Classical probabilistic realisation of quantum double-slit interference

We demonstrate how the interference effects for a quantum particle in the double-slit experiment can be described by classical probabilities. We investigate a classical field theory for a complex scalar field with probabilistic initial conditions. A central element are conserved charges leading to the concept of particles. These are statistical observables which describe properties of the probability distribution for field configurations. The conserved charges define subsystems for particle excitations of a vacuum state. We encode the probabilistic information for the one-particle subsystem in a complex wave function. The Liouville equation for the classical probability distribution implies that the time evolution of this wave function obeys the Schr\"odinger equation for a quantum particle in a potential. The potential arises from a space-dependence of the mass term in the otherwise relativistic classical field theory. It can be chosen arbitrarily, realizing the typical quantum effects of interference, tunneling or discrete energy spectra.