INFORMATION LOSS IN THE CONTINUUM-LIMIT AND SCHRODINGERS EQUATION IN AN ELECTROMAGNETIC-FIELD

Since the time of Einstein's work on Brownian motion it has been known that random walks provide a microscopic model for the diffusion equat ion. Less well known is the fact that some instances of Schrodinger's equation occur naturally in the description of the statistics of these same walks and thus have classical contexts which are distinct from t heir usual association with quantum mechanics. An interesting feature of these models is the fact that the information which relates Schrodi nger's equation to its classical context is not contained in the parti al differential equation itself, but is lost in the continuum limit wh ich gives rise to the equation. In this article we illustrate the abov e by showing that Schrodinger's equation for a particle in an electrom agnetic field in 1 + 1 dimension occurs as a continuum limit of a desc ription of a classical system of point particles on a lattice. The der ivation shows that the information lost in the continuum limit is nece ssary to link the mathematics to the physical context of the equation. (C) 1998 Elsevier Science Ireland Ltd. All rights reserved.