quantization

Contents

Idea

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

Every perturbative quantum field theory has an S-matrix, usually thought of as a perturbation series over Feynman diagrams extracted from the action functional. The rigorous construction of this as an operator-valued distribution is the content of causal perturbation theory.

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

The perception of the relevance of the S-matrix for the foundations of quantum field theory has a convoluted (and ongoing) history, see below.

Details

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 (3) below.)

Now it is clear from the construction and using the product law for differentiation, that $\vert \psi(-)\rangle_S$ satisfies the following differential equation:

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

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”. But in fact this 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:

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

Here time-ordering means

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

(This is abuse of notation: 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

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

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

In perturbative algebraic quantum field 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 point are taken care of by the axiomatization of the “adiabatically switched S-matrix” according to causal perturbation theory.

The analogue of the limit $t \to \infty$ in the construction of the S-matrix (now: adiabatic limit) in general does not exist in field theory (“infrared divergencies”). But in fact it need not be taken: The field algebra in a bounded region of spacetime may be computed with any adiabatic switching that is constant on this region. Moreover, the algebras assigned to regions of spacetime this way satisfy causal locality by the causal ordering in the construction of the S-matrix. Therefore, even without taking the adiabatic limit in causal perturbation theory one obtains a field theory in the form of a local net of observables. This is the topic of locally covariant perturbative quantum field theory.

$\,$

Spacetime and Causality

Throughout, let $\Sigma$ be a time-oriented globally hyperbolic Lorentzian manifold of dimension

$p + 1 \in \mathbb{N}$

This is to model the spacetime over which we consider field theory. Hence we will often refer to $\Sigma$ just as spacetime, for short.

We need to consider the following concepts and constructions related to the causal structure of $\Sigma$.

Definition

(causal past and future)

For $x \in \Sigma$ a point in spacetime, we write

$V^+(x), V^-(x) \subset \Sigma$

for its open future cone and open past cone, respectively, and

$\overline{V}^+(x), \overline{V}^-(x) \subset \Sigma$

for the corresponding closed cones. We write

$J(x) \coloneqq \overline{V}^+(x) \cup \overline{V}^-(x)$

for the full causal cone.

For $S \subset \Sigma$ a subset we write

$\overline{V}^\pm(S) \;\coloneqq\; \underset{x \in S}{\cup} \overline{V}^{\pm}(x)$

for the union of the future/past closed cones of all its points.

Definition

(causal order)

Consider the relation on the set $P(\Sigma)$ of subsets of spacetime which says a subset $S_1 \subset \Sigma$ is not prior to a subset $S_2 \subset \Sigma$, denoted $S_1 \geq S_2$, if $S_1$ does not intersect the causal past of $S_2$ (def. 1), or equivalently that $S_2$ does not intersect the causal future of $S_1$:

\begin{aligned} (S_1 \geq S_2) & \coloneqq S_1 \cap \overline{V}^-(S_2) = \emptyset \\ & \Leftrightarrow S_2 \cap \overline{V}^+(S_1) = \emptyset \end{aligned} \,.

If $S_1 \geq S_2$ and $S_2 \geq S_1$ we say that the two subsets are spacelike separated.

Definition

(causal complement and causal closure of subset of spacetime)

For $S \subset X$ a subset of spacetime, its causal complement $S^\perp$ is the complement of the causal cone:

$S^\perp \;\coloneqq\; S \setminus J_X(S) \,.$

The causal complement $S^{\perp \perp}$ of the causal complement $S^\perp$ is called the causal closure. If

$S = S^{\perp \perp}$

then the subset $S$ is called a causally closed subset.

Given a spacetime $\Sigma$, we write

$CausClsdSubsets(\Sigma) \;\in\; Cat$

for the partially ordered set of causally closed subsets, partially ordered by inclusion $\mathcal{O}_1 \subset \mathcal{O}_2$.

Definition

For a causally closed subset $\mathcal{O} \subset \Sigma$ of spacetime (def. 1) say that an adiabatic switching function or cutoff function for $\mathcal{O}$ is a smooth function $g_{sw}$ of compact support (a bump function) whose restriction to some neighbourhood of $\mathcal{O}$ is the constant function with value $1$:

$Cutoffs(\mathcal{O}) \;\coloneqq\; \left\{ g_{sw} \in \mathcal{C}^\infty_c(\Sigma) \;\vert\; \underset{ {U \supset \mathcal{O}} \atop { \text{neighbourhood} } }{\exists} \left( g_{sw}\vert_S = const_1 \right) \right\} \,.$

Often we consider the vctor space space $C^\infty(\Sigma)\langle g \rangle$ spanned by a formal variable $g$ under multiplication with smooth functions, and consider as adiabatic switching functions the corresponding images in this space, which are hence bump functions constant on $g$ over a neighbourhood of $\mathcal{O}$.

Lemma

(causal partition)

Let $\mathcal{O} \subset \Sigma$ be a causally closed subset (def. 1) and let $f \in C^\infty_{cp}(\Sigma)$ be a compactly supported smooth function which vanishes on a neighbourhood $U \supset \mathcal{O}$, i.e. $f\vert_U = 0$.

Then there exists a causal partition of $f$ in that there exist compactly supported smooth functions $a,r \in C^\infty_{cp}(\Sigma)$ such that

1. they sum up to $f$:

$f = a + r$
2. their support satisfies the following causal ordering (def. 2)

$supp(a) \geq \mathcal{O} \geq supp(r) \,.$
Proof idea

By assumption $\mathcal{O}$ has a Cauchy surface. This may be extended to a Cauchy surface $\Sigma_p$ of $\Sigma$, such that this is one leaf of a foliation of $\Sigma$ by Cauchy surfaces, given by a diffeomorphism $\Sigma \simeq (-1,1) \times \Sigma_p$ with the original $\Sigma_p$ at zero. There exists then $\epsilon \in (0,1)$ such that the restriction of $supp(f)$ to the interval $(-\epsilon, \epsilon)$ is in the causal complement $\overline{\mathcal{O}}$ of the given region (def. 1):

$supp(f) \cap (-\epsilon, \epsilon) \times \Sigma_p \;\subset\; \overline{\mathcal{O}} \,.$

Let then $\chi \colon \Sigma \to \mathbb{R}$ be any smooth function with

1. $\chi\vert_{(-1,0] \times \Sigma_p} = 1$

2. $\chi\vert_{(\epsilon,1) \times \Sigma_p} = 0$.

Then

$r \coloneqq \chi \cdot f \phantom{AAA} \text{and} \phantom{AAA} a \coloneqq (1-\chi) \cdot f$

are smooth functions as required.

Free fields and Propagators

The definition and construction of a perturbative S-matrix below proceeds on the backdrop of a formal of the underlying free field theory, in particular a corresponding perturbative algebra of observables, the Wick algebra $\mathcal{W}$. Here we briefly recall relevant bckground from free field theory.

But just stating the axioms for the perturbative Lagrangian S-matrix (below) and deducing from these axioms the causally local net of quantum observables (further below) only requires knowing that there is a continuous linear functional

$:(-): \;\colon\; \mathcal{F}_{loc} \longrightarrow \mathcal{W}$

from local observables to the coresponding quantum observables, being elements of the Wick algebra.

The actual construction of the S-matrix (further below) and the proof of the main theorem of perturbative renormalization requires more details, which we briefly summarize here.

Definition

(local observables)

Let $\Sigma$ be a globally hyperbolic spacetime, let

$\array{ E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }$

be a smooth fiber bundle regarded as a field bundle and let

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

be a local Lagrangian density. We write

$\Omega \in \Omega^{p,2}_\Sigma(E)$

for the corresponding pre-symplectic current and

$\Omega^{p,0}_{\Sigma, Ham}(E)$

for the induced space of Hamiltonian differential forms. We write

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

for the subspace of solutions to the Euler-Lagrange equations inside the full field configuration space (the shell).

If the Euler-Lagrange operator $j^\infty_\Sigma(-)^\ast \delta_{EL}\mathbf{L}$ is a linear normally hyperbolic differential operator on $\Gamma_\Sigma(E)$ we say that the field theory is a free field theory.

Since $\Sigma$ is assumed to be time-oriented globally hyperbolic, there exists a 1-form $e^0$ which represents the time orientation. The space $\mathcal{F}_{loc}$ of local observables is the image of the Hamiltonian differential forms with compact spacetime support in the smooth functions on the on-shell under the operations of forming the wedge product with $e^0$ followed by transgression

$\mathcal{F}_{loc} \;\coloneqq\; im\left( \Omega^{p+1,0}_{\Sigma,Ham,cp}(E) \overset{e^0 \wedge (-)}{\longrightarrow} \Omega^{p2,0}_{\Sigma, cp}(E) \overset{\tau_{\Sigma}}{\longrightarrow} C^\infty\left( \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \right) \right) \,.$

Since the kernel of the transgression map is the space of forms in the image of the total horizontal derivative (horizontal derivative), this is equivalently

$\mathcal{F}_{loc} \simeq im\left( \Omega^{p+1,0}_{\Sigma,Ham, cp}(E) \overset{e^0 \wedge (-)}{\longrightarrow} \Omega^{p2,0}_{\Sigma, cp}(E) \right) /im(d) \,.$

Due to the restriction to Hamiltonian differential forms in the construction of $\mathcal{F}_{loc}$, the presymplectic current $\Omega$ induces a Poisson bracket on the space of local observables

$\{-,-\} \;\colon\; \mathcal{F}_{loc} \otimes \mathcal{F}_{loc} \longrightarrow \mathcal{F}_{loc} \,.$

Given a formal deformation quantization $\mathcal{W}$ of this Poisson bracket, then we write

$\mathcal{F}_{loc} \overset{:(-):}{\longrightarrow} \mathcal{W}$

for the corresponding quantization map.

In the case of free field theories such $\mathcal{W}$ is given by the Wick algebra. See there for more.

In the following we need to keep track of powers of the coupling constant and the source field. To that end, we write $\langle g,j\rangle$ be the vector space spanned by the elements $g$ and $j$ and

$\mathcal{F}_{loc} \langle g,j\rangle \;\coloneqq\; \mathcal{F}_{loc} \otimes \langle g,j\rangle$

Then every element of $\mathcal{F}_{loc}\langle g,j \rangle$ may be presented as

$\underset{n_1}{\sum} g_{sw,n_1} L_{n_1} + \underset{n_2}{\sum} j_{sw,n_2} A_{n_2}$

where $L_{n_1}, L_{n_2} \in \Omega_{\Sigma,poly}^{p+1,0}(E)$ are polynomial local observables, to be regarded as interaction Lagrangian densities and observables, respectively, and where

$g_{sw, n_1} \in C^\infty_{cp}(\Sigma) \langle g\rangle$

and

$j_{sw,n_2} \in C^\infty_{cp}(\Sigma) \langle j \rangle$

are compactly supported smooth functions (bump functions) on spacetime $\Sigma$ times the coupling constant or source strength, respectively, to be regarded as adiabatically switched coupling constant and source field, respectively.

We will be considering formal power series in $g/\hbar$ and $j/\hbar$ with coefficients in the Wick algebra $\mathcal{W}$. Since the vector space underlying the Wick algebra $\mathcal{W} = (\mathcal{F}_{mc}[ [ \hbar ] ], \cdot_H )$ is itself that of a formal power series in $\hbar$, we have

$\mathcal{W}[ [ g/\hbar, j/\hbar] ] \subset ( \mathcal{F}_{mc}((\hbar)) [ [ g, j ] ], \cdot_H ) \,.$

Perturbative S-Matrix and Time-ordered products

We consider now the axioms for a perturbative S-matrix of a Lagrangian field theory as used in causal perturbation theory (def. 5 below). Since, by definition, the S-matrix is 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. 6 below. That these latter axioms already imply the former is the statement of prop. 4 below. Its proof requires a close look at the “reverse-time ordered products” for the inverse S-matrix (def. 8 below) and their induced reverse-causal factorization (prop. 3 below).

The axioms we consider here are just the bare minimum of causal perturbation theory, sufficient to imply that the induced perturbative quantum observables organize into a causally local net of quantum observables (discussed below).

In applications one considers further axioms, in particular compatibility of the S-matrix with spacetime symmetry. This is needed for the proof of the main theorem of perturbative renormalization (see below).

Definition

(perturbative Lagrangian S-matrix)

Let $\mathcal{W}$ be a Wick algebra encoding the quantization of free fields in $E$, with

$\mathcal{F}_{loc} \overset{:(-):}{\longrightarrow} \mathcal{W}$

the quantization map (def. 4).

Then a Lagrangian S-matrix for fields of type $E$ perturbing the free fields encoded by $\mathcal{W}$, is a functional

$S \;\colon\; \mathcal{F}_{loc}\otimes ( \langle g,j\rangle ) \longrightarrow \mathcal{W}[ [ g/\hbar, j/\hbar ] ]$

(on local observables (def. 4) times the coupling constant $g$ or source strength $j$ with values in the algebra of formal power series in the formal variables $g/\hbar$ and $j/\hbar$ in the given Wick algebra) such that the following conditions hold for fixed $L_{int}, \{ J_n\}_{n = 1}^N$:

1. (perturbation)

There exist distributions (multi- linear continuous functionals) of the form

$T \;\colon\; (\mathcal{F}_{loc} \otimes ( \langle g,j \rangle ) )^{\otimes^k} \longrightarrow \mathcal{W}[ [ g/\hbar, j/\hbar ] ]$

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

1. The unary operation is the quantization map

$T(L + A) = :L: + :A:$
2. The S-matrix is the exponential of “time-ordered products” in that for $L, A \in \mathcal{F}_{loc}$

\begin{aligned} S( g L + j A ) & = T\left( \exp_{\otimes}\left( \tfrac{1}{i \hbar} \left( g L + j A\right) \right) \right) \\ & \coloneqq \underoverset{k = 0}{\infty}{\sum} \tfrac{1}{k!} \frac{1}{(i \hbar)^k} T(\underset{k\, \text{arguments}}{\underbrace{ (g L + j A) \cdots (g L + j A) }}) \end{aligned}
2. (normalization)

$S(0) = 1$

For all $J_1, J_2, L \in \mathcal{F}_{loc}$ we have

$\left( supp(J_1) \geq supp(J_2) \right) \;\; \Rightarrow \;\; \left( \underset{L \in \mathcal{F}_{loc}}{\forall} \left( S(L + J_1 + J_2) = S(L + J_1) S(L)^{-1} S(L + J_2) \right) \right) \,.$

Given such perturbative $S$-matrix, then we say that the generating function (for quantum observables, see def. 9 below) that it induces is the functional

(4)$Z \;\colon\; \mathcal{F}_{loc} \langle g \rangle \times \mathcal{F}_{loc} \langle j \rangle \longrightarrow \mathcal{W}[ [ g/\hbar] ][ [ j/\hbar ] ]$

given by

$Z_{g_{sw}L_{int}}(j_{sw}A) \;\coloneqq\; S(g_{sw}L_{int})^{-1} S( g_{sw}L_{int} + j_{sw}A ) \,.$

Def. 5 is due to (Epstein-Glaser 73 (1)) (in view of lemma 2 below), except that these authors remain a little vague on the nature of the domain. The domain $\mathcal{F}_{loc}$ is made explicit (in terms of axioms for the time-ordered products, see def. 6 below), in (Brunetti-Fredenhagen 99, section 3, DütschFredenhagen 04, appendix E, Hollands-Wald 04, around (20)); for review see (Rejzner 16, around def. 6.7).

Remark

(further axioms)

The list of axioms in def. 5, similarly those for the time-ordered products below in def. 6, is just the bare minimum which implies that the corresponding quantum observables organize into a causally local net (discussed below). In applications such as in discussion of renormalization (below) one considers further axioms, such a unitarity and compatibility with spacetime symmetry.

Remark

(invertibility of the perturbative S-matrix)

The mutliplicative inverse $S(-)^{-1}$ of the perturbative S-matrix in def. 5 always exists: By the axioms “perturbation” and “normalization” this follows with the usual formula for the multiplicative inverse of formal power series that are non-vanishing in degree 0:

If we write

$S(g L + j A) \coloneqq 1 + D(g L + j A)$

then

(5)\begin{aligned} S(g L + j A)^{-1} &= (1 + D(j L + j A))^{-1} \\ & = \underoverset{r = 0}{\infty}{\sum} (-D(g L + j A))^r \\ & = \underoverset{r = 0}{\infty}{\sum} (1 - S(g L + j A))^r \,, \end{aligned}

where the last sum does exist in $\mathcal{W}[ [ g/\hbar, j/\hbar] ]$ because by the axiom “normalization” $D(L)$ has vanishing coefficient in zeroth order, so that only a finite sub-sum of the formal infinite sum contributes in each order.

Remark

(interpretation of the perturbative S-matrix as a path integral)

In def. 5 $g/\hbar$ has the interpretation of the coupling constant divided by Planck's constant. One obtains a formal power series with the expected non-negative powers of $\hbar$ after passing to the quantum observables induced by the S-matrix, see def. 9 below.

The local density $g_{sw}L_{int}$ has the interpretation interaction Lagrangian density $L_{int}$ adiabatically switched by a compactly supported function $g_{sw}$, and $j_{sw}$ has the interpretation of a source field.

In informal heuristic discussion of perturbative quantum field theory the S-matrix is thought of as a path integral, written

$S\left( \tfrac{g}{\hbar} L_{int} + j \right) \;\overset{\text{not really!}}{=}\; \underset{\Phi \in \Gamma_\Sigma(E)_{asmpt}}{\int} \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) + j A(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]$

where the integration is thought to be over the configuration space $\Gamma_\Sigma(E)_{asmpt}$ of fields $\Phi$ (the space of sections of the given field bundle) which satisfy given asymptotic conditions at $x^0 \to \pm \infty$; and as these boundary conditions vary the above is regarded as an integral kernel that defines the required operator in $\mathcal{W}$ (e.g. Weinberg 95, around (9.3.10) and (9.4.1)).

We may think of the axioms in def. 5 as rigorously defining the path integral, not as an actual integration, but “synthetically” by defining the expected causal behaviour of the would-be integration.

Definition 5 suggests to focus on the multilinear operations $T(...)$ which define the perturbative S-matix order-by-order:

Definition

(time-ordered product)

Let $\mathcal{W}$ be a Wick algebra encoding the quantization of free fields in $E$ (def. 4).

A time-ordered product is a sequence of distributions (multi- linear continuous functionals) of the form

$T_k \;\colon\; \mathcal{F}_{loc}^{\otimes^k} \longrightarrow \mathcal{W}[ [ g/\hbar ] ]$

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

1. (perturbation) $T_1(g L + j A) = :g L: + :j A:$

2. (normalization) $T_0 = 1$

3. (symmetry) each $T_k$ is symmetric in its arguments

4. (causal factorization) If $supp(L_1) \cup \cdots \cup supp(L_r) \;\geq\; supp(L_{r+1}) \cup \cdots \cup supp(L_k)$ then

$T((g L_1 + j A_1) \cdots (g L_k + j A_k) ) = T( (g L_1 + j A_1) \cdots (g L_r + j A_r) ) T( (g L_{r+1} + j A_{r+1}) \cdots ( g L_k + j A_{k} ) ) \,.$
Definition

(notation for time-ordered products as generalized functions)

It will be convenient (as in Epstein-Glaser 73) to think of the time-ordered products, being operator-valued distributions, as generalized functions with dependence on spacetime points:

\begin{aligned} & \int_{\Sigma^{r+s}} T_{L_1, \cdots, L_r, A_1, \cdots, A_s}(x_1, \cdots, x_{r}, y_1, \cdots, y_s) g_{sw,1}(x_1) \cdots g_{sw, r}(x_r) j_{sw,1}(y_1) \cdots j_{sw,s}(y_s) dvol_\Sigma(x_1, \cdots, x_r, y_1, \cdots, y_s) \\ & \coloneqq T( g_{sw,1} L_1 \cdots g_{sw,k} L \cdot j_{sw,1} A_1 \cdots j_{sw,s}A_s ) \end{aligned} \,.

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

Finally, due to the “symmetry” axiom in def. 6, a time-ordered product depends 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

$\int_{\Sigma^{r+s}} T(\mathbf{X}, \mathbf{Y}) \, g_{sw,1}(x_1) \cdots g_{sw, r}(x_r) \, j_{sw,1}(y_1) \cdots j_{sw,s}(y_s) dvol_\Sigma(\mathbf{X},\mathbf{Y}) \,.$

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

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

and

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

be power series of distributions in formal power series in $g/\hbar$ as in def. 7. Then the product $W(g) \coloneqq U(g) V(g)$ with expansion

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

is given simply by

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

This is because for fixed cardinality ${\vert \mathbf{I} \vert} = n_1$ this 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 $U(g)V(g)$.

Remark

Naively it might seem that the time-ordered products of def. 6 are given simply by multiplication with step functions, in the notation as generalized functions (def. 7):

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

etc. (for instance Weinberg 95, p. 143, between (3.5.9) and (3.5.10)).

This however is simply a mathematical error, in general: Both $T(-,-)$ as well as $\theta$ are distributions and their product of distributions is in general not defined. The notorious “divergencies which plague quantum field theory” are the signature of this ill defined operation.

On the other hand, when both distributions are restricted to the complement of the diagonal (i.e. restricted away from $x_1 = x_2$) then 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 expression from the complement of the diagonal to the diagonal. Any such extension will produce time-ordered products. There are in general several different such extensions. This freedom of choice is the freedom of renormalization; or equivalently, by the main theorem of perturbative renormalization theory, this is the freedom of choosing “counter terms” for the local interaction. This we discuss below in Feynman diagrams and (re-)normalization.

In order to prove that the axioms for time-ordered products do imply those for a perturbative S-matrix (prop. 4 below) we need to consider the corresponding reverse-time ordere products:

Definition

(reverse-time ordered product)

Given a time-ordered product $T = \{T_k\}_{k \in \mathbb{N}}$ (def. 6), its reverse-time ordered product

$\overline{T}_k \;\colon\; \mathcal{F}_{loc}^{\otimes^k} \longrightarrow \mathcal{W}[ [ g/\hbar ] ]$

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

$\overline{T}( L_1 \cdots L_n ) \;\coloneqq\; \left\{ \array{ \underoverset{r = 1}{n}{\sum} (-1)^r \underset{\sigma \in Unshuffl(n,r)}{\sum} T( L_{\sigma(1)} \cdots L_{\sigma(k_1)} ) \, T( L_{\sigma(k_1 + 1)} \cdots L_{\sigma(k_2)} ) \cdots T( L_{\sigma(k_{r-1}+1)} \cdots L_{\sigma_{k_r}} ) &\vert& k \geq 1 \\ 1 &\vert& k = 0 } \right. \,,$

where the sum is over all unshuffles $\sigma$ of $(1 \leq \cdots \leq n)$ into $r$ non-empty ordered subsequences. Alternatively, as a generalized function as in def. 7, this reads

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

(e.g. Epstein-Glaser 73 (11))

Proposition

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

Given a time-ordered products $T(-)$ (def. 6), then the corresponding reverse time-ordered product $\overline{T}(-)$ (def. 8) expresses the inverse $S(-)^{-1}$ (according to remark 2) of the corresponding perturbative S-matrix $S(L) \coloneqq \underset{k \in \mathbb{N}}{\sum} \tfrac{1}{k!} T(\underset{k\,\text{args}}{\underbrace{L \cdots L}})$:

$S(L)^{-1} = \underset{k \in \mathbb{N}}{\sum} \tfrac{1}{k!} \overline{T}( \underset{k \, \text{args}}{\underbrace{L \cdots L}} ) \,.$
Proof

By definition we have

$\underset{k \in \mathbb{N}}{\sum} \tfrac{1}{k!} \overline{T}( \underset{k \, \text{args}}{\underbrace{L \cdots L}} ) = \underset{ k \in \mathbb{N}}{\sum} \tfrac{1}{k!} \underoverset{r = 1}{k}{\sum} (-1)^r \underset{\sigma \in Unshuffl(k,r)}{\sum} T( L_{\sigma(1)} \cdots L_{\sigma(k_1)} ) T( L_{\sigma(k_1 + 1)} \cdots L_{\sigma(k_2)} ) \cdots T( L_{\sigma(k_{r-1}+1)} \cdots L_{\sigma_{k_r}} )$

where $\underset{k \in \{1 , \cdots, n\}}{\forall} L_k \coloneqq L$.

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:

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

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

Moreover, since all the $L_k$ 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{ L \cdots L }} ) T( \underset{k_2 \, \text{factors}}{\underbrace{ L \cdots L }} ) \cdots T( \underset{k_r \, \text{factors}}{\underbrace{ L \cdots L }} ) \\ & = \underoverset{\infty}{r = 0}{\sum} \left( - \underoverset{k = 0}{\infty}{\sum} T ( \underset{k\,\text{factors}}{\underbrace{L \cdots L}} ) \right)^r \\ & = S(L)^{-1} \,, \end{aligned}

where in the last line we used (5).

In fact prop. 1 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. 6. Then the reverse-time ordered products according to def. 8 satisfies the following inversion relation for all $\mathbf{X} \neq \emptyset$ (in the condensed notation of def. 7)

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

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. 6. Then the reverse-time ordered products according to def. 8 satisfies reverse-causal factorization.

Proof

In the condensed notation of def. 7, we need to show that for $\mathbf{X} = \mathbf{P} \cup \mathbf{Q}$ with $\mathbf{P} \cap \mathbf{Q} = \emptyset$ then

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

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 2.

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 general assumption) and that of $\overline{T}(-)$ (which holds by the induction assumption, using that $\mathbf{J} \neq \mathbf{X}$ hence that ${\vert \mathbf{J}\vert} \lt {\vert \mathbf{X}\vert}$).

2. 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

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. 2 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 perturbative S-matrix)

Let $\{T_k\}_{k \in \mathbb{N}}$ be a system of time-ordered products according to def. 6. Then

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

is indeed a perturbative S-matrix according to def. 5.

Proof

The axiom “perturbation” and “normalization” for the S-matrix are immediate from the corresponding axioms 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 simple causal factorization property of the S-matrix

$(supp(g_{sw_1}L_1) \geq supp(g_{sw,}L_2)) \;\Rightarrow\; \left( S(g_{sw,1}L_1 + g_{sw,2}L_2) = S(g_{sw,1}L_1) S(g_{sw,2}L_2) \right)$

is immediate from the time-ordering axiom of 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 functional $Z$ (4) into powers of $(g/\hbar)$ and $(j/\hbar)$

$Z_{L}(L + A) \;=\; \underoverset{n,m = 0}{\infty}{\sum} \frac{1}{n! m!} R( \underset{n\, \text{factors}}{\underbrace{L \cdots L}}, ( \underset{m \, \text{factors}}{ \underbrace{ A \cdots A } } ) )$

and then compare order-by-order with the given time-ordered product $T$ and its induced reverse-time ordered product (def. 8) via prop. 1. (These $R(-,-)$ are also called the “generating retarded products, discussed in their own right around def. 10 below.)

In the condensed notation of def. 7 and its way of absorbing combinatorial prefactors as in example 2 this yields at order $(g/\hbar)^{\vert \mathbf{Y}\vert} (j/\hbar)^{\vert \mathbf{X}\vert}$ the coefficient

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

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

$\left(J_1 \geq J_2\right) \Rightarrow \left( Z_{L + J_1}(J_2) = Z_L(J_2) \right)$

and by lemma 2 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 $\mathbf{X}$ (def. 2), hence the maximal subset with

$\mathbf{P} \geq \mathbf{X} \,,$

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 $\mathbf{Q}$:

$\mathbf{P} \geq \mathbf{Q} \,.$

With this decomposition of $\mathbf{Y}$, the sum in (6) 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. 6) and $\overline{T}(-)$ (prop. 3) 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. 2.

$\,$

Quantum observables and Retarded products

A genuine local observable should depend on the values of the fields on some compact subset of spacetime. Moreover, a perturbative quantum observable should be a power series in Planck's constant $\hbar$, reducing to the corresponding classical observable at $\hbar = 0$. The perturbative S-matrix constructed above is neither localized in spacetime this way, nor is it a power series in $\hbar$ (it is a Laurent series in $\hbar$). So it is not a local observable. But the actual quantum observables on interacting fields may be expressed in terms of the S-matrix by Bogoliubov's formula (def. 9 below).

This formula is consistent in that it implies that local observables form a causally local net as their spacetime support varies (prop. 6 below). (On deeper grounds, this formula turns out to yield the formal Fedosov deformation quantization of the interacting field theory (Collini 16).)

Namely a key consequence of the “causal additivity” axiom on the S-matrix in def. 5 turns out to be that the perturbative quantum observables on interacting fields with compact spacetime support (def. 9)

1. depend on the adiabatic switching $g_{sw}$ of the interaction Lagrangian density only up to canonical unitary isomorphism (lemma 2 below)

2. form a causally local net of observables in the sense of the Haag-Kastler axioms as the spacetime localization varies (prop. 6 below).

To the extent that a local net of observables may be regarded as defining a quantum field theory, which is the claim of (perturbative AQFT), this proves that the perturbative S-matrices in causal perturbation theory as in def. 5 indeed makes sense, despite the involvement of adiabatic switching of the interaction Lagrangian density which does not make physical sense when interpreted naively: In reality the interaction is of course not (for realistic theories at least) “switched off” outside some bounded region of spacetime; but the result here shows that if we pretend that it does then first of all we get a consistent mathematical formulas and moreover we can then nevertheless compute the correct quantum observables that are localized in this spacetime region. But the local net of observables as the spacetime localization varies is supposed to encode the full quantum field theory. Certainly any given experiment in practice probes a bounded spacetime region, and hence the algebra of observables localized in this region is sufficient to compare the theory to experiment.

$\,$

Definition

(perturbative quantum observables on interacting fields via Bogoliubov's formula)

Let $S$ be a perturbative S-matrix as in def. 5, and $g_{sw} L_{int} \in \mathcal{F}_{loc}\langle g\rangle$ an adiabatically switched interaction Lagrangian density.

Then for $A \in \mathcal{F}_{loc}$ a local observable, the perturbative quantum observable $\widehat{A}$ corresponding to $A$ is the operator-valued distribution

$\widehat{A} \;\colon\; C^\infty_{cp}(\Sigma) \longrightarrow \mathcal{W}[ [ g ] ][ [ \hbar ] ]$

which is the derivative of the generating functional $Z$ ((4) in def. 5) at vanishing source field:

$\widehat{A}(j) \;\coloneqq\; - i \hbar \frac{d}{d \epsilon} Z_{g_{sw} L_{int}}( \epsilon j A)\vert_{\epsilon = 0} \,.$

This definition of $\widehat{A}$ without the adiabatic switching $g_{sw}$ is originally due to Bogoliubov-Shirkov 59, nowadays sometimes called Bogoliubov's formula (e.g. Rejzner 16 (6.12)). The version with adiabatic switching is due to (Epstein-Glaser 73 around (74)). Review includes (Dütsch-Fredenhagen 00, around (17)).

Remark

(interpretation of Bogoliubov’s formula in terms of the path integral

With the perturbative S-matrix informally thought of as a path integral as in remark 3

$S(\tfrac{g}{\hbar} L_{int} + j A) \;\overset{\text{not really!}}{=}\; \int \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) + j A(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi]$

the Bogoliubov formula in def. 9 simimlarly would have the following interpretation:

$\widehat A(j) \;\overset{\text{not really!}}{=}\; \frac{ \int j A(\Phi) \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi] } { \int \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi] }$

If here we were to regard the expression

$\mu(\Phi) \;\overset{\text{not really}}{\coloneqq}\; \frac{ \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\Phi) }D[\Phi] } { \int \exp\left( \int_X \left( \tfrac{g}{i \hbar} L_{int}(\Phi) \right) \right) e^{\tfrac{1}{i \hbar}\int_X L_{free}(\phi) }D[\phi] }$

as a “complex probability measure” on the the configuration space of fields, then this formula would express the expectation value of the functional $A$ under this measure:

$\widehat{A}(j) \overset{\text{not really!}}{=} [j A]_{\mu} = \int j A(\Phi) \mu(\Phi) \,.$

The power series coefficients of the quantum observables on interacting fields are also called the retarded products. For the time being we mention these here just for completeness:

Definition

(retarded products induced from perturbative S-matrix)

It follows from the perturbation axiom in def. 5 that there is a system of continuous linear functionals

$R \;\colon\; \left(\mathcal{F}_{loc}\langle g\rangle\right)^{\otimes^k} \otimes (\mathcal{F}_{loc})^{\otimes^l} \longrightarrow \mathcal{W}[ [ g/\hbar] ] [ [ j/\hbar] ]$

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

$Z_{g_{sw} L}(j_{sw} A) = \underset{k,l \in \mathbb{N}}{\sum} \frac{1}{k! l!} R( \underset{k \,\text{arguments}}{\underbrace{ g_{sw} L \cdots g_{sw} L } }, \underset{l \; \text{arguments}}{\underbrace{ j_{sw} A \cdots j_{sw} A }} ) \,.$

Similarly there is

$R \;\colon\; \left(\mathcal{F}_{loc}\langle g \rangle\right)^{\otimes^k} \otimes \left(\mathcal{F}_{loc}\langle j \rangle\right) \longrightarrow \mathcal{W}[ [ g ] ] [ [ \hbar ] ]$

such that

$\widehat{A}(h) = \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} r( \underset{k \,\text{arguments}}{\underbrace{g_{sw}L_{int} \cdots g_{sw}L_{int}}}, h A ) \,.$

These are called the (generating) retarded products (Glaser-Lehmann-Zimmermann 57, Epstein-Glaser 73, section 8.1).

Direct axiomatization of the retarded products is due to (Dütsch-Fredenhagen 04), see (Collini 16, section 2.2).

$\,$

It is useful now to reformulate the causal additivity-property of the perturbative S-matrix in terms of the generating functions / retarded products:

Lemma

(causal locality of the perturbative S-matrix)

Let $S$ be a perturbative S-matrix according to def. 5 with $Z$ the generating functional (4) it induces

1. The following conditions are equivalent for all $L, J_1, J_2 \in \mathcal{F}_{loc}$:

1. $Z_L(J_1 + J_2) = Z_L(J_1) Z_L(J_2)$

2. $Z_{L + J_1}(J_2) = Z_L(J_2)$

3. $S(L + J_1 + J_2) = S(L + J_1) \, S(L)^{-1} \, S(L + J_2)$

Hence causal additivity in def. 5 implies that all these conditions hold if $supp(J_1) \geq supp(J_2)$.

2. If $supp(J_1)$ is spacelike separted from $supp(J_1)$, hence if the causal ordering (def 2) is $supp(J_1) \geq supp(J_2)$ and $supp(J_2) \geq supp(J_1)$ then

$Z_{L_{int}}(J_1) Z_{L_{int}}(J_2) = Z_{L_{int}}(J_2) J_{L_{int}}(J_1) \,.$

Similarly, if $supp(L_1) \geq supp(L_2)$ and $supp(L_2) \geq supp(L_1)$ then

$S(L_1) \, S(L_2) = S(L_2) \, S(L_1) \,.$
3. If $L_1\vert_{O} = L_2\vert_{O}$ on a causally closed subset $O \subset \mathbb{R}^{d-1,1}$ then there exists an invertible $K \in \mathcal{W}[ [ g/\hbar] ]$ such that for all $J$ with $supp(J) \subset O$ it relates $Z_{L_1}(J)$ to $Z_{L_2}(J)$ by conjugation:

$Z_{L_2}(J) = K^{-1} \, Z_{L_1}(J) \, K \,.$
Proof

The equivalence of the three conditions in the first statement is immediate from the definitions:

Expanding out the definition of $V$, the first expression is equivalent to

$S(L)^{-1} S(L + J_1 + J_2) = S(L)^{-1} S(L + J_1 ) S(L)^{-1} S(L + J_2) \,.$

Multiplying both sides of this equation by $S(L)$, shows that it is equivalent to the third clause.

Multiplying once more with $S(L + J_1)^{-1}$ this third equation is seen to be equivalent to

$S(L + J_1)^{-1} S(L + J_1 + J_2) = S(L)^{-1} S(L + J_2)$

which is equivalently the second clause, by definition of $V$.

Now the first clause of the first item immediately implies the first clause of the second item.

Similarly, setting $L = 0$ and $J_1 = L_1$ and $J_2 = L_2$ in the third clause of the first item it reduces to

$\left( supp(L_1) \geq supp(L_2) \right) \;\Rightarrow\; S(L_1 + L_2) = S(L_1)S(L_2) \,.$

Hence if $supp(L_1) \geq supp(L_2)$ and $supp(L_2) \geq supp(L_1)$ then

$S(L_1) S(L_2) = S(L_1 + L_2) = S(L_2 + L_1) = S(L_2) S(L_1) \,,$

which is the second clause of the second statement to be shown.

For the last statement, notice that by causal closure of $O$ the difference $L_2 - L_2$, which by assumption has $supp(L_2 - L_1) \in X \setminus O$, may, according to lemma 1, be written as

$L_2 - L_1 = a + r$

such that their causal order (def. 2) is

$supp(a) \geq supp(J) \geq supp(r)$

It follows with causal additivity and its equivalent formulations above that

\begin{aligned} Z_{L_2}(J) & = Z_{L_1 + a + r}(J) \\ & = Z_{L_1 + r}(J) \\ & = S(L_1 + r)^{-1} \, S(L_1 + r + J) \\ & = S(L_1 + r)^{-1} \, S(J + L_1) \, S(L_1)^{-1} \, S(L_1 + r) \\ & = S(L_1 + r)^{-1} \underset{= id}{\underbrace{S(L_1) S(L_1)^{-1}}} S(L_1 + J) \, S(L_1)^{-1} \, S(L_1 + r) \\ & = Z_{L_1}(r)^{-1} \, Z_{L_1}(J) \, Z_{L_1}(r) \end{aligned}

and hence the last statement holds for $K \coloneqq Z_{L_1}(r)$.

We now use this fact (lemma 2) to neatly organize the system of localized quantum observables on interacting fields:

Definition

(system of perturbative generating algebras of observables)

Let $S$ be a perturbative S-matrix according to def. 5 and let $L_{int} \in \Omega^{d,0}(E)$ be an interaction Lagrangian density.

For $\mathcal{O} \subset \Sigma$ a causally closed subset of spacetime (def. 1) and for $g_{sw} \in Cutoffs(\mathcal{O})$ an adiabatic switching function (def. 3) which is constant on a neighbourhood of $\mathcal{O}$, write

$Gen_{g_{sw} L_{int}}(\mathcal{O}) \coloneqq \left\langle Z_{g_{sw}L_{int}}(J) \;\vert\; supp(J) \subset \mathcal{O} \right\rangle \;\subset\; \mathcal{W}[ [ g/\hbar] ]$

for the smallest subalgebra of the Wick algebra which contains the generating functions for correlation functions (def. 9) of the form $Z_{g_{sw}L_{int}}(J)$, for all those local observables $J \in \mathcal{F}_{loc}$ with $supp(J) \subset \mathcal{O}$.

Moreover, write

$Gen_{L_{int}}(\mathcal{O}) \;\subset\; \underset{g_{sw} \in Cutoffs(\mathcal{O})}{\prod} Gen_{g_{sw}L_{int}}(\mathcal{O})$

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

$Z_{L_{int}}(J) \;\coloneqq\; \left( Z_{g_{sw}L_{int}} (J) \right)_{g_{sw} \in Cutoffs(\mathcal{O})}$

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

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

$i_{\mathcal{O}_1, \mathcal{O}_2} \;\colon\; Gen_{L_{int}}(\mathcal{O}_1) \longrightarrow Gen_{L_{int}}(\mathcal{O}_2)$

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

This construction defines a functor

$Gen_{L_{int}} \;\colon\; CausClsdSubsets(\Sigma) \longrightarrow StarAlgebras$

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

Remark

(algebra of observables well defined up to canonical isomorphism)

By lemma 2, for every causally closed $\mathcal{O} \subset X$ and every $g_{sw} \in Cutoffs(\mathcal{O})$ the abstract algebra $Gen_{L_{int}}(\mathcal{O})$ from def. 2 is canonically isomorphic to the subalgebra $Gen_{g_{sw}L_{int}}(\mathcal{O}) \subset \mathcal{W}[ [ g/\hbar ] ]$ of formal power series in the Wick algebra.

Beware the slight subtlety in this statement:

The unitary elements $K$ in $\mathcal{W}[ [ g/\hbar] ]$ which exhibit the isomorphisms by conjugation are not unique, since there are many choices of splittings $g_2 - g_1 = a + r$ in the proof of prop. 2. But the induced isomorphisms between the algebras generated by the $Z_{L_{int}}(J)$ is independent of this ambiguity, since, again by the proof of prop. 2, conjugation by each such $K$ gives the same result on the given generators: $Z_{g_1 L_{int}}(J) \mapsto Z_{g_2 L_{int}}(J)$.

Proposition

(system of perturbative generating algebras is causally local net of observables)

Given a perturbative S-matrix according to def. 5 and an interaction Lagrangian density $L_{int}$, then the system of generating algebras of observables $Gen_{L_{int}}$ (def. 11) is a causally 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

$i_{\mathcal{O}_1, \mathcal{O}_2} \;\colon\; Gen_{L_{int}}(\mathcal{O}_1) \hookrightarrow Gen_{L_{int}}(\mathcal{O}_2)$
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 (def. 2) satisfies

$\mathcal{O}_1 \geq \mathcal{O}_2 \;\text{and}\; \mathcal{O}_2 \geq \mathcal{O}_1$

then for $\mathcal{O} \subset X$ 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 $\mathcal{O}$:

$\left[ i_{\mathcal{O}_1,\mathcal{O}}(Gen_{L_{int}}(\mathcal{O}_1)) \;,\; i_{\mathcal{O}_2,\mathcal{O}}(Gen_{L_{int}}(\mathcal{O}_2)) \right] \;=\; 0 \;\;\; \in Gen_{L_{int}}(\mathcal{O}) \,.$
Proof

Isotony is immediate from the definition of the algebra homomorphisms in def. 11.

Causal locality of the system of observables follows from the causal additivity of the S-matrix, by the first clause in the second statement of lemma 2.

In the same kind of way as def. 11 the actual net of algebra of perturbative quantum observables (def. 9) is defined:

Definition

(system of algebras of quantum observables)

Let $S$ be a perturbative S-matrix according to def. 5 and let $L_{int} \in \Omega^{d,0}(E)$ be an interaction Lagrangian density.

For $\mathcal{O} \subset \Sigma$ a causally closed subset of spacetime (def. 1) and for $g_{sw} \in Cutoffs(\mathcal{O})$ an compatible adiabatic switching function (def. 3) write

$Obs_{g_{sw} L_{int}}(\mathcal{O}) \coloneqq \left\langle -i \frac{d}{d \epsilon} Z_{g_{sw}L_{int}}(\epsilon J)\vert_{\epsilon = 0} \;\vert\; supp(J) \subset \mathcal{O} \right\rangle \;\subset\; \mathcal{W}[ [ g ] ] [ [ \hbar ] ]$

for the smallest subalgebra of the Wick algebra which contains the perturbative quantum observables on interacting fields (def. 9) supported in $\mathcal{O}$.

Moreover, let

$Obs_{L_{int}}(\mathcal{O}) \subset \underset{g_{sw} \in Cutoffs(\mathcal{O})}{\prod} Obs_{g_{sw}L_{int}}(\mathcal{O})$

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

$-i \hbar \frac{d}{d \epsilon} Z_{L_{int}}(\epsilon J)\vert_{\epsilon = 0} \;\coloneqq\; \left( - i \hbar \frac{d}{d \epsilon} Z_{g_{sw}L_{int}} (\epsilon J)\vert_{\epsilon = 0} \right)_{g_{sw} \in Cutoffs(\mathcal{O})}$

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

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

$i_{\mathcal{O}_1, \mathcal{O}_2} \;\colon\; Obs_{L_{int}}(\mathcal{O}_1) \longrightarrow Obs_{L_{int}}(\mathcal{O}_2)$

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

This construction defines a functor

$Obs_{L_{int}} \;\colon\; CausClsdSubsets(\Sigma) \longrightarrow StarAlgebras$

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

As a corollary of prop. 5 we then have the key result:

Proposition

(system of algebras of perturbative quantum observables is local net of observables)

Given a perturbative S-matrix according to def. 5 and an interaction Lagrangian density $L_{int}$, then the system of algebras of observables $Obs_{L_{int}}$ (def. 12) 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

$i_{\mathcal{O}_1, \mathcal{O}_2} \;\colon\; Obs_{L_{int}}(\mathcal{O}_1) \hookrightarrow Obs_{L_{int}}(\mathcal{O}_2)$
2. (causal locality) For $\mathcal{O}_1, \mathcal{O}_2 \subset X$ two causally closed subsets which are spacelike separated, in that

$\mathcal{O}_1 \geq \mathcal{O}_2 \;\text{and}\; \mathcal{O}_2 \geq \mathcal{O}_1$

then for $\mathcal{O} \subset X$ 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 $\mathcal{O}$:

$\left[ i_{\mathcal{O}_1,\mathcal{O}}(Obs_{L_{int}}(\mathcal{O}_1)) \;,\; i_{\mathcal{O}_2,\mathcal{O}}(Obs_{L_{int}}(\mathcal{O}_2)) \right] \;=\; 0 \;\;\; \in Obs_{L_{int}}(\mathcal{O}) \,.$
Proof

The first point is again immediate from the definition (def. 12).

For the second point it is sufficient to check the commutativity relation on generators. For these the statement follows with prop. 5:

\begin{aligned} \left[ -i \frac{d}{d \epsilon_1} Z_{g_{sw}L_{int}}(\epsilon_1 J_1)\vert_{\epsilon_1 = 0} \;,\; -i \frac{d}{d \epsilon_2} Z_{g_{sw}L_{int}}(\epsilon_2 J_2)\vert_{\epsilon_2 = 0} \right] & = - \frac{d}{d \epsilon_1} \frac{d}{d \epsilon_2} \underset{ = 0}{ \underbrace{ \left[ Z_{g_{sw}L_{int}}(\epsilon_1 J_1) \;,\; Z_{g_{sw}L_{int}}(\epsilon_2 J_2) \right]}} \vert_{ {\epsilon_1 = 0} \atop {\epsilon_2 = 0}} \\ & = 0 \end{aligned}

for $supp(J_1) \geq supp(J_2)$ and $supp(J_2) \geq supp(J_1)$.

$\,$

Feynman diagrams and Renormalization

So far we considered only the axioms on a consistent perturbative S-matrix /time-ordered products and its formal consequences. Now we discuss the actual construction of time-ordered products, hence of perturbative S-matrices, by the process called renormalization of Feynman diagrams.

We first discuss how time-ordered product, and hence the perturbative S-matrix above, is uniquely determined away from the locus where interaction points coincide (prop. 7 below). Moreover, we discuss how on that locus the time-ordered product is naturally expressed as a sum of products of distributions of Feynman propagators that are labeled by Feynman diagrams: the Feynman perturbation series (prop. 8 below).

This means that the full time-ordered product is an extension of distributions of these scattering amplitudes- to the locus of coinciding vertices. The space of possible such extensions turns out to be finite-dimensional in each order of $g/\hbar, j/\hbar$, parameterizing the choice of point-supported distributions at the interaction points whose scaling degree is bounded by the given Feynman propagators.

Definition

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

$\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}_{pds} \hookrightarrow \left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}$

for the subspace of the $k$-fold tensor product of the space of compactly supported polynomial local densities (def. 4) on those tuples which have pairwise disjoint spacetime support.

Proposition

(time-ordered product away from the diagonal)

Restricted to $\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}_{pds}$ (def. 13) there is a unique time-ordered product (def. 6), given by the Moyal star product that is induced by the Feynman propagator $\omega_F$

$F \star_{\omega_F} G \;\coloneqq\; prod \circ \exp\left( \left\langle \omega_F , \frac{\delta}{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) (F \otimes G)$

in that

$T( L_1 \cdots L_k ) = L_1 \star_{\omega_F} L_2 \star_{\omega_F} \cdots \star_{\omega_F} L_k \,.$
Proof

Since the singular support of the Feynman propagator is on the diagonal, and since the support of elements in $\left(\mathcal{F}_{loc}\langle g,j\rangle\right)^{\otimes^k}_{pds}$ is by definition in the complement of the diagonal, the star product $\star_{\omega_F}$ is well defined. By construction it satisfies the axioms “peturbation” and “normalization” in def. 6. The only non-trivial point to check is that it indeed satisfies “causal factorization”:

Unwinding the definition of the Hadamard state $\omega$ and the Feynman propagator $\omega_F$, we have

\begin{aligned} \omega & = \tfrac{i}{2}( \Delta_R - \Delta_A ) + H \\ \omega_F & = \tfrac{i}{2}( \Delta_R + \Delta_A ) + H \end{aligned}

where the propagators on the right have, in particular, the following properties:

1. the advanced propagator vanishes when its first argument is not in the causal past of its second argument:

$(supp(F) \geq supp(G)) \;\Rightarrow\; \left( \left\langle \Delta_A , \frac{\delta F}{\delta \phi} \otimes \frac{\delta G}{\delta \phi} \right\rangle = 0 \right) \,.$
2. the retarded propagator equals the advanced propagator with arguments switched:

$\left\langle \Delta_R , \frac{\delta F}{\delta \phi} \otimes \frac{\delta G}{\delta \phi} \right\rangle = \left\langle \Delta_A , \frac{\delta G}{\delta \phi} \otimes \frac{\delta F}{\delta \phi} \right\rangle$
3. $H$ is symmetric:

$\left\langle H, \frac{\delta F}{\delta \phi} \otimes \frac{\delta G}{\delta \phi} \right\rangle = \left\langle H, \frac{\delta G}{\delta \phi} \otimes \frac{\delta F}{\delta \phi} \right\rangle$

It follows for $supp(F) \geq supp(G)$ that

\begin{aligned} F \star_{\omega_F} G & = prod \circ \exp\left( \left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \left\langle \tfrac{i}{2}( \Delta_R + \Delta_A ) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \left\langle \tfrac{i}{2}\Delta_R + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \left\langle \tfrac{i}{2}( \Delta_R - \Delta_A ) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \left\langle \omega , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = F \star_{\omega} G \end{aligned}

and for $supp(G) \geq supp(F)$ that

\begin{aligned} F \star_{\omega_F} G & = prod \circ \exp\left( \left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \left\langle \tfrac{i}{2}( \Delta_R + \Delta_A ) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \left\langle \tfrac{i}{2} \Delta_A + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( F \otimes G ) \\ & = prod \circ \exp\left( \left\langle \tfrac{i}{2} \Delta_R + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( G \otimes F ) \\ & = prod \circ \exp\left( \left\langle \tfrac{i}{2} (\Delta_R - \Delta_A) + H , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) ( G \otimes F ) \\ & = G \star_{\omega} F \,. \end{aligned}

This shows that $\star_F$ is a consistent time-ordered product on the subspace of functionals with disjoint support. It is immediate from the above that it is the unique solution on this subspace.

Remark

(time-ordered product is assocativative)

Prop. 7 implies in particular that the time-ordered product is associative, in that

$T( T(V_1 \cdots V_{k_1}) \cdots T(V_{k_{n-1}+1} \cdots V_{k_n} ) ) = T( V_1 \cdots V_{k_1} \cdots V_{k_{n-1}+1} \cdots V_{n_n} ) \,.$

It follows that the problem of constructing time-ordered products, and hence (by prop. 4) the perturbative S-matrix, consists of finding compatible extension of the distribution $prod \circ \exp\left( \left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right)$ to the diagonal.

Moreover, by the nature of the exponential expression, this means in each order to extend products of Feynman propagators labeled by graphs whose vertices correspond to the polynomial factors in $F$ and $G$ and whose edges indicate over which variables the Feynman propagators are to be multiplied.

Definition

(scalar field Feynman diagram)

A scalar field Feynman diagram $\Gamma$ is

1. a natural number $v \in \mathcal{N}$ (number of vertices);

2. a $v$-tuple of elements $(V_r \in \mathcal{F}_{loc} \langle g,j\rangle)_{r \in \{1, \cdots, v\}}$ (the interaction and external field vertices)

3. for each $a \lt b \in \{1, \cdots, v\}$ a natural number $e_{a,b} \in \mathbb{N}$ (“of edges from the $a$th to the $b$th vertex”).

For a given tuple $(V_j)$ of interaction vertices we write

$FDiag_{(V_j)}$

for set of scalar field Feynman diagrams with that tuple of vertices.

Proposition

(Feynman perturbation series away from coinciding vertices)

For $v \in \mathbb{N}$ the $v$-fold time-ordered product away from the diagonal, given by prop. 7

$T_v \;\colon\; \left(\mathcal{F}_{loc}\langle g,j\rangle\right)_{pds}^{\otimes^{v}} \longrightarrow \mathcal{W}[ [ g/\hbar, j/\hbar] ]$

is equal to

$T_k(V_1 \cdots V_v) \;=\; prod \circ \underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v}}}{\sum} \underset{ r \lt s \in \{1, \cdots, v\} }{\prod} \tfrac{1}{e_{r,s}!} \left\langle \hbar \omega_F \,,\, \frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}} \frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} } \right\rangle (V_1 \otimes \cdots \otimes V_v) \,,$

where the edge numbers $e_{r,s} = e_{r,s}(\Gamma)$ are those of the given Feynman diagram $\Gamma$.

Proof

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

\begin{aligned} T(V_1 \cdots V_v V_{v+1}) & = T( T(V_1 \cdots V_v) V_{v+1} ) \\ &= prod \circ \exp\left( \left\langle \hbar \omega_F, \frac{\delta}{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right) \left( prod \circ \underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v}}}{\sum} \underset{ r \gt s \in \{1, \cdots, v\} }{\prod} \frac{1}{e_{r,s}!} \left\langle \hbar \omega_F \,,\, \frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}} \frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} } \right\rangle (V_1 \otimes \cdots \otimes V_v) \right) \;\otimes\; V_{v+1} \\ & = prod \circ \underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v}}}{\sum} \underset{ r \gt s \in \{1, \cdots, v\} }{\prod} \tfrac{1}{e_{r,s}!} \left\langle \hbar \omega_F \,,\, \frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}} \frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} } \right\rangle \left( \underset{e_{1,{v+1}}, \cdots e_{v,v+1} \in \mathbb{N}}{\sum} \underset{t \in \{1, \cdots v\}}{\prod} \tfrac{1}{e_{t,v+1} !} \left( \frac{\delta^{e_{1,v+1}} V_1 }{\delta \phi_{1}^{e_{1,v+1}}} \otimes \cdots \otimes \frac{ \delta^{e_{v,v+1}} V_v}{ \delta \phi_{v}^{e_{v,v+1}} } \right) \;\otimes\; \frac{\delta^{e_{1,v+1} + \cdots + e_{v,v+1}} V_{v+1}}{\delta \phi_{v-1}^{e_{1,v+1} + \cdots + e_{v,v+1}}} \right) \\ &= prod \circ \underset{\Gamma \in \mathcal{G}_{(V_j)_{j = 1}^{v+1}}}{\sum} \underset{ r \lt s \in \{1, \cdots, v+1\} }{\prod} \tfrac{1}{e_{r,s}!} \left\langle \hbar \omega_F \,,\, \frac{\delta^{e_{r,s}}}{\delta \phi_r^{e_{r,s}}} \frac{\delta^{e_{r,s}}}{ \delta \phi_s^{e_{r,s}} } \right\rangle (V_1 \otimes \cdots \otimes V_{v+1}) \end{aligned}

Here in the first step we use the associativity of the time-ordered product (remark 7), in the second step we use the induction assumption, in the third we pass the outer functional derivatives through the pointwise product using the product rule, and in the fourth step we recognize that this amounts to summing in addition over all possible choices of sets of edges from the first $v$ vertices to the new $v+1$st vertex, which yield in total the sum over all diagrams with $v+1$ vertices.

(…)

The main theorem of perturbative renormalization states that

1. For fixed $L_{int}$ the extension of these time-ordered products to the diagonal exists, and in each order there is a finite dimensional space of possible choices.

2. There is a way to make these choices coherently, so that one obtains the $S$-matrix indeed as a function in $L_{int}$. (a “renormalization scheme”).

3. The perturbative S-matrices $S$ and $\tilde S$ for two different such renormalization schemes are related by a transformation $Z \;\colon\; \mathcal{F}_{loc} \longrightarrow \mathcal{F}_{loc}$ as

$\tilde S = S \circ Z \,.$

here $Z(L_{int})$ is $L_{int}$ with “counterterms added”.

4. The transformations $Z \colon \mathcal{T}_{loc} \to \mathcal{T}_{loc}$ form a group, called the Stückelberg-Peterson renormalization group. Hence the renormalization schemes / coherent perturbative S-matrices form a torsor over this group.

$\,$

$\,$

$\,$

In functorial quantum field theory

At least the idea of the S-matrix is very explicit in the Atiyah-Segal picture of functorial QFT (FQFT).

Here a quantum field theory is given by a functor

$Z \colon Bord_d^S \longrightarrow Vect$

from a suitable category of cobordisms to a suitable category of vector spaces.

• To a codimension-1 slice $M_{d-1}$ of space this assigns a vector space $Z(M_{d-1})$ – the (Hilbert) space of quantum states over $M_{d-1}$;

• to a spacetime/worldvolume manifold $M$ with boundaries $\partial M$ one assigns the quantum propagator which is the linear map $Z(M) : Z(\partial_{in} M) \to Z(\partial_{out} M)$ that takes incoming states to outgoing states via propagation along the spacetime/worldvolume $M$. This $Z(M)$ is alternatively known as the the scattering amplitude or S-matrix for propagation from $\partial_{in}M$ to $\partial_{out}M$ along a process of shape $M$.

Now for genuine topological field theories all spaces of quantum states are finite dimensional and hence we can equivalently consider the dual vector space (using that finite dimensional vector spaces form a compact closed category). Doing so the propagator map

$Z(M) : Z(\partial_{in}M) \to Z(\partial_{out}M)$

equivalently becomes a linear map of the form

$\mathbb{C} \to Z(\partial_{out}M) \otimes Z(\partial_{in}M)^\ast = Z(\partial M) \,.$

Notice that such a linear map from the canonical 1-dimensional complex vector space $\mathbb{C}$ to some other vector space is equivalently just a choice of element in that vector space. It is in this sense that $Z(M)$ is equivalently a vector in $Z(\partial_{out}M) \otimes Z(\partial_{in}M)^\ast = Z(\partial M)$.

In this form in physics the propagator is usually called the correlator or n-point function .

Segal’s axioms for FQFT (CFT in his case) were originally explicitly about the propagators/S-matrices, while Atiyah formulated it in terms of the correlators this way. Both perspectives go over into each other under duality as above.

Notice that this kind of discussion is not restricted to topological field theory. For instance already plain quantum mechanics is usefully formulated this way, that’s the point of finite quantum mechanics in terms of dagger-compact categories.

History

In the 1960s there was a prominent proposal, around Geoffrey Chew, that (perturbative) quantum field theory should be defined by axiomatizing properties of the S-matrix. This is a radical perspective where no spacetime geometry and physical fields are made explicit, but where the entire physics is encoded by what quantum particles see that scatter through it.

Historically, the S-matrix “bootstrap” approach fell out of fashion with the success of the quark model and of QCD, which is a local field theory governed by an action functional (Yang-Mills theory).

But later perturbative string theory revived the S-matrix approach. In general, perturbative string theory is not defined by a geometric background. Instead the background is algebraically encoded by a 2d SCFT (“2-spectral triple”) and the string perturbation series is a formula that translates this into an S-matrix. Spacetime physics then is whatever is seen by string scattering processes (see also at string theory FAQ – What are the equations of string theory?)

More recently, the S-matrix perspective becomes fashionable also in Yang-Mills theory, at least in super Yang-Mills theory: one observes that the theory enjoyes good structures in its scattering amplitudes which are essentially invisible in the vast summation of Feynman diagrams that extract the S-matrix from the action functional. Instead there are entirely different mathematical structures that encode at least some sub-class of scattering amplitudes (see at amplituhedron).

The history of physics cannot be well understood without appreciating the unbelievable antagonism between the Chew/Mandelstam/Gribov S-matrix camp, and the Weinberg/Glashow/Polyakov Field theory camp. The two sides hated each other, did not hire each other, and did not read each other, at least not in the west. The only people that straddled both camps were older folks and Russians— Gell-Mann more than Landau (who believed the Landau pole implied S-matrix), Gribov and Migdal more than anyone else in the west other than Gell-Mann and Wilson. Wilson did his PhD in S-matrix theory, for example, as did David Gross (under Chew).

In the 1970s, S-matrix theory just plain died. All practitioners jumped ship rapidly in 1974, with the triple-whammy of Wilsonian field theory, the discovery of the Charm quark, and asymptotic freedom. These results killed S-matrix theory for thirty years. Those that jumped ship include all the original string theorists who stayed employed: notably Veneziano, who was convinced that gauge theory was right when t'Hooft showed that large-N gauge fields give the string topological expansion, and Susskind, who didn’t mention Regge theory after the early 1970s. Everybody stopped studying string theory except Scherk and Schwarz, and Schwarz was protected by Gell-Mann, or else he would never have been tenured and funded.

This sorry history means that not a single S-matrix theory course is taught in the curriculum today, nobody studies it except a few theorists of advanced age hidden away in particle accelerators, and the main S-matrix theory, string theory, is not properly explained and remains completely enigmatic even to most physicists. There were some good reasons for this — some S-matrix people said silly things about the consistency of quantum field theory — but to be fair, quantum field theory people said equally silly things about S-matrix theory.

Weinberg came up with these heuristic arguments in the 1960s, which convinced him that S-matrix theory was a dead end, or rather, to show that it was a tautological synonym for quantum field theory. Weinberg was motivated by models of pion-nucleon interactions, which was a hot S-matrix topic in the early 1960s. The solution to the problem is the chiral symmetry breaking models of the pion condensate, and these are effective field theories.

Building on this result, Weinberg became convinced that the only real solution to the S-matrix was a field theory of some particles with spin. He still says this every once in a while, but it is dead wrong. The most charitable interpretation is that every S-matrix has a field theory limit, where all but a finite number of particles decouple, but this is not true either (consider little string theory). String theory exists, and there are non-field theoretic S-matrices, namely all the ones in string theory, including little string theory in (5+1)d, which is non-gravitational.

From (Weinberg 09, p. 11):

I offered this in my 1979 paper as what Arthur Wightman would call a folk theorem: “if one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with perturbative unitarity, analyticity, cluster decomposition, and the assumed symmetry properties.”

There was an interesting irony in this. I had been at Berkeley from 1959 to 1966, when Geoffrey Chew and his collaborators were elaborating a program for calculating S-matrix elements for strong interaction processes by the use of unitarity, analyticity, and Lorentz invariance, without reference to quantum field theory. I found it an attractive philosophy, because it relied only on a minimum of principles, all well established. Unfortunately, the S-matrix theorists were never able to develop a reliable method of calculation, so I worked instead on other things, including current algebra. Now in 1979 I realized that the assumptions of S-matrix theory, supplemented by chiral invariance, were indeed all that are needed at low energy, but the most convenient way of implementing these assumptions in actual calculations was by good old quantum field theory, which the S-matrix theorists had hoped to supplant.

See also at sigma model the section Exposition of second quantization of sigma-models

product in perturbative QFT$\,\,$ induces
normal-ordered productWick algebra (free field quantum observables)
time-ordered productS-matrix (scattering amplitudes)
retarded productinteracting quantum observables

References

Early work basing perturbative quantum field theory on the concept of the S-matrix is

A textbook account of the traditional heuristic picture is in

• Steven Weinberg, chapter 3 of The quantum theory of fields - Volume I: Foundations, Cambridge 1995

The mathematically rigorous construction in field theory via causal perturbation theory is due to

based on ideas due to (Stückelberg 49, Stückelberg 51) and

Brief introduction to the S-matrix in quantum mechanics and its rigorous construction in field theory via causal perturbation theory

The formulation of causal perturbation theory in terms of Feynman diagrams is due to

• Kai Keller, chapter IV of Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization, PhD thesis (arXxiv:1006.2148)

Construction of the local net of quantum observables from causal perturbation theory was hinted at in

• V. A. Il’in and D. S. Slavnov, Observable algebras in the S-matrix approach, Theor. Math. Phys. 36 (1978) 32. (spire, doi)

then rediscovered in

The axiomatization in terms of retarded products, which as such were maybe introduced in

• Vladimir Glaser, H. Lehmann, W. Zimmermann, Field operators and retarded functions, Il Nuovo Cimento 6, 1122-1128 (1957) (doi:10.1007/bf02747395)

• O. Steinmann, Perturbation Expansions in Axiomatic Field Theory, Lecture Notes in Physics, vol. 11, Springer, Berlin and Heidelberg, 1971.

goes back to

A detailed discussion is in

following

For review and further development in the context of perturbative AQFT see the references there, such as

• Katarzyna Rejzner, Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (pdf)