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