In this book the authors develop and work out applications to gravity and gauge theories and their interactions with generic matter fields, including spinors in full detail. Spinor fields in particular appear to be the prototypes of truly gauge-natural objects, which are not purely gauge nor purely natural, so that they are a paradigmatic example of the intriguing relations between gauge natural geometry and physical phenomenology. In particular, the gauge natural framework for spinors is developed in this book in full detail, and it is shown to be fundamentally related to the interaction between fermions and dynamical tetrad gravity.

In this book the authors develop and work out applications to gravity and gauge theories and their interactions with generic matter fields, including spinors in full detail. Spinor fields in particular appear to be the prototypes of truly gauge-natural objects, which are not purely gauge nor purely natural, so that they are a paradigmatic example of the intriguing relations between gauge natural geometry and physical phenomenology. In particular, the gauge natural framework for spinors is developed in this book in full detail, and it is shown to be fundamentally related to the interaction between fermions and dynamical tetrad gravity.

Based on a highly regarded lecture course at Moscow State University, this is a clear and systematic introduction to gauge field theory. It is unique in providing the means to master gauge field theory prior to the advanced study of quantum mechanics. Though gauge field theory is typically included in courses on quantum field theory, many of its ideas and results can be understood at the classical or semi-classical level. Accordingly, this book is organized so that its early chapters require no special knowledge of quantum mechanics. Aspects of gauge field theory relying on quantum mechanics are introduced only later and in a graduated fashion--making the text ideal for students studying gauge field theory and quantum mechanics simultaneously. The book begins with the basic concepts on which gauge field theory is built. It introduces gauge-invariant Lagrangians and describes the spectra of linear perturbations, including perturbations above nontrivial ground states. The second part focuses on the construction and interpretation of classical solutions that exist entirely due to the nonlinearity of field equations: solitons, bounces, instantons, and sphalerons. The third section considers some of the interesting effects that appear due to interactions of fermions with topological scalar and gauge fields. Mathematical digressions and numerous problems are included throughout. An appendix sketches the role of instantons as saddle points of Euclidean functional integral and related topics. Perfectly suited as an advanced undergraduate or beginning graduate text, this book is an excellent starting point for anyone seeking to understand gauge fields.

Contemporary quantum field theory is mainly developed as quantization of classical fields. Therefore, classical field theory and its BRST extension is the necessary step towards quantum field theory. This book aims to provide a complete mathematical foundation of Lagrangian classical field theory and its BRST extension for the purpose of quantization. Based on the standard geometric formulation of theory of nonlinear differential operators, Lagrangian field theory is treated in a very general setting. Reducible degenerate Lagrangian theories of even and odd fields on an arbitrary smooth manifold are considered. The second Noether theorems generalized to these theories and formulated in the homology terms provide the strict mathematical formulation of BRST extended classical field theory. The most physically relevant field theories OCo gauge theory on principal bundles, gravitation theory on natural bundles, theory of spinor fields and topological field theory OCo are presented in a complete way. This book is designed for theoreticians and mathematical physicists specializing in field theory. The authors have tried throughout to provide the necessary mathematical background, thus making the exposition self-contained.

In the framework of the geometric formulation of field theory, classical fields are represented by sections of fibred manifolds, and their dynamics is phrased in jet manifold terms. The Hamiltonian formalism in fibred manifolds is the multisymplectic generalization of the Hamiltonian formalism in mechanics when canonical momenta correspond to derivatives of fields with respect to all world coordinates, not only to time. This book is devoted to the application of this formalism to fundamental field models including gauge theory, gravitation theory, and spontaneous symmetry breaking. All these models are constraint ones. Their Euler-Lagrange equations are underdetermined and need additional conditions. In the Hamiltonian formalism, these conditions appear automatically as a part of the Hamilton equations, corresponding to different Hamiltonian forms associated with a degenerate Lagrangian density. The general procedure for describing constraint systems with quadratic and affine Lagrangian densities is presented.

Contemporary quantum field theory is mainly developed as quantization of classical fields. Therefore, classical field theory and its BRST extension is the necessary step towards quantum field theory. This book aims to provide a complete mathematical foundation of Lagrangian classical field theory and its BRST extension for the purpose of quantization. Based on the standard geometric formulation of theory of nonlinear differential operators, Lagrangian field theory is treated in a very general setting. Reducible degenerate Lagrangian theories of even and odd fields on an arbitrary smooth manifold are considered. The second Noether theorems generalized to these theories and formulated in the homology terms provide the strict mathematical formulation of BRST extended classical field theory. The most physically relevant field theories — gauge theory on principal bundles, gravitation theory on natural bundles, theory of spinor fields and topological field theory — are presented in a complete way.This book is designed for theoreticians and mathematical physicists specializing in field theory. The authors have tried throughout to provide the necessary mathematical background, thus making the exposition self-contained.

This book incorporates 3 modern aspects of mathematical physics: the jet methods in differential geometry, Lagrangian formalism on jet manifolds and the multimomentum approach to Hamiltonian formalism. Several contemporary field models are investigated in detail.This is not a book on differential geometry. However, modern concepts of differential geometry such as jet manifolds and connections are used throughout the book. Quadratic Lagrangians and Hamiltonians are studied at the general level including a treatment of Hamiltonian formalism on composite fiber manifolds. The book presents new geometric methods and results in field theory.