# A Survey of Hidden-Variables Theories

### International Series of Monographs in Natural Philosophy

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A Survey of Hidden-Variables Theories is a three-part book on the hidden-variable theories, referred in this book as ""theories of the first kind"". Part I reviews the motives in developing different types of hidden-variables theories. The quest for determinism led to theories of the first kind; the quest for theories that look like causal theories when applied to spatially separated systems that interacted in the past led to theories of the second kind. Parts II and III further describe the theories of the first kind and second kind, respectively. This book is written to make the literature on hidden variables comprehensible to those who are confused by the original papers with their controversies, and to average reader of physics papers.

## Table of Contents

Preface

Acknowledgments

Part I. Three Kinds of Hidden-Variables Theories and So-called Proofs of Impossibility

Foreword to Part I

Chapter 1. Introduction

1.1 Some Fundamentals of Quantum Theory

1.2 First Reasons for Suggesting Hidden-Variables Theories

1.3 Hidden-Variables Theories of the First Kind

1.4 Incompleteness of Some Hidden-Variables Theories

1.5 Hidden-Variables Theories of the Second Kind

1.6 Difference between Numbers of Distinct Hidden-Variables States in Theories of the First and of the Second Kind

1.7 Hidden-Variables Theories of the Zeroth Kind

1.8 Philosophical, Religious, and Polemic Motives behind the Quarrels between Proponents of Hidden-Variables Theory and of Pure Quantum Theory

1.9 The "Potential" But "Virtual" Nature of the Predictions Made by Hidden-Variables Theories

1.10 "Ensembles" Corresponding to Hidden-Variables States and "Expectation Values" for Them

Chapter 2. Von Neumann's Impossibility of Hidden-Variables Theories

2.1 The Source of the Trouble with Von Neumann's Definition of Hidden-Variables Theories

2.2 The Nonexistence of a Unique Density Operator for an Ensemble of Systems All in the Same Hidden-Variables State (φ, ξ)

2.3 Dispersionfree States in Hidden-Variables Theory

2.4 Von Neumann's Three "Reasons" for the Nonexistence of Hidden-Variables States

Chapter 3. The Work of Gleason and of Kochen and Specker

3.1 How to Measure Simultaneously J2x, J2y, and J2z in a j = 1 State

3.2 The Kochen-Specker Paradox

3.3 Proofs of the Kochen-Specker Paradox

3.4 Conclusions Drawn from the Kochen-Specker Paradox

3.5 Generalization of the Kochen-Specker Paradox to Hilbert Space

3.6 The Work of Gleason

3.7 Reducing the Impact of the Gleason Troubles in Hidden-Variables Theories

3.8 Troubles with Composite Systems in Hidden-Variables Theories of the First Kind

Chapter 4. The Work of Jauch and Piron

4.1. Jauch-Piron's Hidden-Variables Theory

4.2 Notation Used by Jauch and Piron

4.3 Axioms and Postulates of Jauch and Piron

4.4 Conclusions of Jauch and Piron

4.5 Reinterpretation (w → v) of the Work of Jauch and Piron

4.6 Gudder's Reformulation of Postulate (4)º

Appendices to Part I

Appendix A An Attempt at Writing the Quantum-Mechanical Density Operator U(ψ)op in a Simple Case as a Superposition of Weighted Microstate Density Operators V(ψ, ξ)op

Appendix B A Simple New Proof of the Kochen-Specker Paradox

Appendix C Bell's version of Gleason's Proof of the Kochen-Specker Paradox

Appendix D Generalization of the Kochen-Specker Paradox to Complex Hilbert Vectors in Hilbert Spaces of Three or More Dimensions

Appendix E Invalidity of the Distributive Law for Quantum-Theoretical Propositions

Appendix F Compatibility of Propositions According to Jauch and Piron

Appendix G Validity of Jauch and Piron's Postulate (4)º for incompatible Propositions

Appendix H Coherence of Lattices of Propositions in Quantum Theory

Appendix I An Example How v(a) = v(b) = 1 Does Not Exclude v(a∩b) = 0

Appendix J Angular Eigenfunctions for j = 1

Appendix K Von Neumann's Proof of Impossibility of Dispersionfree States

Part II. Theories of the First Kind

Foreword to Part II

Chapter 1. Introduction

1.1 Hidden-Variables States (Microstates) and Quantum States

1.2 The Main Parts of a Theory of the First Kind

1.3 Theories Discussed in Part II

Chapter 2. Bohm's 1951 Theory

2.1 Ensemble Velocity Field and Particle Velocities

2.2 The Imagined or Actual Particle Momentum and the Quantum-Mechanical Potential

2.3 Possible Alteration of Bohm's Theory

2.4 Comparison with De Broglie's Work

2.5 Solving for the Possible Trajectories of a Particle

2.6 In What Way is Bohm's 1951 Theory a Hidden-Variables Theory?

2.7 How to Tell through Which Slit a Particle Came that is Observed in the Interference Pattern behind Several Slits

2.8 What Keeps Particles Absent from Regions Where ψ = 0?

2.9 Generalization of Bohm's Theory to Interacting Particles

2.10 Inelastic Scattering

2.11 Bohm's Theory of Measurement of Observables

2.12 The Einstein-Podolsky-Rosen Paradox Explained by Bohm's Theory

2.13 Treatment of Photons in Bohm's Theory

2.14 Bell's Hidden Spin Variable for an Electron

2.15 Relativistic Generalizations of Bohm's Theory

2.16 Evaluation of Bohm's Theory

Chapter 3. The Theory of Wiener and Siegel

3.1 The Mathematical Apparatus

3.2 The Role of the Ξi and of Wiener's and Siegel's Δi or δi or αi or α as Hidden Variables

3.3 Some Properties of the Δi(α)

3.4 Transformation of Δi(α) from xi-Space to Ai-Space

3.5 Equilibrium Hidden-Variables Distribution in ξ-Space

3.6 How the Hidden Variables Determine the Result of a Measurement in the Theory of Wiener and Siegel

3.7 Verification That the Wiener-Siegel Theory Meets the Requirements of a Hidden-Variables Theory of the First Kind

3.8 Application of the Polychotomic Algorithm to Spin-½ States

3.9 Time Dependence in the Wiener-Siegel Theory

3.10 Evaluation of the Wiener-Siegel Theory

Chapter 4. The Theory of Bohm and Bub

4.1 Normalization of Wiener-Siegel's Ξ(x)

4.2 Time Dependence and Reduction of Wave Packets in the Bohm-Bub Theory

4.3 Bohm-Bub Modification of the Schrödinger Equation

4.4 Application of the Bohm-Bub Theory to the Collapse of the State Vector ψ during a Measurement

4.5 Application to Quantum Electrodynamics and to Relativistic Field Theory

4.6 Possibility of Experimental Verification

4.7 Evaluation of the Bohm-Bub Theory

Chapter 5. Experiments for Verifying or Falsifying the Polychotomic Algorithm

5.1 The Nature of the Experiments of Papaliolios and of a Possible Variation of it

5.2 Vector Addition of Polarizations

5.3 Omitting the Normalizations

5.4 The Ξ Bias Caused by the First Filter

5.5 The Light Transmitted by the Second Polarization Filter

5.6 The Light Transmitted by the Third Polarization Filter

5.7 Other Simple Possibilities for dΞ/dt

5.8 The Experiment of Papaliolios

5.9 Conclusions

Appendices to Part II

Appendix A Bohm's Quantum-Mechanical Potential

Appendix B How in Bohm's Theory P(x) becomes (ϱ(x)

Appendix C Cancellation of Infinities of U against Those of V

Appendix D Many-Particle Bohm Theory

Appendix E Angular Momenta and Electron Orbits in Atomic States in Bohm's Theory

Appendix F Imagined and Conventional Momentum Distribution in an Estate

Appendix G The Gaussian Wave Packet

Appendix H Bohm's Quantum Electrodynamics

Appendix I De Broglie's Relativistic Theory of the Pilot Wave

Appendix J Lack of Uniqueness of the Polychotomic Algorithm When Degeneracy Makes the Choice of the Set {ϕi} of Possible Results Nonunique

Appendix K Problems Arising when the Polychotomic Algorithm is Applied to Composite Systems or to Single Systems

Appendix L Application of the Polychotomic Algorithm for a Measurement of an Observable When the State Depends also on a Coordinate of a Different System

Appendix M Various Canonical Hidden-Variables Distributions for a Composite System with Factorizable State Vector ψ

Appendix N Reductions of State Vectors in the Theory of Wiener and Siegel

Appendix O "Polarization" of a Photon in Quantum Theory

Appendix P Hidden-Variables Bias behind First Polarization Filter

Appendix Q The Transmission of Light by the Second Polarization Filter

Appendix R The Transmission of Light by the Third Polarization Filter

Appendix S Wiener-Siegel's Application of the Polychotomic Algorithm to Eigenvalues Instead of to Eigenfunctions

Part III. Theories of the Second Kind

Foreword to Part III

Chapter 1. Introduction

Chapter 2. Quantum Theory of Polarization Correlation in Two-Photon Emission

2.1 The 0-1-0 Case and the 1-1-0 Case

2.2 Abbreviated Notation for Polarized Photon States

2.3 Photon Angular Momentum

2.4 Expectation Values for Observables in Polarized Photon States in Momentum Space

2.5 Quantum Numbers of the Photon Wavefunction

2.6 Two-photon Wavefunctions

2.7 Derivation of the Final Wavefunction for Two Photons Emitted by a Cascading Atom

2.8 The Observable Measured by an Ideal Linear Polarization Filter

2.9 Polarization Correlations between Different Photons Passing through Different Polarization Filters

2.10 The Nonlocality Paradox

Chapter 3. Some Examples of Hidden-Variables Theories of the Second Kind for "Explaining" Polarization Correlations

3.1 Minimum Requirements

3.2 Internal and External Hidden Variables

3.3 Photon Polarization as a Possible Hidden Variable

3.4 Correlations between Two Photons in Hidden-Variables Theory

3.5 Coincidence Correlations

3.6 Results for R(a, b), for the Sample Hidden-Variables Theories Considered

Chapter 4. The Experiments of Clauser and of Holt

4.1 The Experiments of Kocher and Commins

4.2 Our Lack of Understanding of Hidden-Variables Theories of the Second Kind

4.3 The idea of Clauser and of Home and Shimony

4.4 The Experimental Results

4.5 Calculation of the Extreme Values of Δ/R0 according to Quantum Theory under Ideal Circumstances

4.6 Predictions of Hidden-Variables Theories for the Experiments of Clauser and of Holt

4.7 A Simple General Proof of - 1 ≤ Δ/R0 ≤ 0 for Perfectly Efficient Experiments, according to Any Hidden-Variables Theory of the Second Kind

4.8 The Case of Imperfect Efficiency

4.9 Bell's Inequality and the Derivation of Δ ≤ 0 by Clauser and by Home and Shimony

Chapter 5. The Experiments of Bleuler and Bradt, Wu and Shaknov, of Langhoff, and of Kasday, Ullman, and Wu

5.1 The Annihilation Two-photon Wavefunction

5.2 The Experimental Setup

5.3 Quantum-Theoretical Dependence of the Number of Coincidence Counts upon the Angles φ and θ

5.4 Experimental Results

5.5 Furry's Solution of the Nonlocality Paradox

5.6 Proofs that Experiments with Compton Scattering of Annihilation Radiation cannot Conclusively Exclude all Hidden-Variables Theories of the Second Kind

5.7 The Kind of Hidden-Variables Theory that is Excluded by the Experiments Using Compton Scattering Annihilation Radiation

Chapter 6. Conclusions

Appendices to Part III

Appendix A The Expectation Value of an Observable Q for One Photon of Given Polarization and Momentum

Appendix B The Expectation Value of a Field Observable Q in a Two-photon State

Appendix C The Photon Spin Observable

Appendix D Commutativity of Photon Spin Components

Appendix E Derivation of Two-photon Polarization Wavefunctions from Conservation of Angular Momentum and of Relative Parity

Appendix F Indeterminacy of the Momenta in Two-photon States

Appendix G Undisprovable ad hoc Theories of the Second Kind of Bell and of Kasday for Compton Scattering of Pairs of Annihilation Photons

References

Author Index

Subject Index

Other Titles in the Series in Natural Philosophy

## Product details

- No. of pages: 376
- Language: English
- Copyright: © Pergamon 1973
- Published: January 1, 1973
- Imprint: Pergamon
- eBook ISBN: 9781483151212

## About the Author

### F. J. Belinfante

## About the Editor

### D. Ter Haar

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