The Cellular Automaton Interpretation of Quantum Mechanics.

• Superior document: Fundamental Theories of Physics Series ; v.185
• Author: 't Hooft, Gerard.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2016.; ©2016.
• Year of Publication: 2016
• Edition: 1st ed.
• Language: English
• Series: Fundamental Theories of Physics Series
• Physical Description: 1 online resource (296 pages)

Table of Contents

Intro -- The Cellular Automaton Interpretation of Quantum Mechanics -- Foreword -- Preface -- Acknowledgements -- Contents -- List of Figures -- Part I: The Cellular Automaton Interpretation as a General Doctrine -- Chapter 1: Motivation for This Work -- 1.1 Why an Interpretation Is Needed -- 1.2 Outline of the Ideas Exposed in Part I -- 1.3 A 19th Century Philosophy -- 1.4 Brief History of the Cellular Automaton -- 1.5 Modern Thoughts About Quantum Mechanics -- 1.6 Notation -- Chapter 2: Deterministic Models in Quantum Notation -- 2.1 The Basic Structure of Deterministic Models -- 2.1.1 Operators: Beables, Changeables and Superimposables -- 2.2 The Cogwheel Model -- 2.2.1 Generalizations of the Cogwheel Model: Cogwheels with N Teeth -- 2.2.2 The Most General Deterministic, Time Reversible, Finite Model -- Chapter 3: Interpreting Quantum Mechanics -- 3.1 The Copenhagen Doctrine -- 3.2 The Einsteinian View -- 3.3 Notions Not Admitted in the CAI -- 3.4 The Collapsing Wave Function and Schrödinger's Cat -- 3.5 Decoherence and Born's Probability Axiom -- 3.6 Bell's Theorem, Bell's Inequalities and the CHSH Inequality -- 3.7 The Mouse Dropping Function -- 3.7.1 Ontology Conservation and Hidden Information -- 3.8 Free Will and Time Inversion -- Chapter 4: Deterministic Quantum Mechanics -- 4.1 Introduction -- 4.2 The Classical Limit Revisited -- 4.3 Born's Probability Rule -- 4.3.1 The Use of Templates -- 4.3.2 Probabilities -- Chapter 5: Concise Description of the CA Interpretation -- 5.1 Time Reversible Cellular Automata -- 5.2 The CAT and the CAI -- 5.3 Motivation -- 5.3.1 The Wave Function of the Universe -- 5.4 The Rules -- 5.5 Features of the Cellular Automaton Interpretation (CAI) -- 5.5.1 Beables, Changeables and Superimposables -- 5.5.2 Observers and the Observed -- 5.5.3 Inner Products of Template States -- 5.5.4 Density Matrices.

5.6 The Hamiltonian -- 5.6.1 Locality -- 5.6.2 The Double Role of the Hamiltonian -- 5.6.3 The Energy Basis -- 5.7 Miscellaneous -- 5.7.1 The Earth-Mars Interchange Operator -- 5.7.2 Rejecting Local Counterfactual De niteness and Free Will -- 5.7.3 Entanglement and Superdeterminism -- 5.7.4 The Superposition Principle in Quantum Mechanics -- 5.7.5 The Vacuum State -- 5.7.6 A Remark About Scales -- 5.7.7 Exponential Decay -- 5.7.8 A Single Photon Passing Through a Sequence of Polarizers -- 5.7.9 The Double Slit Experiment -- 5.8 The Quantum Computer -- Chapter 6: Quantum Gravity -- Chapter 7: Information Loss -- 7.1 Cogwheels with Information Loss -- 7.2 Time Reversibility of Theories with Information Loss -- 7.3 The Arrow of Time -- 7.4 Information Loss and Thermodynamics -- Chapter 8: More Problems -- 8.1 What Will Be the CA for the SM? -- 8.2 The Hierarchy Problem -- Chapter 9: Alleys to Be Further Investigated and Open Questions -- 9.1 Positivity of the Hamiltonian -- 9.2 Second Quantization in a Deterministic Theory -- 9.3 Information Loss and Time Inversion -- 9.4 Holography and Hawking Radiation -- Chapter 10: Conclusions -- 10.1 The CAI -- 10.2 Counterfactual De niteness -- 10.3 Superdeterminism and Conspiracy -- 10.3.1 The Role of Entanglement -- 10.3.2 Choosing a Basis -- 10.3.3 Correlations and Hidden Information -- 10.4 The Importance of Second Quantization -- Part II: Calculation Techniques -- Chapter 11: Introduction to Part II -- 11.1 Outline of Part II -- 11.2 Notation -- 11.3 More on Dirac's Notation for Quantum Mechanics -- Chapter 12: More on Cogwheels -- 12.1 The Group SU(2), and the Harmonic Rotator -- 12.2 In nite, Discrete Cogwheels -- 12.3 Automata that Are Continuous in Time -- Chapter 13: The Continuum Limit of Cogwheels, Harmonic Rotators and Oscillators -- 13.1 The Operator phiop in the Harmonic Rotator.

13.2 The Harmonic Rotator in the x Frame -- Chapter 14: Locality -- Chapter 15: Fermions -- 15.1 The Jordan-Wigner Transformation -- 15.2 `Neutrinos' in Three Space Dimensions -- 15.2.1 Algebra of the Beable `Neutrino' Operators -- 15.2.2 Orthonormality and Transformations of the `Neutrino' Beable States -- 15.2.3 Second Quantization of the `Neutrinos' -- 15.3 The `Neutrino' Vacuum Correlations -- Chapter 16: PQ Theory -- 16.1 The Algebra of Finite Displacements -- 16.1.1 From the One-Dimensional In nite Line to the Two-Dimensional Torus -- 16.1.2 The States |Q,P&gt -- in the q Basis -- 16.2 Transformations in the PQ Theory -- 16.3 Resume of the Quasi-periodic Phase Function phi(xi,kappa) -- 16.4 The Wave Function of the State |0,0&gt -- -- Chapter 17: Models in Two Space-Time Dimensions Without Interactions -- 17.1 Two Dimensional Model of Massless Bosons -- 17.1.1 Second-Quantized Massless Bosons in Two Dimensions -- 17.1.2 The Cellular Automaton with Integers in 2 Dimensions -- 17.1.3 The Mapping Between the Boson Theory and the Automaton -- 17.1.4 An Alternative Ontological Basis: The Compacti ed Model -- 17.1.5 The Quantum Ground State -- 17.2 Bosonic Theories in Higher Dimensions? -- 17.2.1 Instability -- 17.2.2 Abstract Formalism for the Multidimensional Harmonic Oscillator -- 17.3 (Super)strings -- 17.3.1 String Basics -- 17.3.2 Strings on a Lattice -- 17.3.3 The Lowest String Excitations -- 17.3.4 The Superstring -- 17.3.5 Deterministic Strings and the Longitudinal Modes -- 17.3.6 Some Brief Remarks on (Super)string Interactions -- Chapter 18: Symmetries -- 18.1 Classical and Quantum Symmetries -- 18.2 Continuous Transformations on a Lattice -- 18.2.1 Continuous Translations -- 18.2.2 Continuous Rotations 1: Covering the Brillouin Zone with Circular Regions -- 18.2.3 Continuous Rotations 2: Using Noether Charges and a Discrete Subgroup.

18.2.4 Continuous Rotations 3: Using the Real Number Operators p and q Constructed Out of P and Q -- 18.2.5 Quantum Symmetries and Classical Evolution -- 18.2.6 Quantum Symmetries and Classical Evolution 2 -- 18.3 Large Symmetry Groups in the CAI -- Chapter 19: The Discretized Hamiltonian Formalism in PQ Theory -- 19.1 The Vacuum State, and the Double Role of the Hamiltonian (Cont'd) -- 19.2 The Hamilton Problem for Discrete Deterministic Systems -- 19.3 Conserved Classical Energy in PQ Theory -- 19.3.1 Multi-dimensional Harmonic Oscillator -- 19.4 More General, Integer-Valued Hamiltonian Models with Interactions -- 19.4.1 One-Dimensional System: A Single Q, P Pair -- 19.4.2 The Multi-dimensional Case -- 19.4.3 The Lagrangian -- 19.4.4 Discrete Field Theories -- 19.4.5 From the Integer Valued to the Quantum Hamiltonian -- Chapter 20: Quantum Field Theory -- 20.1 General Continuum Theories-The Bosonic Case -- 20.2 Fermionic Field Theories -- 20.3 Standard Second Quantization -- 20.4 Perturbation Theory -- 20.4.1 Non-convergence of the Coupling Constant Expansion -- 20.5 The Algebraic Structure of the General, Renormalizable, Relativistic Quantum Field Theory -- 20.6 Vacuum Fluctuations, Correlations and Commutators -- 20.7 Commutators and Signals -- 20.8 The Renormalization Group -- Chapter 21: The Cellular Automaton -- 21.1 Local Time Reversibility by Switching from Even to Odd Sites and Back -- 21.1.1 The Time Reversible Cellular Automaton -- 21.1.2 The Discrete Classical Hamiltonian Model -- 21.2 The Baker Campbell Hausdorff Expansion -- 21.3 Conjugacy Classes -- Chapter 22: The Problem of Quantum Locality -- 22.1 Second Quantization in Cellular Automata -- 22.2 More About Edge States -- 22.3 Invisible Hidden Variables -- 22.4 How Essential Is the Role of Gravity? -- Chapter 23: Conclusions of Part II.

Appendix A: Some Remarks on Gravity in 2+1 Dimensions -- A.1 Discreteness of Time -- Appendix B: A Summary of Our Views on Conformal Gravity -- Appendix C: Abbreviations -- References.