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arXiv:2010.05945 · International Journal of Modern Physics D

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

Tanuman Ghosh, Banibrata N. Mukhopadhyay

Published Aug 4, 2021• 6 citations
arXivHTMLPDF

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.

DOI
10.1142/s0218271821500905

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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.
It is well-known that Dirac particles gain geometric phase, namely Berry phase, while moving in an electromagnetic field. Researchers have already shown covariant formalism for the Berry connection due to an electromagnetic field. A similar effect is expected to happen due to the presence of Gravity. We use WKB approximation to develop a covariant formalism of Berry-like connection in the presence of Einstein gravity, which can be further used to describe the Berry-like phase or simply Berry phase. We also extend this formalism for massless Dirac particles (Weyl particles).Then we further show that this connection can be split into two parts, one of which vanishes when the metric is spherically symmetric and thus can be linked to the Aharonov-Bohm-like effect in the 3 + 1 formalism. At the same time, the other term can be related to the Pancharatnam-Berry like effect.
Abstract We study the Hawking effect in terms of the geometric phase for a two-level atom coupled to vacuum fluctuations in the background of Kerr black hole in a gedanken experiment. Since the atom interacts with a bath of fluctuating massless scalar field, its geometric phase for the nonunitary evolution turns out to be affected by the spacetime curvature that backscatters the vacuum field modes. We examine the geometric phase for two kinds of vacua, i.e. the Unruh vacuum and the Candelas–Chrzanowski–Howard vacuum. We find that for both of these two cases, the geometric phase of the static atom locating outside the infinite redshift surface (or the stationary atom locating inside the ergosphere) exposes that the atom behaves as if it were immersed in a thermal radiation at the Hawking temperature from the Kerr black hole. Therefore, our results show that one in principle can reveal the Hawking effect in the Kerr black hole by analyzing the correction of the geometric phase as opposed to that in a flat spacetime.
The Berry phase [3] is traditionally understood as a geometric phase acquired during cyclic adiabatic evolution, often interpreted as the flux of an effective “magnetic field” through a solid angle in parameter space. In this thesis, we investigate situations in which a geometric phase arises even when no spatial solid angle is enclosed, revealing the limitations of the standard purely spatial interpretation. We first review the Berry phase for a spin-½ particle in a slowly varying magnetic field and analyze scenarios in which the system’s eigenstate is changed during the adiabatic evolution. We show that the resulting phase can be understood through a spacetime formulation analogous to the electrodynamic Aharonov–Bohm effect [1], where both “analogous magnetic” and “analogous electric” fields contribute through a spacetime surface integral. This framework clarifies how nonzero geometric phases can occur even when the projected spatial solid angle vanishes. We then extend this perspective to the Jahn–Teller 𝐸 ⊗ 𝑒 system. While a closed loop around the conical intersection yields the familiar π phase, we show that nontrivial geometric phases can also emerge in paths with zero net enclosed solid angle. Advisors: Herman Batelaan and Ilya Kravchenko
We study a relativistic charged Dirac particle moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Dirac equation with the time-dependent Hamiltonian can be reduced to a Dirac-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Dirac equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explictly available. For a slowly rotating magnetic field, the eigenvalue problem of the effective Hamiltonian is solved approximately and the geometric phases are calculated. The same problem for a charged or neutral Dirac particle with an anomalous magnetic moment is discussed briefly.
We study a relativistic charged Dirac particle moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Dirac equation with the time-dependent Hamiltonian can be reduced to a Dirac-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Dirac equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explictly available. For a slowly rotating magnetic field, the eigenvalue problem of the effective Hamiltonian is solved approximately and the geometric phases are calculated. The same problem for a charged or neutral Dirac particle with an anomalous magnetic moment is discussed briefly.
In this contribution, we study the quantum dynamics of a neutral particle in the presence of a topological defect. We investigate the appearance of a geometric phase in the relativistic quantum dynamics of a neutral particle which possesses permanent magnetic and electric dipole moments in the presence of an electromagnetic field in this curved background. We also study the influence of noninertial effects of a rotating frame and and obtain several contributions to the relativistic geometric phase due to the noninertial effects and the topology of spacetime. The analogous Aharonov-Casher and He-Mckellar-Wilkens effects are investigated in the nonrelativistic dynamics with the presence of a topological defect and under the influence of noninertial effects. We also obtain effects analogous to the Sagnac effect and Mashhoon effect due to the presence of the topological defect.