Contents

Contents

Idea

What is called causal perturbation theory is a mathematically rigorous construction of perturbative (gauge-) quantum field theory, such as quantum electrodynamics, based on a mathematical formulation of renormalization by Stückelberg-Bogoliubov-Epstein-Glaser (Epstein-Glaser 73).

Causal perturbation theory may be regarded as providing a well-defined consistent generalization from quantum mechanics to quantum field theory on Lorentzian spacetimes of the construction of the S-matrix via the Dyson formula (“time-ordered products”) in the interaction picture .

A key idea of causal perturbation theory is that the interaction term $V$ is considered multiplied with some smooth function $g$ which has compact support on spacetime. This hence serves as a spacetime-dependent “coupling constant” which “switches off” the interaction outside a compact region, but not discontinuously as in many other schemes, but smoothly, hence “adiabatically” in terminology borrowed from thermodynamics. Therefore this is often called the “adiabatic switching” function, or similar.

The corresponding S-matrix would naively be given by the Dyson formula

$S_g \coloneqq T \exp\left( \int_X g(x) V(x) dvol(x) \right)$

for $V$ the interaction term, and “$T$” indicating the time-ordered product. Causal perturbation theory proceeds by axiomatizing the key structural properties of this “adiabatically switchedS-matrix, in particular its causal additivity, making sense of the “time-ordered product” by appropriate causal ordering (whence the name of the approach) and then proving by induction that solutions to these axioms exist.

It turns out that at each step in the induction (corresponding to each loop order) a distribution has to be extended to a point, namely to the point at which the interaction takes place. A key theorem (Epstein-Glaser 73, section 5, this prop.) states that this extension is parameterized by a finite-dimensional space of point-supported distributions at that point. One may organize the distributions that need to be extended into Feynman diagrams (Dütsch-Fredenhagen-Keller-Rejzner 14). This way the freedom in extension of distributions is identified with the traditional renormalization freedom in perturbative quantum field theory, see at main theorem of perturbative renormalization for more on this point.

(It may be argued, vividly so in Scharf 95, p. 181-182, that the notorious “infinities” that “plague” quantum field theory in other approaches are nothing but the result of incorrectly dealing with the issue of extension of distributions.)

In fact the interacting field algebra induced by the S-matrix constructed via causal perturbation theory this way is a model for Fedosov's formal deformation quantization of the given local Lagrangian density (Collini 16, Hawkins-Rejzner 16), thus justifying the method from first principles of (perturbative) quantization.

The key axiom imposed on the S-matrix in causal perturbation theory is causal locality, whence the name of the approach. This axiom asks that if adiabatic switching functions $g$ and $h$ have spacelike separated supports then the S-matrix factors

$S_{g + h} = S_g S_h \,.$

The time-ordered products $T(..)$ of fields are handled by splitting of distributions using tools from microlocal analysis. This is the mathematically rigorous step that takes care of what in other approaches are the “ultraviolet divergences”.

Originally the idea was that in the end the limit $g \to 1$ had to be taken, removing the adiabatic switching of the the compact support of the interaction. In general this limit does not exist (“infrared divergences”, e.g. AAS 10, section 6).

But in (Il’in-Slavnov 78) it was observed that in fact the algebra of observables on any bounded region may be computed with a $g$ whose compact support contains the causal closure of that region. Later in (Brunetti-Fredenhagen 00) it was pointed out that this means that causal perturbation theory in fact serves to construct the causally local net of observables of the perturbative quantum field theory (see this prop for details), as in the axioms for AQFT and in fact as in the axioms of AQFT on curved spacetimes, but with values in formal power series algebras (as befits a perturbation theory) instead of C*-algebras (suitable for non-perturbative quantum field theory).

(This takes care of the “algebraic adiabatic limit”, which defines the quantum observables. It does not yet in itself define the “weak adiabatic limit” for the would-be vacuum quantum state.)

This way causal perturbation theory leads to a unification of AQFT AQFT on curved spacetimes with perturbative quantum field theory. This unification is now known as locally covariant perturbative quantum field theory, see there for more.

Properties

Main theorem of renormalization

A central result in causal perturbation theory is called the main theorem of perturbative renormalization theory. This says that any two renormalization schemes, hence any two solutions to the inductive construction of the S-matrix $V \mapsto S(V)$, as indicated above, for interaction terms $V$, are related by a unique natural transformation $Z \colon V \to V'$

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

The collection of these operations $Z$ forms a group called the Stückelberg-Petermann renormalization group.

This is a mathematical reflection of the idea that renormalization is about regarding a perturbative quantum field theory with interaction $V$ as a effective field theory at some energy scale and then discovering that at higher energy there is a more fundamental interaction $Z(V)$ which effectively looks like $V$ at lower energy.

Lecture notes in

The method is due to

with precursors in

whence sometimes called the Stückelberg-Bogoliubov-Epstein-Glaser method.

The expression of causal perturbation theory in terms of Feynman diagram techniques is due to

That causal perturbation theory (in the generality of curved spacetimes, see below) is equivalently the (Fedosov-)formal deformation quantization of the interacting Lagrangian density was shown for the scalar field phi^4 theory in

and for the interacting scalar field in the toy example of regular non-local interactions in

On Minkowski spacetime

Causal perturbation theory has been worked out in detail for the example of quantum electrodynamics on Minkowski spacetime in

Non-technical exposition of this includes

Technical review includes

• Andreas Aste, Cyrill von Arx, Günter Scharf, Regularization in quantum field theory from the causal point of view, Prog. Part. Nucl. Phys.64:61-119, 2010 (arXiv:0906.1952)

• Christian Pöselt, The method of Epstein and Glaser I, 2002 (pdf)

On curved spacetimes

The generalization of causal perturbation theory to quantum field theory on curved spacetimes is developed in

Review is in

Local covariance

The observation that the method of causal perturbation theory naturally leads to locally covariant perturbative quantum field theory is due to

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

and was re-discovered and then popularized in

Review includes

For more on this see the references at locally covariant perturbative quantum field theory.