WHAT IS A QUANTUM FIELD THEORY?

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DescriptionContentsResourcesCoursesAbout the Authors 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. Maintains balance between mathematical methods and the efficient but non-rigorous methods of physics, clarifying at every step the status of the arguments with respect to rigor Keeps the advanced mathematical tools to a minimum and builds them from the ground up, providing easy access to a broad audience Covers in detail certain fundamental topics, such as renormalization, which are barely touched in most physics books Includes a self-contained introduction to all the physics needed, and in particular the basics of quantum mechanics and special relativity Table of Contents 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.