Geometric phase for Dirac Hamiltonian under gravitational fields in the nonrelativistic regime

Abstract

We show the appearance of geometric phase in a Dirac particle traversing in nonrelativistic limit in a time-independent gravitational field. This turns out to be similar to the one originally described as a geometric phase in magnetic fields. We explore the geometric phase in the Kerr and Schwarzschild geometries, which have significant astrophysical implications. Nevertheless, the work can be extended to any spacetime background including that of time-dependent. In the Kerr background, i.e. around a rotating black hole, geometric phase reveals both the Aharonov–Bohm effect and Pancharatnam–Berry phase. However, in a Schwarzschild geometry, i.e. around a nonrotating black hole, only the latter emerges. We expect that our assertions can be validated in both the strong gravity scenarios, like the spacetime around black holes, and weak gravity environment around Earth.

Pseudo-supersymmetric approach to the Dirac operator in the Schwarzschild spacetime

“…and m is the mass of the fermion. In order not to change equation ( 8), the left handside of it can be multipled by γ 0 , γ 0 γ 0 = g 00 , then, it can be divided by g −1 00 i∂ 0 ψ = Hψ (10) where…”

“…There are no bound state solutions of the Dirac equation in Schwarzschild spacetime regardless of the value of the mass [7]. In some studies, approximative analytic solutions for the bound states are found [8], tunneling spectrum of vector particles and the Hawking temperature is investigated in [9], geometric phase in pseudo-Hermitian quantum mechanics is investigated [10], stationary solutions are discussed in [11]. QNMs can be described as the eigenmodes that show dissipative oscillations through spacetime and they can be produced by the perturbation theory [12].…”