A coupling prescription for post-Newtonian corrections in Quantum Mechanics

The interplay between quantum theory and general relativity remains one of the main challenges of modern physics. A renewed interest in the low-energy limit is driven by the prospect of new experiments that could probe this interface. Here we develop a covariant framework for expressing post-Newtonian corrections to Schrödinger's equation on arbitrary gravitational backgrounds based on a 1/c² expansion of Lorentzian geometry, where c is the speed of light. Our framework provides a generic coupling prescription of quantum systems to gravity that is valid in the intermediate regime between Newtonian gravity and General Relativity, and that retains the focus on geometry. At each order in 1/c² this produces a nonrelativistic geometry to which quantum systems at that order couple. By considering the gauge symmetries of both the nonrelativistic geometries and the 1/c² expansion of the complex Klein--Gordon field, we devise a prescription that allows us to derive the Schrödinger equation and its post-Newtonian corrections on a gravitational background order-by-order in 1/c². We also demonstrate that these results can be obtained from a 1/c² expansion of the complex Klein--Gordon Lagrangian. We illustrate our methods by performing the 1/c² expansion of the Kerr metric up to mathcal{O}\(c^-2\), which leads to a special case of the Hartle--Thorne metric. The associated Schrödinger equation captures novel and potentially measurable effects.