Talagrand M. What Is a Quantum Field Theory...Introduction..2022

Textbook in PDF format

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.
Introduction
Basics
Preliminaries
Basics of Non-relativistic Quantum Mechanics
Non-relativistic Quantum Fields
The Lorentz Group and the Poincaré Group
The Massive Scalar Free Field
Quantization
The Casimir Effect
Spin
Representations of the Orthogonal and the Lorentz Group
Representations of the Poincaré Group
Basic Free Fields
Interactions
Perturbation Theory
Scattering, the Scattering Matrix and Cross-Sections
The Scattering Matrix in Perturbation Theory
Interacting Quantum Fields
Renormalization
Prologue: Power Counting
The Bogoliubov–Parasiuk–Hepp–Zimmermann Scheme
Counter-terms
Controlling Singularities
Proof of Convergence of the BPHZ Scheme
Complements
Appendix A Complements on Representations
Appendix B End of Proof of Stone’s Theorem
Appendix C Canonical Commutation Relations
Appendix D A Crash Course on Lie Algebras
Appendix E Special Relativity
Appendix F Does a Position Operator Exist?
Appendix G More on the Representations of the Poincaré Group
Appendix H Hamiltonian Formalism for Classical Fields
Appendix I Quantization of the Electromagnetic Field through the Gupta–Bleuler Approach
Appendix J Lippmann–Schwinger Equations and Scattering States
Appendix K Functions on Surfaces and Distributions
Appendix L What Is a Tempered Distribution Really?
Appendix M Wightman Axioms and Haag’s Theorem
Appendix N Feynman Propagator and Klein-Gordon Equation