Geometry, particles, and fields
The present book consists of two parts. The first part is a self-contained introduction to field theory, leading up to recent important concepts like solitons and instantons. There are two main themes in Part one: on the one hand, the author discusses the structure of gauge theory, exemplified by ordinary electromagnetism. This includes a derivation of the Aharonov-Bohm effect and the flux quantization of magnetic vortices in a superconductor. On the other hand, the structure of nonlinear field theory, exemplified by the \(\phi^4\)-model and the sine-Gordon model in \((1+1)\)-dimensional spacetime, are also discussed. This includes the construction of a topological charge, the particle interpretation of the kink solution; and finally, the relevance of the kink solution for the tunnel effect in quantum mechanics is pointed out. Although the present book deals mainly with the classical aspects of field theory, the quantum-mechanical aspects using path-integral techniques are also touched. The second part is a self-contained introduction to differential geometry. The main emphasis is placed on the so-called exterior calculus of differential forms, which permits, on the one hand, the construction of various differential operators -- the exterior derivative, the codifferential, and the Laplacian -- and, on the other hand, the construction of a covariant integral. Metrics and various related concepts, especially Christoffel fields, geodesics, and conformal mappings, are also investigated. The second part also contains a number of illustrative applications. The Lagrangian formalism is put on covariant form. A detailed discussion of magnetic monopoles, including the Lagrangian formalism for monopoles and the quantization of magnetic charges, continues the investigation of gauge theories initiated in Part one. The present book is written mainly for students, and for this reason it presupposes little advance knowledge except for a standard course in calculus (on the level of multiple integrals) and a standard course in classical physics (including classical mechanics, special relativity, and electrodynamics). This textbook will prove useful to high-energy physicists who want to get acquainted with the basic concepts of differential geometry. It may also be useful to mathematicians who want to know more about the application of central concepts from differential geometry in theoretical physics. The inclusion of numerous exercises, worked out, renders the book also useful for independent study. Topics covered include: Basic Properties of Particles and Fields; Electromagnetism; Interaction of Fields and Particles; Dynamics of Classical Fields; Solitons; Path Integrals and Instantons; Basic Principles and Applications of Differential Geometry; Differentiable Manifolds and Tensor Analysis; Differential Forms and the Exterior Algebra; Integral Calculus on Manifolds; Dirac Monopoles; Smooth Maps and Winding Numbers; Symmetries and Conservation Laws.
scientific article; zbMATH DE number 1138367