A main theme of the book outlines the role of the quantum potential in quantum mechanics and general relativity and one of its origins via fluctuations formulated in terms of Fisher information. Another theme is the description of various approaches to Bohmian mechanics and their role in quantum mechanics and general relativity. Along the way various approaches to, for instance, the Dirac equation, the Einstein equations, the Klein-Gordon equation, the Maxwell equations and the Schr?dinger equations are described. Statistics and geometry are intertwined in various ways and, among other matters, the aether, cosmology, entropy, fractals, quantum Kaehler geometry, the vacuum and the zero point field are discussed. There is also some speculative material and some original work along with material extracted from over 1000 references and the work is current up to April 2005.
Author(s): R.W. Carroll
Series: Fundamental Theories of Physics 148
Edition: 1
Publisher: Springer
Year: 2006
Language: English
Pages: 444
Contents......Page 6
Preface......Page 9
1. Diffusion and Stochastic Processes......Page 13
2. Scale Relativity......Page 26
3. Remarks on Fractal Spacetime......Page 34
3.1. Comments on Cantor Sets......Page 35
3.2. Comments on Hydrodynamics......Page 39
4. Remarks on Fractal Calculus......Page 42
5. A Bohmian Approach to Quantum Fractals......Page 46
2. DEBROGLIE-BOHM IN VARIOUS CONTEXTS......Page 50
1. The Klein-Gordon and Dirac Equations......Page 52
1.1. Electromagnetism and the Dirac Equation......Page 56
2. Bertoldi-Faraggi-Matone Theory......Page 59
3.1. Emergence of Particles......Page 66
3.2. Bosonic Bohmian Theory......Page 72
3.3. Fermionic Theory......Page 75
4. DeDonder, Weyl, and Bohm......Page 82
5. QFT and Stochastic Jumps......Page 86
6. Bohmian Mechanics in QFT......Page 98
1. Introduction......Page 102
2. Sketch of Debroglie-Bohm-Weyl Theory......Page 105
2.1. Dirac-Weyl Action......Page 112
2.2. Remarks on Conformal Gravity......Page 115
3. The Schrödinger Equation in Weyl Space......Page 122
3.1. Fisher Information Revisited......Page 128
3.2. The KG Equation......Page 130
4. Scale Relativity and KG......Page 137
5. Quantum Measurement and Geometry......Page 143
5.1. Measurement on a Biconformal Space......Page 148
1. Dirac-Weyl Geometry......Page 154
2. Remarks on Cosmology......Page 162
3. WDW Equation......Page 168
3.1. Constraints in Ashtekar Variables......Page 173
4. Remarks on Regularization......Page 175
5. Pilot Wave Cosmology......Page 181
5.1. Euclidean Quantum Gravity......Page 183
6. Bohm and Noncommutative Geometry......Page 188
7. Exact Uncertainty and Gravity......Page 201
1. The Zero Point Field......Page 214
1.1. Remarks on the Aether and Vacuum......Page 221
1.2. A Version of the Dirac Aether......Page 223
1.3. Massless Particles......Page 227
1.4. Einstein Aether Waves......Page 229
2. Stochastic Electrodynamics......Page 232
3. Photons and EM......Page 238
4. Quantum Geometry......Page 242
4.1. Probability Aspects......Page 248
1. The Dynamics of Uncertainty......Page 250
1.1. Information Dynamics......Page 255
1.2. Information Measures for QM......Page 256
1.3. Phase Transitions......Page 257
1.4. Fisher Information and Hamilton's Equations......Page 258
1.5. Uncertainty and Fluctuations......Page 260
2. A Touch of Chaos......Page 263
2.1. Chaos and the Quantum Potential......Page 265
3. Generalized Thermostatistics......Page 267
3.1. Nonextensive Statistical Thermodynamics......Page 270
4. Fisher Physics......Page 273
4.1. Legendre Thermodynamics......Page 279
4.2. First and Second Laws......Page 282
1.1. The Schrödinger Equation......Page 285
1.2. Debroglie-Bohm......Page 289
1.3. Geometry, Gravity, and QM......Page 296
1.4. Geometric Phases......Page 301
2. Hydrodynamics and Geometry......Page 303
2.1. Particle and Wave Pictures......Page 306
2.2. Electromagnetism and the SE......Page 311
3. Some Speculations on the Aether......Page 317
3.1. Discussion of a Putative Psi Aether......Page 319
4. Remarks on Trajectories......Page 325
1. Introduction and Background......Page 328
1.1. Wick's Theorem and Renormalization......Page 330
1.2. Products and Relations to Physics......Page 332
2. Quantum Fields and Quantum Groups......Page 336
3. Renormalization and Algebra......Page 341
3.1. Various Algebras......Page 348
4. Tau Function and Free Fermions......Page 350
4.1. Symmetric Functions......Page 352
4.2. PSDO on a Circle......Page 356
5. Intertwining......Page 358
6. Vertex Operators and Symmetric Functions......Page 367
APPENDIX A: DeDonder-Weyl Theory......Page 374
APPENDIX B: Relativity and Electromagnetism......Page 387
APPENDIX C: Remarks on Quantum Gravity......Page 394
APPENDIX D: Dirac on Weyl Geometry......Page 401
APPENDIX E: Biconformal Geometry......Page 410
Bibliography......Page 416
C......Page 441
D......Page 442
G......Page 443
L......Page 444
P......Page 445
R......Page 446
U......Page 447
Z......Page 448
