A coupling prescription for post-Newtonian corrections in Quantum Mechanics

TL;DR

This paper develops a covariant framework for deriving post-Newtonian corrections to the Schrödinger equation in curved spacetime, enabling systematic coupling of quantum systems to gravity in the intermediate regime.

Contribution

It introduces a generic coupling prescription for quantum systems to gravity using a $1/c^2$ expansion, bridging Newtonian gravity and General Relativity with geometric focus.

Findings

Derived Schrödinger equation with post-Newtonian corrections

Applied framework to Kerr metric, obtaining a Hartle–Thorne-like metric

Identified potentially measurable quantum gravitational effects

Abstract

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 Schr\"odinger's equation on arbitrary gravitational backgrounds based on a $1/c^2$ 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 $1/c^2$ 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 $1/c^2$ expansion of the complex Klein--Gordon field, we devise a prescription that allows us to derive the Schr\"odinger equation and its post-Newtonian corrections on a gravitational background order-by-order in $1/c^2$. We also demonstrate that these results can be obtained from a $1/c^2$ expansion of the complex Klein--Gordon Lagrangian. We illustrate our methods by performing the $1/c^2$ expansion of the Kerr metric up to $\mathcal{O}(c^{-2})$, which leads to a special case of the Hartle--Thorne metric. The associated Schr\"odinger equation captures novel and potentially measurable effects.