On reformulating quantum mechanics and stochastic theory

Sheldon Goldstein

Kunio Yasue

A time-symmetric stochastic control theory is proposed as one of the representatives of quantum mechanics. The main idea is based on Nelson’s probability theoretical approach to quantum mechanics. His approach is reformulated as a time-symmetric stochastic control problem. Several different control constraints equivalent to Nelson’s are obtained. One of them has a close connection to the Lagrangian formalism of classical mechanics. This suggests to us the use of stochastic calculus of variations. Within the realm of this time-symmetric stochastic control theory it is shown why Schrödinger’s original variational method of quantization was successful. Several advantageous points of the stochastic control theoretical approach to quantum mechanics, including the analysis of the classical limit, are also discussed.

Takeshi Kodama, Tomoi Koide C

In this note, we discuss the stochastic variational method (SVM) to derive the action principle of Schrödinger equation. In this formalism, the wave function is introduced as one of convenient ways to express the action. This work is a reformulation of Nelson—Yasue’s stochastic quantization scheme and its variational formulation by Yasue, and will furnish possible way to deal with more complicated system such as quantum field theory, suggesting an origin of quantum mechanics.

L. de la Peña-Auerbach

Hervé Bergeron

In a previous article [H. Bergeron, J. Math. Phys. 42, 3983 (2001)], we presented a method to obtain a continuous transition from classical to quantum mechanics starting from the usual phase space formulation of classical mechanics. This procedure was based on a Koopman-von Neumann approach where classical equations are reformulated into a quantumlike form. In this article, we develop a different derivation of quantum equations, based on purely classical stochastic arguments, taking some elements from the Bohm-Fenyes-Nelson approach. This study starts from a remark noticed by different authors [M. Born, Physics in My Generation (Pergamon Press, London, 1956); E. Prugovecki, Stochastic Quantum Mechanics and Quantum Spacetime (Reidel, Dordrecht, 1986)], suggesting that physical continuous observables are stochastic by nature. Following this idea, we study how intrinsic stochastic properties can be introduced into the framework of classical mechanics. Then we analyze how the quantum theory can emerge from this modified classical framework. This approach allows us to show that the transition from classical to quantum formalism (for a spinless particle) does not require real postulates, but rather soft generalizations.

Luis de la Peña, Ana Marı́a Cetto