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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Repeated measurements in stochastic mechanics

Philippe Blanchard, Simon Golin, Maurizio Serva

Published Dec 15, 1986• 29 citations
Publisher

Abstract

Stochastic mechanics provides a probabilistic scheme for the description of quantum systems. Grabert, Haaumlnggi, and Talkner, and Nelson have pointed out that its multitime correlations seem to be in disagreement with quantum-mechanical predictions. We show that these difficulties are removed upon a careful analysis of repeated measurements in stochastic mechanics. The wave-packet reduction is naturally described in the stochastic framework, and the predictions for repeated measurements consequently agree in both theories.

DOI
10.1103/physrevd.34.3732

Related papers

Reply to ‘‘Comment on ‘Repeated measurements in stochastic mechanics’ ’’

Philippe Blanchard, Maurizio Serva

In a previous paper we showed that predictions about the results of repeated measurements coincide in quantum and stochastic mechanics when wave-packet collapse is taken into acocunt. Our result removed some apparent paradoxes pointed out by Grabert, H\"anggi, Talkner, and Nelson. Recently Wang and Liang argued that our analysis was incorrect and they produced an explicit counterexample in order to confute our results. We think that their comment is due to a misunderstanding of our paper. For this reason we discuss here, in more detail, the controversial point of our work, concerning the regularization of the quantum-mechanical propagator. After stating more clearly our point, we show that the criticism of Wang and Liang is supported by an improper treatment of the divergences in their counterexample.

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Comment on "Repeated measurements in stochastic mechanics"

M. S. Wang, Wei‐Kuang Liang

An example is given to show explicitly that the quantum-mechanical multitime correlations, in general, do not agree with the stochastic-mechanical multitime correlations with the wave function reduction taken into account. This result contradicts the previous assertion made by Blanchard, Golin, and Serva.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Aug 15, 1993•15 citations

The measurement problem in the stochastic formulation of quantum mechanics.

Blanchard Ph, Cini M, Maurizio Serva

The description of physical systems obtained in the classical limit of quantum mechanics is not consistent with the one given by classical statistical mechanics. In qantum mechanics the complete description of a physical system is given by specifying the wave function, which evolves completely deterministically. Probability comes into play only when a measurement is made. Classical statistical mechanics, on the contrary, concerns systems whose description is not complete and where probability enters as a consequence of a lack of knowledge. We show that Nelson's stochastic mechanics, whose predictions are identical with those of quantum mechanics, provides nevertheless a unified description of the deterministic and random aspects of quantum mechanics which eliminates the inconsistency outlined above and gives a simple and satisfactory solution to the measurement problem.

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Measurement in stochastic mechanics

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Quantum Mechanics and Stochastic Mechanics for compatible observables at different times

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Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum Mechanics in presence of non-interacting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with Quantum Mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of non interacting systems at different times.

•Jan 1, 2001•23 citations
Stochastic mechanics provides a probabilistic scheme for the description of quantum systems. Grabert, Haaumlnggi, and Talkner, and Nelson have pointed out that its multitime correlations seem to be in disagreement with quantum-mechanical predictions. We show that these difficulties are removed upon a careful analysis of repeated measurements in stochastic mechanics. The wave-packet reduction is naturally described in the stochastic framework, and the predictions for repeated measurements consequently agree in both theories.
In a previous paper we showed that predictions about the results of repeated measurements coincide in quantum and stochastic mechanics when wave-packet collapse is taken into acocunt. Our result removed some apparent paradoxes pointed out by Grabert, H\"anggi, Talkner, and Nelson. Recently Wang and Liang argued that our analysis was incorrect and they produced an explicit counterexample in order to confute our results. We think that their comment is due to a misunderstanding of our paper. For this reason we discuss here, in more detail, the controversial point of our work, concerning the regularization of the quantum-mechanical propagator. After stating more clearly our point, we show that the criticism of Wang and Liang is supported by an improper treatment of the divergences in their counterexample.
The description of physical systems obtained in the classical limit of quantum mechanics is not consistent with the one given by classical statistical mechanics. In qantum mechanics the complete description of a physical system is given by specifying the wave function, which evolves completely deterministically. Probability comes into play only when a measurement is made. Classical statistical mechanics, on the contrary, concerns systems whose description is not complete and where probability enters as a consequence of a lack of knowledge. We show that Nelson's stochastic mechanics, whose predictions are identical with those of quantum mechanics, provides nevertheless a unified description of the deterministic and random aspects of quantum mechanics which eliminates the inconsistency outlined above and gives a simple and satisfactory solution to the measurement problem.
Nelson's stochastic mechanics may be understood as a stochastic underpinning, or reconstruction, of nonrelativistic quantum mechanics, once the diffusion scale is fixed by ℏ\hbarℏ and the admissible states are restricted by the usual single-valuedness condition on the wavefunction. In this note I briefly indicate what this route achieves and why it remains conceptually attractive. Four advantages are emphasized. First, it supplies a clear configuration-space stochastic picture of the underlying processes. Second, the Born rule is built in from the outset, with ∣ψ∣2|ψ|^2∣ψ∣2 arising as the probability density ρρρ of the underlying diffusion process rather than as an independent postulate. Third, it offers a markedly different perspective on measurement and nonlocality: in particular, collapse need not be treated as an extra axiom, and the nonlocality associated with entangled states is softened relative to the deterministic Bohmian guidance picture. Fourth, by tying quantumness to a diffusion scale, it naturally suggests a continuum of physical descriptions ranging from the strictly classical to the strictly quantum-mechanical regime. I conclude by proposing a natural distance scale in stochastic mechanics and examining its implications for testing possible limits of Bell correlations.
Bohm Mechanics and Nelson Stochastic Mechanics are confronted with Quantum Mechanics in presence of non-interacting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with Quantum Mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of non interacting systems at different times.