Geometry, particles, and fields
type: Item
instance of: scholarly article
title: Geometry, particles, and fields
zbMATH Open document ID: 0897.53001
author: Bjørn Felsager
published in: Graduate Texts in Contemporary Physics
publication date: 1998-04-02T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 53-01
Mathematics Subject Classification ID: 70Sxx
Mathematics Subject Classification ID: 81-01
Mathematics Subject Classification ID: 58D30
zbMATH DE Number: 1138367
zbMATH Keywords: field theory
zbMATH Keywords: electromagnetism
zbMATH Keywords: Aharonov-Bohm effect
zbMATH Keywords: gauge theory
zbMATH Keywords: sine-Gordon model
zbMATH Keywords: path-integral techniques
zbMATH Keywords: differential forms
zbMATH Keywords: exterior derivative
zbMATH Keywords: Laplacian
zbMATH Keywords: covariant integral
zbMATH Keywords: differential geometry
zbMATH Keywords: kink solution
zbMATH Keywords: Christoffel fields
zbMATH Keywords: metrics
zbMATH Keywords: geodesics
zbMATH Keywords: conformal mappings
zbMATH Keywords: abelian Higgs' model
zbMATH Keywords: monopoles
zbMATH Keywords: instantons
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: superconductors
zbMATH Keywords: Dirac string
reviewed by: Viktor Abramov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1138367