Some Elementary Gauge Theory Concepts
Gauge theory, which underlies modern particle physics as well as the theory of gravity, and hence all of physics as we know it today, is itself based on a few fundamental concepts, the consequences of which are often as beautiful as they are deep. Unfortunately, in view of the pressure to cover aspects of the theory that are necessary for its many important applications, very little space is usually devoted in textbooks and graduate courses to the treatment of these concepts. The present small volume is an attempt to help in some degree to redress this imbalance in the literature.The topics covered are elementary in the sense of being basic, not in the sense of being shallow or easy. Although all will already feature at the classical field level, and most even before the introduction of an action principle, they often lead one to pose some quite profound questions, so that much of the material treated is by necessity at the front line of research. The approach adopted is physically motivated, although there is no hesitation in introducing mathematical concepts when they are a help to understanding. In the presentation, little is assumed of the reader, and no pains has been spared to make the whole volume understandable to researchers in other fields and to graduate students, provided that the reader is willing to devote sufficient effort required by the subject matter. On the other hand, neither has there been any conscious attempt to avoid essential difficulties, or to trivialise concepts which are intrinsically abstruse. It is thus hoped that the result will be enjoyable reading for researchers and students alike.
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Loop Space Formulation
Generalized Gauge Structures
abelian theory base space Bianchi identity classical closed curve cohomology commute compact complex components concepts connection consider constraint construct continuous deformation coordinates corresponding curvature deduce defined denotes Dirac displacement algebra dual field dynamics electromagnetic field element equations of motion equivalent example fibre bundles field tensor Figure formulation gauge field gauge group gauge invariance gauge potential gauge structure gauge theory gauge transformation given group G Hence Higgs field holonomy homotopy class homotopy groups infinitesimal internal symmetry isomorphic isospin Lie algebra Lie group line integral loop space loop variables magnetic charge magnetic flux magnetic monopole manifold mathematical matrices monopole charge neighbouring point nonabelian monopole nonabelian theory obtained parallel transport parametrized loop particle patching function path phase factor physical Poincare lemma principal bundle quantized quantum represents rotation satisfying Section 3.3 specified sphere surface theorem theory with gauge topological vector field wave function world-line Yang-Mills theory zero