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Physical Review

Simultaneous Measurement in the Bohm-Bub Hidden-Variable Theory

J. H. Tutsch

Published Jul 25, 1969• 13 citations
Publisher

Abstract

A recently proposed theory of measurement for nonrelativistic quantum mechanics, due to Bohm and Bub, is generalized, and a mathematical analysis of the theory is presented. The fact that there are always theories of the Bohm-Bub type, given the standard formulation of quantum mechanics, is discussed. The phenomenological coefficient $\ensuremath{\gamma}$ of the original theory is defined to be the absolute value of the energy change of the system during measurement divided by $\ensuremath{\hbar}$. This definition leads to a Heisenberg-like relation between the average energy change of the system during the measurement process and the collapse time, but the new relation depends on the so-called hidden variables, and hence could lead to non-quantum-mechanical effects. The time dependence of the hidden variables is considered, and they are then assumed to be constant for each individual system. An experiment is proposed which would test this assumption. A positive result would disagree significantly with the quantum-mechanical prediction. The question of whether or not the new variables must remain unobservable is analyzed mathematically. A rigorous definition of simultaneous measurement is made, and the resulting interference is given a mathematical description.

DOI
10.1103/physrev.183.1116

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A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I

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The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of ${10}^{\ensuremath{-}13}$ cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.

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Journal of Mathematical Physics•Sep 1, 1968•7 citations
A recently proposed theory of measurement for nonrelativistic quantum mechanics, due to Bohm and Bub, is generalized, and a mathematical analysis of the theory is presented. The fact that there are always theories of the Bohm-Bub type, given the standard formulation of quantum mechanics, is discussed. The phenomenological coefficient \ensuremathγ\ensuremath{\gamma}\ensuremathγ of the original theory is defined to be the absolute value of the energy change of the system during measurement divided by \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ. This definition leads to a Heisenberg-like relation between the average energy change of the system during the measurement process and the collapse time, but the new relation depends on the so-called hidden variables, and hence could lead to non-quantum-mechanical effects. The time dependence of the hidden variables is considered, and they are then assumed to be constant for each individual system. An experiment is proposed which would test this assumption. A positive result would disagree significantly with the quantum-mechanical prediction. The question of whether or not the new variables must remain unobservable is analyzed mathematically. A rigorous definition of simultaneous measurement is made, and the resulting interference is given a mathematical description.
An abstract definition of a general hidden-variables theory is given, and it is shown that such a theory is always possible in the present framework of quantum mechanics and is, in fact, unique in a certain sense. It is noted that the Bohm-Bub hidden-variables example is contained in this theory and an attempt is made to clarify the position of this theory with respect to hidden-variable impossibility proofs. The general definition is used in the consideration of quantum-mechanical ordering and the measurement process.
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.
The paper argues that far from challenging-or even refuting-Bohm's quantum theory, the no-hidden-variables theorems in fact support the Bohmian ontology for quantum mechanics. The reason is that (i) all measurements come down to position measurements; and (ii) Bohm's theory provides a clear and coherent explanation of the measurement outcome statistics based on an ontology of particle positions, a law for their evolution and a probability measure linked with that law. What the no-hidden-variables theorems teach us is that (i) one cannot infer the properties that the physical systems possess from observables; and that (ii) measurements, being an interaction like other interactions, change the state of the measured system.
The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of ${10}^{\ensuremath{-}13}$ cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.