Foundations of Quantum Mechanics I

Contents......Page 8
I The Problem: An Axiomatic Basis for Quantum Mechanics......Page 12
1 The Axiomatic Formulation of a Physical Theory......Page 13
2 The Fundamental Domain for Quantum Mechanics......Page 15
3 The Measurement Problem......Page 21
II Microsystems, Preparation, and Registration Procedures......Page 23
1 The Concept of a Physical Object......Page 24
2 Selection Procedures......Page 26
3 Statistical Selection Procedures......Page 29
4.1 Preparation Procedures......Page 32
4.2 Registration Procedures......Page 33
4.3 The Dependence of Registration upon Preparation......Page 35
4.4 The Concept of a Physical System......Page 37
4.5 The Structure of Probability Fields for Physical Systems......Page 39
III Ensembles and Effects......Page 52
1 Combinations of Preparation and Registration Methods......Page 53
2 Mixtures and Decompositions of Ensembles and Effects......Page 58
3 General Laws: Preparation and Registration of Microsystems......Page 67
4.1 Properties and Physical Objects......Page 71
4.2 Pseudoproperties......Page 80
5 Ensembles and Effects in Quantum Mechanics......Page 84
6 Decision Effects and Faces of K......Page 86
IV Coexistent Effects and Coexistent Decompositions......Page 94
1.1 Coexistent Registrations......Page 95
1.2 Coexistent Effects......Page 97
1.3 Commensurable Decision Effects......Page 101
1.4 Observables......Page 107
2 Structures in the Class of Observables......Page 117
2.1 The Spaces B(\Sigma) and B'(\Sigma)......Page 118
2.2 Mixture Morphisms Corresponding to an Observable......Page 133
2.3 The Kernel of an Observable; Mixture of Effects for an Observable......Page 134
2.4 Mixtures and Decompositions of Observables......Page 139
2.5 Measurement Scales for Observables......Page 150
3 Coexistent and Complementary Observables......Page 163
4 Realizations of Observables......Page 165
5 Coexistent Decompositions of Ensembles......Page 167
6 Complementary Decompositions of Ensembles......Page 177
7 Realizations of Decompositions......Page 183
8.1 Objective Properties of Microsystems and Superselection Rules......Page 184
8.2 Pseudoproperties of Microsystems......Page 188
8.3 Logic of Decision Effects?......Page 192
1 Morphisms for Selection Procedures......Page 210
2 Morphisms of Statistical Selection Procedures......Page 212
3 Morphisms of Preparation and Registration Procedures......Page 214
4.1 Morphisms of Ensembles......Page 217
4.2 Morphisms of Effects......Page 222
4.3 Coexistent Operations and Coexistent Effects Morphisms......Page 225
5 Isomorphisms and Automorphisms of Ensembles and Effects......Page 227
1 Homomorphic Maps of a Group G in the Group A of B-continuous Effect Automorphisms......Page 242
1.1 Generation of a Representation of G in A by Means of a Representation of G by r-Automorphisms......Page 243
1.2 Some General Properties of a Representation of G in A......Page 248
1.3 Topologies on the Group A......Page 256
1.4 The Representation of G in Phase Space \Gamma......Page 258
2 The G-invariant Structure Corresponding to a Group Representation......Page 259
3.1 The Topological Structure of the Group A_{(B)}......Page 260
3.2 The Topological Properties of a Representation of G......Page 263
3.3 Unitary and Anti-unitary Representations Up to a Factor......Page 265
1 The Galileo Group as a Set of Transformations of Registration Procedures Relative to Preparation Procedures......Page 269
2 Irreducible Representations of the Galileo Group and Their Physical Meaning......Page 273
3 Irreducible Representations of the Rotation Group......Page 283
4 Position and Momentum Observables......Page 295
5 Energy and Angular Momentum Observables......Page 303
6 Time Observable?......Page 304
7 Spatial Reflections (Parity Transformations)......Page 310
8 The Problem of the Space D for Elementary Systems......Page 313
9 The Problem of Differentiability......Page 315
1 Registrations and Effects of the Inner Structure......Page 318
2 Composite Systems Consisting of Two Different Elementary Systems......Page 321
3 Composite Systems Consisting of Two Identical Elementary Systems......Page 331
4 Composite Systems Consisting of Electrons and Atomic Nuclei......Page 334
5 The Hamiltonian Operator......Page 339
6 Microsystems in External Fields......Page 343
7 Criticism of the Description of Interaction in Quantum Mechanics and the Problem of the Space D......Page 350
1 Definition of a Lattice......Page 354
2 Orthomodularity......Page 357
3 Boolean Rings......Page 359
4 Set Lattices......Page 363
1 Topological Spaces......Page 364
2 Uniform Spaces......Page 366
4 Connectedness......Page 368
1 Linear Vector Spaces......Page 370
3 The Dual Space for a Banach Space......Page 371
4 Weak Topologies......Page 372
5 Linear Maps of Banach Spaces......Page 373
6 Ordered Vector Spaces......Page 374
1 The Hilbert Space Structure Type......Page 377
2 Orthogonal Systems and Closed Subspaces......Page 380
3 The Banach Space of Bounded Operators......Page 383
4 Bounded Linear Forms......Page 384
5 The Banach Space L_r(H)......Page 386
6 Projection Operators......Page 388
7 Isometric and Unitary Operators......Page 390
8 Spectral Representation of Self-adjoint and Unitary Operators......Page 391
9 The Spectrum of Compact Self-adjoint Operators......Page 395
10 Spectral Representation of Unbounded Self-adjoint Operators......Page 396
11 The Trace as a Bilinear Form......Page 401
12 Gleason's Theorem......Page 408
13 Isomorphisms and Anti-isomorphisms......Page 415
14 Products of Hilbert Spaces......Page 416
15 The Spaces B(H_1, H_2, ...) and B'(H_1, H_2,...)......Page 422
References......Page 426
List of Frequently Used Symbols......Page 432
List of Axioms......Page 433
Index......Page 434