Towards the Mathematics of Quantum Field Theory

Quantum Field Theory is a general framework for the description of the physics of relativistic quantum systems, especially of elementary particles. There used to be a standard way of considering QFT in terms of an action principle, somehow mimicking the classical field theories, but nowadays there are so many different meanings of the QFT that physicist use or mention that one can peruse several monographs and find almost nothing in common in these different approaches.

Moreover, from the mathematical point of view, there are so many vaguely defined terms and inconsistencies (e.g., divergent integrals) that the QFT that physicists use cannot be properly called mathematics. However the success of some QFTs, quantum electrodynamics or QED being the most widely known, is more than tempting, it is a challenge to a mathematician. This has led to some attempts to create a theory that is rigorous from the mathematical point of view. The most interesting are some topological or geometrical approaches to QFT developed in the last few years. Topological quantum field theory is one of these approaches, but until now it was severely restricted to a 2-dimensional toy model.

Derived differential geometry is a relatively new formalism to study QFT from a rigorous mathematical point of view. The general framework involves the theory of D-modules, for a version of the analysis of linear partial differential equations that arise in this context, and symmetrical monoidal categories and the functor of points viewpoint for the several spaces involved in the analysis of the non-linear algebraic partial differential equations that arise in this context.

The book under review is devoted to this second approach. All general machinery is developed in several survey chapters on homological and homotopical algebra, from monoidal categories to derived categories, Verdier derived functors, higher categories, deformation theory, and a (mostly classical) chapter on linear algebraic groups and Hopf algebras. There are also some chapters surveying the parametrized differential geometry and the Lagrangian geometric formalism.

Once this is developed in the first part of the book, the second part takes time to examine the variational formulation of classical and quantum physical systems. The third part of the book starts with some classical staples, such as the quantization of the harmonic oscillator, and rapidly turns to the mathematical difficulties of the perturbative functional integrals, Alain Connes’ (and others’) approach to renormalization, non-perturbative QFT, deformation theory and quantization.

In the spirit of Springer’s Ergebnisse series, 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 is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.