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 [1], Lämmerzahl [2] and Giulini and Großardt [3] 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. [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.

28 pages, published in Classical and Quantum Gravity. v2: corrected typos, revised section 2 and beginning of section 3, extended discussion of results in section 3. v3: corrected reference [10]