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Journal of Physics A Mathematical and General

Stochastic Schrodinger and Heisenberg equations: a martingale problem in quantum stochastic processes

Hiroshi Hasegawa, R. F. Streater

Published Dec 21, 1983• 9 citations
Publisher

Abstract

The relation between a quantum dynamical semigroup and the associated quantum stochastic processes can be formulated by an analogy in operator algebras to the martingale problem of Stroock and Varadhan (1969) for classical diffusions. It is shown that under simple circumstances the formulation yields the explicit solution. The authors give three such examples.

DOI
10.1088/0305-4470/16/18/004

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The relation between a quantum dynamical semigroup and the associated quantum stochastic processes can be formulated by an analogy in operator algebras to the martingale problem of Stroock and Varadhan (1969) for classical diffusions. It is shown that under simple circumstances the formulation yields the explicit solution. The authors give three such examples.
A brief presentation is given of the basic concepts in quantum probability theory in analogy to the classical ones, as well as the necessary information about noncommutative stochastic integration and its explicit representation in the Fock space. Algebraic differential relations that define quantum diffusion motion with nondemolition observation are derived. In the Markov case we obtain the stochastic equation of quantum diffusion filtering which is analogous to the Zakai equation of classical nonlinear filtering. A quantum linear stochastic model with continuous observation is given for which this equation is reduced to a linear stochastic quantum filtering equation of Kalman-Busi type and to the operator Riccati equation. We also consider an example of observing the coordinate of a free quantum particle, for which the solution of a stationary quantum filtering problem is obtained.
The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.
Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.
Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.
"Quantum trajectories" are solutions of stochastic differential equations also called Belavkin or Stochastic Schrödinger Equations. They describe random phenomena in quantum measurement theory. Two types of such equations are usually considered, one is driven by a one-dimensional Brownian motion and the other is driven by a counting process. In this article, we present a way to obtain more advanced models which use jump-diffusion stochastic differential equations. Such models come from solutions of martingale problems for infinitesimal generators. These generators are obtained from the limit of generators of classical Markov chains which describe discrete models of quantum trajectories. Furthermore, stochastic models of jump-diffusion equations are physically justified by proving that their solutions can be obtained as the limit of the discrete trajectories.
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.