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Journal of Mathematical Physics

Stochastic lengths

Marc-Thierry Jaekel

Published May 1, 1992• 1 citation
Publisher

Abstract

The autocorrelations of the line element for the sample paths of diffusion processes are computed. Lengths are defined for the sample paths of diffusions which depend upon an arbitrary function f and reduce to the classical relativistic length for differentiable trajectories. Their mean values are computed for a general diffusion and provide a family of stochastic actions labeled by f. In a variational principle, these actions are extremized over the drift and probability density and the equations characterizing the critical diffusions are derived. The Klein–Gordon equation is obtained from the variation of arbitrary f-dependent stochastic extensions of the length of trajectories.

DOI
10.1063/1.529699

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The autocorrelations of the line element for the sample paths of diffusion processes are computed. Lengths are defined for the sample paths of diffusions which depend upon an arbitrary function f and reduce to the classical relativistic length for differentiable trajectories. Their mean values are computed for a general diffusion and provide a family of stochastic actions labeled by f. In a variational principle, these actions are extremized over the drift and probability density and the equations characterizing the critical diffusions are derived. The Klein–Gordon equation is obtained from the variation of arbitrary f-dependent stochastic extensions of the length of trajectories.
The stochastic variational principle which applies to diffusions and extremises the mean value of the classical action is generalised to include non-Markovian diffusions. A whole family of different stochastic actions, labelled by a time-dependent parameter, is obtained and all of them are shown to lead to the Shrodinger equation. As a result, the well known degeneracy of the class of stochastic processes which correspond to a single quantum state is recovered. It is also related to an arbitrariness in the choice of the noise which can occur in two places: in the definition of the noise, through the time-dependent parameter, and in the auto-correlations of the noise which, although constant during the variational method, remain undetermined. The whole scheme is given a covariant form.
We propose an extension to the relativistic case of the stochastic variational principles both of Lagrangian and Eulerian type. The action we use is the mean classical action evaluated on the paths of relativistic covariant diffusions. The resulting equations of motion are the relativistic stochastic Lagrange equations Nous proposons une extension au cas relativiste des principes variationnels de type lagrangien et eulerien. Nous utilisons comme action l'action moyenne classique evaluee sur les trajectoires des diffusions covariantes relativistes. Les equations du mouvement obtenues sont les equations de Lagrange stochastiques relativistes
Markov diffusions corresponding to solutions of the Klein-Gordon equation are introduced in a similar way as in Nelson's stochastic mechanics. A relativistic version of the Nelson-Newton law for Markov diffusions is established. Stochastic generalizations of the proper time and of the relativistic action functional are proposed and a stochastic version of the relativistic principle of least action is stated. The formalism can be regarded as a quantization procedure leading from classical special relativity to the Klein-Gordon equation.
In this paper we establish three variational principles that provide new foundations for Nelson’s stochastic mechanics in the case of nonrelativistic particles without spin. The resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature. We first develop two stochastic variational principles whose Hamilton–Jacobi-like equations are precisely the two coupled partial differential equations that are obtained from the Schrödinger equation (Madelung equations). The two problems are zero-sum, noncooperative, stochastic differential games that are familiar in the control theory literature. They are solved here by means of a new, absolutely elementary method based on Lagrange functionals. For both games the saddle-point equilibrium solution is given by the Nelson’s process and the optimal controls for the two competing players are precisely Nelson’s current velocity v and osmotic velocity u, respectively. The first variational principle includes as special cases both the Guerra–Morato variational principle [Phys. Rev. D 27, 1774 (1983)] and Schrödinger original variational derivation of the time-independent equation. It also reduces to the classical least action principle when the intensity of the underlying noise tends to zero. It appears as a saddle-point action principle. In the second variational principle the action is simply the difference between the initial and final configurational entropy. It is therefore a saddle-point entropy production principle. From the variational principles it follows, in particular, that both v(x,t) and u(x,t) are gradients of appropriate principal functions. In the variational principles, the role of the background noise has the intuitive meaning of attempting to contrast the more classical mechanical features of the system by trying to maximize the action in the first principle and by trying to increase the entropy in the second. Combining the two variational principles, we get the quantum Hamilton principle, i.e., a variational characterization of the logarithm of the wave function ψ. The Lagrangian is the Lagrangian of classical mechanics with the complex-valued velocity v−iu replacing the classical velocity. The dynamics is given by a stochastic differential equation for real-valued diffusions with complex-valued drift and driving noise processes. From the variational principle we derive a Newton-type law. We finally define the momentum process and show that its mean and variance coincide with those of the quantum momentum operator.
Given a positive energy solution of the Klein–Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2=−(1/c2)dXν dXν. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an M4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein–Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.