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Journal de physique

Covariance and stochastic mechanics

Marc-Thierry Jaekel

Published Jan 1, 1988• 5 citations
Publisher

Abstract

Covariance of stochastic mechanics under changes of time and simultaneous time-dependent changes of coordinates is made explicit in the context of diffusion processes, and extended to the operator formulation. Covariant time derivatives are defined in both representations and used to express the equations of motion in a manifestly covariant way. They also provide general constraints on the correspondence between quantum states and stochastic processes.

DOI
10.1051/jphys:01988004906092700

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Covariance of stochastic mechanics under changes of time and simultaneous time-dependent changes of coordinates is made explicit in the context of diffusion processes, and extended to the operator formulation. Covariant time derivatives are defined in both representations and used to express the equations of motion in a manifestly covariant way. They also provide general constraints on the correspondence between quantum states and stochastic processes.
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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.
We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component ϕjϕ_jϕj​ of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated to a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector which turns out to be of the Lindblad type. When the noise changes only the phase of ϕjϕ_jϕj​, the Schrödinger and the quantum Liouville equation are obtained. The component ψjψ_jψj​ of the wave vector obeying the Schrödinger equation is related to stochastic wave vector by ∣ψj∣2=⟨∣ϕj∣2⟩|ψ_j|^2=\langle|ϕ_j|^2\rangle∣ψj​∣2=⟨∣ϕj​∣2⟩.
Stochastic mechanics develops a formulation of quantum physics by assuming that a classical particle is also subjected to Brownian-like motion. Because this presumes that a particle's path is non-differentiable, making this transformation often involves implicitly replacing time-derivatives in classical mechanics by finite differences. In this paper, we will develop a stochastic mechanics by explicitly replacing time-derivatives in classical mechanics by finite differences, deriving the resulting dynamics and then introducing stochasticity. Like Nelson's original version of stochastic mechanics, our stochastic mechanics is also consistent with the Schrödinger Equation.
Abstract We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component φ j of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated with a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector, which turns out to be of the Lindblad type. When the noise changes only the phase of φ j , the Schrödinger and the quantum Liouville equations are obtained. The component ψ j of the wave vector obeying the Schrödinger equation is related to the stochastic wave vector by | ψ j | 2 = ⟨ | ϕ j | 2 ⟩ .
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.