28 1. INTRODUCTION 11.2. Another, related, approach to perturbative quantum field the- ory on Riemannian space-times was developed by Hollands (Hol09) and Hollands-Olbermann (HO09). In this approach the field theory is encoded in a vertex algebra on the space-time manifold. This seems to be philosophically very closely related to my joint work with Owen Gwilliam (CG10), which uses the renormalization techniques developed in this paper to produce a factorization algebra on the space-time manifold. Thus, the quantization results proved by Hollands-Olbermann should be close analogues of the results presented here. 11.3. D. Tamarkin (Tam03) also develops an approach to renormaliza- tion of quantum field theory based on the theory of vertex algebras and on the Batalin-Vilkovisky formalism. Tamarkin’s approach, again, seems to be closely related to both the work of Hollands-Olbermann and my joint work with Gwilliam. 11.4. In a lecture at the conference “Renormalization: algebraic, geo- metric and probabilistic aspects” in Lyon in 2010, Maxim Kontsevich pre- sented an approach to perturbative renormalization which he developed some years before. The output of Kontsevich’s construction is (as in (HO09) and (CG10)) a vertex algebra on the space-time manifold. The form of Kont- sevich’s theorem is very similar to the main theorem of this book: order by order in , the space of possible quantizations is a torsor for an Abelian group constructed from certain Lagrangians. Kontsevich’s work relies on a new construction of counterterms which, like the Epstein-Glaser construc- tion and the construction developed here, relies on working in real space rather than on momentum space. Again, it is natural to speculate that there is a close relationship between Kontsevich’s work and the construction of factorization algebras presented in (CG10). 11.5. Let me finally mention an approach to perturbative renormaliza- tion developed initially by Connes and Kreimer (CK98 CK99), and further developed by (among others) Connes-Marcolli (CM04 CM08b) and van Suijlekom (vS07). The first result of this approach is that the Bogoliubov- Parasiuk-Hepp-Zimmermann (BP57 Hep66) algorithm has a beautiful in- terpretation in terms of the Birkhoff decomposition for loops in a certain pro-algebraic group constructed combinatorially from graphs. In this book, however, counterterms have no intrinsic importance: they are simply a technical tool used to prove the main results. Thus, it is not clear to me if there is any relationship between Connes-Kreimer Hopf algebra and the results of this book. Acknowledgements Many people have contributed to the material in this book. Without the constant encouragement and insightful editorial suggestions of Lauren

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.