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.

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