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.

22+3+13 pages (main text, references, appendix), to appear in Classical and Quantum Gravity. v2: corrected author spelling in arXiv metadata. v3: added references, extended introductory discussion, corrected typos

References in corpus (12)

• The Schrödinger-Newton equation as non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields
• Review on Non-Relativistic Gravity
• Coupling Quantum Matter and Gravity
• Teleparallel Newton--Cartan gravity
• Inertial and gravitational effects on a geonium atom
• A Wavefunction Description for a Localized Quantum Particle in Curved Spacetimes
• Classical perspectives on the Newton--Wigner position observable
• Post-Newtonian Description of Quantum Systems in Gravitational Fields
• Geometric post-Newtonian description of spin-half particles in curved spacetime
• qBounce: Systematic shifts of transition frequencies of gravitational states of ultra-cold neutrons using Ramsey gravity resonance spectroscopy
• UGR tests with atomic clocks and atom interferometers
• Gravitational waves in scalar-tensor theory to one-and-a-half post-Newtonian order

Cited by in corpus (4)

• Coupling Quantum Matter and Gravity
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• Atom interferometers in weakly curved spacetimes using Bragg diffraction and Bloch oscillations
• Finite Pulse-Time Effects in Long-Baseline Quantum Clock Interferometry