Saller Operational Quantum Theory I

Nonrelativistic Structures
1. Auflage 2007
ISBN: 978-0-387-34643-4
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 408 Seiten

Reihe: Operational Physics

Operational Quantum Theory I 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 these objects as the secondary focus is reversed with the operational interactions as basic quantum structures. Quantum theory, specifically nonrelativistic quantum mechanics, is developed from the theory of Lie group and Lie algebra operations acting on both finite and infinite dimensional vector spaces. In this book, time and space related finite dimensional representation structures and simple Lie operations, and as a non-relativistic application, the Kepler problem which has long fascinated quantum theorists, are dealt with in some detail. Operational Quantum Theory I features many structures which allow the reader to better understand the applications of operational quantum theory, and to provide conceptually appropriate descriptions of the subject. Operational Quantum Theory I aims to understand more deeply on an operational basis what one is working with in nonrelativistic quantum theory, but also suggests new approaches to the characteristic problems of quantum mechanics.

Autoren/Hrsg.

Saller, Heinrich

Inhaltsverzeichnis

1;Contents;7
2;INTRODUCTION;14
3;1 SPACETIME TRANSLATIONS;29
3.1;1.1 Time Translations;30
3.2;1.2 Position Translations;32
3.3;1.3 Spacetime Translations;38
3.4;1.4 Decompositions of Spacetime;44
3.5;1.5 Summary;52
3.6;1.6 Relations and Mappings;52
3.7;1.7 Equivalence and Order;53
3.8;1.8 Numbers;55
3.9;1.9 Monoids and Groups;57
3.10;1.10 Vector Space Duality;60
3.11;1.11 Bilinearity and Tensor Product;61
3.12;1.12 Algebras;63
3.13;1.13 Reflections (Conjugations);68
3.14;1.14 Equivalent Vector Space Bases;74
3.15;1.15 Matrix Diagonalization and Orientation Manifolds;78
3.16;1.16 Reflections in Orthogonal Groups;79
3.17;Bibliography;81
4;2 TIME REPRESENTATIONS;82
4.1;2.1 The Time Group;83
4.2;2.2 Representations of the Complex Numbers;84
4.3;2.3 Time Representations and Unitarity;85
4.4;2.4 Causal Time Representations;87
4.5;2.5 Nondecomposable Hamiltonians;88
4.6;2.6 Time Orbits and Equations of Motion;89
4.7;2.7 Self-Dual Time Representations;90
4.8;2.8 Compact Time Representations;92
4.9;2.9 Noncompact Time Representations;93
4.10;2.10 Invariants and Weights;95
4.11;2.11 Summary;97
4.12;2.12 Group Realizations and Klein Spaces;97
4.13;2.13 Group and Lie Algebra Representations;103
4.14;2.14 Invariant Inner Products and Self- Dual Representations;108
4.15;2.15 Characters of Groups;110
4.16;2.16 Representations of Ordered Monoids;111
4.17;2.17 Minimal Polynomials;112
4.18;2.18 The Hausdorff Product;118
4.19;2.19 (Semi)Simple and Decomposable Endomorphisms;118
4.20;2.20 Representations of Compact (Finite) Groups;120
4.21;2.21 Algebra Representations and Modules;121
4.22;2.22 Characteristic and Minimal Polynomial;127
4.23;Bibliography;133
5;3 SPIN, ROTATIONS, AND POSITION;135
5.1;3.1 Linear Operations on the Alternative;136
5.2;3.2 Pauli Spinors;137
5.3;3.3 Spin Group;139
5.4;3.4 Spinor Reflections;139
5.5;3.5 Spin Representations;141
5.6;3.6 Position Translations from Adjoint Spin Structures;144
5.7;3.7 Polynomials with Spin Group Action;146
5.8;3.8 Spin Representation Matrix Elements;149
5.9;3.9 Spin Invariants and Weights;151
5.10;3.10 Summary;152
5.11;3.11 Derivations of Algebras;153
5.12;3.12 Differentiable Manifolds;156
5.13;3.13 Exponential and Logarithmic Mappings;157
5.14;3.14 (Semi)Simple Lie Algebras;161
5.15;3.15 Lie Algebra Inner Products;162
5.16;3.16 Lie Algebra Decompositions;164
5.17;3.17 Multilinearity and Tensor Algebra;164
5.18;3.18 Enveloping Algebra;171
5.19;Bibliography;175
6;4 ANTISTRUCTURES: The Real in the Complex;177
6.1;4.1 Anticonjugation;178
6.2;4.2 The Complex Quartet;180
6.3;4.3 Antidoubling;182
6.4;4.4 Dual and Antirepresentations;184
6.5;4.5 Particles and Antiparticles;186
6.6;4.6 Summary;189
6.7;4.7 Twin Vector Spaces;190
6.8;4.8 Complexification of Real Vector Spaces;190
6.9;Bibliography;191
7;5 SIMPLE LIE OPERATIONS;192
7.1;5.1 Diagonalization of Operations;193
7.2;5.2 Abelian, Nilpotent, and Solvable;196
7.3;5.3 The Basic Lie Operations;200
7.4;5.4 Spectral Decompositions of Lie Algebras;203
7.5;5.5 Spin Structure of Simple Lie Algebras;208
7.6;5.6 Roots and Weights;215
7.7;5.7 Classification of Complex Simple Lie Algebras and Dynkin Diagrams;223
7.8;5.8 Simple Complex and Compact Lie Groups;225
7.9;5.9 Simple Root Systems;226
7.10;5.10 Real Simple Lie Algebras;232
7.11;Bibliography;238
8;6 RATIONAL QUANTUM NUMBERS;239
8.1;6.1 Simple Representations of Simple Lie Symmetries;240
8.2;6.2 Representation Invariants and Weights of Simple Lie Algebras;241
8.3;6.3 Representations of Simple Lie Algebras;247
8.4;6.4 Centrality of Representations;257
8.5;Bibliography;262
9;7 QUANTUM ALGEBRAS;263
9.1;7.1 Quantization;264
9.2;7.2 Actions in Quantum Algebras;269
9.3;7.3 Quantum Algebras with Conjugation;275
9.4;7.4 Grading of Quantum Algebras;277
9.5;7.5 Symmetry and Statistics;280
9.6;7.6 Fundamental Spin Quantum Algebra;281
9.7;7.7 Adjoint Quantum Algebras;283
9.8;7.8 The Quantum Algebra for Position Translations;284
9.9;7.9 Quantum Implemented Time Action;286
9.10;7.10 Classical Lagrangians;292
9.11;7.11 Summary;294
9.12;7.12 Graded Algebras;294
9.13;7.13 Algebras with Bilinear Forms;297
9.14;7.14 Clifford Algebras;299
9.15;Bibliography;308
10;8 QUANTUM PROBABILITY;309
10.1;8.1 From Operator Algebra to Hilbert Spaces;310
10.2;8.2 Probability Amplitudes;314
10.3;8.3 Time Translation Eigenalgebras with Probability Interpretation;316
10.4;8.4 Tensor Algebra Forms;318
10.5;8.5 Fock States and Fock Spaces;321
10.6;8.6 Position Representation;325
10.7;8.7 The Irreducible Nonabelian Form for a Noncompact Time Representation;332
10.8;8.8 Summary;333
10.9;8.9 Algebra Forms;334
10.10;8.10 Topologies;335
10.11;8.11 Ordered Vector Spaces;339
10.12;8.12 Normed Vector Spaces;341
10.13;8.13 Banach Algebras;346
10.14;Bibliography;352
11;9 THE KEPLER FACTOR;353
11.1;9.1 Center of Mass Transformation;354
11.2;9.2 Intrinsic and ad hoc Units;355
11.3;9.3 Symmetries of the Kepler Dynamics;356
11.4;9.4 Classical Time Orbits;359
11.5;9.5 Kepler Bound State Vectors;364
11.6;9.6 Position Representations;369
11.7;9.7 Orbits of 1-Dimensional Position;371
11.8;9.8 Scattering Orbits of 3- Dimensional Position;374
11.9;9.9 Bound Orbits of 3-Dimensional Position;378
11.10;9.10 Scattering;385
11.11;9.11 Summary;390
11.12;9.12 Lattices and Logics;391
11.13;9.13 Measure Rings and Borel Spaces;392
11.14;9.14 Disjoint-Additive Mappings (Measures);394
11.15;9.15 Generalized Mappings (Distributions);399
11.16;9.16 Lebesgue Function Spaces;402
11.17;9.17 Direct Integral Vector Spaces;406
11.18;9.18 Linear Lattices ( Birkhoff- von Neumann Logics);408
11.19;Bibliography;410
12;Index;411