Geometrodynamics of Gauge Fields: On the Geometry of Yang-Mills and Gravitational Gauge Theories (Mathematical Physics Studies)

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. Front Matter....Pages i-xvii Historical Background....Pages 1-12 Geometry of Gauge Fields....Pages 13-35 Maxwell and Yang–Mills Theory....Pages 37-63 Gravitation as a Gauge Theory....Pages 65-94 Einstein–Cartan Theory....Pages 95-107 Teleparallelism....Pages 109-136 Yang’s Theory of Gravity....Pages 137-159 BRST Quantization of Gravity....Pages 161-179 Gravitational Instantons....Pages 181-195 Three-Dimensional Gravity....Pages 197-225 Spinor Bundles....Pages 227-259 Chiral Anomalies....Pages 261-273 Topological \(\mathrm {SL} (5,\mathbb {R})\) Gauge-Invariant Action....Pages 275-291 Geometrodynamics and Its Extensions....Pages 293-328 Color Geometrodynamics....Pages 329-345 Geometric Model of Quark Confinement?....Pages 347-358 Back Matter....Pages 359-373 This monograph is aiming at 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 mathemarical notions of fibre bundles and exterior forms. Topics such as the dynamics of Yang-Mills theories, instanton solutions and topological invariants are included as applications. By transferring these concepts to local space-time symmetries, generalizations of Einstein' theory of gravity arise in a Riemann-Cartan space with curvature and torsion. It provides the framework in which the Poincarè gauge theory, the Rainich geometrization of the Einstein-Maxwell system, and higher-dimensional, non-abelian Kaluza-Klein theories are treated.