Post-Newtonian corrections to Schrödinger equations in gravitational fields

Abstract

In this paper we extend the WKB-like 'non-relativistic' expansion of the minimally coupled Klein-Gordon equation after (Kiefer and Singh 1991 Phys. Rev. D 44 1067-76; Lammerzahl 1995 Phys. Lett. A 203 12-7; Giulini and Großardt 2012 Class. Quantum Grav. 29 215010) to arbitrary order in c -1, leading to Schrödinger 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 1997 Phys. Rev. D 55 1964-70). 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.