11. OTHER APPROACHES TO PERTURBATIVE QUANTUM FIELD THEORY 27

This calculation uses the following strange “coincidence” in Lie algebra

cohomology: although

H5(su(3))

is one-dimensional, the outer automor-

phism group of su(3) acts on this space in a non-trivial way. A more direct

construction of Yang-Mills theory, not relying on obstruction theory, is de-

sirable.

10. Observables and correlation functions

The key quantities one wants to compute in a quantum field theory are

the correlation functions between observables. These are the quantities that

can be more-or-less directly related to experiment.

The theory of observables and correlation functions is addressed in the

work in progress (CG10), written jointly with Owen Gwilliam. In this sequel,

we will show how the observables of a quantum field theory (in the sense of

this book) form a rich algebraic structure called a factorization algebra. The

concept of factorization algebra was introduced by Beilinson and Drinfeld

(BD04), as a geometric formulation of the axioms of a vertex algebra. The

factorization algebra associated to a quantum field theory is a complete

encoding of the theory: from this algebraic object one can reconstruct the

correlation functions, the operator product expansion, and so on.

Thus, a proper treatment of correlation functions requires a great deal

of preliminary work on the theory of factorization algebras and on the fac-

torization algebra associated to a quantum field theory. This is beyond the

scope of the present work.

11. Other approaches to perturbative quantum field theory

Let me finish by comparing briefly the approach to perturbative quantum

field theory developed here with others developed in the literature.

11.1. In the last ten years, the perturbative version of algebraic quan-

tum field theory has been developed by Brunetti, D¨ utsch, Fredenhagen,

Hollands, Wald and others: see (BF00; BF09; DF01; HW10). In this work,

the authors investigate the problem of constructing a solution to the axioms

of algebraic quantum field theory in perturbation theory. These authors

prove results which have a very similar form to those proved in this book:

term by term in , there is an ambiguity in quantization, described by a

certain class of Lagrangians.

The proof of these results relies on a version of the Epstein-Glaser

(EG73) construction of counterterms. This construction of counterterms re-

lies, like the approach used in this book, on working directly on real space,

as opposed to on momentum space. In the Epstein-Glaser approach to

renormalization, as in the approach described here, the proof that the coun-

terterms are local is easy. In contrast, in momentum-space approaches to

constructing counterterms – such as that developed by Bogoliubov-Parasiuk

(BP57) and Hepp (Hep66) – the problem of constructing local counterterms

involves complicated graph combinatorics.