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A coupling prescription for post-Newtonian corrections in Quantum Mechanics

by Jelle Hartong, Emil Have, Niels A. Obers, Igor Pikovski

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Submission summary

Authors (as registered SciPost users): Emil Have · Niels Obers
Submission information
Preprint Link: scipost_202402_00005v2  (pdf)
Date accepted: 2024-03-11
Date submitted: 2024-03-05 19:30
Submitted by: Have, Emil
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical

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.

Author comments upon resubmission

We have attached the response to the second referee as a reply.

List of changes

See the response, which is attached as a reply to the second referee.

Published as SciPost Phys. 16, 088 (2024)


Reports on this Submission

Report #1 by Philip Schwartz (Referee 1) on 2024-3-7 (Invited Report)

Report

I thank the authors for their work in addressing my comments. I recommend publication of the revised manuscript in its current form.

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