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Operational Spacetime Interactions and Particles

:	Heinrich Saller
:	Operational Spacetime Interactions and Particles
:	Springer-Verlag
:	9781441908988
:	1
:	CHF 132.90
:

:	Theoretische Physik
:	English

:	344
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

The book provides readers with an understanding of the mutual conditioning of spacetime and interactions and matter. The spacetime manifold will be looked at to be a reservoir for the parametrization of operation Lie groups or subgroup classes of Lie groups. With basic operation groups or Lie algebras, all physical structures can be interpreted in terms of corresponding realizations or representations. Physical properties are related eigenvalues or invariants. As an explicit example of operational spacetime is proposed, called electroweak spacetime, parametrizing the classes of the internal hypercharge - isospin group in the general linear group in two complex dimensions, i.e., the Lorentz cover group, extended by the casual (dilation) and phase group. Its representations and invariants will be investigated with the aim to connect them, qualitatively and numerically, with the properties of interactions and particles as arising in the representations of its tangent Minkowski spaces.

	Contents	6
	0 Introduction and Orientation	12
	1 Einstein's Gravity	27
	1.1 Geometrization of Gravity	27
	1.2 Schwarzschild--Kruskal Spacetime	30
	1.3 Friedmann and de Sitter Universes	34
	2 Riemannian Manifolds	39
	2.1 Differentiable Manifolds	39
	2.1.1 External Derivative	42
	2.2 Riemannian Operation Groups	43
	2.2.1 Metric-Induced Isomorphisms	43
	2.2.2 Tangent Euclidean and Poincaré Groups	44
	2.2.3 Global and Local Invariance Groups	46
	2.2.4 Riemannian Connection	49
	2.3 Affine Connections	50
	2.3.1 Torsion, Curvature, and Ricci Tensor	51
	2.3.2 Cartan's Stuctural Equations	53
	2.4 Lie Groups as Manifolds	54
	2.4.1 Lie Group Operations	54
	2.4.2 Lie Algebra Operations	55
	2.4.3 The Poincaré Group of a Lie Group	56
	2.4.4 Lie--Jacobi Isomorphisms for Lie Groups	56
	2.4.5 Examples	57
	2.4.6 Adjoint and Killing Connection of Lie Groups	59
	2.5 Riemannian Manifolds	61
	2.5.1 Lorentz Covariant Derivatives	61
	2.5.2 Laplace--Beltrami Operator	62
	2.5.3 Riemannian Curvature	63
	2.5.4 Einstein Tensor and Conserved Quantities	64
	2.6 Tangent and Operational Metrics	65
	2.6.1 Invariants	66
	2.7 Maximally Symmetric Manifolds	67
	2.7.1 Spheres and Hyperboloids	68
	2.7.2 Constant-Curvature Manifolds	69
	2.8 Rotation-Symmetric Manifolds	70
	2.9 Basic Riemannian Manifolds	71
	2.9.1 Manifolds with Dimension 1	72
	2.9.2 Manifolds with Dimension 2	72
	2.9.3 Manifolds with Dimension 3	74
	2.9.4 Rotation-Invariant Four-Dimensional Spacetimes	77
	2.9.5 Robertson--Walker Metrics	80
	2.10 Covariantly Constant-Curvature Manifolds	83
	2.10.1 Orthogonal Symmetric Lie Algebras	83
	2.10.2 Real Simple Lie Algebras	84
	2.10.3 Globally Symmetric Riemannian Manifolds	86
	2.10.4 Curvature of Globally Symmetric RiemannianManifolds	87
	2.10.5 Examples	89
	3 Mass Points	91
	3.1 Nonrelativistic Classical Interactions	92
	3.2 The Symmetries of the Kepler Dynamics	94
	3.3 Electrodynamics for Charged Mass Points	96
	3.4 Einstein Gravity for Mass Points	97
	3.5 Geodesics of Static Spacetimes	98
	3.6 Gravity for Charged Mass Points	101
	4 Quantum Mechanics	103
	4.1 Nonrelativistic Wave Mechanics	104
	4.2 Harmonic Oscillator	106
	4.2.1 Position Representation	107
	4.2.2 Color SU(3) for 3-Position	107
	4.2.3 Harmonic Fermi Oscillator	109
	4.2.4 Bose and Fermi Oscillators	109
	4.3 Kepler Dynamics	110
	4.3.1 Position Representation	111
	4.3.2 Orthogonal Lenz--Runge Symmetry	112
	4.4 Particles and Ghosts	115
	4.4.1 Definite Metric, Fock Space, and Particles	116
	4.4.2 Indefinite Metric and Ghosts	118
	5 Quantum Fields of Flat Spacetime	121
	5.1 Electrodynamics of Fields	123
	5.2 Gravity of Fields	125
	5.3 Gravity and Electrodynamics	127
	5.4 Linearized Einstein Equations	128
	5.5 Free Particles for Flat Spacetime	129
	5.6 Massive Particles with Spin 1 and Spin 2	133
	5.7 Massless Polarized Photons and Gravitons	136
	5.8 Quantum Gauge Fields	140
	5.8.1 Fadeev--Popov Ghosts in Quantum Mechanics	141
	5.8.2 Fadeev--Popov Ghosts for Quantum Gauge Fields	143
	5.8.3 Particle Analysis of Massless Vector Fields	144
	5.9 Hilbert Representations of the Poincaré Group	146
	5.10 Normalizations and Coupling Constants	148
	5.11 Renormalization of Gauge Fields	150
	5.11.1 Perturbative Corrections of Normalizations	151
	5.11.2 Lie Algebra Renormalization by Vacuum Polarization	154
	6 External and Internal Operations	156
	6.1 Fiber Bundles	157
	6.1.1 Fibers and Base	157
	6.1.2 Structural and Gauge Groups	158
	6.2 Nonrelativistic and Relativistic Bundles	159
	6.3 Connections of Vector Space Bundles	161
	6.4 Pure Gauges, Distinguished Frames, and CompositeGauge Fields	163
	6.5 Chargelike Internal Connections	165
	6.5.1 Currents as Lie Algebra Densities	165
	6.5.2 Normalizations of Gauge Fields	167
	6.5.3 Gauge Interactions in the Standard Model	167<