Operational Spacetime (eBook)

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
6.6 Ground-State Degeneracy 170
6.6.1 Electroweak Symmetry Reduction 170
6.6.2 Transmutation from Hyperisospin to Electromagnetic Symmetry 174
6.7 Lie Group Coset Bundles 176
6.8 Electroweak Spacetime 177
7 Relativities and Homogeneous Spaces 181
7.1 Parametrization of Relativity Manifolds 184
7.1.1 Goldstone Manifold for Weak Coordinates 184
7.1.2 Orientation Manifolds of Metrical Tensors 185
7.1.3 Orientation Manifold of Metrical Hyperboloids 185
7.1.4 2-Sphere for Perpendicular Relativity 186
7.1.5 Mass-Hyperboloid for Rotation Relativity 186
7.1.6 Spacetime Future Cone for Unitary Relativity 187
7.2 Relativity Transitions by Transmutators 188
7.2.1 From Interactions to Particles 188
7.2.2 Pauli Transmutator 189
7.2.3 Weyl Transmutators 190
7.2.4 Higgs Transmutators 191
7.2.5 Real Tetrads (Vierbeins) 192
7.2.6 Complex Dyads (Zweibeins) 192
7.3 Linear Representations of Relativities 193
7.3.1 Rectangular Transmutators 194
7.3.2 Representations of Perpendicular Relativity 195
7.3.3 Representations of Electromagnetic Relativity 196
7.3.4 Representations of Rotation Relativity 198
7.3.5 Representations of Unitary Relativity 199
7.3.6 Representations of Lorentz Group Relativity 199
7.4 Relativity Representations by Induction 202
7.4.1 Induced Representations 202
7.4.2 Transmutators as Induced Representations 203
7.5 Hilbert Spaces of Compact Relativities 204
7.6 Flat Spaces as Orthogonal Relativities 206
7.6.1 Particle Analysis of Special Relativity 206
8 Representation Coeffficients 209
8.1 Finite-dimensional Representations 210
8.1.1 Metrics of Representation Spaces 210
8.1.2 Metrics of Lie Algebras 212
8.2 Group Algebras and Representation Spaces 213
8.2.1 Finite Groups 213
8.2.2 Algebras and Vector Spaces for LocallyCompact Groups 214
8.3 Schur Product of Group Functions 217
8.3.1 Duality for Group Function Spaces 217
8.3.2 Hilbert Metrics of Cyclic Representation Spaces 218
8.3.3 Induced Positive-Type Measures 220
8.4 Harmonic Analysis of Representations 221
8.5 Schur Orthogonality for Compact Groups 223
8.6 Translation Representations 225
8.6.1 Fourier Transformation 225
8.6.2 Cyclic Translation Representations 227
8.6.3 Spherical and Hyperbolic Positive-Type Functions 228
8.6.4 Breit--Wigner Functions 230
8.6.5 Gaussian Functions 231
8.7 Harmonic Representations of Orthogonal Groups 233
8.8 Hilbert Metrics for Flat Manifolds 234
8.8.1 Euclidean Position for Nonrelativistic Scattering 235
8.8.2 Minkowski Spacetime for Relativistic Particles 237
8.9 Parabolic Subgroups 239
8.9.1 Discrete and Continuous Invariants 241
8.10 Eigenfunctions of Homogeneous Spaces (Spherical Functions) 241
8.10.1 Simple Examples 243
8.10.2 Eigenfunctions of Compact Groups 244
8.10.3 Eigenfunctions of Euclidean Groups 244
8.10.4 Eigenfunctions of Noncompact Groups 244
8.11 Hilbert Metrics for Hyperboloids and Spheres 246
8.11.1 Hyperbolic Position in the Hydrogen Atom 246
8.11.2 Representations of Hyperboloids and Spheres 248
8.12 Spherical, Hyperbolic, Feynman, and Causal Distributions 252
8.13 Residual Distributions 254
8.13.1 Macdonald, Bessel, and Neumann Functions 254
8.13.2 Some Special Cases 257
8.14 Hypergeometric Functions 260
9 Convolutions and Product Representations 262
9.1 Composite Nambu--Goldstone Bosons 263
9.1.1 Chirality 263
9.1.2 Chiral Degeneracy 265
9.1.3 Massless Chiral Boson 266
9.2 Convolutions for Abelian Groups 268
9.2.1 Convolutions with Linear Invariants 269
9.2.2 Convolutions with Self-Dual Invariants 269
9.3 Convolutions for Position Representations 270
9.3.1 Convolutions for Euclidean Spaces 270
9.3.2 Convolutions for Odd-Dimensional Hyperboloids 271
9.3.3 Convolutions for Odd-Dimensional Spheres 272
9.4 Residual Normalization 273
9.5 Convolution of Feynman Measures 274
9.6 Convolutions for Even-Dimensional Spacetimes 276
9.7 Feynman Propagators 278
9.7.1 Convolutions of Feynman Propagators 279
9.7.2 Convolutions for Free Particles 280
9.7.3 Off-Shell Convolution Contributions 281
10 Interactions and Kernels 284
10.1 Invariant Differential Operators 286
10.2 Kernels 288
10.2.1 Kernels for Bilinear Forms 288
10.2.2 Group Kernels 289
10.3 Green's Kernels 290
10.3.1 Linear Kernels for Spacetime 291
10.3.2 Laplace--Beltrami Kernels 292
10.4 Tangent Kernels and Interactions 294
10.4.1 Position Interactions 294
10.5 Duality and Normalization 295
10.5.1 Schur Products of Feynman Measures 296
10.5.2 Representation Normalizations for Spheresand Hyperboloids 297
10.6 Kernel Resolvents and Eigenvalues 298
10.6.1 Spacetime Normalization in the Nambu--Jona-Lasinio Model 300
11 Electroweak Spacetime 302
11.1 Operational Spacetime 302
11.2 Representations of Electroweak Spacetime 304
11.2.1 Harmonic Analysis of the Causal Cartan Plane 306
11.2.2 Harmonic Analysis of Even-DimensionalCausal Spacetimes 308
11.3 Time and Position Subrepresentations 311
11.3.1 Projection to Subgroup Representations 312
11.3.2 Projection to and Embedding of Particlesand Interactions 313
11.4 Hilbert Spaces for Causal Spacetimes 314
11.4.1 Hardy Spaces for the Future and Past 315
11.4.2 Hardy Spaces for Spacetime Cones 315
11.5 Spacetime Interactions (Kernels) 317
11.6 Normalization of Electroweak Spacetime 318
11.6.1 Central Correlation of Hypercharge and Isospin 318
11.6.2 The Mass Ratio of Spacetime 319
11.6.3 Internal Multiplicities 321
12 Masses and Coupling Constants 323
12.1 Harmonic Coefficients of Spacetime 325
12.2 Translation Invariants as Particle Masses 327
12.3 Correlation of Spin and Masses 329
12.4 Massless Interactions 330
12.4.1 Massless Vector Modes 331
12.4.2 Massless Tensor Modes 332
12.5 Spacetime Masses and Normalizations 333
Bibliography 338
Index 342