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Overview

Now available in paperback! 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 English mathematical physicist and philosopher of science. The topics range from black holes and quantum information to the very nature of mathematical cognition itself.

Product Details

ISBN-13: 9788378862871
Publisher: Copernicus Center Press
Publication date: 08/14/2017
Edition description: Reprint
Pages: 292
Product dimensions: 6.50(w) x 9.50(h) x 0.80(d)

Table of Contents

1 From geometric quantum mechanics to quantum information Paolo Aniello Jesús Clemente-Gallardo Giuseppe Marmo George F. Volkert 1

1.1 Introduction 1

1.2 Geometrical formulation of the Hilbert space picture 3

1.2.1 From Hermitian operators to real-valued function 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 Some applications: 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 Abhay Ashtekar 23

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 Katarzyna Grabowska Jerzy Kijowski 51

3.1 Introduction 51

3.2 Symplectic relations and their generating functions 54

3.3 Langrangian 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 HGeneral relativity and von Neumann algebras Michael Heller Zdzistaw Odrzygózdz Leszek Pysiak Wieslaw Sasin 75

4.1 Introduction 75

4.2 Space-time as a noncommutative space 77

4.3 Algebra of random operators 79

4.4 Differential 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 Stanislaw Krajewski 87

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 Donald Lynden-Bell 105

6.1 Introduction 105

6.2 A Newtonian non-relativistic mechanics without absolute space 107

6.3 Mechian 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 Shahh MAjid 117

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-yeat old's insight into quantum gravity 132

7.2 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 de Morgan 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 Leszek M. Sokolowski 179

8.1 Introduction 180

8.2 The ways NGL 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 curvature hypothesis and conformally-cyclic cosmology Paul Tod 201

9.1 Introduction 201

9.2 The Weyl curvature hypothesis 202

9.3 Conformally-cyclic cosmology 206

10 Can empirical facts become mathematical truths? Krzyszof Wójtowicz 213

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 Nick Woodhouse 231

11.1 Introduction 231

11.2 The Penrose transform 232

11.3 Generalizations 235

11.4 Affine 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 Schesinger and Painlevé equations 243

11.10 The Coulomb field 246

11.11 Global transform 248

Bibliography 255

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