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
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