Quantum Leap: From Dirac and Feynman, Across the Universe, to Human Body and Mind

Contents......Page 9
Preface......Page 7
Acknowledgments......Page 8
1.1 Soft Introduction to Quantum Mechanics......Page 17
1.2.1 Quantum Hilbert Space......Page 23
1.2.2 Formal Hilbert Space......Page 24
1.3 Human Intelligence, Mind and Reason......Page 27
1.3.0.1 Human Reason......Page 29
2.1 Basics of Non–Relativistic Quantum Mechanics......Page 35
2.1.1 Canonical Quantization......Page 39
2.1.2 Quantum States and Operators......Page 40
2.1.3 Quantum Pictures......Page 46
2.1.4 Spectrum of a Quantum Operator......Page 48
2.1.5 General Representation Model......Page 51
2.1.6 Direct Product Space......Page 52
2.1.7 State–Space for n Quantum Particles......Page 53
2.2.1 Difficulties of the Relativistic Quantum Mechanics......Page 56
2.2.2 Particles of Half–Odd Integral Spin......Page 59
2.2.3 Particles of Integral Spin......Page 61
2.2.4 Dirac’s Electrodynamics Action Principle......Page 70
2.2.5 Dirac Equation and Formal QED in Brief......Page 76
2.2.6 Lorentzian Space-Time and Gravity......Page 78
2.2.7 Unification of Fundamental Interactions......Page 79
2.2.7.1 First Unification......Page 80
3.1 Path Integrals: Sums Over Histories......Page 83
3.1.1.1 Classical Probability Concept......Page 84
3.1.1.2 Quantum Probability Concept......Page 89
3.1.1.3 Quantum Coherent States......Page 91
3.1.1.4 Dirac’s hbra | keti Transition Amplitude......Page 92
3.1.1.5 Feynman’s Sum–over–Histories......Page 93
3.1.1.6 The Basic Form of a Path Integral......Page 95
3.1.1.7 Adaptive Path Integral......Page 96
3.1.2.1 Extract from Feynman’s Nobel Lecture......Page 97
3.1.2.2 Lagrangian Path Integral......Page 101
3.1.2.3 Hamiltonian Path Integral......Page 102
3.1.2.4 Feynman–Kac Formula......Page 103
3.1.3.1 Canonical versus Path–Integral Quantization......Page 105
3.1.3.3 Sources......Page 110
3.1.3.4 Fields......Page 111
3.1.3.5 Gauges......Page 112
3.1.3.6 Riemannian–Symplectic Geometries......Page 113
3.1.4 Statistical Mechanics via Path Integrals......Page 114
3.1.5 Path–Integral Monte–Carlo Simulation......Page 117
3.1.6 Sum over Geometries and Topologies......Page 124
3.1.6.1 Simplicial Quantum Geometry......Page 126
3.1.6.2 Discrete Gravitational Path Integrals......Page 128
3.1.6.3 Regge Calculus......Page 130
3.1.6.4 Lorentzian Path Integral......Page 133
3.2.1 Path Integrals and Green’s Functions......Page 138
3.2.2 Topological Quantum Field Theory......Page 144
3.2.3 TQFT and Seiberg–Witten Theory......Page 149
3.2.3.1 SW Invariants and Monopole Equations......Page 150
3.2.3.2 Topological Lagrangian......Page 151
3.2.3.3 Quantum Field Theory......Page 154
3.2.3.4 Dimensional Reduction and 3D Field Theory......Page 158
3.2.3.5 Geometrical Interpretation......Page 161
3.2.4 TQFTs Associated with SW–Monopoles......Page 164
3.2.4.1 Dimensional Reduction......Page 168
3.2.4.2 TQFTs of 3D Monopoles......Page 170
3.2.4.3 Non–Abelian Case......Page 181
3.3.1.1 Strings......Page 186
3.3.1.2 Interactions......Page 187
3.3.1.3 Loop Topology of Closed Surfaces......Page 188
3.3.2 Transition Amplitudes for Strings......Page 190
3.3.3 Weyl Invariance and Vertex Operator Formulation......Page 193
3.3.5 Transition Amplitude for a Single Point Particle......Page 194
3.3.6 Witten’s Open String Field Theory......Page 195
3.3.6.1 Operator Formulation of String Field Theory......Page 196
3.3.6.2 Open Strings......Page 198
3.3.6.3 Construction of Overlap Vertices......Page 201
3.3.6.4 Transformation of String Fields......Page 210
3.3.7 Topological Strings......Page 215
3.3.8 Geometrical Transitions......Page 232
3.3.9 Topological Strings and Black Hole Attractors......Page 236
3.4.1 Stochastic Optimal Control......Page 243
3.4.1.1 Path–Integral Formalism......Page 244
3.4.1.2 Monte Carlo Sampling......Page 246
3.4.2 Nonlinear Dynamics of Option Pricing......Page 248
3.4.2.1 Theory and Simulations of Option Pricing......Page 249
3.4.2.2 Option Pricing via Path Integrals......Page 252
3.4.2.3 Continuum Limit and American Options......Page 258
3.4.3.1 Continuum Limit of the Kuramoto Net......Page 260
3.4.3.2 Path–Integral for Complex Networks......Page 261
3.4.4 Cerebellum as a Neural Path–Integral......Page 262
3.4.4.1 Spinal Autogenetic Reflex Control......Page 264
3.4.4.2 Cerebellum – the Comparator......Page 266
3.4.4.3 Hamiltonian Action and Neural Path Integral......Page 268
3.4.5.1 Phase Transitions in Hamiltonian Systems......Page 269
3.4.5.2 Geometry of the Largest Lyapunov Exponent......Page 272
3.4.5.3 Euler Characteristics of Hamiltonian Systems......Page 276
4.1.1 What Is Quantum Gravity?......Page 281
4.1.2.1 String Theory......Page 282
4.1.2.2 Loop Quantum Gravity......Page 285
4.1.3.1 Discrete Approaches......Page 289
4.1.3.2 Hawking’s Euclidean Quantum Gravity......Page 291
4.1.3.4 QFT in Curved Space–Time......Page 292
4.1.3.5 New Approaches to Quantum Gravity......Page 293
4.1.3.6 Hawking’s Black Hole Entropy......Page 299
4.2.1 Introduction to Loop Quantum Gravity......Page 300
4.2.2 Formalism of Loop Quantum Gravity......Page 307
4.2.3 Loop Algebra......Page 308
4.2.4 Loop Quantum Gravity......Page 309
4.2.5 Loop States and Spin Network States......Page 311
4.2.6 Diagrammatic Representation of the States......Page 314
4.2.7 Quantum Operators......Page 315
4.2.8 Loop v.s. Connection Representation......Page 316
4.3 Quantum Cosmology......Page 317
4.3.1 Hawking’s Cosmology in ‘Plain English’......Page 318
4.3.2 Theories of Everything, Anthropic Principle and Wave Function of the Universe......Page 324
4.3.2.1 Quantum Mechanics......Page 327
4.3.2.2 Scientific Reduction......Page 328
4.3.2.3 Anthropic Reasoning in Quantum Cosmology......Page 329
4.3.2.4 Quantum State of the Universe......Page 336
4.3.2.5 From Quantum Mechanics to Quantum Gravity......Page 341
4.3.3 Quantum Gravity and Black Holes......Page 346
4.3.4 Generalized Quantum Mechanics......Page 364
4.3.4.1 Quantum Mechanics Today......Page 367
4.3.4.2 Spacetime and Quantum Theory......Page 368
4.3.4.3 The Quantum Mechanics of Closed Systems......Page 369
4.3.4.4 Quantum Theory in 3+1 Form......Page 372
4.3.4.5 Generalized Quantum Theory......Page 374
4.3.4.6 A Quantum Theory of Spacetime Geometry......Page 375
4.3.4.7 Beyond Spacetime......Page 378
4.3.4.8 Emergence/Excess Baggage......Page 379
4.3.4.9 Emergence of Signature......Page 380
4.3.4.10 Beyond Quantum Theory......Page 382
4.3.5 Anthropic String Landscape......Page 383
4.3.5.1 The Landscape......Page 389
4.3.5.2 The Trouble with de Sitter Space......Page 394
4.3.5.3 De Sitter Space is Unstable......Page 397
4.3.5.4 Bubble Cosmology......Page 398
4.3.5.5 Cosmology as a Resonance......Page 400
4.3.6 Top–Down Cosmology......Page 404
4.3.6.1 Trace Anomaly Driven Inflation......Page 407
4.3.6.2 Instability of Anomaly–Induced Inflation......Page 417
4.3.6.3 Origin of Inflation......Page 421
4.3.7 Cosmology in the String Landscape......Page 425
4.3.7.1 Quantum State......Page 427
4.3.7.2 Prediction in Quantum Cosmology......Page 429
4.3.7.3 Anthropic Reasoning......Page 433
4.3.7.4 Models of Inflation......Page 434
4.3.7.5 Prediction in a Potential Landscape......Page 437
4.3.7.6 Alternative Proposals......Page 438
4.3.7.7 Interpretations......Page 439
4.3.8 Brane Cosmology......Page 441
4.3.8.1 The Randall–Sundrum Brane World......Page 444
4.3.8.2 Including a Bulk Scalar Field......Page 457
4.3.8.3 Moving Branes in a Static Bulk......Page 467
4.3.8.4 Cosmology of a Two–Brane System......Page 472
4.3.8.5 Brane Collision......Page 478
4.3.8.6 Open Questions......Page 484
4.3.9 Hawking’s Brane New World......Page 485
4.3.9.1 RS Scenario from AdS/CFT......Page 489
4.3.9.2 CFT on the Domain Wall......Page 494
4.3.9.3 The Total Action......Page 500
4.3.9.4 Comparison with 4D Gravity......Page 503
4.3.9.5 Summary on Hawking’s Brane New World......Page 510
4.3.10 Brane–World Quantum Cosmology......Page 511
5.1 Basics of Electrical Muscular Stimulation......Page 515
5.2 EMS Functor......Page 519
5.2.1 Global macro–level of EMStotal......Page 520
5.2.2 Local Micro–Level of EMStotal......Page 522
5.2.3 Micro–Level Adaptation and Muscular Training......Page 523
5.3.1 External Smooth Maxwell Electrodynamics......Page 524
5.3.2 Internal Cellular Bio–Quantum Electrodynamics......Page 526
5.4.1 External Anatomical Muscular Mechanics......Page 527
5.4.2 Internal Myofibrillar Bio–Quantum Mechanics......Page 528
5.5.1.1 Facial Curvatures and Their Deviations......Page 530
5.5.1.2 Facial Riemannian Patch–Manifold......Page 534
5.5.2 Cellular Muscle–Fat Geometry......Page 536
6.1 Nash’s Classical Game Theory......Page 539
6.1.1 Basics of Classical Game Theory......Page 540
6.1.2.1 Background to Nash Equilibrium......Page 545
6.1.2.2 The Rationalistic Interpretation......Page 551
6.1.2.3 The Mass–Action Interpretation......Page 552
6.2 Quantum Games......Page 558
6.2.1 Quantum Strategies......Page 560
6.2.2 Quantum Games......Page 564
6.2.3 Two–Qubit Quantum Games......Page 566
6.2.3.1 Prisoners’ Dilemma......Page 569
6.2.3.2 Chicken Game......Page 577
6.2.4 Quantum Cryptography and Gambling......Page 579
6.3 Formal Quantum Games......Page 587
6.3.1 Formal Framework......Page 588
6.3.2 Various Types of Formal Games......Page 589
6.3.2.2 Quantum Simultaneous Games......Page 590
6.3.2.3 Quantum Sequential Games......Page 592
6.3.3 Equivalence of Formal Games......Page 593
6.3.5 Summary on Formal Quantum Games......Page 598
6.4.1 Entanglement, Teleportation and Information......Page 601
6.4.2 Quantum Computing......Page 604
6.4.2.2 Quantum Computers......Page 607
6.4.2.3 The Circuit Model for Quantum Computers......Page 608
6.4.2.4 Elementary Quantum Algorithms......Page 610
6.5 The Hardware for Quantum Computers......Page 611
6.5.1.1 Josephson Effect......Page 615
6.5.1.2 Pendulum Analog......Page 616
6.5.2 Dissipative Josephson Junction......Page 618
6.5.2.1 Junction Hamiltonian and its Eigenstates......Page 620
6.5.2.2 Transition Rate......Page 622
6.5.3 Josephson Junction Ladder (JJL)......Page 624
6.5.3.1 Underdamped JJL......Page 630
6.5.4 Synchronization in Arrays of Josephson Junctions......Page 635
6.5.4.1 Phase Model for Underdamped JJL......Page 638
6.5.4.2 Comparison of LKM2 and RCSJ Models......Page 644
6.5.4.3 ‘Small–World’ Connections in JJL Arrays......Page 646
7.1.1 Human Mind......Page 651
7.1.1.1 The Mind–Body Problem......Page 656
7.1.2.1 EPR Paradox......Page 671
7.1.2.2 Bell’s Theorem......Page 678
7.1.3 Orchestrated Reduction and Penrose Paradox......Page 683
7.1.3.1 Orchestrated Objective Reduction......Page 685
7.1.3.2 The Orch OR Model......Page 687
7.1.4 Physical Science and Consciousness......Page 689
7.1.4.1 Reducing the Consciousness to Its Elements......Page 690
7.1.4.2 Manipulation of Consciousness......Page 698
7.1.4.3 A Need for Paradigm–Shift......Page 701
7.1.4.4 Quantum Synapses......Page 702
7.1.5.1 Biochemistry of Microtubules......Page 703
7.1.5.2 Kink Soliton Model of MT–Dynamics......Page 705
7.1.5.3 Liouville–Hamiltonian Neurodynamics......Page 708
7.1.5.4 Dissipative Quantum Brain Model......Page 714
7.1.5.5 QED Brain......Page 722
7.1.6 Theory of Mind and Matter......Page 730
7.1.6.1 Matter......Page 731
7.1.6.2 Mind......Page 737
7.1.6.3 Unification......Page 742
7.1.7 Quantum Consciousness......Page 745
7.1.7.1 The Experiment......Page 749
7.1.7.2 Bio–Active Peptides......Page 752
7.1.7.3 Drugs......Page 753
7.1.8 Quantum–Like Psychodynamics......Page 754
7.1.8.1 Life Space Foam......Page 755
7.1.9 Joint Action Psychodynamics......Page 771
7.1.10 Feynman–Finsler Approach to Adaptation......Page 775
Bibliography......Page 779
Index......Page 843