What Is a Quantum Field Theory?

A lively and erudite introduction for readers with a background in undergraduate mathematics but no previous knowledge of physics.

Michel Talagrand (Author)

9781316510278, Cambridge University Press

Hardback, published 17 March 2022

756 pages
25 x 17.4 x 4.2 cm, 1.41 kg

'The text has many exercises and sixteen (!) appendices from which one can learn quite a bit. This shows the dedication of the author to the subject and his wish to share his knowledge with others. The book hits the point between mathematics and physics where the first is not too abstract and the second not too phenomenological … In short, the book is exceptional and might set standards.' Marek Nowakowski, MathSciNet

Quantum field theory (QFT) is one of the great achievements of physics, of profound interest to mathematicians. Most pedagogical texts on QFT are geared toward budding professional physicists, however, whereas mathematical accounts are abstract and difficult to relate to the physics. This book bridges the gap. While the treatment is rigorous whenever possible, the accent is not on formality but on explaining what the physicists do and why, using precise mathematical language. In particular, it covers in detail the mysterious procedure of renormalization. Written for readers with a mathematical background but no previous knowledge of physics and largely self-contained, it presents both basic physical ideas from special relativity and quantum mechanics and advanced mathematical concepts in complete detail. It will be of interest to mathematicians wanting to learn about QFT and, with nearly 300 exercises, also to physics students seeking greater rigor than they typically find in their courses. Erratum for the book can be found at michel.talagrand.net/erratum.pdf.

Introduction
Part I. Basics: 1. Preliminaries
2. Basics of non-relativistic quantum mechanics
3. Non-relativistic quantum fields
4. The Lorentz group and the Poincaré group
5. The massive scalar free field
6. Quantization
7. The Casimir effect
Part II. Spin: 8. Representations of the orthogonal and the Lorentz group
9. Representations of the Poincaré group
10. Basic free fields
Part III. Interactions: 11. Perturbation theory
12. Scattering, the scattering matrix and cross sections
13. The scattering matrix in perturbation theory
14. Interacting quantum fields
Part IV. Renormalization: 15. Prologue – power counting
16. The Bogoliubov-Parasiuk-Hepp-Zimmermann scheme
17. Counter-terms
18. Controlling singularities
19. Proof of convergence of the BPHZ scheme.

Subject Areas: Mathematical physics [PHU]