Introducing graduate students and researchers to mathematical physics, this book discusses two recent developments: the demonstration that causality can be defined on discrete space-times; and Sewell's measurement theory, in which the wave packet is reduced without recourse to the observer's conscious ego, nonlinearities or interaction with the rest of the universe. The definition of causality on a discrete space-time assumes that space-time is made up of geometrical points. Using Sewell's measurement theory, the author concludes that the notion of geometrical points is as meaningful in quantum mechanics as it is in classical mechanics, and that it is impossible to tell whether the differential calculus is a discovery or an invention. Providing a mathematical discourse on the relation between theoretical and experimental physics, the book gives detailed accounts of the mathematically difficult measurement theories of von Neumann and Sewell.
Author(s): R. N. Sen
Series: Cambridge Monographs on Mathematical Physics
Publisher: CUP
Year: 2010
Language: English
Pages: 413
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
Preface......Page 17
Acknowledgement......Page 19
To the reader......Page 21
Prologue......Page 23
Part I Causality and differentiable structure......Page 31
Introduction to Part I......Page 33
1 Mathematical structures on sets of points......Page 37
1.2 Mathematical structures on countably infinite sets......Page 38
1.3 Mathematical structures on uncountable sets......Page 40
1.4 Global geometrical structures on Rn......Page 41
1.5.1 Riemannian and Lorentz structures......Page 42
1.5.3 The Weyl projective structure......Page 43
2.2 The order axiom......Page 45
2.3 The identification axiom......Page 48
2.4 Light cones......Page 49
2.4.1 Timelike points......Page 50
2.4.2 Extension of order......Page 51
2.5 The cone axiom......Page 52
2.6 D-sets......Page 54
2.7 Properties of D-sets......Page 58
2.7.1 D-sets and timelike order......Page 59
2.7.2 Light rays from T-interiors of D-sets......Page 60
2.7.3 Incidence of light rays on cone boundaries......Page 61
2.7.5 New D-sets from old......Page 62
2.7.7 Spacelike separation in D-sets......Page 63
2.7.8 Timelike order and D-subsets......Page 65
2.8 The local structure axiom......Page 66
2.9 Ordered spaces......Page 67
3.1 The order topology......Page 68
3.1.1 The Tychonoff property......Page 71
3.1.2 Order equivalence......Page 74
3.2.1 The standard maps......Page 75
3.2.2 Three or more dimensions......Page 77
4.1 Spaces in which light rays are complete......Page 82
4.2 Metric and uniform completions......Page 83
4.2.1 Uniformizability of ordered spaces......Page 84
4.2.2 Complete uniformizability......Page 86
4.3 The concept of order completion......Page 87
4.3.1 The order uniformity on D-sets......Page 89
4.4 Extension of order: notations and definitions......Page 91
4.5 Extension of order: topological results......Page 93
4.5.1 Symmetry properties......Page 94
4.5.3 Density lemmas......Page 95
4.6.1 The segment l[a, η]......Page 96
4.6.1.1 Properties of the segment l[a, η]......Page 97
4.6.2.1 Properties of the segment l[ξ, η]......Page 98
4.7.1 The order axiom......Page 99
4.7.3 The cone axiom......Page 100
4.7.4 D-sets in......Page 101
4.7.5 The local structure axiom......Page 102
5.1 Timelike curves in M......Page 103
5.2.1 2 -cells in D-intervals......Page 106
5.2.2 Cylindrical coordinates on D-intervals......Page 109
5.3.2 The local differentiable structure......Page 110
Part II Geometrical points and measurement theory......Page 113
Introduction to Part II......Page 115
6.1 The impact of quantum theory......Page 123
6.2 Precise measurements in classical mechanics......Page 125
6.2.1 Space, time and measurement in classical mechanics......Page 126
6.3 Discussion......Page 127
6.4 The role of the experimentalist......Page 128
6.5 Born's probability interpretation of classical mechanics......Page 129
7.1.1 Pure states......Page 132
7.1.2 Mixed states......Page 133
7.1.3 Partial traces......Page 136
7.1.4 Dynamics......Page 137
7.1.5 Observables......Page 138
7.2 The probability interpretation of quantum mechanics......Page 139
7.2.2 Remarks on notations......Page 141
7.3 Superselection rules......Page 142
7.4 The Galilei group......Page 143
7.4.1.1 Factor systems and group exponents......Page 145
7.4.1.2 The extended Galilei group......Page 147
7.4.1.3 Mass in nonrelativistic quantum mechanics......Page 148
7.4.2 Bargmann’s superselection rule; conservation of mass......Page 149
7.5 Theorems of von Neumann and Stone; Reeh's example......Page 150
7.5.1 The Stone–von Neumann uniqueness theorem......Page 151
7.5.2 Reeh’s example......Page 153
7.5.3 Stone’s theorem......Page 155
8.1 Overview......Page 157
8.2.1 The inference from observation......Page 160
8.2.2 Measurement of operators with continuous spectra......Page 163
8.2.3 The quantum measurement problem......Page 164
8.3 Von Neumann's Chapter VI......Page 165
8.3.1 Formulation of the problem......Page 166
8.3.2 Composite systems......Page 167
8.3.3 Discussion of the measuring process......Page 177
8.4 Wigner's reservations......Page 179
8.5 Reconsideration of von Neumann's theory......Page 180
8.5.1 Entanglement......Page 181
8.5.2 Description of the measuring apparatus......Page 182
9.1 Commuting self-adjoint operators......Page 183
9.1.1 Commuting observables with discrete spectra......Page 185
9.2.1 The uncertainty principle......Page 186
9.2.3 Determination of…......Page 188
9.2.4 Von Neumann’s commuting operators…......Page 189
9.2.5 The von Neumann–Wigner characterization of macroscopic measurements......Page 191
9.3 Answers to Wigner, I......Page 192
10 Sewell's theory of measurement......Page 195
10.1 Preliminary discussion......Page 197
10.2 The object–apparatus interaction......Page 198
10.3 The macroscopic observables of A......Page 202
10.4 Expectations and conditional expectations of observables......Page 203
10.4.1 Explicit expressions......Page 207
10.4.2 Properties of F r,s;alpha......Page 208
10.5.1 Interpretation......Page 209
10.6 Consistency and robustness of Sewell's generic model......Page 210
10.6.2 The question of robustness......Page 211
10.7 Answers to Wigner, II......Page 212
11.1 Bell's example......Page 214
11.2.1 Degenerate eigenvalues......Page 217
11.2.2 The case of dim…......Page 218
11.3 The Araki–Yanase theorem......Page 219
11.4 Impossibility theorems of Shimony and Busch......Page 223
11.5 The Heisenberg cut......Page 224
11.6 Measurement of continuous spectra......Page 225
11.7 Adequacy of Sewell's scheme......Page 226
11.8 Answers to Wigner, III......Page 227
11.8.1 The tension with the theory of relativity......Page 228
12 Large quantum systems......Page 230
12.1 Elementary excitations in superfluid helium......Page 231
12.2 Hilbert bundles......Page 232
12.3 Bundle representations......Page 233
12.3.1 Landau excitations......Page 234
12.3.1.1 Unitary representations of E3 and H......Page 235
12.3.1.2 Landau excitations as bundle representations......Page 236
12.3.1.3 Unstable excitations......Page 237
12.4 Dynamics on Banach bundles......Page 238
12.5.1 C-algebras......Page 240
12.5.2 States on C-algebras......Page 241
12.6.1 The Gelfand–Naimark–Segal theorem......Page 242
12.6.2 The weak operator topology on a representation......Page 244
12.7 Algebraic description of infinite systems......Page 245
12.7.1 C-dynamical systems......Page 246
12.8 Temporal evolution of reduced descriptions......Page 247
Causal automorphisms of an ordered space......Page 250
Sets, real numbers and physics......Page 251
Interaction of quantum and classical systems......Page 252
Superseparability and noninterferometry......Page 253
Is nature law abiding?......Page 255
Mathematical appendices......Page 257
A1.1 Sets......Page 259
A1.2.1 Union, intersection and difference......Page 260
A1.2.2 Subsets......Page 261
A1.3 Maps......Page 262
A1.4 Finite and infinite sets......Page 265
A1.5.1 Cardinal numbers......Page 267
A1.6 Products of sets......Page 268
A1.8 Russell's paradox......Page 270
A2.1 Irrational numbers......Page 272
A2.2.1 Dedekind’s construction......Page 275
A2.2.3 Cantor’s procedure......Page 277
A2.2.4 Well-ordering and real numbers......Page 278
A3.1 Topological spaces......Page 279
A3.1.1 Closed sets, interior, boundary......Page 281
A3.1.2 Comparison of topologies......Page 282
A3.2.1 Homeomorphisms......Page 283
A3.3.2 Topological products......Page 284
A3.3.3 Quotient spaces......Page 285
A3.4.1 Second countability and separability......Page 286
A3.4.2 Separation properties......Page 287
A3.4.2.1 The T0-separation property......Page 288
A3.4.3 Separation by continuous functions......Page 289
A3.5 Metric spaces and the metric topology......Page 290
A3.5.1 Metrizability......Page 291
A3.6.1 Some properties of compact spaces......Page 292
A3.7 Connectedness and path-connectedness......Page 294
A3.8 The theory of convergence......Page 295
A3.8.1.1 Filters on topological spaces......Page 296
A3.9 Topological groups......Page 297
A4.1 Metric completion......Page 299
A4.2 Uniformities......Page 302
A4.2.1 Definition and general properties......Page 303
A4.3 Complete uniform spaces......Page 305
A4.3.1 Uniform completion......Page 306
A4.3.3 Complete uniformizability......Page 307
A4.3.4 Total boundedness......Page 308
A5 Measure and integral......Page 310
A5.1 Normed spaces......Page 311
A5.2.1 The sup norm......Page 312
A5.2.2 The 1-norm. Incompleteness of the Riemann integral......Page 314
A5.2.3 The p-norms; the 2-norm......Page 315
A5.3 The Riemann–Stieltjes integral......Page 317
A5.3.1 Step functions as integrators......Page 319
A5.4 alpha-algebras, measures and integrals......Page 321
A5.4.1 Measurable sets......Page 322
A5.4.2 Measures......Page 323
A5.4.3 Integrals......Page 325
A5.5 Haar measure......Page 328
A5.6.2 The Lebesgue measure on......Page 329
A5.6.3 The space L1([a, b], dx)......Page 330
A5.6.5 The space Lp......Page 331
A5.7 Noninvariant measures on R......Page 332
A5.7.1 Signed measures; the Radon–Nikodym theorem......Page 334
A5.8 Differentiation......Page 336
A6.1 Hilbert space......Page 338
A6.1.1 Direct sums......Page 341
A6.1.2 Tensor products......Page 343
A6.2 Operators on Hilbert space......Page 345
A6.3 Bounded operators......Page 348
A6.4.1 The finite-dimensional case......Page 350
A6.4.2 Spectral theorem: compact operators......Page 351
A6.4.3 Resolutions of the identity......Page 353
A6.4.4.1 The spectrum of an operator......Page 356
A6.4.4.2 Approximate eigenvectors......Page 357
A6.5 Unbounded operators......Page 358
A6.6 The spectral theorem for unbounded operators......Page 360
A7.1 Recapitulation of basic notions......Page 363
A7.1.1 Conditioning......Page 365
A7.2 Probability: general theory......Page 366
A7.3 Conditioning......Page 368
A8.1 Fibre bundles......Page 371
A8.1.1 Definition of fibre bundles......Page 372
A8.1.2 Principal bundles; the bundle structure theorem......Page 373
A8.1.4 Bundle maps......Page 374
A8.2.1 Definition of a differentiable manifold......Page 375
A8.2.2 Functions on manifolds......Page 378
A8.3 Tangent vectors and the tangent bundle......Page 379
A8.4 Maps of manifolds......Page 380
A8.5 Vector fields and derivations......Page 381
A8.6 One-parameter groups of transformations......Page 383
A8.7 Lie groups and Lie algebras......Page 384
A8.7.1 Left-invariant vector fields and Lie algebras......Page 385
A8.7.2 The exponential map......Page 386
A8.7.3 Some examples and counterexamples......Page 387
A8.7.4 Complex Lie groups and algebras; complexification......Page 388
List of symbols for Part I......Page 389
References......Page 392
Index......Page 402
