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.
Highlights
Suppose we are given a quantum-mechanical system whose time evolution in the absence of gravity is known in terms of the ordinary time-dependent Schrödinger equation
We have shown how to derive a Schrödinger equation with post-Newtonian correction terms describing a single quantum particle in a general curved background spacetime by means of a WKB-like formal expansion of the minimally coupled Klein–Gordon equation
We extended this method to account for, in principle, terms of arbitrary orders in c−1, it gets recursive at higher orders, making it computationally more difficult to handle than methods based on formal quantisation of the classical description of the particle
Summary
Suppose we are given a quantum-mechanical system whose time evolution in the absence of gravity is known in terms of the ordinary time-dependent Schrödinger equation. The second, fundamentally different approach takes a field-theoretic perspective and derives the Schrödinger equation as an equation for the positive frequency solutions of the minimally coupled classical Klein–Gordon equation3 This is accomplished by Kiefer and Singh [1], Lämmerzahl [2] and Giulini and Großardt [3] by making a WKB-like ansatz for the Klein–Gordon field and (formally) expanding the Klein–Gordon equation in powers of c−1, in the end viewing the Klein–Gordon theory as a (formal) deformation of the Schrödinger theory, implementing the deformation of Galilei to Poincaré symmetry well-known at the level of Lie algebras [17]. Since we are concerned mostly with conceptual questions, we will generally not be mathematically very rigorous, and in particular not mention domains of definition of operators
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