# Post-Newtonian corrections to Schr\"odinger equations in gravitational fields

## TL;DR

This paper develops higher-order post-Newtonian corrections to Schr"odinger equations for quantum particles in gravitational fields, extending previous approximations and comparing different quantization methods.

## Contribution

It introduces a systematic expansion of the Klein-Gordon equation to arbitrary order in 1/c, unifying different approaches to quantum particles in curved spacetime.

## Key findings

- Derived Schr"odinger equations with higher-order gravitational corrections.
- Showed equivalence of Klein-Gordon and canonical quantization results in stationary spacetimes.
- Extended previous non-relativistic expansions to arbitrary order in 1/c.

## Abstract

In this paper we extend the WKB-like `non-relativistic' expansion of the minimally coupled Klein--Gordon equation after Kiefer and Singh [1], L\"ammerzahl [2] and Giulini and Gro{\ss}ardt [3] to arbitrary order in $c^{-1}$, leading to Schr\"odinger equations describing a quantum particle in a general gravitational field, and compare the results with canonical quantisation of a free particle in curved spacetime, following Wajima et al. [4]. Furthermore, using a more operator-algebraic approach, the Klein--Gordon equation and the canonical quantisation method are shown to lead to the same results for some special terms in the Hamiltonian describing a single particle in a general stationary spacetime, without any `non-relativistic' expansion.