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Journal of Mathematical Physics

A system theoretic representation of mechanical systems. II. Stochastic interpretation

Dipak K. Saraswati, Hans Primas

Published Dec 1, 1978• 0 citations
Publisher

Abstract

In a previous paper by the authors it was shown that mechanical systems, classical or quantal, can be realized as stationary linear dynamical systems. In this state space representation of mechanical systems, the input space, the output space, and the state space are Hilbert spaces and to make a connection with the language of a mathematical system theory, in which the corresponding spaces are topological vector spaces, it is shown here that the input, the output, and the state of the dynamical systems representing mechanical systems can be interpreted as second order stochastic processes. An orthogonal decomposition of mechanical motion into a deterministic and a purely nondeterministic motion, suggested by the stochastic interpretation, is discussed and it is pointed out that by iterating the decomposition procedure one can obtain a simple model which approximates the true mechanical motion.

DOI
10.1063/1.523623

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In a previous paper by the authors it was shown that mechanical systems, classical or quantal, can be realized as stationary linear dynamical systems. In this state space representation of mechanical systems, the input space, the output space, and the state space are Hilbert spaces and to make a connection with the language of a mathematical system theory, in which the corresponding spaces are topological vector spaces, it is shown here that the input, the output, and the state of the dynamical systems representing mechanical systems can be interpreted as second order stochastic processes. An orthogonal decomposition of mechanical motion into a deterministic and a purely nondeterministic motion, suggested by the stochastic interpretation, is discussed and it is pointed out that by iterating the decomposition procedure one can obtain a simple model which approximates the true mechanical motion.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics . In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics , deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.
This author has been engaged over the past several years in the effort to develop certain basic intuitions into a fully mathematical theory. These evolving intuitions first began with the idea that the economist's notion of utility might more truly be represented mathematically by a normed measure on the commodity space rather than by a point function, the value of the measure on a particular set of goods having the sense of a relative propensity of the individual to select a good in that set. With this idea there came in turn, quite naturally, the insight that utility and probability are one and the same thing. And accordingly it emerged that the theory we were seeking would not be restricted to the economic behavior of human beings, nor even to the behavior in general of human beings: it would pertain, in utmost generality, to all behavior of all systems. Thus, it must affirm that all behavior is the evolution of stochastic processes, and that all systems are stochastic processes. Consistent with this the structural elements of stochastic processes are seen as the fundamental, objective, real entities, the ultimate building-blocks of reality. These are eventualities and acts and-according to the traditional mathematical representation of stochastic processes, also-probabilities. This structural aspect of the theory under development puts into sharp evidence its wide divergence from the classical scientific conception of reality. Our intensive search failed to discover a dynamical law for the proposed new theory within the mathematical context of the traditional, σ-field-real-valued-measure representation of stochastic processes. We therefore have gone on to a higher level of possible mathematical representation. This particular representation would have collections of eventualities represented by lattices of certain configurations in a unitary space. Closed linear manifolds are among these configurations, and indeed the formalism of quantum mechanics falls into the context of this representation. But the general lattice element in the kinds of lattices being investigated is more complicated than a linear manifold, being heuristically comparable with an acute-angle cone of vectors. These configurations are formulated by means of a ƒlanking relation between linear manifolds. One such lattice has been developed. Others are currently being sought through variations on the notion of flanking. The fact that probabilities present themselves in the unitary space representation-as is familiar from quantum mechanics-as parameters of the relationship between linear manifolds suggests that probability is not a fundamental structural concept. It is consequently indicated that eventualities and acts alone are the basic elements of real structure. Accordingly, the dynamical law must be formulable in terms of these elements only. There is a natural suggestion of what the law might look like. Even at the present early stage of development of the new theory being sought it is possible to cite many pieces of empirical evidence vouching for its exclusive validity. Such evidence is particularly noteworthy in the domain of human behavior.
Employing a formulation of quantum statistical ensembles in terms of classical probability distributions on projective Hilbert space, it is shown that the wave function of an open quantum system represents a stochastic process. The stochastic dynamics of the state vector is obtained from a microscopic system-plus-reservoir model by deriving within the Markov approximation the differential Chapman-Kolmogorov equation for the classical distribution of the reduced system. The realizations of the stochastic process are found to be similar to those of the Monte Carlo wave function simulation method proposed, in general form, by Zoller et al. [Phys. Rev. A 46, 4363 (1992)].
An estimation method for unknown parameters in the initial conditions and the potential of a quantal system using the stochastic interpretation of quantum mechanics and some results in system theory are presented. According to this interpretation the possible trajectories of a particle through coordinate space may be represented by the realization of a stochastic process that satisfies a stochastic differential equation. The drift term in this equation is derived from the wave function and consequently contains all unknown parameters in the initial conditions and the potential. The main assumption of the paper is that a continuous sequence of position measurements on the trajectory of the particle can be identified with a realization of this stochastic process over the corresponding period of time. An application of the stochastic filtering theorems subsequently provides a minimum variance estimate of the unknown parameters in the drift conditional on this continuous sequence of measurements. As simple illustrations, this method is used to obtain estimates for the initial momentum of a free particle given measurements on its trajectory and to construct an estimator for the unknown parameters in a harmonic potential. It is shown that an optimal estimator exists if the stochastic processes are associated with a wave function from a potential of the Rellich type. In addition the a posteriori probability density of the parameters in the quantal system is calculated, assuming that all parameters involved prescribe a Rellich potential.