Information loss in the continuum limit and Schrödinger's 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 equation. Less well known is the fact that some instances of Schrödinger's equation occur naturally in the description of the statistics of these same walks and thus have classical contexts which are distinct from their usual association with quantum mechanics. An interesting feature of these models is the fact that the information which relates Schrödinger's equation to its classical context is not contained in the partial differential equation itself, but is lost in the continuum limit which gives rise to the equation. In this article we illustrate the above by showing that Schrödinger's equation for a particle in an electromagnetic field in 1 + 1 dimension occurs as a continuum limit of a description of a classical system of point particles on a lattice. The derivation shows that the information lost in the continuum limit is necessary to link the mathematics to the physical context of the equation.