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; Lämmerzahl 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.
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