Andreas Wipf - Statistical Approach to Quantum Field Theory: An Introduction

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This new expanded second edition has been totally revised and corrected. The reader finds two completely new chapters. One covers the exact solution of the finite temperature Schwinger model with periodic boundary conditions. This simple model supports instanton solutionssimilarly as QCDand allows for a detailed discussion of topological sectors in gauge theories, the anomaly-induced breaking of chiral symmetry, and the intriguing role of fermionic zero modes. The other new chapter is devoted to interacting fermions at finite fermion density and finite temperature. Such low-dimensional models are used to describe long-energy properties of Dirac-type materials in condensed matter physics. The large-N solutions of the Gross-Neveu, Nambu-Jona-Lasinio, and Thirring models are presented in great detail, where N denotes the number of fermion flavors. Towards the end of the book, corrections to the large-N solution and simulation results of a finite number of fermion flavors are presented. Further problems are added at the end of each chapter in order to guide the reader to a deeper understanding of the presented topics. This book is aimed at advanced students and young researchers who want to acquire the necessary tools and experience to produce research results in the statistical approach to quantum field theory.

It is a great pleasure to thank again the many collaborators, teachers, and colleagues already mentioned in the first edition of this book. In addition, I would like to thank the group in Frankfurt (Laurin Pannullo, Marc Wagner, and Marc Winstel) and my more recent PhD students and postdocs for a fruitful collaboration on interacting Fermi systems in the continuum and on the lattice. Several new sections in this second edition are based on an early collaboration with I. Sachs and an ongoing collaboration with the group in Frankfurt and with J. Lenz, M. Mandl, D. Schmidt, and B. Wellegehausen. I would like to thank Holger Gies and Felix Karbstein for many inspiring discussions about interacting Fermions, and Julian Lenz, Michael Mandl, and Ingrid Wipf for proofreading the new chapters.

Statistical field theory deals with the behavior of classical or quantum systems consisting of an enormous number of degrees of freedom in and out of equilibrium. Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems with an infinite number of degrees of freedom. It is the natural language of particle physics and condensed matter physics. In the past decades the powerful methods in statistical physics and Euclidean quantum field theory have come closer and closer, with common tools based on the use of path integrals. The interpretation of Euclidean field theories as particular systems of statistical physics opened up new avenues to understand strongly coupled quantum systems or quantum field theories at zero or finite temperature. The powerful methods of statistical physics and stochastics can be applied to study for example the vacuum sector, effective action, thermodynamic potentials, correlation functions, finite size effects, nature of phase transitions or critical behavior of quantum systems.