TOWARDS THE MATHEMATICS OF QUANTUM FIELD THEORY

1. Introduction
2. A Categorical Toolbox
3. Parametrized and Functional Differential Geometry
4. Functorial Analysis
5. Linear Groups
6. Hopf Algebras
7. Connections and Curvature
8. Lagrangian and Hamiltonian Systems
9. Homotopical Algebra
10. A Glimpse at Homotopical Geometry
11. Algebraic Analysis of Linear Partial Differential Equations
12. Algebraic Analysis of Non-linear Partial Differential Equations
13. Gauge Theories and Their Homotopical Poisson Reduction
14. Variational Problems of Experimental Classical Physics
15. Variational Problems of Experimental Quantum Physics
16. Variational Problems of Theoretical Physics
17. Quantum Mechanics
18. Mathematical Difficulties of Perturbative Functional Integrals
19. The Connes-Kreimer-van Suijlekom View of Renormalization
20. Nonperturbative Quantum Field Theory
21. Perturbative Renormalization à la Wilson
22. Causal Perturbative Quantum Field Theory
23. Topological Deformation Quantizations
24. Factorization Spaces and Quantization

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 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.

Features
• Introduces new concepts
• Provides a clear overview
• Gives a complete mathematical toolbox for the coordinate-free treatment of classical and quantum field theories

Author
Frédéric Paugam is a pure mathematician working at the University Pierre et Marie Curie. He started his career in arithmetic geometry, working on Galois representations and abelian varieties. He first became interested in the mathematics of quantum physics through the study of quantum statistical mechanics. He then approached quantum field theory with the categorical methods that he had learned from the work of Grothendieck’s school. He has since held various courses on this subject, allowing him to develop the tools and content of this book with the aim of teaching in mind.