Schrödinger-Poisson systems under gradient fields

A singularity-free generalisation of Newtonian gravity can be constructed (Lazar in Phys Rev D 102:096002, 2020) within the framework of gradient field theory. This procedure offers a straightforward regularisation of Newtonian gravity and remains equally well applicable to other fields, such as electromagnetic fields. Here, with the aim of finding potentially measurable effects of gradient fields on the dispersion properties of various media, we present a quantum kinetic treatment of matter under gradient fields. The method is based on the application of the Wigner-Moyal procedure to the modified Schrödinger-Poisson equation emerging in the framework of gradient field theory. This allows us to treat, on equal footing, three different scenarios, namely self-gravitating systems, plasmas, and cold atoms in magneto-optical traps. We address the signature of gradient fields in the elementary excitations of these media. In particular, we estimate this effect to be accessible in state-of-the-art plasma-based experiments. We discuss in detail the classical kinetic and hydrodynamic limits of our approach and obtain a class of generalised Lane-Emden equations, in the context of gradient field theory, which remain valid in the three scenarios discussed here.