Saller Operational Quantum Theory II

Relativistic Structures
1. Auflage 2006
ISBN: 978-0-387-34644-1
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 333 Seiten

Reihe: Operational Physics

Operational Quantum Theory II is a distinguished work on quantum theory at an advanced algebraic level. The classically oriented hierarchy with objects such as particles as the primary focus, and interactions of the objects as the secondary focus is reversed with the operational interactions as basic quantum structures. Quantum theory, specifically relativistic quantum field theory is developed the theory of Lie group and Lie algebra operations acting on both finite and infinite dimensional vector spaces. This book deals with the operational concepts of relativistic space time, the Lorentz and Poincaré group operations and their unitary representations, particularly the elementary articles. Also discussed are eigenvalues and invariants for non-compact operations in general as well as the harmonic analysis of noncompact nonabelian Lie groups and their homogeneous spaces. In addition to the operational formulation of the standard model of particle interactions, an attempt is made to understand the particle spectrum with the masses and coupling constants as the invariants and normalizations of a tangent representation structure of a an homogeneous space time model. Operational Quantum Theory II aims to understand more deeply on an operational basis what one is working with in relativistic quantum field theory, but also suggests new solutions to previously unsolved problems.

Autoren/Hrsg.

Saller, Heinrich

Inhaltsverzeichnis

1;Contents;7
2;INTRODUCTION;13
2.1;MATHEMATICAL TOOLS;24
3;1 LORENTZ OPERATIONS;28
3.1;1.1 Spacetime Lie Algebras;29
3.2;1.2 Left- and Right-Handed Weyl Spinors;31
3.3;1.3 Finite-Dimensional Representations of the Lorentz Operations;33
3.4;1.4 Spacetime Translations as Spinor Transformations;37
3.5;1.5 Minkowski Cli.ord Algebras;42
3.6;1.6 Dirac Spinors and Dirac Algebra;43
3.7;1.7 Re.ections for Position and Time;45
3.8;1.8 Dirac Equation;50
3.9;1.9 Polynomials with Lorentz Group Action;51
3.10;1.10 Summary;54
3.11;1.11 Doubled Lie Algebra;56
3.12;1.12 Conjugate-Adjoint Representations;57
4;2 SPACETIME AS UNITARY OPERATION CLASSES;58
4.1;2.1 Spacetime Translations;58
4.2;2.2 Nonlinear Spacetime;62
4.3;2.3 Spacetime and Hyperisospin;64
4.4;2.4 Orbits and Fixgroups of Hyperisospin;67
4.5;2.5 Orbits and Fixgroups in Spacetime;71
4.6;2.6 Summary;78
4.7;2.7 Fixgroups of Representations;79
4.8;2.8 Orbits with Signatures;79
4.9;2.9 Fix- and Stabil-Lie Algebras;80
4.10;2.10 Transmutators as Coset Representations;81
5;3 PROPAGATORS;84
5.1;3.1 Point Measures for Energies;84
5.2;3.2 Relativistically Distributed Time Representations;86
5.3;3.3 Fourier Transforms of Energy- Momentum Distributions;87
5.4;3.4 Scattering Waves (on Shell);90
5.5;3.5 Macdonald, Neumann, and Bessel Functions;91
5.6;3.6 Yukawa Potential and Force (o. Shell);93
5.7;3.7 Feynman Propagators;95
5.8;3.8 Summary;96
5.9;3.9 Distributions;97
5.10;3.10 Fourier Transformation;100
5.11;3.11 Measures of Symmetric Spaces;101
5.12;Bibliography;103
6;4 MASSIVE PARTICLE QUANTUM FIELDS;105
6.1;4.1 Quantum Bose and Fermi Oscillators;107
6.2;4.2 Relativistic Distribution of Time Representations;111
6.3;4.3 Quantum Fields for Massive Particles;112
6.4;4.4 Lorentz Group Embedding of Spin;116
6.5;4.5 Massive Spin-;119
6.6;Particle Fields;119
6.7;4.6 Massive Spin-1 Particle Fields;122
6.8;4.7 Massive Spin-1/2 Dirac Particle Fields;124
6.9;4.8 Massive Spin-1/2 Majorana Particle Fields;127
6.10;4.9 Spacetime Re.ections of Spinor Fields;128
6.11;4.10 Representation Currents;129
6.12;4.11 Relativistic Scattering;134
6.13;4.12 Summary;138
6.14;Bibliography;140
7;5 MASSLESS QUANTUM FIELDS;141
7.1;5.1 Noncompact Time Representations in Quantum Algebras;142
7.2;5.2 Inde.nite Metric in Quantum Algebras;146
7.3;5.3 Relativistic Distributions of Noncompact Time Representations;148
7.4;5.4 The Hilbert Spaces for Massless Particles;150
7.5;5.5 Massless Scalar Bose Particle Fields;151
7.6;5.6 Massless Scalar Fermi Fields ( Fadeev- Popov Fields);153
7.7;5.7 Polarization (Helicity) in Spacetime;154
7.8;5.8 Massless Weyl Particle Fields;156
7.9;5.9 Massless Vector Bose Fields ( Gauge Fields);157
7.10;5.10 Eigenvectors and Nilvectors in a Gauge Dynamics;162
7.11;5.11 Summary;166
7.12;Bibliography;166
8;6 GAUGE INTERACTIONS;167
8.1;6.1 Classical Maxwell Equations;168
8.2;6.2 The Electromagnetic Gauge Field;171
8.3;6.3 The Charged Relativistic Mass Point;175
8.4;6.4 Electrodynamics as U(1)-Representation;176
8.5;6.5 Quantum Gauge Fields;180
8.6;6.6 Representation Currents;180
8.7;6.7 Lie-Algebra-Valued Gauge Fields;181
8.8;6.8 Lie Algebras of Spacetime and Gauge Group;184
8.9;6.9 Electroweak and Strong Gauge Interactions;186
8.10;6.10 Ground State Degeneracy;188
8.11;6.11 From Interactions to Particles;192
8.12;6.12 Reflections in the Standard Model;200
8.13;6.13 Summary;203
8.14;6.14 Fadeev- Popov Degrees of Freedom;204
8.15;6.15 Gauge and BRS-Vertices;208
8.16;6.16 Cartan Tori;210
8.17;Bibliography;214
9;7 HARMONIC ANALYSIS;216
9.1;7.1 Representations on Group Functions;219
9.2;7.2 Harmonic Analysis of Finite Groups;222
9.3;7.3 Algebras and Vector Spaces for Locally Compact Groups;225
9.4;7.4 Harmonic Analysis of Compact Groups;227
9.5;7.5 Hilbert Representations and Scalar- Product- Inducing Functions;231
9.6;7.6 Harmonic Analysis of NoncompactGroups;235
9.7;7.7 Induced Group Representations;238
9.8;7.8 Harmonic Analysis of Symmetric Spaces;247
9.9;7.9 Induced Representations of Compact Groups;250
9.10;7.10 Representations of A.ne Groups;254
9.11;7.11 Group Representations on Homogeneous Functions;264
9.12;7.12 Harmonic Analysis of Hyperboloids;269
9.13;7.13 Convolutions;273
9.14;7.14 Abelian Convolution of Functions and Distributions;274
9.15;7.15 Parabolic Subgroups;276
9.16;Bibliography;277
10;8 RESIDUAL SPACETIME REPRESENTATIONS;279
10.1;8.1 Linear and Nonlinear Spacetime;280
10.2;8.2 Residual Representations;282
10.3;8.3 Residual Representations of the Reals;290
10.4;8.4 Residual Representations of Tangent Groups;292
10.5;8.5 Residual Representations of Position;294
10.6;8.6 Residual Representations of Causal Spacetime;298
10.7;8.7 Time and Position Subgroup Representations;302
10.8;Bibliography;305
11;9 SPECTRUM OF SPACETIME;307
11.1;9.1 Convolutions for Abelian Groups;308
11.2;9.2 Convolutions for Position Representations;310
11.3;9.3 Convolution of Singularity Hyperboloids;313
11.4;9.4 Convolutions for Spacetime;315
11.5;9.5 Tangent Structures for Spacetime;322
11.6;9.6 Translation Invariants as Particle Masses;329
11.7;9.7 Normalization of Translation Representations;333
11.8;MATHEMATICAL TOOLS 9.8 Divergences in Feynman Integrals;336
11.9;Bibliography;338
12;Index;339