Geometrodynamics of Gauge Fields: On the Geometry of Yang-Mills and Gravitational Gauge Theories

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Springer, Jan 22, 2017 - Science - 373 pages

This monograph aims to provide a unified, geometrical foundation of gauge theories of elementary particle physics. The underlying geometrical structure is unfolded in a coordinate-free manner via the modern mathematical notions of fibre bundles and exterior forms. Topics such as the dynamics of Yang-Mills theories, instanton solutions and topological invariants are included. By transferring these concepts to local space-time symmetries, generalizations of Einstein's theory of gravity arise in a Riemann-Cartan space with curvature and torsion. It provides the framework in which the (broken) Poincaré gauge theory, the Rainich geometrization of the Einstein-Maxwell system, and higher-dimensional, non-abelian Kaluza-Klein theories are developed.

Since the discovery of the Higgs boson, concepts of spontaneous symmetry breaking in gravity have come again into focus, and, in this revised edition, these will be exposed in geometric terms. Quantizing gravity remains an open issue: formulating it as a de Sitter type gauge theory in the spirit of Yang-Mills, some new progress in its topological form is presented. After symmetry breaking, Einstein’s standard general relativity with cosmological constant emerges as a classical background. The geometrical structure of BRST quantization with non-propagating topological ghosts is developed in some detail.

 

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Contents

1 Historical Background
1
2 Geometry of Gauge Fields
13
3 Maxwell and YangMills Theory
36
5 EinsteinCartan Theory
95
6 Teleparallelism
109
7 Yangs Theory of Gravity
137
8 BRST Quantization of Gravity
161
9 Gravitational Instantons
180
11 Spinor Bundles
226
13 Topological SL 5mathbbR GaugeInvariant Action
274
14 Geometrodynamics and Its Extensions
293
15 Color Geometrodynamics
329
16 Geometric Model of Quark Confinement?
347
Appendix ANotation
359
Appendix B Calculus of Exterior Differential Forms
363
Appendix C Lie Groups
370

10 ThreeDimensional Gravity
197

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About the author (2017)

Eckehard Mielke has been Professor in the Department of Physics, Universidad Autónoma Metropolitana-Iztapalapa, Mexico since 1997. His work has covered the Wheeler-DeWitt equation, knot theory and wormholes, conformal changes of metrics, Poincaré gauge theory and its metric-affine generalizations, double dual solutions, topological models of gravity with Chern-Simons terms, Ashtekar variables, boson stars, dark matter, and inflationary cosmology.

He is the author of the Springer book "Sonne, Mond und .... Schwarze L ocher" (1997) (Sun, Moon and...Black Holes), a popular book offering a short trip through recent developments in astronomy.

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