The Probabilistic World: A Fundamental Approach to Quantum Mechanics and Probabilistic Computing

Dedication
Preface
Contents
Chapter 1  Introduction
Part I  Time, Wave Functions, and Quantum Field Theory from Fundamental Probabilities
	Chapter 2  Fundamental Probabilities
	Chapter 3  Fundamental Probabilism
		3.1 Probabilistic Description of Nature
		3.2 Probabilistic Realism
		3.3 Basic Concepts
			3.3.1 Probabilities
			3.3.2 Axiomatic Setting
			3.3.3 Observables
			3.3.4 Correlations
	Chapter 4  Probabilistic Time
		4.1 Classical Statistics
			4.1.1 Observables and Probabilities
			4.1.2 Ising Spins, Occupation Numbers, or Classical Bits
			4.1.3 Unique Jump Chains
			4.1.4 Local Chains
			4.1.5 Transfer Matrix
			4.1.6 Operators for Local Observables
		4.2 Time and Evolution
			4.2.1 Time as Ordering Structure
			4.2.2 Evolution
			4.2.3 Classical Wave Functions
			4.2.4 Step Evolution Operator
			4.2.5 Influence of Boundary Conditions
			4.2.6 Classical Density Matrix
			4.2.7 Independence from the Future
			4.2.8 Clock Systems
		4.3 Physical Time
			4.3.1 Continuous Time
			4.3.2 Properties of Physical Time
	Chapter 5  Fermions
		5.1 Quantum Field Theory for Free Fermions in Two Dimensions
		5.2 Complex Structure
		5.3 Particles and Holes
		5.4 Conserved Quantities and Symmetries
		5.5 Reference Frames and Lorentz Symmetry
	Chapter 6  Probabilistic and Deterministic Evolution
		6.1 Orthogonal and Unitary Step Evolution Operators
		6.2 Probabilistic Cellular Automata
		6.3 Static Memory Materials
		6.4 Partial Loss of Memory and the Emergence of Quantum Mechanics
		6.5 Markov Chains
	Chapter 7  Quantum Field Theory
		7.1 Fermionic Quantum Field Theory with Interactions
			7.1.1 Interacting Dirac Automaton
			7.1.2 Creation and Annihilation Operators
			7.1.3 Interaction Part of the Step Evolution Operator
			7.1.4 Time Evolution
			7.1.5 Particles
		7.2 Change of Basis and Similarity Transformations
		7.3 Fourier Transform for Cellular Automata
			7.3.1 Fourier Basis for Clock Systems
			7.3.2 Transport Automata in Momentum Space
		7.4 Particles and Antiparticles
	Chapter 8  Subsystems
		8.1 Subsystems and Environment
			8.1.1 Subsystems and Correlation with the Environment
			8.1.2 Time-Local Subsystems
		8.2 Observables and Operators
			8.2.1 Local Observables and Non-Commuting Operators
			8.2.2 Algebras of Local Observables and Operators
			8.2.3 Classical Correlations and the Continuum Limit
			8.2.4 Averaged Observables
			8.2.5 Probabilistic Observables and Incomplete Statistics
		8.3 Classes of Subsystems
			8.3.1 Correlation Subsystems
			8.3.2 Integrating out Variables
			8.3.3 Subtraces
			8.3.4 General Local Subsystems
			8.3.5 Incomplete Statistics for Subsystems
	Chapter 9  Discussion and Conclusions
Part II  Quantum Mechanics from Classical
Statistics
	Chapter 10  The Classical and the QuantumWorld
	Chapter 11  Qubit Automaton
		11.1 Discrete Qubit Chain
		11.2 Classical Wave Function and Step Evolution Operator
		11.3 Quantum Subsystems
		11.4 Incomplete Statistics
		11.5 Quantum Condition
		11.6 Unitary Evolution
		11.7 Probabilistic Observables
		11.8 Bit-Quantum Map
		11.9 Simple Quantum System from Classical Statistics
	Chapter 12  Entanglement in Classical and Quantum Statistics
		12.1 Entanglement in Classical Statistics and Quantum Mechanics
		12.2 Two-Qubit Quantum Systems
		12.3 Classical Probabilistic Systems for Two Qubits
		12.4 Correlation Map
		12.5 Classical Entanglement
		12.6 Classical Wave Function and Entanglement
		12.7 Bell’s Inequalities
		12.8 Completeness of the Correlation Map
		12.9 Many Qubits
	Chapter 13 Continuous Classical Variables
		13.1 Continuous Variables and Ising Spins
		13.2 Quantum Clock System
		13.3 Deterministic Evolution with Continuous Variables
		13.4 Classical Wave Function and Quantum Particles
	Chapter 14  Quantum Mechanics
		14.1 Classical Ising Spins and Quantum Spin
		14.2 Unitary Evolution for a One-Qubit Quantum System
		14.3 Time Reversal and Complex Conjugation
		14.4 Quantum Mechanics in Continuous Time
		14.5 Quantum Particle in a Harmonic Potential
		14.6 Dynamical Selection of Quantum Subsystems
		14.7 Particle–Wave Duality
	Chapter 15  Correlated Computing
		15.1 Deterministic and Probabilistic Computing
		15.2 Quantum Computing by Probabilistic Automata
		15.3 Artificial Neural Networks
		15.4 Neuromorphic Computing
	Chapter 16  Conditional Probabilities and Measurements
		16.1 Conditional Probabilities
		16.2 Sequences of Measurements
		16.3 Ideal Measurements for Subsystems
		16.4 Reduction of the Wave Function
		16.5 Decoherence and Syncoherence
	Chapter 17  The “Paradoxes” of Quantum Mechanics
		17.1 Classical Correlation Functions and Bell’s Inequalities
		17.2 Kochen–Specker Theorem
		17.3 Einstein–Podolsky–Rosen Paradox
	Chapter 18  Embedding Quantum Mechanics in Classical Statistics
		18.1 Towards a Probabilistic Model of the World
		18.2 Short Answers to Quantum Questions
Appendix A Matrix Chains
Appendix B Positivity of Overall Probability Distribution
Appendix C Weyl Complex Structure for Two-ParticleWave Function
Appendix D Complex Structure Based on Sublattices
Appendix E Complex Fermionic Operators
Appendix F Subtraces for the Two-Bit Local Chain
References