Foldy--Wouthuysen Transformation of the Generalized Dirac Equation in Symmetric Teleparallel Gravity
Abstract
We investigate the non-relativistic limit of the generalized Dirac equation in a weak, static, and spherically symmetric background of symmetric teleparallel gravity. The underlying generalized spinor connection incorporates the complete Clifford-algebra basis and introduces additional couplings to the non-metricity sector beyond those of the conventional Dirac theory. Working in the coincident gauge and adopting the weak-field Schwarzschild geometry in isotropic coordinates, we derive the corresponding generalized Dirac Hamiltonian and perform successive Foldy--Wouthuysen transformations up to order 1/m2, retaining terms to first order in the gravitational potential and its spatial derivatives. The resulting block-diagonal Hamiltonian contains not only the expected gravitational counterparts of the kinetic, spin--orbit, and Darwin interactions, but also additional operator structures generated by the generalized spinor connection. In particular, direct spin--gravity, anisotropic spin--momentum--gravity, and tidal spin--momentum couplings arise naturally from the generalized metric-affine interaction. We further perform an order-of-magnitude analysis for an electron in the Earth's weak gravitational field to justify the adopted truncation of the inverse-mass expansion. These results demonstrate that the generalized Dirac equation in a symmetric teleparallel background gives rise to new low-energy interaction channels involving the fermion spin, momentum, and spatial derivatives of the gravitational field. The resulting effective Hamiltonian provides a framework for exploring phenomenological constraints on the additional couplings entering the generalized spinor connection.
Versions
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BibTeX
@misc{ugur2026foldywouthuysen,
title = {{Foldy--Wouthuysen Transformation of the Generalized Dirac Equation in Symmetric Teleparallel Gravity}},
author = {Sabutay Ugur and Muzaffer Adak and Ali Bagci and Caglar Pala},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.27889},
eprint = {2607.27889},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.27889},
}
RIS
TY - UNPB
TI - Foldy--Wouthuysen Transformation of the Generalized Dirac Equation in Symmetric Teleparallel Gravity
AU - Sabutay Ugur
AU - Muzaffer Adak
AU - Ali Bagci
AU - Caglar Pala
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2607.27889
UR - https://arxiv.org/abs/2607.27889
AB - We investigate the non-relativistic limit of the generalized Dirac equation in a weak, static, and spherically symmetric background of symmetric teleparallel gravity. The underlying generalized spinor connection incorporates the complete Clifford-algebra basis and introduces additional couplings to the non-metricity sector beyond those of the conventional Dirac theory. Working in the coincident gauge and adopting the weak-field Schwarzschild geometry in isotropic coordinates, we derive the corresponding generalized Dirac Hamiltonian and perform successive Foldy--Wouthuysen transformations up to order 1/m^2, retaining terms to first order in the gravitational potential and its spatial derivatives. The resulting block-diagonal Hamiltonian contains not only the expected gravitational counterparts of the kinetic, spin--orbit, and Darwin interactions, but also additional operator structures generated by the generalized spinor connection. In particular, direct spin--gravity, anisotropic spin--momentum--gravity, and tidal spin--momentum couplings arise naturally from the generalized metric-affine interaction. We further perform an order-of-magnitude analysis for an electron in the Earth's weak gravitational field to justify the adopted truncation of the inverse-mass expansion. These results demonstrate that the generalized Dirac equation in a symmetric teleparallel background gives rise to new low-energy interaction channels involving the fermion spin, momentum, and spatial derivatives of the gravitational field. The resulting effective Hamiltonian provides a framework for exploring phenomenological constraints on the additional couplings entering the generalized spinor connection.
ER -