What Is a Quantum Field Theory?

Michel Talagrand has received the Loève Prize, the Fermat Prize, the Shaw Prize and the Abel Prize. He was a plenary speaker at the International Congress of Mathematicians and is currently a member of the Académie des Sciences (Paris). He has written several books in probability theory and well over 200 research papers.

• 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; Part V. Complements: A. Complements on representations; B. End of proof of Stone's theorem; C. Canonical commutation relations; D. A crash course on Lie algebras; E. Special relativity; F. Does a position operator exist?; G. More on the representations of the Poincaré group; H. Hamiltonian formalism for classical fields; I. Quantization of the electromagnetic field through the Gupta–Bleuler approach; J. Lippmann–Schwinger equations and scattering states; K. Functions on surfaces and distributions; L. What is a tempered distribution really?; M. Wightman axioms and Haag's theorem; N. Feynman propagator and Klein–Gordon equation; O. Yukawa potential; P. Principal values and delta functions; Solutions to selected exercises; Reading suggestions; References; Index.