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.

arXiv comment22+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 (27)

• Universal decoherence due to gravitational time dilation
• Quantum interferometric visibility as a witness of general relativistic proper time
• Phase shift in atom interferometry due to spacetime curvature
• Action Principle for Newtonian Gravity
• The Schrödinger-Newton equation as non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields
• Non-Relativistic Gravity and its Coupling to Matter
• Gravitational redshift in quantum-clock interferometry
• Review on Non-Relativistic Gravity
• Post-Newtonian extension of the Newton-Cartan theory
• Post-Newtonian Hamiltonian description of an atom in a weak gravitational field
• Interference of Clocks: A Quantum Twin Paradox
• The Dirac equation in general relativity, a guide for calculations
• Covariant formulation of the post-1-Newtonian approximation to General Relativity
• Coupling Quantum Matter and Gravity
• Teleparallel Newton--Cartan gravity
• Post-Newtonian corrections to Schrödinger equations in gravitational fields
• Quantification of GR effects in muon g-2, EDM and other spin precession experiments
• Gravitational Effects in g Factor Measurements and High-Precision Spectroscopy: Limits of Einstein's Equivalence Principle
• 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
• Gravitational effects on geonium and free electron ${\mathrm{g}}_s$-factor measurements in a Penning trap
• Geometric post-Newtonian description of spin-half particles in curved spacetime
• Gravitational waves in scalar-tensor theory to one-and-a-half post-Newtonian order
• 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

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