Operational Quantum Theory II - Relativistic Structures

Contents

INTRODUCTION

MATHEMATICAL TOOLS

1 LORENTZ OPERATIONS

1.1 Spacetime Lie Algebras

1.2 Left- and Right-Handed Weyl Spinors

1.3 Finite-Dimensional Representations of the Lorentz Operations

1.4 Spacetime Translations as Spinor Transformations

1.5 Minkowski Cli.ord Algebras

1.6 Dirac Spinors and Dirac Algebra

1.7 Re.ections for Position and Time

1.8 Dirac Equation

1.9 Polynomials with Lorentz Group Action

1.10 Summary

1.11 Doubled Lie Algebra

1.12 Conjugate-Adjoint Representations

2 SPACETIME AS UNITARY OPERATION CLASSES

2.1 Spacetime Translations

2.2 Nonlinear Spacetime

2.3 Spacetime and Hyperisospin

2.4 Orbits and Fixgroups of Hyperisospin

2.5 Orbits and Fixgroups in Spacetime

2.6 Summary

2.7 Fixgroups of Representations

2.8 Orbits with Signatures

2.9 Fix- and Stabil-Lie Algebras

2.10 Transmutators as Coset Representations

3 PROPAGATORS

3.1 Point Measures for Energies

3.2 Relativistically Distributed Time Representations

3.3 Fourier Transforms of Energy- Momentum Distributions

3.4 Scattering Waves (on Shell)

3.5 Macdonald, Neumann, and Bessel Functions

3.6 Yukawa Potential and Force (o. Shell)

3.7 Feynman Propagators

3.8 Summary

3.9 Distributions

3.10 Fourier Transformation

3.11 Measures of Symmetric Spaces

Bibliography

4 MASSIVE PARTICLE QUANTUM FIELDS

4.1 Quantum Bose and Fermi Oscillators

4.2 Relativistic Distribution of Time Representations

4.3 Quantum Fields for Massive Particles

4.4 Lorentz Group Embedding of Spin

4.5 Massive Spin-

Particle Fields

4.6 Massive Spin-1 Particle Fields

4.7 Massive Spin-1/2 Dirac Particle Fields

4.8 Massive Spin-1/2 Majorana Particle Fields

4.9 Spacetime Re.ections of Spinor Fields

4.10 Representation Currents

4.11 Relativistic Scattering

4.12 Summary

Bibliography

5 MASSLESS QUANTUM FIELDS

5.1 Noncompact Time Representations in Quantum Algebras

5.2 Inde.nite Metric in Quantum Algebras

5.3 Relativistic Distributions of Noncompact Time Representations

5.4 The Hilbert Spaces for Massless Particles

5.5 Massless Scalar Bose Particle Fields

5.6 Massless Scalar Fermi Fields ( Fadeev- Popov Fields)

5.7 Polarization (Helicity) in Spacetime

5.8 Massless Weyl Particle Fields

5.9 Massless Vector Bose Fields ( Gauge Fields)

5.10 Eigenvectors and Nilvectors in a Gauge Dynamics

5.11 Summary

Bibliography

6 GAUGE INTERACTIONS

6.1 Classical Maxwell Equations

6.2 The Electromagnetic Gauge Field

6.3 The Charged Relativistic Mass Point

6.4 Electrodynamics as U(1)-Representation

6.5 Quantum Gauge Fields

6.6 Representation Currents

6.7 Lie-Algebra-Valued Gauge Fields

6.8 Lie Algebras of Spacetime and Gauge Group

6.9 Electroweak and Strong Gauge Interactions

6.10 Ground State Degeneracy

6.11 From Interactions to Particles

6.12 Reflections in the Standard Model

6.13 Summary

6.14 Fadeev- Popov Degrees of Freedom

6.15 Gauge and BRS-Vertices

6.16 Cartan Tori

Bibliography

7 HARMONIC ANALYSIS

7.1 Representations on Group Functions

7.2 Harmonic Analysis of Finite Groups

7.3 Algebras and Vector Spaces for Locally Compact Groups

7.4 Harmonic Analysis of Compact Groups

7.5 Hilbert Representations and Scalar- Product- Inducing Functions

7.6 Harmonic Analysis of NoncompactGroups

7.7 Induced Group Representations

7.8 Harmonic Analysis of Symmetric Spaces

7.9 Induced Representations of Compact Groups

7.10 Representations of A.ne Groups

7.11 Group Representations on Homogeneous Functions

7.12 Harmonic Analysis of Hyperboloids

7.13 Convolutions

7.14 Abelian Convolution of Functions and Distributions

7.15 Parabolic Subgroups

Bibliography

8 RESIDUAL SPACETIME REPRESENTATIONS

8.1 Linear and Nonlinear Spacetime

8.2 Residual Representations

8.3 Residual Representations of the Reals

8.4 Residual Representations of Tangent Groups

8.5 Residual Representations of Position

8.6 Residual Representations of Causal Spacetime

8.7 Time and Position Subgroup Representations

Bibliography

9 SPECTRUM OF SPACETIME

9.1 Convolutions for Abelian Groups

9.2 Convolutions for Position Representations

9.3 Convolution of Singularity Hyperboloids

9.4 Convolutions for Spacetime

9.5 Tangent Structures for Spacetime

9.6 Translation Invariants as Particle Masses

9.7 Normalization of Translation Representations

MATHEMATICAL TOOLS 9.8 Divergences in Feynman Integrals

Bibliography

Index