Operational Quantum Theory I (eBook)

Contents 7
INTRODUCTION 14
1 SPACETIME TRANSLATIONS 29
1.1 Time Translations 30
1.2 Position Translations 32
1.3 Spacetime Translations 38
1.4 Decompositions of Spacetime 44
1.5 Summary 52
1.6 Relations and Mappings 52
1.7 Equivalence and Order 53
1.8 Numbers 55
1.9 Monoids and Groups 57
1.10 Vector Space Duality 60
1.11 Bilinearity and Tensor Product 61
1.12 Algebras 63
1.13 Reflections (Conjugations) 68
1.14 Equivalent Vector Space Bases 74
1.15 Matrix Diagonalization and Orientation Manifolds 78
1.16 Reflections in Orthogonal Groups 79
Bibliography 81
2 TIME REPRESENTATIONS 82
2.1 The Time Group 83
2.2 Representations of the Complex Numbers 84
2.3 Time Representations and Unitarity 85
2.4 Causal Time Representations 87
2.5 Nondecomposable Hamiltonians 88
2.6 Time Orbits and Equations of Motion 89
2.7 Self-Dual Time Representations 90
2.8 Compact Time Representations 92
2.9 Noncompact Time Representations 93
2.10 Invariants and Weights 95
2.11 Summary 97
2.12 Group Realizations and Klein Spaces 97
2.13 Group and Lie Algebra Representations 103
2.14 Invariant Inner Products and Self- Dual Representations 108
2.15 Characters of Groups 110
2.16 Representations of Ordered Monoids 111
2.17 Minimal Polynomials 112
2.18 The Hausdorff Product 118
2.19 (Semi)Simple and Decomposable Endomorphisms 118
2.20 Representations of Compact (Finite) Groups 120
2.21 Algebra Representations and Modules 121
2.22 Characteristic and Minimal Polynomial 127
Bibliography 133
3 SPIN, ROTATIONS, AND POSITION 135
3.1 Linear Operations on the Alternative 136
3.2 Pauli Spinors 137
3.3 Spin Group 139
3.4 Spinor Reflections 139
3.5 Spin Representations 141
3.6 Position Translations from Adjoint Spin Structures 144
3.7 Polynomials with Spin Group Action 146
3.8 Spin Representation Matrix Elements 149
3.9 Spin Invariants and Weights 151
3.10 Summary 152
3.11 Derivations of Algebras 153
3.12 Differentiable Manifolds 156
3.13 Exponential and Logarithmic Mappings 157
3.14 (Semi)Simple Lie Algebras 161
3.15 Lie Algebra Inner Products 162
3.16 Lie Algebra Decompositions 164
3.17 Multilinearity and Tensor Algebra 164
3.18 Enveloping Algebra 171
Bibliography 175
4 ANTISTRUCTURES: The Real in the Complex 177
4.1 Anticonjugation 178
4.2 The Complex Quartet 180
4.3 Antidoubling 182
4.4 Dual and Antirepresentations 184
4.5 Particles and Antiparticles 186
4.6 Summary 189
4.7 Twin Vector Spaces 190
4.8 Complexification of Real Vector Spaces 190
Bibliography 191
5 SIMPLE LIE OPERATIONS 192
5.1 Diagonalization of Operations 193
5.2 Abelian, Nilpotent, and Solvable 196
5.3 The Basic Lie Operations 200
5.4 Spectral Decompositions of Lie Algebras 203
5.5 Spin Structure of Simple Lie Algebras 208
5.6 Roots and Weights 215
5.7 Classification of Complex Simple Lie Algebras and Dynkin Diagrams 223
5.8 Simple Complex and Compact Lie Groups 225
5.9 Simple Root Systems 226
5.10 Real Simple Lie Algebras 232
Bibliography 238
6 RATIONAL QUANTUM NUMBERS 239
6.1 Simple Representations of Simple Lie Symmetries 240
6.2 Representation Invariants and Weights of Simple Lie Algebras 241
6.3 Representations of Simple Lie Algebras 247
6.4 Centrality of Representations 257
Bibliography 262
7 QUANTUM ALGEBRAS 263
7.1 Quantization 264
7.2 Actions in Quantum Algebras 269
7.3 Quantum Algebras with Conjugation 275
7.4 Grading of Quantum Algebras 277
7.5 Symmetry and Statistics 280
7.6 Fundamental Spin Quantum Algebra 281
7.7 Adjoint Quantum Algebras 283
7.8 The Quantum Algebra for Position Translations 284
7.9 Quantum Implemented Time Action 286
7.10 Classical Lagrangians 292
7.11 Summary 294
7.12 Graded Algebras 294
7.13 Algebras with Bilinear Forms 297
7.14 Clifford Algebras 299
Bibliography 308
8 QUANTUM PROBABILITY 309
8.1 From Operator Algebra to Hilbert Spaces 310
8.2 Probability Amplitudes 314
8.3 Time Translation Eigenalgebras with Probability Interpretation 316
8.4 Tensor Algebra Forms 318
8.5 Fock States and Fock Spaces 321
8.6 Position Representation 325
8.7 The Irreducible Nonabelian Form for a Noncompact Time Representation 332
8.8 Summary 333
8.9 Algebra Forms 334
8.10 Topologies 335
8.11 Ordered Vector Spaces 339
8.12 Normed Vector Spaces 341
8.13 Banach Algebras 346
Bibliography 352
9 THE KEPLER FACTOR 353
9.1 Center of Mass Transformation 354
9.2 Intrinsic and ad hoc Units 355
9.3 Symmetries of the Kepler Dynamics 356
9.4 Classical Time Orbits 359
9.5 Kepler Bound State Vectors 364
9.6 Position Representations 369
9.7 Orbits of 1-Dimensional Position 371
9.8 Scattering Orbits of 3- Dimensional Position 374
9.9 Bound Orbits of 3-Dimensional Position 378
9.10 Scattering 385
9.11 Summary 390
9.12 Lattices and Logics 391
9.13 Measure Rings and Borel Spaces 392
9.14 Disjoint-Additive Mappings (Measures) 394
9.15 Generalized Mappings (Distributions) 399
9.16 Lebesgue Function Spaces 402
9.17 Direct Integral Vector Spaces 406
9.18 Linear Lattices ( Birkhoff- von Neumann Logics) 408
Bibliography 410
Index 411