Perturbative Algebraic Quantum Field Theory - Rejzner, Kasia

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Résumé :
Perturbative Algebraic Quantum Field Theory (pAQFT), the subject of this book, is a complete and mathematically rigorous treatment of perturbative quantum field theory (pQFT) that doesn?t require the use of divergent quantities and works on a large class of Lorenzian manifolds. We discuss in detail the examples of scalar fields, gauge theories and the effective quantum gravity. pQFT models describe a wide range of physical phenomena and have remarkable agreement with experimental results. Despite this success, the theory suffers from many conceptual problems. pAQFT is a good candidate to solve many, if not all, of these conceptual problems. Chapters 1-3 provide some background in mathematics and physics. Chapter 4 concerns classical theory of the scalar field, which is subsequently quantized in chapters 5 and 6. Chapter 7 covers gauge theory and chapter 8 discusses effective quantum gravity. The book aims to be accessible to researchers and graduate students, who are interested in the mathematical foundations of pQFT.

Sommaire:
Introduction.- Algebraic approach to quantum theory.- Algebraic quantum mechanics.- Causality.- Haag-Kastler axioms.- pAQFT axioms.- LCQFT.- Kinematical structure.- The space of field configurations.- Functionals on the configuration space.- Fermionic field configurations.- Vector fields.- Functorial interpretation.- Classical theory.- Dynamics.- Natural Lagrangians.- Homological characterization of the solution space.- The net of topological Poisson algabras.- Analogy with classical mechanics.- Deformation quantization.- Star products.- The star product on the space of multivector fields.- K?hler structure.- Interaction.- Outline of the approach.- Scatering matrix and time ordered products.- Renormalization group.- Interacting local nets.- Explicit construction.- Gauge theories.- Classical gauge theory.- Gauge-fixing.- BV formalism.- Effective quantum gravity.- From LCQFT to quantum gravity.- Dynamics and symmetries.- Linearized theory.- Quantization.- Relational observables.- Background independence.