Semiclassical Scalar Propagators in Curved Backgrounds : Formalism and Ambiguities

The phenomenology of quantum systems in curved space-times is among the most fascinating fields of physics, allowing –often at the gedankenexperiment level– constraints on tentative theories of quantum gravity. Determining the dynamics of fields in curved backgrounds remains however a complicated task because of the highly intricate partial differential equations involved, especially when the space metric exhibits no symmetry. In this article, we provide –in a pedagogical way– a general formalism to determine this dynamics at the semiclassical order. To this purpose, a generic expression for the semiclassical propagator is computed and the equation of motion for the probability four-current is derived. Those results underline a direct analogy between the computation of the propagator in general relativistic quantum mechanics and the computation of the propagator for stationary systems in non-relativistic quantum mechanics. A possible application of this formalism to curvature-induced quantum interferences is also discussed. PACS numbers: 04.62.+v, 11.15.Kc