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Physics of Particles and Nuclei

Stochastic variational quantization and maximum entropy principle

Takeshi Kodama, Tomoi Koide C

Published Sep 1, 2015• 1 citation
Publisher

Abstract

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.

DOI
10.1134/s1063779615050135

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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.
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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 that 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.
The article aims to develop the stochastic interpretation of quantum mechanics by E. Nelson based on balancing the intra-systemic contradiction (i.e., antisymmetry) between “order” and “chaos”. For the set task, it is proposed to combine two mutually opposite system-forming principles: “the principle of least action” and “the principle of maximum entropy” into one, the “principle of averaged efficiency extremum”. In a detailed consideration of the averaged states of a chaotically wandering particle, the time-independent (stationary) and time-dependent stochastic Schrödinger-Euler-Poisson equations are obtained as conditions for finding the extremals of the globally averaged efficiency functional of the stochastic system under study. The resulting stochastic equations coincide with the corresponding Schrödinger equations up to coefficients. In this case, the ratio of the reduced Planck constant to the particle mass is expressed through the averaged characteristics of a three-dimensional random process in which the considered wandering particle participates. The obtained stochastic equations are suitable for describing the quantum states of stochastic systems of any scale.
Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.