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Journal of Mathematical Physics

Quantumlike formulation of stochastic problems

Emilio Santos

Published Nov 1, 1974• 11 citations
Publisher

Abstract

It is shown that a stochastic process can be viewed as a set of states (normalized positive linear functionals) over an Abelian C*-algebra. Alternatively, the stochastic process can be associated with a set of representations of the algebra as a subalgebra of the (noncommutative) C*-algebra of bounded operators in a Hilbert space. Then, an operator equation can be associated with every stochastic equation in some general conditions. The formalism is applied to Brownian motion. Then, we study the nonrelativistic motion of a single particle in stochastic electrodynamics, a theory which has been proposed as a possible alternative to quantum mechanics. The equations of motion, which are derived, coincide with the basic ones of quantum mechanics. The differences between this theory and quantum mechanics are summarized.

DOI
10.1063/1.1666564

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It is shown that a stochastic process can be viewed as a set of states (normalized positive linear functionals) over an Abelian C*-algebra. Alternatively, the stochastic process can be associated with a set of representations of the algebra as a subalgebra of the (noncommutative) C*-algebra of bounded operators in a Hilbert space. Then, an operator equation can be associated with every stochastic equation in some general conditions. The formalism is applied to Brownian motion. Then, we study the nonrelativistic motion of a single particle in stochastic electrodynamics, a theory which has been proposed as a possible alternative to quantum mechanics. The equations of motion, which are derived, coincide with the basic ones of quantum mechanics. The differences between this theory and quantum mechanics are summarized.
The theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrödinger's equation for specific values of certain parameters. A generalized Fokker-Planck-Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrödinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum-mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non-Markoffian terms are briefly studied.
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.
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.
In Paper I of this investigation a new calculus for functions of noncommuting operators was developed, based on the notion of mapping of operators onto ccc-number functions. With the help of this calculus, a general theory is formulated, in the present paper, of phase-space representation of quantum-mechanical systems. It is shown that there is a whole class of such representations, one associated with each type of mapping, the simplest one being the well-known representation due to Weyl. For each representation, the quantum-mechanical expectation value of an operator is found to be expressible in the form of a phase-space average of classical statistical mechanics. The phase-space distribution functions are, however, not true probabilities, in general. The phase-space forms of the main quantum-mechanical equations of motion are obtained and are found to have the form of a generalized Liouville equation. The phase-space form of the Bloch equation for the density operator of a quantum system in thermal equilibrium is also derived. The generalized characteristic functions of boson systems are defined and their main properties are studied. The equations of motion for the characteristic functions are also derived. As an illustration of the theory, a generalized stochastic description of a quantized electromagnetic field is obtained.
In Feynman's space-time approach to nonrelativistic quantum mechanics, the wave function may be considered to be the sum of path integrals over Brownian-motion trajectories. This formalism is entirely equivalent to the Schr\"odinger-equation treatment with its intrinsic indeterminacy. Pursuing the analogy with Brownian motion further, one may introduce a hidden variable corresponding to particle velocity in order to represent the wave function as a sum of path integrals taken over phase-space trajectories. By means of a hidden parameter corresponding to the time scale of fictitious interactions of the particle with the vacuum, one can define generalized propagators which over very short times produce wave functions which are localizable in phase space and hence are deterministic. For long times the conventional Feynman formalism is recovered. This short-time causality does not violate the von Neumann injunction against hidden variables since the new velocity variable turns out to be non-Hermitian over the long time scales normally considered. In contrast to the models proposed by Bohm and others, the present Brownian model is linear and preserves the usual statistical interpretation of the wave function for sufficiently long times.
Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities.