From Classical to Quantum Fields

## Abstract Quantum field theory has become the universal language of most modern theoretical physics. This book is meant to provide an introduction to this subject with particular emphasis on the physics of the fundamental interactions and elementary particles. It is addressed to advanced undergraduate, or beginning graduate, students, who have majored in physics or mathematics. The ambition is to show how these two disciplines, through their mutual interactions over the past hundred years, have enriched themselves and have both shaped our understanding of the fundamental laws of nature. The subject of this book, the transition from a classical field theory to the corresponding Quantum Field Theory through the use of Feynman’s functional integral, perfectly exemplifies this connection. It is shown how some fundamental physical principles, such as relativistic invariance, locality of the interactions, causality and positivity of the energy, form the basic elements of a modern physical theory. The standard theory of the fundamental forces is a perfect example of this connection. Based on some abstract concepts, such as group theory, gauge symmetries, and differential geometry, it provides for a detailed model whose agreement with experiment has been spectacular. The book starts with a brief description of the field theory axioms and explains the principles of gauge invariance and spontaneous symmetry breaking. It develops the techniques of perturbation theory and renormalisation with some specific examples. The last Chapters contain a presentation of the standard model and its experimental successes, as well as the attempts to go beyond with a discussion of grand unified theories and supersymmetry. "Quantum field theory has become the universal language of most modern theoretical physics. This book is meant to provide an introduction to this subject with particular emphasis on the physics of the fundamental interactions and elementary particles. It is addressed to advanced undergraduate, or beginning graduate, students, who have majored in physics or mathematics. The ambition is to show how these two disciplines, through their mutual interactions over the past hundred years, have enriched themselves and have both shaped our understanding of the fundamental laws of nature. The subject of this book, the transition from a classical field theory to the corresponding Quantum Field Theory through the use of Feynman's functional integral, perfectly exemplifies this connection. It is shown how some fundamental physical principles, such as relativistic invariance, locality of the interactions, causality and positivity of the energy, form the basic elements of a modern physical theory. The standard theory of the fundamental forces is a perfect example of this connection. Based on some abstract concepts, such as group theory, gauge symmetries, and differential geometry, it provides for a detailed model whose agreement with experiment has been spectacular. The book starts with a brief description of the field theory axioms and explains the principles of gauge invariance and spontaneous symmetry breaking. It develops the techniques of perturbation theory and renormalisation with some specific examples. The last Chapters contain a presentation of the standard model and its experimental successes, as well as the attempts to go beyond with a discussion of grand unified theories and supersymmetry."--Résumé de l'éditeur 1. Introduction 2. Relativistic Invariance 3. The Electromagnetic Field 4. General Relativity: A Field Theory of Gravitation 5. The Physical States 6. Relativistic Wave Equations 7. Towards a Relativistic Quantum Mechanics 8. Functional Integrals and Probabilistic Amplitudes 9. Functional Integrals and Quantum Mechanics: Formal Developments 10. The Euclidean Functional Integrals 11. Fermions and Functional Formalism 12. Relativistic Quantum Fields 13. Applications 14. Geometry and Quantum Dynamics 15. Broken Symmetries 16. Quantum Field Theory at Higher Orders 17. A First Glance at Renormalisation and Symmetry 18. Renormalisation of Yang-Mills Theory and BRST Symmetry 19. Some Consequences of the Renormalisation Group 20. Analyticity Properties of Feynman Diagrams 21. Infrared Singularities 22. Coherent States and Classical Limit of Quantum Electrodynamics 23. Quantum Field Theories with a Large Number of Fields 24. The Existence of Field Theories beyond the Perturbation Expansion 25. Fundamental Interactions 26. Beyond the Standard Model 27. Supersymmetry, or the Defense of Scalars Appendix A. Tensor Calculus Appendix B. Differential Calculus Appendix C. Groups and Lie Algebras Appendix D. A Collection of Useful Formulae Appendix E. Extract from Maxwell's A Treatise on Electricity and Magnetism Quantum Field Theory has become the universal language of most modern theoretical physics. This introductory textbook shows how this beautiful theory offers the correct mathematical framework to describe and understand the fundamental interactions of elementary particles. The book begins with a brief reminder of basic classical field theories, electrodynamics and general relativity, as well as their symmetry properties, and proceeds with the principles of quantisation following Feynman's path integral approach. Special care is used at every step to illustrate the correct mathematical formulation of the underlying assumptions. Gauge theories and the problems encountered in their quantisation are discussed in detail. The last chapters contain a full description of the Standard Model of particle physics and the attempts to go beyond it, such as grand unified theories and supersymmetry. Written for advanced undergraduate and beginning graduate students in physics and mathematics, the book could also serve as a reference for active researchers in the field.