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Description
This new expanded second edition has been totally revised and corrected. The reader finds two complete new chapters. One covers the exact solution of the finite temperature Schwinger model with periodic boundary conditions. This simple model supports instanton solutions – similarly as QCD – and 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 meant for 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.
Table of Contents
Introduction.- Path Integrals in Quantum and Statistical Mechanics.- High-Dimensional Integrals.- Monte Carlo Simulations in Quantum Mechanics.- Scalar Fields at Zero and Finite Temperature.- Classical Spin Models: An Introduction.- Mean Field Approximation.- Transfer Matrices, Correlation Inequalities and Roots of Partition Functions.- High-Temperature and Low-Temperature Expansions.- Peierls Argument and Duality Transformations.- Renormalization Group on the Lattice.- Functional Renormalization Group.- Lattice Gauge Theories.- Two-Dimensional Lattice Gauge Theories and Group Integrals.- Fermions on a Lattice.- Finite Temperature Schwinger Model.- Interacting fermions.
