From classical mechanics to unitary propagation without quantization

In the 1970s Marc Kac demonstrated that a simple stochastic model involving random walks could provide a microscopic basis for the telegraph equations in the same way that Brownian motion provided a stochastic basis for the diffusion equation. This article demonstrates that a modification of the space-time geometry of the random walks in the Kac model provides the Dirac equation with a similar microscopic basis. The central feature of the new model is that space-time paths occur in ‘entwined pairs’; for each path in the Kac model there is a corresponding ‘return-path’ which is traversed in the opposite direction with respect to macroscopic time. The special pairing of paths automatically ‘quantizes’ the system eliminating the need for formal analytic continuation and providing an explicit mechanism for ‘wave-particle duality’. The result brings the Dirac equation into the domain of classical statistical mechanics where it can be studied as a phenomenology with a known stochastic basis.