Towards the Mathematics of Quantum Field Theory

Beschreibung

The aim of this book is to introduce mathematicians (and, in particular, graduate students) to the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in play. This should in turn promote interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, even if the mathematical one is the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature.
The first part of the book introduces the mathematical methods needed to work with the physicists' spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to generalgauge theories. The second part presents a large family of examples of classical field theories, both from experimental and theoretical physics, while the third part provides an introduction to quantum field theory, presents various renormalization methods, and discusses the quantization of factorization algebras. The book is primarily intended for pure mathematicians (and in particular graduate students) who would like to learn about the mathematics of quantum field theory.

This ambitious and original book sets out to introduce to mathematicians (even including graduate students ) the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in use. This in turn promotes the interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, though mathematicians are the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature.
The first part of the book introduces the mathematical methods needed to work with the physicists' spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications togeneral gauge theories. The second part presents a large family of examples of classical field theories, both from experimental and theoretical physics, while the third part provides an introduction to quantum field theory, presents various renormalization methods, and discusses the quantization of factorization algebras.

Autor*in

Frédéric Paugam

Themen in »Towards the Mathematics of Quantum Field Theory«

81-02, 81T70, 81T13, 58A03, 58A20, 18-02, 18G55, 81T20, 81T17 category theory functional analysis functional geometry homotopical geometry quantum field theory and renormalization

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“The book under review is a valuable witness to a wide range of recent developments that aim at unraveling the mathematics of quantum field theory. … the book is well written with an original organization of the material. Its reductionist viewpoint makes it a valuable textbook, especially for the more experienced practitioners in this domain of mathematical physics.” (Walter D. van Suijlekom, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 117, 2015)

“This is a highly specialized monograph exposing current research in a systematic way, aimed at researchers and graduate students in this area. The author has tried to include several examples to illustrate the general methods being treated and ease the reader through the journey.” (Felipe Zaldivar, MAA Reviews, July, 2014)

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