Semiclassical causal geodesics: Minkowski spacetime case

Abstract

We use an integral quantization model based on the Heisenberg–Weyl group to describe the motion of a spinless particle in the Minkowski background spacetime. This work is a sequel to a previous paper, devoted to mathematical aspects of our model: construction of the space of coherent states and properties of elementary observables. We compute transition amplitudes corresponding to the free motion of a particle between two coherent states. These amplitudes are then used to model quantum random walks of free relativistic particles. Our quantization scheme allows us to recover interference patterns occurring in a standard double-slit experiment, known from the classical approach. This result is obtained by modeling the slits in terms of eigenstates of the position operator and computing transition amplitudes between position and coherent states. We design our model in a way which allows for a future generalization to a semiclassical quantization of the geodesic motion in curved spacetimes.