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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
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