Paugam F. Towards the Mathematics of Quantum Field Theory

Springer International Publishing, 2014. — 485 p. — ISBN: 9783319045634This 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.Mathematical Preliminaries
A Categorical Toolbox
Parametrized and Functional Differential Geometry
Functorial Analysis
Linear Groups
Hopf Algebras
Connections and Curvature
Lagrangian and Hamiltonian Systems
Homotopical Algebra
A Glimpse at Homotopical Geometry
Algebraic Analysis of Linear Partial Differential Equations
Algebraic Analysis of Non-linear Partial Differential Equations
Gauge Theories and Their Homotopical Poisson Reduction
Classical Trajectories and Fields
Variational Problems of Experimental Classical Physics
Variational Problems of Experimental Quantum Physics
Variational Problems of Theoretical Physics
Quantum Trajectories and Fields
Quantum Mechanics
Mathematical Difficulties of Perturbative Functional Integrals
The Connes-Kreimer-van Suijlekom View of Renormalization
Nonperturbative Quantum Field Theory
Perturbative Renormalization à la Wilson
Causal Perturbative Quantum Field Theory
Topological Deformation Quantizations
Factorization Spaces and Quantization