Variational Principle for Stochastic Mechanics Based on Information Measures

TL;DR

This paper introduces a new variational principle for stochastic mechanics that incorporates information measures as constraints, providing insights into quantum mechanics and unifying physical and informational aspects of dynamics.

Contribution

It proposes a novel variational framework that recovers Nelson's theory, explains multiple Lagrangians, and integrates information measures into the Lagrangian structure.

Findings

Recovers Nelson's stochastic mechanics and Schrödinger equation

Explains the use of multiple Lagrangians in variational methods

Highlights the role of informational terms in the Lagrangian

Abstract

Stochastic mechanics is regarded as a physical theory to explain quantum mechanics with classical terms such that some of the quantum mechanics paradoxes can be avoided. Here we propose a new variational principle to uncover more insights on stochastic mechanics. According to this principle, information measures, such as relative entropy and Fisher information, are imposed as constraints on top of the least action principle. This principle not only recovers Nelson's theory and consequently, the Schr\"{o}dinger equation, but also clears an unresolved issue in stochastic mechanics on why multiple Lagrangians can be used in the variational method and yield the same theory. The concept of forward and backward paths provides an intuitive physical picture for stochastic mechanics. Each path configuration is considered as a degree of freedom and has its own law of dynamics. Thus, the variation principle proposed here can be a new tool to derive more advanced stochastic theory by including additional degrees of freedom in the theory. The structure of Lagrangian developed here shows that some terms in the Lagrangian are originated from information constraints. This suggests a Lagrangian may need to include both physical and informational terms in order to have a complete description of the dynamics of a physical system.