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, 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.
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