# Towards the Mathematics of Quantum Field Theory

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics book series (MATHE3, volume 59)

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 general 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. 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.

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 general 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. 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.

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

- DOI https://doi.org/10.1007/978-3-319-04564-1
- Copyright Information Springer International Publishing Switzerland 2014
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-319-04563-4
- Online ISBN 978-3-319-04564-1
- Series Print ISSN 0071-1136
- Series Online ISSN 2197-5655
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