The Geometric Universe Science, Geometry, and the Work of Roger Penrose

, by Huggett, S. A.; Mason, L. J.; Tod, K. P.; Tsou, S. T.; Woodhouse, N. M. J.

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ISBN: 9780198500599 | 0198500599
Cover: Hardcover
Copyright: 7/16/1998

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This collection has been inspired by the work of Roger Penrose. It gives an overview of current work on the interaction between geometry and physics, from which many important developments in research have emerged. This volume collects together the contributions of many important researchers, including Sir Roger himself, and gives an overview of the many applications of geometrical ideas and techniques across mathematics and the physical sciences. From the area of pure mathematics papers are included on the topics of classical differential geometry and non-commutative geometry, knot invariants, and the applications of gauge theory. Contributions from applied mathematics cover the topics of integrable systems and general relativity. Current research in experimental and theoretical physics inspired chapters on string theory, quantum gravity, the foundations of quantum mechanics, quasi-crystals and astrophysics. The collection also includes articles on quantum computation, quantum cryptography and the possible role of micro-tubules in a theory of consciousness.

List of contributors

xvii

I PLENARY LECTURES

(254)

1 Roger Penrose--A Personal Appreciation

(6)

Michael Atiyah

1 Personal and historical remarks

(1)

2 Twistors

(1)

3 Integrable systems and solitons

(1)

4 Rival philosophies

(1)

5 Other topics

(1)

6 Conclusion

(1)

Bibliography

(2)

2 Hypercomplex Manifolds and the Space of Framings

(22)

Nigel Hitchin

1 Introduction

(1)

2 Framings

(3)

3 SU(2)-invariance

(1)

4 Hypercomplex manifolds

(6)

5 Twistor spaces and isomonodromic deformations

(4)

6 Holonomy and hypergeometric functions

(5)

Bibliography

(2)

3 Gauge Theory in Higher Dimensions

(18)

S.K. Donaldson

R.P. Thomas

1 Introduction

(1)

2 The familiar theory

(2)

3 The complex analogy

(4)

4 Exceptional holonomy

(1)

5 The two-dimensional picture

(2)

6 Adiabatic limits and dimension reduction

(2)

7 An example: quadrics in P(5)

(1)

8 Vanishing cycles and pseudoholomorphic curves

(2)

9 Submanifolds

(2)

Bibliography

(2)

4 Noncommutative Differential Geometry and the Structure of Space-Time

(32)

Alain Connes

Foreword

(1)

1 Generalities

(5)

2 Infinitesimal calculus

(10)

3 The local index formula and the transverse fundamental class

(5)

4 The notion of manifold and the axioms of geometry

(6)

5 The spectral geometry of space-time

(3)

Bibliography

(3)

5 Einstein's Equation and Conformal Structure

(18)

Helmut Friedrich

1 Introduction

(1)

2 Asymptotic simplicity and conformal Einstein equations

(2)

3 De Sitter-type space-times

(1)

4 Anti-de Sitter-type space-times

(2)

5 Minkowski-type space-times

(8)

5.1 Conformal Minkowski space

(1)

5.2 Some existence results

(1)

5.3 Difficulties

(1)

5.4 Assumptions on the data

(1)

5.5 Gauge conditions and conformal field equations

(1)

5.6 The finite regular initial value problem near space-like infinity

(1)

5.7 The total characteristic at space-like infinity

(1)

5.8 Comments on our procedure

(1)

6 Concluding remarks

(3)

Bibliography

(2)

6 Twistors, Geometry, and Integrable Systems

(10)

R.S. Ward

1 Introduction

(1)

2 Twistors for 3-dimensional space-time

(2)

3 An integrable Yang-Mills-Higgs system

101

(2)

4 SU(2) bundles over XXX

103

(1)

5 Soliton solutions

104

(2)

6 Concluding remarks

106

(1)

Bibliography

107

(2)

7 On Four-Dimensional Einstein Manifolds

109

(14)

Claude LeBrun

1 Introduction

109

(1)

2 The curvature of 4-manifolds

110

(1)

3 The Hitchin-Thorpe inequality

111

(2)

4 Recent results

113

(1)

5 Seiberg-Witten techniques

114

(3)

6 Entropy inequalities

117

(3)

7 Concluding remarks

120

(1)

Bibliography

120

(3)

8 Loss of Information in Black Holes

123

(12)

Stephen Hawking

1 Personal and historical remarks

123

(1)

2 Information loss

124

(11)

9 Funda-mental Geometry: the Penrose-Hameroff `Orch OR' Model of Consciousness

135

(26)

Stuart Hameroff

1 Introduction: on the trail of an enigma

135

(1)

2 Philosophy: a panexperiential `funda-mentality'

135

(3)

3 Physics: objective reduction (OR)

138

(2)

4 Biology: quantum coherence in microtubules?

140

(10)

4.1 Microtubules

140

(1)

4.2 Frohlich's biological coherence

141

(1)

4.3 Quantum isolation--avoiding environmental interaction and decoherence

142

(2)

4.4 Macroscopic quantum coherence and gap junctions

144

(2)

4.5 Evolution, Orch OR and the Cambrian explosion

146

(4)

5 Summary of the `Orch OR' model of consciousness

150

(2)

6 Assumptions and testable predictions of Orch OR

152

(2)

7 Conclusion: Penrose's Platonic world

154

(1)

Bibliography

155

(6)

10 Implications of Transience for Spacetime Structure

161

(12)

Abner Shimony

Bibliography

170

(3)

11 Geometric Issues in Quantum Gravity

173

(22)

Abhay Ashtekar

1 Introduction

173

(4)

1.1 Setting the stage

173

(2)

1.2 Quantum geometry

175

(2)

2 Quantum states

177

(5)

2.1 Phase space

177

(1)

2.2 Quantum configuration space

178

(2)

2.3 Kinematical Hilbert space

180

(2)

3 Quantum geometry

182

(7)

3.1 Preliminaries

182

(1)

3.2 Triad operators

183

(2)

3.3 Area operators

185

(2)

3.4 Properties of area operators

187

(2)

4 Discussion

189

(3)

Acknowledgements

192

(1)

Bibliography

192

(3)

12 From Quantum Code-making to Quantum Code-breaking

195

(20)

Artur Ekert

1 What is wrong with classical cryptography?

195

(2)

2 Is the Bell theorem of any practical use?

197

(2)

3 Quantum key distribution

199

(2)

4 Quantum eavesdropping

201

(2)

5 Public key cryptosystems

203

(2)

6 Fast and slow algorithms

205

(1)

7 Quantum computers

206

(3)

8 Quantum code-breaking

209

(2)

9 Concluding remarks

211

(1)

Bibliography

212

(3)

13 Penrose Tilings and Quasicrystals Revisited

215

(12)

Paul J. Steinhardt

1 Introduction

215

(3)

2 New approach to Penrose tiling: single tile/matching rule

218

(2)

3 New approach to Penrose tiling: maximizing cluster density

220

(3)

4 Implications

223

(2)

Bibliography

225

(2)

14 Decaying Neutrinos and the Geometry of the Universe

227

(8)

D.W. Sciama

1 Introduction

227

(1)

2 Relic neutrinos as dark matter

228

(2)

3 Decaying neutrinos and the ionisation of the universe

230

(2)

4 A new observational test of the decaying neutrino theory

232

(1)

Bibliography

233

(2)

15 Quantum Geometric Origin of All Forces in String Theory

235

(10)

Gabriele Veneziano

1 Introduction

235

(1)

2 Forces and local symmetries

235

(1)

3 Field-theoretic Kaluza-Klein

236

(2)

4 Quantum string magic

238

(2)

5 String-theoretic Kaluza-Klein

240

(1)

6 S-duality and the big fix

241

(2)

Bibliography

243

(2)

16 Space from the Point of View of Loop Groups

245

(12)

Graeme Segal

1 Loop groups and quantum field theory

246

(3)

2 Loop groups and low-dimensional topology

249

(1)

3 String theory

250

(2)

Bibliography

252

(5)

II PARALLEL SESSION I: QUANTUM THEORY AND BEYOND

257

(50)

17 The Twistor Diagram Programme

257

(8)

Andrew P. Hodges

Bibliography

262

(3)

18 Geometric Models for Quantum Statistical Inference

265

(12)

Dorje C. Brody

Lane P. Hughston

1 Introduction

265

(1)

2 Information geometry

266

(3)

3 Classical estimation

269

(1)

4 Quantum geometry

270

(1)

5 Quantum statistical estimation

271

(1)

6 Higher order variance bounds

272

(2)

7 Quantum geometry vs information geometry

274

(1)

Bibliography

274

(3)

19 Spin Networks and Topology

277

(14)

Louis H. Kauffman

1 Introduction

277

(1)

2 Networks and discrete space

277

(1)

3 The bracket state summation and the Jones polynomial

278

(4)

3.1 The Reidemeister moves

279

(3)

4 Spin networks

282

(5)

Bibliography

287

(4)

20 The Physics of Spin Networks

291

(16)

Lee Smolin

1 Introduction

291

(1)

2 Spin networks in non-perturbative quantum gravity

292

(7)

3 Future directions

299

(2)

Bibliography

301

(6)

III PARALLEL SESSION II: GEOMETRY AND GRAVITY

307

(42)

21 The Sen Conjecture for Distinct Fundamental Monopoles

307

(10)

Gary Gibbons

1 Introduction

307

(1)

2 S-duality

308

(1)

3 Weak XXX strong coupling

308

(1)

4 Bound states at threshold

309

(1)

5 The moduli space

309

(3)

6 Harmonic forms

312

(1)

7 Uniqueness

313

(1)

8 Geodesics

314

(1)

9 Bound states in the continuum

315

(1)

Bibliography

315

(2)

22 An Unorthodox View of GR via Characteristic Surfaces

317

(8)

Simonetta Frittelli

E.T. Newman

Carlos Kozameh

1 Introduction

317

(2)

2 The null surface formulation of GR

319

(3)

2.1 Kinematics

319

(1)

2.2 Imposing the Einstein equations

319

(1)

2.3 Self-dual Einstein equations

320

(1)

2.4 Asymptotically flat GR

321

(1)

3 Discussion and applications

322

(1)

Bibliography

323

(2)

23 Amalgamated Codazzi-Raychaudhuri Identity for Foliation

325

(12)

Brandon Carter

1 Introduction

325

(3)

2 The deformation tensor

328

(2)

3 The adapted foliation connection

330

(1)

4 The amalgamated foliation curvature tensor

331

(3)

Bibliography

334

(3)

24 Abstract/Virtual/Reality/Complexity

337

(12)

George Sparling

1 Introduction

337

(1)

2 Complexity

338

(2)

3 Reality

340

(2)

4 Abstract

342

(1)

5 Virtual

343

(2)

Bibliography

345

(4)

IV PARALLEL SESSION III: FUNDAMENTAL QUESTIONS IN QUANTUM MECHANICS

349

(34)

25 Interaction-Free Measurements

349

(8)

Lev Vaidman

1 The Penrose bomb testing problem

349

(2)

2 The Elitzur-Vaidman bomb testing problem

351

(1)

3 Experimental realization of the IFM

352

(1)

4 Generalized IFM

353

(1)

5 Applications of the IFM

354

(1)

6 The IFM as counterfactuals

354

(1)

Bibliography

355

(2)

26 Quantum Measurement Problem and the Gravitational Field

357

(12)

Jeeva Anandan

1 Introduction

357

(1)

2 Quantum measurement problem

358

(2)

3 Efforts to resolve the measurement problem

360

(3)

4 Gravitational reduction of the wave packet

363

(4)

Bibliography

367

(2)

27 Entanglement and Quantum Computation

369

(14)

Richard Jozsa

1 Introduction

369

(1)

2 Quantum computation and complexity

370

(3)

3 Superposition and entanglement in quantum computation

373

(3)

4 Entanglement and the super-fast quantum Fourier transform

376

(1)

5 Concluding remarks

377

(1)

Bibliography

378

(5)

V PARALLEL SESSION IV: MATHEMATICAL ASPECTS OF TWISTOR THEORY

383

(40)

28 Penrose Transform for Flag Domains

383

(12)

Simon Gindikin

1 The Penrose transform in formulas

383

(3)

1.1 Twistor geometry

383

(1)

1.2 Penrose transform

384

(1)

1.3 Inverse Penrose transform

385

(1)

1.4 Holomorphic cohomology

385

(1)

1.5 Boundary integral formula

386

(1)

2 Generalized Penrose transform (geometrical problems)

386

(4)

2.1 Holomorphic cohomology

386

(1)

2.2 Flag domains

387

(1)

2.3 Dual manifolds

388

(1)

2.4 Basic conjectures

389

(1)

3 Generalized Penrose transform (analytic problems)

390

(2)

Bibliography

392

(3)

29 Twistor Solution of the Holonomy Problem

395

(8)

S.A. Merkulov

L.J. Schwachhofer

1 Introduction

395

(1)

2 Main result

395

(2)

3 Twistor theory of holonomy groups

397

(4)

Bibliography

401

(2)

30 The Penrose Transform and Real Integral Geometry

403

(8)

Toby N. Bailey

1 Introduction

403

(1)

2 The twistor programme

403

(1)

3 The holomorphic Penrose transform

404

(1)

4 The connection with integral geometry

405

(1)

5 A new method in real integral geometry

406

(3)

5.1 Pull-back from RP(3) to F

406

(1)

5.2 Push-down from F to Gr(2)(R(4))

407

(1)

5.3 Results

408

(1)

Bibliography

409

(2)

31 Pythagorean Spinors and Penrose Twistors

411

(12)

Andrzej Trautman

1 Introduction

411

(1)

2 Pythagorean spinors

411

(2)

3 Projective quadrics and twistors

413

(5)

3.1 The complex case

414

(3)

3.2 The real case

417

(1)

Bibliography

418

(5)

VI AFTERWORD

423

32 Afterword

423

Roger Penrose

1 Geometry, and the roots and aims of twistor theory

423

(1)

2 Towards a twistor description of Einsteinian physics

424

(5)

3 Further issues of physics and biology

429

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The Geometric Universe Science, Geometry, and the Work of Roger Penrose

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