Classical particles and the Dirac equation with an electromagnetic field

The Einstein-Kac lattice model of Brownian motion in one dimension is extended to include massless particles which do not themselves scatter, but which facilitate scattering of Brownian particles. The model is completely classical and no formal quantization is employed. However by observing second-order effects in the distribution of Brownian particles we recover directly the components of Dirac wave functions. Furthermore the probability densities describing the massless particles are shown to form ‘fields’ which obey Maxwellʹs equations. This result extends the context of the Dirac equation to include the physics of ensembles of classical particles with contact interactions. In this context all ‘quantum objects’, including the wave function itself, are observable in terms of distributions of particles on a lattice.