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Polarization States as Hidden Variables

Jean Louis Van Belle

Published Nov 1, 2018• 0 citations
OpenAlex

Abstract

This paper explores a possible physical interpretation of the wavefunction by examining if it can be used to provide a hidden-variable theory for explaining quantum-mechanical interference. The hidden variable is the polarization state of the photon. The outcome is as expected: the theory does not work. Hence, this paper clearly shows the limits of such physical interpretations.

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This paper continues to explore a possible physical interpretation of the wavefunction but with a focus on the wavefunction(s) of a single photon in the Mach-Zehnder experiment. It focuses, in particular, on how one might visualize linear and circular polarization states for photon waves, and how beam splitters may or may not split a circular polarization state into two independent linear polarization states or – vice versa – recombine two linear polarization states into one circular state. As such, it attempts to provide a more refined approach to the rather crude hidden-variable theory for explaining quantum-mechanical interference that was presented in a previous paper (http://vixra.org/pdf/1811.0005v1.pdf). The outcome is the same, however: the theory does not work. Hence, this paper again shows the limit of such physical interpretations, thereby confirming the intuition behind Bell’s Theorem.

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Comment on "How to make quantum mechanics look like a hidden-variable theory and vice versa"

Peter W. Milonni

The hidden-variable theory considered by Scully is discussed in the context of photon polarization correlations.Received 17 January 1984DOI:https://doi.org/10.1103/PhysRevD.32.1040©1985 American Physical Society

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Aug 15, 1985•5 citations

The EPR paradox, hidden variables, and Bell's theorem

Marlan O. Scully, Muhammad Suhail Zubairy

Quantum mechanics is an immensely successful theory, occupying a unique position in the history of science. It has solved mysteries ranging from macroscopic superconductivity to the microscopic theory of elementary particles and has provided deep insights into the nature of vacuum on the one hand and the description of the nucleon on the other. Whole new fields such as quantum optics and quantum electronics owe their very existence to this body of knowledge. However, despite the stunning successes of quantum mechanics, there is no general agreement on the conceptual foundations and interpretation of the subject. The theory provides unambiguous information about the outcome of a measurement of a physical object. However, many feel that it does not provide a satisfactory answer to the nature of the “reality” we should attribute to the physical objects between the acts of measurement. The conceptual difficulty comes about because the wave function |ψ〉 is usually given by a coherent superposition of various distinguishable experimental outcomes. If we denote the collection of states that represent the possible outcomes of an experiment by |ψ j 〉, then |ψ〉 = ∑ j c j |ψ j 〉 where c j = 〈ψ j |ψ〉. The probability of the outcome |ψ j 〉 is P j =|c j | 2 . In the process of measurement, the so called collapse of the wave function takes place and a single, definite state |ψ i 〉 of the physical object is chosen. The difficulty comes about in the interpretation of the mechanism by which this definite state is chosen from amongst all the possible outcomes.

Cambridge University Press eBooks•Sep 4, 1997•1 citation
This paper explores a possible physical interpretation of the wavefunction by examining if it can be used to provide a hidden-variable theory for explaining quantum-mechanical interference. The hidden variable is the polarization state of the photon. The outcome is as expected: the theory does not work. Hence, this paper clearly shows the limits of such physical interpretations.
This paper continues to explore a possible physical interpretation of the wavefunction but with a focus on the wavefunction(s) of a single photon in the Mach-Zehnder experiment. It focuses, in particular, on how one might visualize linear and circular polarization states for photon waves, and how beam splitters may or may not split a circular polarization state into two independent linear polarization states or – vice versa – recombine two linear polarization states into one circular state. As such, it attempts to provide a more refined approach to the rather crude hidden-variable theory for explaining quantum-mechanical interference that was presented in a previous paper (http://vixra.org/pdf/1811.0005v1.pdf). The outcome is the same, however: the theory does not work. Hence, this paper again shows the limit of such physical interpretations, thereby confirming the intuition behind Bell’s Theorem.
The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.
In this paper, we shall show how the theory of measurements is to be understood from the point of view of a physical interpretation of the quantum theory in terms of "hidden" variables, developed in a previous paper. We find that in principle, these "hidden" variables determine the precise results of each individual measurement process. In practice, however, in measurements that we now know how to carry out, the observing apparatus disturbs the observed system in an unpredictable and uncontrollable way, so that the uncertainty principle is obtained as a practical limitation on the possible precision of measurements. This limitation is not, however, inherent in the conceptual structure of our interpretation. We shall see, for example, that simultaneous measurements of position and momentum having unlimited precision would in principle be possible if, as suggested in the previous paper, the mathematical formulation of the quantum theory needs to be modified at very short distances in certain ways that are consistent with our interpretation but not with the usual interpretation.We give a simple explanation of the origin of quantum-mechanical correlations of distant objects in the hypothetical experiment of Einstein, Podolsky, and Rosen, which was suggested by these authors as a criticism of the usual interpretation.Finally, we show that von Neumann's proof that quantum theory is not consistent with hidden variables does not apply to our interpretation, because the hidden variables contemplated here depend both on the state of the measuring apparatus and the observed system and therefore go beyond certain of von Neumann's assumptions.In two appendixes, we treat the problem of the electromagnetic field in our interpretation and answer certain additional objections which have arisen in the attempt to give a precise description for an individual system at the quantum level.
Quantum mechanics is an immensely successful theory, occupying a unique position in the history of science. It has solved mysteries ranging from macroscopic superconductivity to the microscopic theory of elementary particles and has provided deep insights into the nature of vacuum on the one hand and the description of the nucleon on the other. Whole new fields such as quantum optics and quantum electronics owe their very existence to this body of knowledge. However, despite the stunning successes of quantum mechanics, there is no general agreement on the conceptual foundations and interpretation of the subject. The theory provides unambiguous information about the outcome of a measurement of a physical object. However, many feel that it does not provide a satisfactory answer to the nature of the “reality” we should attribute to the physical objects between the acts of measurement. The conceptual difficulty comes about because the wave function |ψ〉 is usually given by a coherent superposition of various distinguishable experimental outcomes. If we denote the collection of states that represent the possible outcomes of an experiment by |ψ j 〉, then |ψ〉 = ∑ j c j |ψ j 〉 where c j = 〈ψ j |ψ〉. The probability of the outcome |ψ j 〉 is P j =|c j | 2 . In the process of measurement, the so called collapse of the wave function takes place and a single, definite state |ψ i 〉 of the physical object is chosen. The difficulty comes about in the interpretation of the mechanism by which this definite state is chosen from amongst all the possible outcomes.