Relativity, Groups, Particles

1 The Lorentz Transformation.- 1.1 Inertial Systems.- 1.2 The Principle of Relativity.- 1.3 Consequences from the Principle of Relativity.- Appendix 1: Reciprocity of Velocities.- Appendix 2: Some Orthogonal Concomitants of Vectors.- 1.4 Invariance of the Speed of Light. Lorentz Transformation.- 1.5 The Line Element.- 1.6 Michelson, Lorentz, Poincare, Einstein.- 2 Physical Interpretation.- 2.1 Geometric Representation of Lorentz Transformations.- 2.2 Relativity of Simultaneity. Causality.- 2.3 Faster than Light.- 2.4 Lorentz Contraction.- 2.5 Retardation Effects: Invisibility of Length Contraction and Apparent Superluminal Speeds.- 2.6 Proper Time and Time Dilation.- 2.7 The Clock or Twin Paradox.- 2.8 On the Influence of Acceleration upon Clocks.- 2.9 Addition of Velocities.- 2.10 Thomas Precession.- 2.11 On Clock Synchronization.- 3 Lorentz Group, Poincare Group, and Minkowski Geometry.- 3.1 Lorentz Group and Poincare Group.- 3.2 Minkowski Space. Four-Vectors.- 3.3 Passive and ActiveTransformations. Reversals.- 3.4 Contravariant and Covariant Components. Fields.- 4 Relativistic Mechanics.- 4.1 Kinematics.- Appendix: Geometry of Relativistic Velocity Space.- 4.2 Collision Laws. Relativistic Mass Increase.- 4.3 Photons: Doppler Effect and Compton Effect.- 4.4 Conversion of Mass into Energy. Mass Defect.- 4.5 Relativistic Phase Space.- Appendix: Invariance of Rn(q).- 5 Relativistic Electrodynamics.- 5.1 Forces.- 5.2 Covariant Maxwell Equations.- 5.3 Lorentz Force.- 5.4 Tensor Algebra.- 5.5 Invariant Tensors, Metric Tensor.- 5.6 Tensor Fields and Tensor Analysis.- 5.7 The Full System of Maxwell Equations. Charge Conservation.- 5.8 Discussion of the Transformation Properties.- 5.9 Conservation Laws. Stress-Energy-Momentum Tensor.- 5.10 Charged Particles.- 6 The Lorentz Group and Some of Its Representations.- 6.1 The Lorentz Group as a Lie Group.- 6.2 The Lorentz Group as a Quasidirect Product.- 6.3 Some Subgroups of the Lorentz Group.- Appendix 1: Active Lorentz Transformations.- Appendix 2: Simplicity of the Lorentz Group L++.- 6.4 Some Representations of the Lorentz Group.- 6.5 Direct Sums and Irreducible Representations.- 6.6 Schur's Lemma.- 7 Representation Theory of the Rotation Group.- 7.1 The Rotation Group SO(3,R).- 7.2 Infinitesimal Transformations.- 7.3 Lie Algebra and Representations of SO(3).- 7.4 Lie Algebras of Lie Groups.- 7.5 Unitary Irreducible Representations of SO(3).- 7.6 SU(2), Spinors, and Representation of Finite Rotations.- 7.7 Representations on Function Spaces.- 7.8 Description of Particles with Spin.- 7.9 The Full Orthogonal Group 0(3).- 7.10 On Multivalued and Ray Representations.- 8 Representation Theory of the Lorentz Group.- 8.1 Lie Algebra and Representations of L++.- 8.2 The Spinor Representation.- 8.3 Spinor Algebra.- Appendix: Determination of the Lower Clebsch-Gordan Terms.- 8.4 The Relation between Spinors and Tensors.- Appendix 1: Spinors and Lightlike 4-Vectors.- Appendix 2: Intrinsic Classification of LorentzTransformations.- 8.5 Representations of the Full Lorentz Group.- 9 Representation Theory of the Poincaré Group.- 9.1 Fields and Field Equations. Dirac Equation.- Appendix: Dirac Spinors and Clifford-Dirac Algebra.- 9.2 Relativistic Covariance in Quantum Mechanics.- 9.3 Lie Algebra and Invariants of the Poincare Group.- 9.4 Irreducible Unitary Representations of the Poincare Group.- 9.5 Representation Theory of P++ and Local Field Equations.- 9.6 Irreducible Semiunitary Ray Representations of P.- 10 Conservation Laws in Relativistic Field Theory.- 10.1 Action Principle and Noether's Theorem.- 10.2 Application to Poincaré-Covariant Field Theory.- 10.3 Relativistic Hydrodynamics.- Appendices.- A Basic Concepts from Group Theory.- A.1 Definition of Groups.- A.2 Subgroups and Factor Groups.- A.3 Homomorphisms, Extensions, Products.- A.4 Transformation Groups.- B Abstract Multilinear Algebra.- B.1 Semilinear Maps.- B.2 Dual Space.- B.3 Complex-Conjugate Space.- B.4 Transposition, Complex,and Hermitian Conjugation.- B.5 Bi- and Sesquilinear Forms.- B.6 Real and Complex Structures.- B.7 Direct Sums.- B.8 Tensor Products.- B.9 Complexification.- B.10 The Tensor Algebra over a Vector Space.- B.11 Symmetric and Exterior Algebra.- B.12 Inner Product. Creation and Annihilation Operators.- B.13 Duality in Exterior Algebra.- C Majorana Spinors, Charge Conjugation, and Time Reversal in Dirac Theory.- C.1 Dirac Algebra Reconsidered.- C.2 Majorana Spinors, Charge Conjugation, Time Reversal.- D Poincaré Covariance in Second Quantization.- D.l The One-Particle Space.- D.2 Fock Space and Field Operator.- D.3 Poincaré Covariance and Conserved Quantities.- Notation.- Author Index.