ROAD TO REALITY WITH ROGER PENROSE LADYMAN J.PRESNELL S.MCCABE G.ECKSTEIN M..J. SZYBKA S. EDITORS Księgarnia

Road to Reality with Roger Penrose

Where does the road to reality lie?

This fundamental question is addressed in this collection of essays by physicists and philosophers, inspired by the original ideas of Sir Roger Penrose.

The topics range from black holes and quantum information to the very nature of mathematical cognition itself.

Contributors

Paolo Aniello, Abhay Ashtekar, Jesús Clemente-Gallardo, Katarzyna Grabowska, Michael Heller, Jerzy Kijowski, Stanisław Krajewski, Donald Lynden-Bell, Shahn Majid, Giuseppe Marmo, Zdzisław Odrzygóźdź, Leszek Pysiak, Wiesław Sasin, Leszek M. Sokołowski, Paul Tod, Georg F. Volkert, Krzysztof Wójtowicz, Nick Woodhouse.

1. Fromgeometric quantum mechanics to quantum information .....1
Paolo Aniello, Jesús Clemente-Gallardo, Giuseppe Marmo, Georg F. Volkert
1.1 Introduction ..... 1
1.2 Geometrical formulation of the Hilbert space picture ..... 3
1.2.1 From Hermitian operators to real-valued functions ..... 4
1.2.2 The Fubini–Study metric seen from the Hilbert space ..... 5
1.2.3 From Hermitian inner products to classical tensor fields ..... 6
1.2.4 Pull-back structures on submanifolds of H ..... 10
1.3 Someapplications : composite systems, entanglement and separability ..... 12
1.3.1 Separable and maximal entangled pure states ..... 12
1.3.2 Quantitative statements ..... 14
1.3.3 Mixed states entanglement and invariant operator valued tensor fields ..... 14
1.4 From quantum to classical information ..... 17
1.5 Conclusions and outlook ..... 21

2. Black holes in general relativity ..... 23
Abhay Ashtekar
2.1 Introduction ..... 23
2.1.1 Newtonian considerations ..... 24
2.1.2 General relativity ..... 26
2.2 Black holes in general relativity ..... 28
2.2.1 Early history ..... 28
2.2.2 Uniqueness ..... 31
2.3 Event horizons and their unforeseen properties ..... 33
2.3.1 Event horizons ..... 34
2.3.2 An unexpected treasure trove ..... 37
2.4 Epilogue ..... 41
2.4.1 Spookiness of event horizons ..... 42
2.4.2 Quasi-local horizons ..... 44

3. Gravitational energy: a quasi-local, Hamiltonian approach ..... 51
Katarzyna Grabowska & Jerzy Kijowski
3.1 Introduction ..... 51
3.2 Symplectic relations and their generating functions ..... 54
3.3 Lagrangian and Hamiltonian formulations of mechanics ..... 57
3.4 Field dynamics as a symplectic relation ..... 59
3.5 Example: symmetric versus canonical energy in Maxwell electrodynamics ..... 62
3.6 Homogeneous Hamiltonian identity in canonical relativity ..... 65
3.7 Examples of the gravitational boundary control and corresponding Hamiltonians ..... 71
3.8 Concluding remarks ..... 74

4. General relativity and von Neumann algebras ..... 75
Michael Heller, Zdzisław Odrzygózdz, Leszek Pysiak & Wiesław Sasin
4.1 Introduction ..... 75
4.2 Space-time as a noncommutative space ..... 77
4.3 Algebra of random operators ..... 79
4.4 Dierential algebra ..... 80
4.5 Generalized space-time geometry ..... 81
4.6 General relativity on random operators ..... 82
4.7 Concluding remarks ..... 83
Appendix ..... 83

5. Penrose’s metalogical argument is unsound ..... 87
Stanisław Krajewski
5.1 Introduction ..... 87
5.2 Necessary conditions for out-Gödeling ..... 89
5.3 Inconsistency/unsoundness of the antimechanist ..... 93
5.4 A relevant discovery: Gödel’s unknowability thesis ..... 97
5.5 Penrose’s new argument ..... 99
5.6 Evolution of machines: robots and the mind ..... 100
5.7 A “natural” view of mathematics ..... 103

6. Mach’s Principle within general relativity ..... 105
Donald Lynden-Bell
6.1 Introduction ..... 105
6.2 A Newtonian non-relativistic mechanics without absolute space ..... 107
6.3 Machian phenomena predicted by general relativity ..... 109
6.3.1 Accelerated inertial frames ..... 109
6.3.2 Rotating inertial frames ..... 110
6.3.3 Induced centrifugal force ..... 111
6.3.4 Mass induction ..... 113
6.4 Closed Universes rotation and the cosmological constant ..... 114

7. Algebraic approach to quantum gravity I: relative realism ..... 117
Shahn Majid
7.1 Introduction ..... 118
7.1.1 Do theoretical physicists need to get out more? ..... 119
7.1.2 Some answers ..... 123
7.2 Relative realism ..... 125
7.2.1 A mathematician’s view ..... 125
7.2.2 In everyday life ..... 128
7.2.3 Is this a chair? ..... 130
7.2.4 My 2-year old’s insight into quantum gravity ..... 132
7.3 Plato’s cave revisited: representation and represented ..... 135
7.3.1 Meta-equation of physics within mathematics ..... 137
7.3.2 Metaphysical dynamics ..... 138
7.3.3 Solutions of the self-duality meta-equation ..... 140
7.3.4 The Abelian groups paradigm and Born reciprocity ..... 143
7.4 The Boolean paradigm—de Morgan duality and vacuum energy ..... 146
7.4.1 Extending deMorgan duality along the self-dual axis ..... 148
7.4.2 The birth of geometry and the birth of quantum logic ..... 153
7.5 The quantum groups paradigm ..... 155
7.5.1 Why is there quantum mechanics and why is there gravity? ..... 158
7.5.2 Graphical picture of bicrossproduct quantum groups ..... 161
7.5.3 Quantum spacetime and quantum Born reciprocity ..... 163
7.5.4 3D quantum gravity and the cosmological constant ..... 165
7.5.5 Quantum anomalies and why do things evolve? ..... 169
7.6 The monoidal functor paradigm and beyond ..... 173
7.6.1 3D quantum gravity revisited ..... 176
7.6.2 4D quantum gravity ..... 177

8. On the abuse of gravity theories in cosmology ..... 179
Leszek M. Sokołowski
8.1 Introduction ..... 180
8.2 The ways NLG theories are used in cosmology ..... 183
8.3 Criticism of the typical cosmological approach to the search
for a correct theory ..... 186
8.3.1 Wealth of diverse solutions ..... 186
8.3.2 Issue of approximations ..... 187
8.3.3 Fundamental nonuniqueness ..... 189
8.4 What then to do? ..... 191
8.4.1 Stability of the ground state ..... 195
8.4.2 Existence of the Newtonian limit ..... 196
8.5 Conclusions ..... 198
8.6 Further reading ..... 198

9. Penrose’s Weyl curvaturehypothesis and conformally-cyclic cosmology ..... 201
Paul Tod
9.1 Introduction ..... 201
9.2 The Weyl curvature hypothesis ..... 202
9.3 Conformally-cyclic cosmology ..... 206

10. Can empirical facts become mathematical truths? ..... 213
Krzysztof Wójtowicz
10.1 Introductory remarks ..... 213
10.2 The received view ..... 214
10.3 Empirical elements in mathematics ..... 216
10.4 Computer Assisted Proofs (CAPs) ..... 216
10.5 Intractable problems ..... 218
10.6 Quantum computation ..... 218
10.7 Very quick computers ..... 221
10.8 Hypercomputation ..... 224
10.9 Quasi-empiricism in mathematics? ..... 228
10.10 Conclusions ..... 229

11. Twistors and special functions ..... 231
Nick Woodhouse
11.1 Introduction ..... 231
11.2 The Penrose transform ..... 232
11.3 Generalizations ..... 235
11.4 Ane bundles ..... 237
11.5 Globality ..... 237
11.6 Conformal reductions ..... 238
11.7 Gauss hypergeometric and sixth Painlevé equations ..... 240
11.8 Hypergeometric functions ..... 241
11.9 Schlesinger and Painlevé equations ..... 243
11.10 The Coulomb eld ..... 246
11.11 Global transform ..... 248

Bibliography ..... 255

292 pages, Hardcover

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