Towards the mathematics of quantum field theory. (English) Zbl 1287.81002

Paugam, Frédéric

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 59. Cham: Springer (ISBN 978-3-319-04563-4/hbk; 978-3-319-04564-1/ebook). xvi, 487 p. (2014).

This is a complete guide to the mathematics underlying the laws of physics in general and quantum physics in particular. By now most physicists agree that one cannot get any deep understanding of the physical world without entering the world of mathematics. In his book Frederic Paugam has given us a work of enormous scope and shows students a road to a coordinate-free presentation of mathematical objects underlying classical and quantum field theories. Part I of the book, the largest, gives a short account of tools from category theory used throughout the following chapters. Some details here refer to the theory of Grothendieck’s universes. There are chapters of linear groups, Hopf algebras, homotopical algebras, and an algebraic analysis of non-linear partial differential equations. Part II is devoted to the study of classical trajectories and fields. Variational principles are discussed using a Lagrangian, followed by a brief description of their physical interpretation. Finally, Part III recalls the historical development of quantization due to Heisenberg, von Neumann and Dirac. It emphasizes the deformation quantization program of Klaus Fredenhagen and his collaborators. There is an important chapter on the mathematical difficulties of perturbation functional integrals and another one on renormalization. Important approaches due to Dyson, Schwinger and Wilson are also discussed. The book will be useful to graduate student, but also to pure and applied mathematicians interested in modern mathematical developments related to quantum theory.

Reviewer: Gert Roepstorff (Aachen)

MSC:

81-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81Q70	Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
81R05	Finite-dimensional groups and algebras motivated by physics and their representations
00A06	Mathematics for nonmathematicians (engineering, social sciences, etc.)
81T70	Quantization in field theory; cohomological methods

Keywords:

mathematics of field theory; categorical tools; functional differential geometry; Lie groups; Hopf algebras; classical trajectories; quantum trajectories; quantization; renormalization