Geometric post-Newtonian description of massive spin-half particles in curved spacetime

authored by: Ashkan Alibabaei, Philip K. Schwartz, Domenico Giulini
Abstract: We consider the Dirac equation coupled to an external electromagnetic field in curved four-dimensional spacetime with a given timelike worldline \(\gamma\) representing a classical clock. We use generalised Fermi normal coordinates in a tubular neighbourhood of \(\gamma\) and expand the Dirac equation up to, and including, the second order in the dimensionless parameter given by the ratio of the geodesic distance to the radii defined by spacetime curvature, linear acceleration of \(\gamma\), and angular velocity of rotation of the employed spatial reference frame along \(\gamma\). With respect to the time measured by the clock \(\gamma\), we compute the Dirac Hamiltonian to that order. On top of this `weak-gravity' expansion we then perform a post-Newtonian expansion up to, and including, the second order of \(1/c\), corresponding to a `slow-velocity' expansion with respect to \(\gamma\). As a result of these combined expansions we give the weak-gravity post-Newtonian expression for the Pauli Hamiltonian of a spin-half particle in an external electromagnetic field. This extends and partially corrects recent results from the literature, which we discuss and compare in some detail.
Organisation(s): CRC 1227 Designed Quantum States of Matter (DQ-mat)
Institute of Quantum Optics
Institute of Theoretical Physics
External Organisation(s): Center of Applied Space Technology and Microgravity (ZARM)
University of Bremen
Type: Article
Journal: Classical and Quantum Gravity
Volume: 40
ISSN: 0264-9381
Publication date: 07.11.2023
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Physics and Astronomy (miscellaneous)
Electronic version(s): https://arxiv.org/abs/2307.04743 (Access: Open)
https://doi.org/10.1088/1361-6382/ad079c (Access: Open)

BibTeX

@article{a555dd4027a24346b3208f082725a1c7,
title = "Geometric post-Newtonian description of massive spin-half particles in curved spacetime",
abstract = "We consider the Dirac equation coupled to an external electromagnetic field in curved four-dimensional spacetime with a given timelike worldline \(\gamma\) representing a classical clock. We use generalised Fermi normal coordinates in a tubular neighbourhood of \(\gamma\) and expand the Dirac equation up to, and including, the second order in the dimensionless parameter given by the ratio of the geodesic distance to the radii defined by spacetime curvature, linear acceleration of \(\gamma\), and angular velocity of rotation of the employed spatial reference frame along \(\gamma\). With respect to the time measured by the clock \(\gamma\), we compute the Dirac Hamiltonian to that order. On top of this `weak-gravity' expansion we then perform a post-Newtonian expansion up to, and including, the second order of \(1/c\), corresponding to a `slow-velocity' expansion with respect to \(\gamma\). As a result of these combined expansions we give the weak-gravity post-Newtonian expression for the Pauli Hamiltonian of a spin-half particle in an external electromagnetic field. This extends and partially corrects recent results from the literature, which we discuss and compare in some detail.",
keywords = "quantum matter in gravity, post-Newtonian expansion, Dirac equation, generalised Fermi normal coordinates, formal WKB-like expansion",
author = "Ashkan Alibabaei and Schwartz, {Philip K.} and Domenico Giulini",
note = "This work was supported by the Deutsche Forschungsgemeinschaft via the Collaborative Research Centre 1227 'DQ-mat'—Project Number 274200144, Project A05. A A acknowledges funding from Trinity College, Cambridge via a Rouse Ball Travelling Studentship.",
year = "2023",
month = nov,
day = "7",
doi = "10.1088/1361-6382/ad079c",
language = "English",
volume = "40",
journal = "Classical and Quantum Gravity",
issn = "0264-9381",
publisher = "IOP Publishing Ltd.",
number = "23",
}