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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