms, and to “complete the square” we added the terms on the left of

$\frac{1}{2} \underset{= 0}{\underbrace{\{S', S'\}_{\mathcal{T}}}} - i \hbar \underset{ = 0}{\underbrace{ \Delta_{BV}(S')}} = 0$

which vanish because, by definition of gauge fixing (this def.), the free gauge-fixed action functional $S'$ is independent of antifields.

But since the operation $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)$ has the inverse $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)$, this implies the claim.

Next we show that the quantum master equation implies the quantum master Ward identities.

We use that the BV-differential $\{-S',-\}$ is a derivation of the Wick algebra product $\star_H$ (lemma ).

First of all this implies that with $\{-S', \mathcal{S}(S_{int})\} = 0$ also $\{-S', \mathcal{S}(S_{int})^{-1}\} = 0$.

Thus we compute as follows:

$\begin{array}{rl}\left\{-S\prime ,-\right\}\circ {ℛ}^{-1}\left(A\right)& =\left\{-S\prime ,{ℛ}^{-1}\left(A\right)\right\}\\ & =\left\{\genfrac{}{}{0}{}{\phantom{\rule{thinmathspace}{0ex}}}{\phantom{\rule{thinmathspace}{0ex}}}-S\prime ,𝒮\left({S}_{\mathrm{int}}{\right)}^{-1}{\star }_{H}\left(𝒮\left({S}_{\mathrm{int}}\right){\star }_{F}a\right)\genfrac{}{}{0}{}{\phantom{\rule{thinmathspace}{0ex}}}{\phantom{\rule{thinmathspace}{0ex}}}\right\}\\ & =\phantom{+}\underset{}{\underset{}{\left\{-S\prime }}\end{array}$