Symmetries in Physics: Philosophical Reflections

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Contributors......Page 9
Acknowledgements......Page 11
Copyright acknowledgements......Page 13
1 The meanings of symmetry......Page 15
2 Symmetry in the history of physics......Page 17
2.1 Symmetry principles......Page 18
2.2 Symmetry arguments......Page 23
3 Symmetries of modern physics: their status and significance......Page 25
4 Structure of the book......Page 29
References......Page 31
Part I Continuous symmetries......Page 33
Symmetry......Page 35
Symmetry and conservation laws......Page 37
Events, laws of nature, invariance principles......Page 38
Geometrical principles of invariance and conservation laws......Page 39
Dynamic principles of invariance......Page 40
Events, laws of nature, and invariance principles......Page 41
1 Introduction......Page 43
2.1 Groups, invariants, and conservation......Page 44
2.2 Relativity, spacetime and gravitation......Page 46
2.3 Symmetry and quantum theory......Page 50
2.3.1 The ascendancy of symmetry: interaction from symmetry......Page 51
3.1 Gauge principle – gauge argument – gauge heuristic......Page 56
3.2 Questioning the gauge logic of nature......Page 57
3.3 Other logics of nature......Page 60
Aside: On describing gauge theories......Page 61
4.1 Gauge invariance: profundity, redundancy, or both?......Page 63
4.1.2 A generalized Kretschmannian objection......Page 66
4.2 How gauge principles got their oomph – a Just So Story?......Page 67
References......Page 71
1 Introduction......Page 75
2 The ‘Introduction’ to Raum-Zeit-Materie......Page 78
3.1 ‘The natural attitude’, ‘transcendental subjectivity’......Page 81
3.2 Evidenz......Page 83
3.3 Essence (Wesen/Eidos), essential seeing (Wesenerschauung)......Page 84
3.5 Eidetic science; regional ontology......Page 86
4 Constituting field physics from ‘purely infinitesimal’ comparison relations......Page 88
4.1 First stage: continuous connection (topology)......Page 91
4.2 Second stage: affine connected manifold......Page 92
4.3 Third stage: metrically connected manifold......Page 94
4.4 The transition to physics......Page 97
5 Concluding remarks......Page 98
References......Page 99
1 Introduction......Page 103
2 Noether’s variational problem......Page 105
3 Transformations, invariance of the action, and symmetries in physics......Page 107
4 Noether’s first theorem: global symmetries and conservation laws......Page 111
The Boundary theorem......Page 114
5.2 Noether’s second theorem......Page 117
Noether’s second theorem......Page 118
5.3 Illustrations......Page 119
References......Page 121
…that we need not choose between......Page 124
Einstein…......Page 125
…and Kretschmann......Page 126
Active general covariance......Page 127
Its physical consequences......Page 128
The context in which Kretschmann’s objection succeeds......Page 129
The context in which the diffeomorphism gauge freedom has physical content......Page 130
5 Gauge theories in particle physics......Page 131
The substitution trick…......Page 132
…yields geometric objects......Page 133
The coordinates as scalars trick......Page 134
Appendix 2: From passive to active covariance......Page 135
References......Page 136
1 Introduction......Page 138
2 The ambiguity of mathematical representation......Page 139
3 Symmetry......Page 141
4 Surplus structure......Page 142
5 Constrained Hamiltonian systems......Page 143
6 Yang–Mills gauge theories......Page 144
7 The case of general relativity......Page 147
8 BRST symmetry......Page 149
9 Conclusion......Page 151
References......Page 152
1 Introduction......Page 154
2 Noether’s theorems, constrained Hamiltonian systems, and all that......Page 155
3 Gauge transformations and first-class Hamiltonian constraints......Page 158
4 A toy example......Page 163
5 The General Theory of Relativity as a gauge theory......Page 165
7 The reach of the constraint apparatus......Page 168
8 The quantization of gauge theories......Page 170
9 The magical gauge argument......Page 171
10.3 Future extensions and modifications of the theory......Page 172
11 Conclusion......Page 173
References......Page 174
1 Introduction......Page 177
2 Symmetries of the action......Page 178
3 The definition of a symmetry......Page 179
4 Global and local symmetries......Page 180
5 Symmetries and the breakdown of determinism......Page 181
6 Two ways to repair determinism......Page 183
7 Local symmetries from the Hamiltonian perspective......Page 184
8 Conclusion......Page 186
References......Page 187
1.1 The effect......Page 188
1.2 The three approaches towards the A–B effect......Page 189
1.2.1 The first approach......Page 190
1.2.3 The third approach......Page 191
2 The fibre bundle formulation of theories with gauge symmetries......Page 192
2.1 What a fibre bundle is......Page 193
2.2 Cross-sections......Page 194
2.3.1 The tangent bundle: a special example of a vector bundle......Page 197
2.3.2 The bundle of frames: a special example of a principal fibre bundle......Page 198
2.3.3 Connection on the bundles or moving around......Page 199
2.4 Gauge transformations......Page 200
2.5 Associations......Page 202
3.1 The holistic character of the topological explanation......Page 203
3.1.1 Holonomies, homotopy, and the U (1) group of electromagnetism......Page 204
3.1.2 Topological explanation (1): the manifold first......Page 205
3.1.3 Topological explanation (2): the bundle first......Page 206
4 Assessing the topological explanation......Page 207
4.1 Assessment of topological explanation (1)......Page 209
4.2 Assessment of topological explanation (2)......Page 210
Acknowledgements......Page 212
References......Page 213
Part II Discrete symmetries......Page 215
Leibniz’s third letter to Clarke, paragraph 6......Page 217
Concerning the ultimate ground of the differentiation of directions in space......Page 218
The identity of indiscernibles......Page 220
1 Introduction......Page 226
2 The mathematics and physics of permutation symmetry......Page 227
4 Permutation invariance and the topological approach to particle identity......Page 233
5.2 Challenges to the Received View......Page 235
6 Individuality and the identity of indiscernibles......Page 239
7 PI is neither sufficient nor necessary for non-individuality......Page 242
8 Underdetermination and the structuralist view of particles......Page 243
9 The experimental and theoretical status of PI......Page 244
10.1 Permutation symmetry and structural realism......Page 247
10.2 Diffeomorphisms, permutations, and the structuralist conception of spacetime......Page 248
References......Page 249
1 Introduction......Page 253
2 Generalizations......Page 256
3 Quarticles......Page 258
4 Concluding remarks......Page 262
References......Page 263
1 Introduction......Page 264
2 Incongruent counterparts......Page 266
3 The challenge of parity violation......Page 276
4 A relational account of parity violation......Page 279
5 Orientation fields......Page 284
Appendix: The weak interaction......Page 289
References......Page 293
1 Introduction......Page 295
2 The problems......Page 296
3 The answer......Page 299
References......Page 302
16 Physics and Leibniz’s principles......Page 303
1 Identity......Page 305
2 The generalist picture......Page 309
3 Possible worlds......Page 311
4 Leibniz Equivalence......Page 312
5 The Principle of Sufficient Reason......Page 317
6 Close......Page 318
References......Page 319
Part III Symmetry breaking......Page 323
I. Introduction......Page 325
VII. Conclusion......Page 326
Symmetry......Page 327
1 Breaking gauge and chiral invariance......Page 329
2 The effective action......Page 332
References......Page 333
1 Preliminaries – I......Page 335
2 Symmetry breaking and Curie’s analysis......Page 336
3 Preliminaries – II......Page 338
4.1 Explicit symmetry breaking......Page 339
4.2 Spontaneous symmetry breaking......Page 341
5 Symmetry breaking and philosophical questions......Page 344
References......Page 347
1 Laws and symmetries......Page 349
2 Spontaneous symmetry breaking in classical physics and QFT......Page 351
3 Using the algebraic formulation of QFT to explain spontaneous symmetry breaking......Page 352
4 Resolving the puzzles......Page 355
5 Issues of interpretation......Page 356
6 Conclusion......Page 358
References......Page 359
1 Introduction......Page 361
2 Mathematics and the physical world......Page 362
3 Spontaneous symmetry breaking: how it all began......Page 366
4 At sea in the vacuum......Page 369
5 What’s wrong with this picture?......Page 371
References......Page 376
Part IV General interpretative issues......Page 379
Initial conditions, laws of nature, invariance......Page 381
lnvariance......Page 382
What is the role and proper place of invariance principles in the framework of the physical sciences?......Page 383
1 What is superfluous structure?......Page 385
2 What is a theory?......Page 386
2.1 Theoretical ontology......Page 387
2.2 Laws and physical possibilities......Page 388
3.1 Qualitative structure......Page 389
3.2 Qualitative vs. measurable......Page 390
3.3 Measurable but non-qualitative features......Page 391
4 Symmetries: kinds of structure-preservation......Page 392
5 The signs of superfluous structure......Page 393
5.2 A sophistical argument......Page 394
6 Symmetries of the laws......Page 396
6.1 Qualitative indiscernibility with dynamic differences......Page 398
6.2 Symmetries of a world, and identity of indiscernibles......Page 401
6.3 A new sophistical argument......Page 402
7 The wider theoretical context......Page 404
8 Conclusion......Page 405
References......Page 406
1 Symmetries......Page 407
2 Symmetry arguments......Page 408
3.1 Platonic cosmology......Page 409
3.2 Simultaneity in special relativity......Page 410
3.3 Time in dust cosmology......Page 411
3.4 Huygens on collision......Page 412
3.5 Symmetries in classical mechanics......Page 414
4 Symmetries of solutions and symmetries of laws......Page 415
4.1 Structuring the space of solutions......Page 416
4.2 Symmetries of equations vs. symmetries of solutions......Page 417
5 Quotienting out symmetries......Page 419
5.1 Quotienting solutions......Page 420
5.2 Quotienting the space of solutions......Page 421
References......Page 425
2 Symmetry and objectivity......Page 427
3 Symmetry and design......Page 433
4 Conclusion......Page 436
References......Page 437
1 Symmetry and equality......Page 439
2 Symmetry, equivalence, and group......Page 441
3 Equivalence and irrelevance......Page 443
4.1 Gauge freedom and constraints......Page 445
4.2 Symmetries and scales......Page 448
References......Page 450
Index......Page 451