Post-Newtonian Description of Quantum Systems in Gravitational Fields

TL;DR

This thesis systematically derives the post-Newtonian coupling of quantum systems to gravity from first principles, compares quantisation methods, and explores the position observable in relativistic quantum mechanics.

Contribution

It provides a first-principles systematic approach to quantum-gravity coupling, develops a WKB-like expansion of the Klein-Gordon equation, and characterizes the Newton--Wigner position in relativistic systems.

Findings

Post-Newtonian quantum Hamiltonian for two-particle systems derived

Agreement between Klein-Gordon and canonical quantisation Hamiltonians up to linear order

Uniqueness theorems for Newton--Wigner position in classical and quantum systems

Abstract

This thesis deals with the systematic treatment of quantum-mechanical systems in post-Newtonian gravitational fields. Starting from clearly spelled-out assumptions, employing a framework of geometric background structures defining the notion of a post-Newtonian expansion, 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 more heuristic approaches that are commonly employed, for example, in the description of quantum-optical experiments under gravity. 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 QFT in curved spacetimes. Specifically, we develop a general WKB-like post-Newtonian expansion of the KG equation to arbitrary order in $c^{-1}$. Furthermore, for stationary spacetimes, we show that the Hamiltonians arising from expansions of the KG equation and from canonical quantisation agree up to linear order in particle momentum, independent of any expansion in $c^{-1}$. Concerning 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 situated in external electromagnetic and gravitational fields, the latter being described by the Eddington-Robertson PPN metric. In the last, independent part of the thesis, we prove two uniqueness results characterising the Newton--Wigner position observable for Poincar\'e-invariant classical Hamiltonian systems: one is a direct classical analogue of the 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.