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Journal of Mathematical Physics

Noether’s theorem in symmetric stochastic calculus of variations

Tetsuya Misawa

Published Oct 1, 1988• 20 citations
Publisher

Abstract

Quite recently, a symmetric stochastic calculus of variations was proposed to formulate canonical stochastic dynamics, which is an extension of Nelson’s stochastic mechanics. In this article a ‘‘Noether’s theorem’’ is formulated within this calculus of variations. Conservation laws of momentum, angular momentum, and energy are proved, which are related with the same laws in quantum mechanics.

DOI
10.1063/1.528145

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Quite recently, a symmetric stochastic calculus of variations was proposed to formulate canonical stochastic dynamics, which is an extension of Nelson’s stochastic mechanics. In this article a ‘‘Noether’s theorem’’ is formulated within this calculus of variations. Conservation laws of momentum, angular momentum, and energy are proved, which are related with the same laws in quantum mechanics.
Within the framework of stochastic calculus of variations for time-symmetric semimartingales X(t,ω), we consider two different stochastic versions of Maupertuis’ least action principle, in Lagrangian and Hamiltonian terms. The general results are applied to classical statistical mechanics, where they coincide with those of classical calculus of variations, and to Nelson’s stochastic mechanics, an approach to quantum mechanics where a time-symmetric semimartingale represents the position of a particle and the dynamics is expressed by a stochastic version of Hamilton’s principle of least action. Some historic examples of old quantum theory are discussed.
A canonical stochastic dynamical system for time-symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations. A certain class of stochastic dynamical systems gives a Hamiltonian formalism of Nelson’s stochastic mechanics. In a manner analogous to classical mechanics, the notions of a stochastic Poisson bracket and canonical transformation are introduced to the stochastic dynamical systems. It is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation.
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.
A quantal-dissipative system described by the Schr\"odinger equation with an additional nonlinear term is formulated in the Lagrangian stochastic mechanics on the basis of the symmetric stochastic calculus of variations (SSCV). A certain class of stochastic Lagrangian dynamical systems defined through the least-action principle in SSCV is associated with solutions of the nonlinear Schr\"odinger equations, and thereby the quantal-dissipative system described by the equation is connected with a stochastic Lagrangian dynamical system. The long-time behavior of momentum, angular momentum, and energy, and the virial theorem for the stochastic Lagrangian system corresponding to the quantal-dissipative system, are investigated through the conserved quantities and the first variations of the functional induced by one-parameter transformations. Examples are given for a free particle and for the harmonic oscillator.
Abstract The discussion of stochasticity inherent in physical phenomena and processes occurring on our world is continued actively at present. For this purpose, various physical models and mathematical descriptions that are based most often on postulated are proposed. As a rule, stochasticity introduced in physical models phenomenologically, as well as its origin and mechanisms of emergence are not considered. However, the problem concerning the origin of stochasticity is far from being solved both in physical and mathematical aspects. In this article, the idea put forth by D.I. Blokhintsev and L.E. Ballentine concerning the emergence of stochasticity in classical and quantum field systems as a consequence of action of the surrounding stochastic background is developed. We are using the Lagrangian formalism and the categorically topological method of description for correct definition of the Lagrangian, action, the Euler–Lagrange equations, Noether currents, and generalized charges in stochastic surroundings. It is shown that Noether currents and conserved quantities become stochastic, and their probabilistic properties are determined by the background characteristics. The change in the action of the background on the observation interval determines the deviation of conserved quantities from their mean values, and the Heisenberg uncertainty relation can be treated as a manifestation of this stochasticity. As an illustration of the operability of a mathematical formulation of the stochasticity origin, we consider a classical scalar field. The proposed approach is naturally generalized to quantum fields. It can be used to obtain stochastic equations describing the behavior of material macroscopic systems in random fields and measurement processes.