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Renormalization in quantum field theory (after R. Borcherds). (English) Zbl 1425.81005

Astérisque 412. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-910-4/pbk). xvi, 186 p. (2019).
The book under review scrutinizes the Borcherds’ approach to perturbative quantum field theory [R. E. Borcherds, Algebra Number Theory 5, No. 5, 627–658 (2011; Zbl 1243.22021)]. This exposition is quite technical and the prerequisites suggested by the author include differential geometry, categorical algebra, functional analysis, algebra and global analysis. So the author collects the necessary background on algebra and functional analysis, vector bundles, tensor products and distributions to prepare for a complete and precise (re)formulation of the renormalization picture for perturbative quantum field theory.
In the first chapter, the author provides some basic results on (bornological) tensor products of (bornological) locally convex spaces, and bornological algebras and their bornological locally convex modules. Reviewing vector bundles in chapter two from an algebraic point of view the author also studies the natural topologies on the spaces of sections of such vector bundles, especially the sections of compact support. In chapter 3, the author introduces the notion of framed symmetric 2-monoidal category and proves that the category of modules over a certain tensor algebra construction on a commutative algebra is a framed 2-monoidal category. The basics of distributions on manifolds are recalled in the forth chapter.
The author explains the underlying algebraic structure of the Borcherds’ approach to quantum field theory in chapter 5, by making use of the results of Chapter 3 especially the notion of framed symmetric 2-monoidal category. He states and proves, completes or corrects some of the Borcherds’ results. In chapter 6 the author gives the complete proof of the free and transitive action of the group of renormalization on the set of Feynman measures associated with a local precut propagator, and in chapter 7 he proves that the set of Feynman measures associated with a local propagator is nonempty if the latter is assumed to be manageable and of cut type.

MSC:

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
16T10 Bialgebras
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46N50 Applications of functional analysis in quantum physics
46T30 Distributions and generalized functions on nonlinear spaces
58D30 Applications of manifolds of mappings to the sciences
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds

Biographic References:

Borcherds, Richard Ewen

Citations:

Zbl 1243.22021
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