Variational Principle for Stochastic Mechanics Based on Information Measures

TL;DR

The paper introduces an information-measure-constrained variational principle for stochastic mechanics, unifying forward and backward path dynamics to recover Nelson's formulation and the Schrödinger equation. By imposing relative entropy constraints between forward and backward path configurations, it derives coupled PDEs whose reduction yields the familiar quantum dynamics, and it shows that incorporating Fisher information through alternative Lagrangians yields the same results. The approach clarifies why different Lagrangians can lead to the same theory and suggests that information terms are intrinsic to the dynamical description, potentially enabling extensions to more degrees of freedom and relativistic theories. Overall, it provides a principled link between physical dynamics and information theory within stochastic mechanics, with implications for foundational questions in quantum mechanics and beyond.

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 Schrödinger 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.

Figures (2)

• Figure 1: Plots of trajectory path of a point particle in phase space. (a) In classical mechanics, a point particle moves from point $a$ to point $b$ with a path configuration $\gamma_{ab}$ determined by least action principle. (b) In stochastic mechanics, diffusion of a point particle is described with forward and backward path configurations $\gamma^{\pm}_{ab}$ in the respective phase space $\Gamma^{\pm}$.
• Figure 2: (a) In classical mechanics, a point particle moves from point $a$ to point $b$ with a path configuration $\gamma_{ab}$ determined by least action principle. (b) In stochastic mechanics, diffusion of a point particle is described with forward and backward path configurations $\gamma^{\pm}_{ab}$. Each path configuration follows its own stochastic differential equation, but connected through the relative entropy constraint. Combing the two PDEs results in the Schrödinger equation. (c) Conjecture: By introducing rotational degrees of freedom $\sigma^{\pm}$, there are four path configurations $\gamma^i$ in the phase spaces $\Gamma^i$$(i=1,2,3,4)$. Can the variation approach developed here lead to the Dirac equation for spin once it is extended into the relativistic framework?

Theorems & Definitions (4)

• Theorem 1
• Corollary 1.1
• Corollary 1.2
• Theorem 2