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arXiv:2212.04763

Covariant formalism for the Berry connection due to gravity

Achal Kumar, Banibrata N. Mukhopadhyay

Published Dec 9, 2022• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2212.04763

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hep-thCross-listedarXiv:2610.41138•Nov 2, 2026
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.
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.
A geometric interpretation of Berry’s phase, its Wilczek–Zee non-Abelian generalization, and Aharonov–Bohm effect is given in terms of connections on principal fiber bundles. It is shown that a principal fiber bundle in all cases can be trivial while the connection and its holonomy group are nontrivial. Therefore the main role is played not by topological effects but geometrical ones. The Berry phase is also shown not to be connected to the adiabatic changes of the Hamiltonian parameters. 1
We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.
In this work, we make a review of the literature that the so-called Berry’s geometric phase. A recent issue, which has been employed in various areas of physics. For this, we use, in a number of references, in which many were critical to the theoretical foundation of this work. At first, we focus on the study of the time evolution of quantum systems governed by Hamiltonians operators whose time dependence is manifested by a set of parameters called classical fields. The adiabatic approximation, is justified mathematically when demonstrated the adiabatic theorem. Under these conditions, we obtain geometric phases - Berry’s phase - associated with these systems in different representations, each displaying its particularity. Finally, as a particular case of geometric phase, we studi from the viewpoint of duality transformations of Maxwell’s equations, the dual Aharonov-Bohm effect, extracting the geometric Berry’s phase associated with the dynamics of the magnetic monopole in the presence of the electric vector potential
Quantum geometry is conventionally understood in parameter or momentum space and need not be related to a geometry in configuration space. Here we show for Dirac fermions in a background electromagnetic field that an induced Lorentzian geometry is generated from the fermionic quantum metric. We study a Lorentz covariant phase space worldline path integral construction, whose application of the adiabatic theorem onto spin degrees of freedom leads to the appearance of the Berry phase at O(ℏ)O(\hbar)O(ℏ), and the quantum metric at O(ℏ2)O(\hbar^2)O(ℏ2) in the worldline action. Both the Feynman and retarded/advanced propagators are examined in 3+1 and 2+1 dimensions, whose different on-shell implementations lead to related quantum metrics; QED/QED3{}_33​ and condensed matter Dirac theory are considered. An effective theory is found by integrating out the momenta degrees of freedom, which gives rise to an emergent Lorentzian metric in configuration space, whose perturbation to the flat metric is dependent on the quantum metric and the background field strength. Apart from the spin degrees of freedom encoded into the Berry phase, the effective theory describes a scalar particle moving in the induced quantum geometry. Phenomenological applications to the covariant geometry are further studied using spectral holonomy in a gapped 3-dimensional Dirac material.