Deep Beauty: Understanding the Quantum World through Mathematical Innovation

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Contributors......Page 11
Preface......Page 13
Acknowledgments......Page 15
Introduction......Page 17
I.1 Beyond Hilbert Space......Page 18
I.2.1 n-Categorical Physics......Page 19
I.2.2 Quantum Theory in Monoidal Categories......Page 20
I.2.3 Quantum Theory in Toposes......Page 21
I.3 Operator Algebras......Page 22
I.4 Behind the Hilbert Space Formalism......Page 23
References......Page 26
PART ONE: Beyond the Hilbert Space Formalism: Category Theory......Page 27
1.1 Introduction......Page 29
1.2 Road Map......Page 30
1.3.1 Maxwell (1876)......Page 33
1.3.4 Poincar (1905)......Page 34
1.3.6 Minkowski (1908)......Page 35
1.3.9 Von Neumann (1932)......Page 36
1.3.10 Wigner (1939)......Page 37
1.3.11 Eilenberg–Mac Lane (1945)......Page 39
1.3.12 Feynman (1947)......Page 41
1.3.13 Yang–Mills (1953)......Page 43
1.3.14 Mac Lane (1963)......Page 44
1.3.16 Bénabou (1967)......Page 49
1.3.17 Penrose (1971)......Page 51
1.3.18 Ponzano–Regge (1968)......Page 59
1.3.19 Grothendieck (1983)......Page 63
1.3.21 Joyal–Street (1985)......Page 66
1.3.22 Jones (1985)......Page 71
1.3.23 Freyd–Yetter (1986)......Page 74
1.3.24 Drinfel’d (1986)......Page 80
1.3.25 Segal (1988)......Page 81
1.3.26 Atiyah (1988)......Page 83
1.3.27 Dijkgraaf (1989)......Page 86
1.3.28 Doplicher–Roberts (1989)......Page 88
1.3.29 Reshetikhin–Turaev (1989)......Page 89
1.3.32 Kashiwara and Lusztig (1990)......Page 96
1.3.33 Kapranov–Voevodsky (1991)......Page 98
1.3.34 Reshetikhin–Turaev (1991)......Page 100
1.3.35 Turaev–Viro (1992)......Page 101
1.3.36 Fukuma–Hosono–Kawai (1992)......Page 102
1.3.37 Barrett–Westbury (1993)......Page 107
1.3.38 Turaev (1992)......Page 112
1.3.39 Kontsevich (1993)......Page 115
1.3.40 Lawrence (1993)......Page 116
1.3.41 Crane–Frenkel (1994)......Page 118
1.3.43 Kontsevich (1994)......Page 120
1.3.44 Gordon–Power–Street (1995)......Page 123
1.3.45 Baez–Dolan (1995)......Page 124
1.3.46 Khovanov (1999)......Page 131
References......Page 134
2.1 Introduction......Page 145
2.2.1 To Measure or Not to Measure......Page 147
2.2.2 Measurement among Other Processes......Page 149
2.2.3 Systems from Processes......Page 150
2.2.4 The Logic of Interacting Processes......Page 151
2.2.5 Processes as Relations......Page 152
2.2.6 Mathematical Rigor......Page 153
2.2.7 The Continuous or the Discrete?......Page 154
2.3 Systems ← Relations ← Composition......Page 155
2.3.1 Systems......Page 156
2.3.2 Processes and Their Composition......Page 157
2.3.2.1 Sequential Composition......Page 158
2.3.2.3 Independence Constraint on Separate Composition......Page 159
2.3.2.5 Nonisolated Systems and Probabilistic Weights of States......Page 160
2.3.3 Graphical Representation of Processes......Page 161
2.3.4 Physical Scenarios, Snapshots, and Subsystems......Page 162
2.3.5 Symmetry Relations......Page 164
2.3.6 Vacuous Relations: Correcting Denotational Artifacts......Page 168
2.3.7 Summary of This Section......Page 169
2.4.2 Axiomatics......Page 170
2.4.3 Concrete Models......Page 173
2.4.4 Where Axioms and Models Meet: A Theorem......Page 175
2.4.5 Nonisolation in the von Neumann Quantum Model......Page 178
2.5 Classicality and Measurement......Page 181
2.5.1 Classicality......Page 182
2.5.2 Measurement......Page 185
2.5.3 Classicality in the von Neumann Quantum Model......Page 187
2.5.4.1 Commutative Monoids and Commutative Comonoids......Page 188
2.5.4.2 Commutative Dagger Frobenius Algebras......Page 190
2.5.4.3 Varying the Coordinate System......Page 192
Acknowledgments......Page 193
References......Page 194
3.1 Introduction......Page 203
3.2 The Problem of Using the Real Numbers in Quantum Gravity......Page 206
3.3.1 The Realism of Classical Physics......Page 207
3.3.2 A Categorial Generalization of the Representation of Physical Quantities......Page 209
3.4.1 The Nature of a Topos......Page 210
3.4.2 The Mathematics of “Neorealism”......Page 212
3.4.3 The Idea of a Pseudo-State......Page 213
3.4.3.1 Option 1 (The Truth-Object Option)......Page 214
3.4.3.2 Option 2 (The Pseudo-State Option)......Page 215
3.5.1 The Kochen–Specker Theorem and Contextuality......Page 216
3.5.2 The Topos of Presheaves SetsV(H)op......Page 218
3.6 Conclusions......Page 219
References......Page 220
4.1.1 The Topos Approach......Page 223
4.1.2 Daseinisation......Page 224
4.2 Propositions and Their Representation in Classical Physics and Standard Quantum Theory......Page 225
4.2.2 Representation of Propositions in Standard Quantum Theory......Page 226
4.3.1 Contexts......Page 227
4.3.2.2 Gelfand Spectra......Page 230
4.3.2.3 The Spectral Presheaf......Page 232
4.4.1 Coarse-Graining of Propositions......Page 233
4.4.2 Daseinisation of Projections......Page 235
4.5.1 Physical Quantities as Arrows......Page 238
4.5.1.1 Approximation in the Spectral Order......Page 239
4.5.1.2 The Presheaf of “Values”......Page 240
4.5.1.3 Daseinisation of a Self-Adjoint Operator......Page 241
4.5.1.4 Pure States and Assignment of “Values” to Physical Quantities......Page 242
4.5.2.1 Filters......Page 243
4.5.2.2 Elements of the Gelfand Spectrum and Filters......Page 244
4.5.2.3 Vector States from Elements of Gelfand Spectra......Page 247
4.5.2.4 The Eigenstate-Eigenvalue Link......Page 248
4.6 Conclusion......Page 251
References......Page 252
5.1 The Stone Spectrum of a Lattice......Page 255
5.2 Stone Spectra of von Neumann Algebras......Page 257
5.3 Observable Functions......Page 259
5.4.1 Continuous Functions......Page 267
5.4.2 Measurable Functions......Page 269
5.5 Topological Characterization of Observable Functions......Page 271
5.6.1 The Spatial Presheaf of Spectral Families......Page 272
5.6.2.1 Algebraic Restriction of Spectral Families......Page 274
5.6.2.2 The Algebraic Presheaf......Page 279
5.6.3.2 The State Presheaf......Page 281
5.6.3.3 Contextual Physical Quantities......Page 282
5.7 Concluding Remarks......Page 283
References......Page 284
6.1 Introduction......Page 287
6.2 Locales and Toposes......Page 289
6.3 C*-Algebras......Page 297
6.4 Bohrification......Page 302
6.5 Projections......Page 308
6.6 States and Observables......Page 317
References......Page 325
PART TWO: Beyond the Hilbert Space Formalism: Operator Algebras......Page 331
7.1 Introduction......Page 333
7.2 The Mathematical Framework......Page 334
7.3 Immediate Consequences......Page 337
7.4 Vacuum Correlations......Page 339
7.5 Geometric Modular Action......Page 343
7.6 Intrinsic Characterization of the Vacuum State......Page 346
7.7 Deriving Spacetime from States and Observables......Page 348
7.8 Concluding Remarks......Page 351
References......Page 352
8.1 Main Claim......Page 359
8.2 Preliminary Historical Comments......Page 360
8.3 Einstein’s Contrasting Standard Nonrelativistic Quantum Mechanics with Field Theory in 1948......Page 362
8.4 Some Notions of Algebraic Quantum Mechanics......Page 364
8.5 Algebraic Quantum Field Theory......Page 366
8.6 Independence......Page 367
8.7 Local Operations......Page 370
8.8 Are Operational Separability and Operational Independence Equivalent?......Page 373
8.9 Closing Comments......Page 374
References......Page 376
PART THREE III: Behind the Hilbert Space Formalism......Page 379
9.1 Introduction......Page 381
9.2 Basic Ideas and the Axioms......Page 384
9.3 Basic Notions......Page 386
9.4 Elementary System: System of Information Capacity of One Bit......Page 388
9.5 Composite System and the Notion of Locality......Page 391
9.6 The Main Proofs......Page 393
9.6.2 Ruling Out the d > 3 Case......Page 394
9.7 ``Two\'\' Quantum Mechanics......Page 396
9.8 Higher-Dimensional Systems and State Update Rule in Measurement......Page 400
9.9 What the Present Reconstruction Tells Us about Quantum Mechanics......Page 401
Acknowledgments......Page 402
Appendix......Page 403
References......Page 404
10.1 Introduction......Page 409
10.2 The Kochen--Specker Proof: Klyachko Version......Page 410
10.3 Bell\'s Proof......Page 415
10.4 Von Neumann\'s Proof......Page 418
References......Page 422
11.2 Introduction......Page 425
11.3 Related Work......Page 427
11.4.1 Operations and Circuits......Page 429
11.4.2 Time in a Circuit......Page 431
11.4.3 Probability......Page 432
11.4.4 Can No Signalling Be an Axiom?......Page 434
11.5.1 Circuit Fragments......Page 436
11.5.2 States......Page 437
11.5.3 Transformations......Page 438
11.5.4 The Identity Transformation......Page 439
11.6.1 Forming Mixtures......Page 440
11.6.2 Homogeneous, Pure, Mixed, and Extremal States......Page 441
11.7.1 Preliminaries and Notation......Page 442
11.7.2 Commutation......Page 444
11.7.3 Homogeneous States and Composite Systems......Page 445
11.7.4 Fiducial Measurements for Composite Systems......Page 447
11.7.5 Homogeneity and Uncorrelatability Are Equivalent Notions......Page 449
11.7.6 Probabilities for Disjoint Circuit......Page 450
11.8.1.1 Operation Locality......Page 451
11.8.2 Classical Probability Theory......Page 452
11.8.3 Quantum Theory......Page 453
11.8.4 Reasonable Postulates for Quantum Theory......Page 454
11.9 Conclusions......Page 455
References......Page 456
12.1 Introduction......Page 459
12.2.1 The SPIN Axiom and the Kochen–Specker Paradox......Page 460
12.2.3 The MIN Axiom, Relativity, and Free Will......Page 462
12.3 The (Strong) Free Will Theorem......Page 463
12.4 Locating the Response......Page 464
12.5 Free Will Versus Determinism......Page 465
Appendix. Can There Be a Mechanism for Wave Function Collapse?......Page 467
References......Page 469
Index......Page 471