Relativistic currents on ideal Aharonov–Bohm cylinders

Abstract

The relativistic theory of the Dirac fermions moving on cylinders in external Aharonov-Bohm field is built starting with a suitably restricted Dirac equation whose spin degrees of freedom are not affected. The exact solutions of this equation on finite or infinite Aharonov-Bohm cylinders allow one to derive the relativistic circular and longitudinal currents pointing out their principal features. It is shown that all the circular currents are related to the energy in the same manner on cylinders or rings either in the relativistic approach or in the non-relativistic one. The specific relativistic effect is the saturation of the circular currents for high values of the total angular momentum. Based on this property some approximative closed formulas are deduced for the total persistent current at T = 0 on finite Aharonov-Bohm cylinders. Moreover, it is shown that all the persistent currents on finite cylinders or rings have similar non-relativistic limits.

Relativistic Persistent Currents in Ideal Aharonov-Bohm Rings and Cylinders

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In this report we revisit the results obtained in [1,2] where the relativistic Aharonov-Bohm was studied for the first time. The method is based on the exact solutions of the complete (1+3)-dimensional Dirac equation of fermions moving in ideal Aharonov-Bohm (AB) rings and cylinders which are used for deriving the exact expressions of the relativistic partial currents.

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“…[1,2]. Herein we have shown first that the solutions of the Dirac equation in AB rings and cylinders may be determined as common eigenspinors of a complete systems of commuting operators including the energy, total angular momentum and a specific operator analogous to the well-known Dirac spherical operator of the relativistic central problems [36].…”

“…The relativistic partial currents we have obtained are related to the derivative of the relativistic energies as in the non-relativistic case but, in contrast with this, there appears a crucial difference, i. e. in the non-relativistic theory the partial currents are proportional to the angular momentum while in our approach the relativistic currents tend to saturation in the limit of high total angular momenta. For this reason we have reconsidered the problem of the relativistic persistent currents at T = 0 proposing an approximative analytic formula that matches the numerical calculations with a satisfactory accuracy [1,2].…”

“…Notice that this report brings together the texts of Refs. [1,2] with some improvements and unitary notations. …”

Aharonov–Bohm rings in de Sitter expanding universe

“…In the non-relativistic approach as well as in the relativistic theory of AB rings in Minkowski spacetime [31,32] the partial currents are proportional with the derivative of energy with respect to β. Therefore, the first step is to investigate if this fundamental relation holds in the dS geometry too.…”