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Journal of Physics A Mathematical and General

Stochastic formulation of quantum mechanics based on a complex Langevin equation

H Okamoto

Published Dec 7, 1990• 13 citations
Publisher

Abstract

It is shown that a stochastic process described by a complex Langevin equation leads to real-time quantum mechanics. The author also derives a relation between a transition probability in the theory and a transition probability amplitude in real-time quantum mechanics in the framework of path-integral formulations. Finally, taking a harmonic oscillator case as an example, the Fokker-Planck equation is solved exactly, and a non-dissipation property of the stochastic process is pointed out.

DOI
10.1088/0305-4470/23/23/025

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It is shown that a stochastic process described by a complex Langevin equation leads to real-time quantum mechanics. The author also derives a relation between a transition probability in the theory and a transition probability amplitude in real-time quantum mechanics in the framework of path-integral formulations. Finally, taking a harmonic oscillator case as an example, the Fokker-Planck equation is solved exactly, and a non-dissipation property of the stochastic process is pointed out.
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.
The stochastic representation of quantum mechanics is used to give a purely stochastic treatment of the quantum Langevin equation. A general coupling involving positions and velocities of the particle and harmonic modes, with arbitrary dependence on the particle's position, is dealt with. Summing over the modes in the path integral and using stochastic integrals, the author obtains explicitly the reduced measure, that is, the stochastic process representing the time evolution of the small system in the presence of noise. Contact is made with the operatorial representation, and a relation is established between the correlations of the noise in both representations. A general condition on the coupling constants is given for recovering time locality in the continuum limit, and the effects of renormalisation are re-analysed within this purely stochastic approach.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.
Abstract We propose a Langevin equation to describe the quantum Brownian motion of bounded particles based on a distinctive formulation concerning both the fluctuation and dissipation forces. The fluctuation force is similar to that employed in the classical case. It is a white noise with a variance proportional to the temperature. The dissipation force is not restricted to be proportional to the velocity and is determined in a way as to guarantee that the stationary state is given by a density operator of the Gibbs canonical type. To this end we derived an equation that gives the time evolution of the density operator, which turns out to be a quantum Fokker–Planck–Kramers equation. The approach is applied to the harmonic oscillator in which case the dissipation force is found to be non Hermitian and proportional to the velocity and position.
A study of the non-dissipative Brownian motion in vacuum is presented. The noise source associated to the stochastic process assumed in this work is vacuum fluctuations of some quantum field capable of interact with a massive particle. For the associated Fokker-Planck equation we do a perturbation theory which has terms presenting dynamics similar to that satisfied by the probability amplitude in Quantum Theory even though the physical interpretation be classical. In particular classical observables present expected values that coincide with those calculated using Quantum Mechanics.
An extended and more rigorous version of a recent proposal for an objective stochastic formulation of quantum mechanics along with its extension to the relativistic case without spin is presented. The relativistic Klein-Gordon equation is shown to be a particular form of the relativistic Kolmogorov-Fokker-Planck equation which is derived from a covariant formulation of the Chapman-Kolmogorov condition. Complexification of probability amplitudes is again achieved only through a conformal rotation of Minkowski space-time M 4 .