Operational Quantum Theory II (eBook)

Contents 7
INTRODUCTION 13
MATHEMATICAL TOOLS 24
1 LORENTZ OPERATIONS 28
1.1 Spacetime Lie Algebras 29
1.2 Left- and Right-Handed Weyl Spinors 31
1.3 Finite-Dimensional Representations of the Lorentz Operations 33
1.4 Spacetime Translations as Spinor Transformations 37
1.5 Minkowski Cli.ord Algebras 42
1.6 Dirac Spinors and Dirac Algebra 43
1.7 Re.ections for Position and Time 45
1.8 Dirac Equation 50
1.9 Polynomials with Lorentz Group Action 51
1.10 Summary 54
1.11 Doubled Lie Algebra 56
1.12 Conjugate-Adjoint Representations 57
2 SPACETIME AS UNITARY OPERATION CLASSES 58
2.1 Spacetime Translations 58
2.2 Nonlinear Spacetime 62
2.3 Spacetime and Hyperisospin 64
2.4 Orbits and Fixgroups of Hyperisospin 67
2.5 Orbits and Fixgroups in Spacetime 71
2.6 Summary 78
2.7 Fixgroups of Representations 79
2.8 Orbits with Signatures 79
2.9 Fix- and Stabil-Lie Algebras 80
2.10 Transmutators as Coset Representations 81
3 PROPAGATORS 84
3.1 Point Measures for Energies 84
3.2 Relativistically Distributed Time Representations 86
3.3 Fourier Transforms of Energy- Momentum Distributions 87
3.4 Scattering Waves (on Shell) 90
3.5 Macdonald, Neumann, and Bessel Functions 91
3.6 Yukawa Potential and Force (o. Shell) 93
3.7 Feynman Propagators 95
3.8 Summary 96
3.9 Distributions 97
3.10 Fourier Transformation 100
3.11 Measures of Symmetric Spaces 101
Bibliography 103
4 MASSIVE PARTICLE QUANTUM FIELDS 105
4.1 Quantum Bose and Fermi Oscillators 107
4.2 Relativistic Distribution of Time Representations 111
4.3 Quantum Fields for Massive Particles 112
4.4 Lorentz Group Embedding of Spin 116
4.5 Massive Spin- 119
Particle Fields 119
4.6 Massive Spin-1 Particle Fields 122
4.7 Massive Spin-1/2 Dirac Particle Fields 124
4.8 Massive Spin-1/2 Majorana Particle Fields 127
4.9 Spacetime Re.ections of Spinor Fields 128
4.10 Representation Currents 129
4.11 Relativistic Scattering 134
4.12 Summary 138
Bibliography 140
5 MASSLESS QUANTUM FIELDS 141
5.1 Noncompact Time Representations in Quantum Algebras 142
5.2 Inde.nite Metric in Quantum Algebras 146
5.3 Relativistic Distributions of Noncompact Time Representations 148
5.4 The Hilbert Spaces for Massless Particles 150
5.5 Massless Scalar Bose Particle Fields 151
5.6 Massless Scalar Fermi Fields ( Fadeev- Popov Fields) 153
5.7 Polarization (Helicity) in Spacetime 154
5.8 Massless Weyl Particle Fields 156
5.9 Massless Vector Bose Fields ( Gauge Fields) 157
5.10 Eigenvectors and Nilvectors in a Gauge Dynamics 162
5.11 Summary 166
Bibliography 166
6 GAUGE INTERACTIONS 167
6.1 Classical Maxwell Equations 168
6.2 The Electromagnetic Gauge Field 171
6.3 The Charged Relativistic Mass Point 175
6.4 Electrodynamics as U(1)-Representation 176
6.5 Quantum Gauge Fields 180
6.6 Representation Currents 180
6.7 Lie-Algebra-Valued Gauge Fields 181
6.8 Lie Algebras of Spacetime and Gauge Group 184
6.9 Electroweak and Strong Gauge Interactions 186
6.10 Ground State Degeneracy 188
6.11 From Interactions to Particles 192
6.12 Reflections in the Standard Model 200
6.13 Summary 203
6.14 Fadeev- Popov Degrees of Freedom 204
6.15 Gauge and BRS-Vertices 208
6.16 Cartan Tori 210
Bibliography 214
7 HARMONIC ANALYSIS 216
7.1 Representations on Group Functions 219
7.2 Harmonic Analysis of Finite Groups 222
7.3 Algebras and Vector Spaces for Locally Compact Groups 225
7.4 Harmonic Analysis of Compact Groups 227
7.5 Hilbert Representations and Scalar- Product- Inducing Functions 231
7.6 Harmonic Analysis of NoncompactGroups 235
7.7 Induced Group Representations 238
7.8 Harmonic Analysis of Symmetric Spaces 247
7.9 Induced Representations of Compact Groups 250
7.10 Representations of A.ne Groups 254
7.11 Group Representations on Homogeneous Functions 264
7.12 Harmonic Analysis of Hyperboloids 269
7.13 Convolutions 273
7.14 Abelian Convolution of Functions and Distributions 274
7.15 Parabolic Subgroups 276
Bibliography 277
8 RESIDUAL SPACETIME REPRESENTATIONS 279
8.1 Linear and Nonlinear Spacetime 280
8.2 Residual Representations 282
8.3 Residual Representations of the Reals 290
8.4 Residual Representations of Tangent Groups 292
8.5 Residual Representations of Position 294
8.6 Residual Representations of Causal Spacetime 298
8.7 Time and Position Subgroup Representations 302
Bibliography 305
9 SPECTRUM OF SPACETIME 307
9.1 Convolutions for Abelian Groups 308
9.2 Convolutions for Position Representations 310
9.3 Convolution of Singularity Hyperboloids 313
9.4 Convolutions for Spacetime 315
9.5 Tangent Structures for Spacetime 322
9.6 Translation Invariants as Particle Masses 329
9.7 Normalization of Translation Representations 333
MATHEMATICAL TOOLS 9.8 Divergences in Feynman Integrals 336
Bibliography 338
Index 339