Post-Newtonian Description of Quantum Systems in Gravitational Fields

Abstract

This thesis deals with the systematic treatment of quantum-mechanical systems situated in post-Newtonian gravitational fields. At first, we develop a framework of geometric background structures that define the notions of a post-Newtonian expansion and of weak gravitational fields. Next, we consider the description of single quantum particles under gravity, before continuing with a simple composite system. Starting from clearly spelled-out assumptions, our systematic approach allows to properly derive the post-Newtonian coupling of quantum-mechanical systems to gravity based on first principles. This sets it apart from other, more heuristic approaches that are commonly employed, for example, in the description of quantum-optical experiments under gravitational influence.

Regarding single particles, we compare simple canonical quantisation of a free particle in curved spacetime to formal expansions of the minimally coupled Klein–Gordon equation, which may be motivated from the framework of quantum field theory in curved spacetimes. Specifically, we develop a general WKB -like post-Newtonian expansion of the Klein–Gordon equation to arbitrary order in the inverse of the velocity of light. Furthermore, for stationary spacetimes, we show that the Hamiltonians arising from expansions of the Klein–Gordon equation and from canonical quantisation agree up to linear order in particle momentum, independent of any expansion in the inverse of the velocity of light.

Concerning the topic of composite systems, we perform a fully detailed systematic derivation of the first order post-Newtonian quantum Hamiltonian describing the dynamics of an electromagnetically bound two-particle system which is situated in external electromagnetic and gravitational fields. This calculation is based on previous work by Sonnleitner and Barnett, which we significantly extend by the inclusion of a weak gravitational field as described by the Eddington–Robertson parametrised post-Newtonian metric.

In the last, independent part of the thesis, we prove two uniqueness results characterising the Newton–Wigner position observable for Poincaré-invariant classical Hamiltonian systems: one is a direct classical analogue of the well-known quantum Newton–Wigner theorem, and the other clarifies the geometric interpretation of the Newton–Wigner position as ‘centre of spin’, as proposed by Fleming in 1965.