Obtaining the Schrödinger and Dirac Equations from the Einstein/KAC Model of Brownian Motion by Projection

Abstract.

It is well known that the Einstein and Kac stochastic models of Brownian motion provide microscopic models for the Diffusion and Telegraph equations respectively. Furthermore these classical partial differential equations may be transformed to the Schrödinger and Dirac Equations by analytically continuing them to imaginary time. However the formal nature of the analytic continuation removes any chance of using the Brownian models as microscopic models of quantum mechanics.

Some recent results have however shown that random walk models may be used to derive the quantum equations without invoking a formal analytic continuation. The quantum equations arise as projections and thus describe real measurable aspects of the space-time geometry of particle trajectories. This shows that there are in fact microscopic models which give rise to the quantum equations, although close inspection of the derivations show that these models provide many-particle simulations of quantum mechanics and do not qualify as microscopic models of quantum mechanics (equations plus interpretation) itself.

There is one intriguing exception to this. By allowing Brownian motion in time as well as space the Dirac equation may be derived as the equation governing a net charge. This constitutes a derivation of the Dirac equation which may possibly provide aspects of an interpretation of quantum mechanics, as well as the equation itself.