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In quantum electrodynamics, local U(1) gauge invariance is implemented through the covariant derivative, which introduces a corresponding electromagnetic gauge field.
Similarly, we consider a geometric U(1) gauge symmetry in reciprocal space to derive a repulsive effective interaction for a Hamiltonian containing a term that linearly couples to the position operator.
This framework is motivated by work on inertial effects in rotating ions,
where the Hamiltonian in the rest frame of an electron bound to the ion was shown to contain terms linear in the position operator, coupling the Coriolis force due to rotation to the position of the electron [1, 2]. We demonstrate that this results in an emergent, repulsive Coulomb-like potential in reciprocal space.
In three spatial dimensions, this corresponds to an effective dipole-dipole force experienced by the electron, highlighting the geometric origin of effective forces in non-inertial quantum systems.
[1] R. Matthias Geilhufe. “Dynamic electron-phonon and spin-phonon interactions due to inertia”. In: Physical Review Research 4.1 (2022), p. L012004.
[2] Friedrich W. Hehl and Wei-Tou Ni. “Inertial effects of a Dirac particle”. In: Physical Review D 42.6 (1990), p. 2045.
