A new formulation of stochastic mechanics

Michele Pavon

L. de la Peña-Auerbach

B. Schulz

Abstract In 1985, Edward Nelson, who formulated the theory of stochastic mechanics, made an interesting remark about Bell's theorem. Nelson analysed the latter in the light of classical fields that behave randomly. He found that if a stochastic hidden variable theory fulfils certain conditions, the inequality of Bell can be violated. Moreover, Nelson was able to prove that this may happen without any instantaneous communication between the two spatially separated measurement stations. Since Nelson's article got almost overlooked by physicists, we try to review his comments on the theorem. We argue that a modification of stochastic mechanics published recently by Fritsche and Haugk can be extended to a theory which fulfils the requirements of Nelson's analysis. The article proceeds to derive the quantum mechanical formalism of spinning particles and the Pauli equation from this version of stochastic mechanics. Then, we investigate Bohm's version of the EPR experiment. Additionally, other setups, like entanglement swapping or time and position correlations, are shortly explained from the viewpoint of our local hidden‐variable model. Finally, we mention that this theory could perhaps be relativistically extended and useful for the formulation of quantum mechanics in curved space‐times.

Gian Fabrizio De Angelis

D Falco

The problem of a formulation of Nelson's (1967) stochastic mechanics of scalar fields in the context of general relativity is considered. The simple example of the scalar field in the Wightman vacuum state on the Rindler wedge is examined, the stochastic counterpart of the Fulling ambiguity of canonical quantisation is formulated, and the role of the stochastic mechanics of thermal mixtures, as formulated by Guerra and Loffredo (1981), is analysed, in the spirit of Davies (1975) and Unruh (1976), in the solution of the above ambiguity. An overall picture emerges which, both in the explicit example considered here and in its straightforward generalisations to static submanifolds of more general spacetimes, confirms Smolin's point of view that stochastic quantisation is a very natural conceptual frame in which to study the general non-covariance of the distinction between quantum and thermal fluctuations.

Philippe Blanchard, Ph. Combe, Wan Zheng