Part I The Cellular Automaton Interpretation as a General Doctrine
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 Modem Thoughts About Quantum Mechanics
1.6 Notation
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
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 Schrrdinger'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
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
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 Definiteness 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
6 Quantum Gravity
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
8 More Problems
8.1 What Will Be the CA for the SM
8.2 The Hierarchy Problem
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
10 Conclusions
10.1 The CAI
10.2 Counterfactual Definiteness
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
11 Introduction to Part i
11.1 Outline of Part
11.2 Notation
11.3 More on Dirac's Notation for Quantum Mechanics
12 More on Cogwheels
12.1 The Group SU(2), and the Harmonic Rotator
12.2 Infinite, Discrete Cogwheels
12.3 Automata that Are Continuous in Time
13 The Continuum Limit of Cogwheels, Harmonic Rotators and
Oscillators
13.1 The Operator ~Pop in the Harmonic Rotator
13.2 The Harmonic Rotator in the x Frame
14 Locality
15 Fermions
15.1 The Jordan-Wigner Transformation
15.2 'Neutrinos' in Three Space Dimensions
15.2.1 Algebra of the Beable 'Neutrino' Operators
15.2.20 rthonormality and Transformations of the 'Neutrino
Beable States
15.2.3 Second Quantization of the 'Neutrinos
15.3 The 'Neutrino' Vacuum Con'elations
16 PQ Theory
16.1 The Algebra of Finite Displacements
16.1.1 From the One-Dimensional Infinite Line to the Two-Dimensional Torus
16.1.2 The States IQ, P) in the q Basis
16.2 Transformations in the PQ Theory
16.3 Resume of the Quasi-periodic Phase Function
16.4 The Wave Function of the State 10, 0)
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 Compactified 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
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
18.3 Large Symmetry Groups in the CAI
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 Theo
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
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
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
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
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