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
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abelian action already Au(x base space Bianchi identity bundle called Chapter classes classical complex components concepts condition connection consider constraint construct continuous coordinates corresponding course cover curvature defined definition denotes depends derivative differential direction displacement dual electromagnetic element equations equivalent example exist fact fibre field field tensor Figure flux follows formulation function Further gauge group gauge potential gauge theory gauge transformation give given Hence homotopy identity integral interaction interesting introduce invariant Lie algebra loop manifold mathematical matrices means monopole charge namely nonabelian obtained operator parallel transport parametrized particle particular patching path phase factor physical present principal problem quantity reference regarded region represents result rotation satisfying seen space space-time specified sphere standard string structure surface symmetry tangent tangent vector topological transformation usual variables vector wave function Yang-Mills