Remarks on connections between quantum mechanics and stochastic quantization

西岡, 道夫

Steven M. Moore

Sheldon Goldstein

Luis de la Peña, Ana Marı́a Cetto, Andrea Valdés-Hernández

The connection is established between two theories that have developed independently with the aim to describe quantum mechanics as a stochastic process, namely stochastic quantum mechanics (sqm) and stochastic electrodynamics (sed). Important commonalities and complementarities between the two theories are identified, notwithstanding their dissimilar origins and approaches. Further, the dynamical equation of sqm is completed with the radiation terms that are an integral element in sed. The central problem of the transition to the quantum dynamics is addressed, pointing to the key role of diffusion in the emergence of quantization.

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.

Harish Parthasarathy