The bound-state Aharonov–Bohm effect around a cosmic string revisited

Abstract

In this article we observe that the self-adjoint extension of the Hamiltonian of a particle moving around a shielded cosmic string gives rise to a gravitational analogue of the bound state AharonovBohm effect.PACS numbers: 98.80. Cq,11.27.+d,03.65.Vf When a quantum particle is confined between two impenetrable concentric cylindrical walls, threaded by a magnetic flux tube along their common axis, its energy spectrum presents a dependence on the flux. This is the so-called bound state Aharonov-Bohm effect [1]. A gravitational analogue of this effect, with a cosmic string replacing the flux tube, has been studied in references [2,3]. This is a topological effect which also appears around disclinations in elastic solids [4]. In this article we revisit the bound state gravitational Aharonov-Bohm effect from the point of view of self-adjoint extensions of the Hamiltonian, without need of a confining wall.Fermions in the presence of an Aharonov-Bohm field may have a bound state appearing from the self-adjoint extension of the Dirac Hamiltonian, as shown in [5], for a specific range of the extension parameter. As shown in the detailed analysis of Kay and Studer [6], a bound state may appear for a quantum particle moving about a cosmic string due to the self-adjoint extension of its Hamiltonian. On the other hand, the presence of walls may also lead to interesting effects related to the selfadjoint extensions as well [7]. In this article we analyze the idealized situation of a free non-relativistic quantum particle in the presence of a cosmic string and therefore, moving in the background spacetime of metricshielded by an impenetrable cylindrical wall, of radius a, concentric with the string. In equation (1) the angle θ may vary in the range [0, 2π] and the parameter α = 1 − 4Gµ < 1, where µ is the string linear mass density, characterizes the string. α effectively introduces an angular deficit 2π(1 − α) in the Minkowski geometry. The motion of the particle is governed by the Hamiltonian in cylindrical coordinatesSince we are effectively excluding a cylindrical portion of the space accessible to the particle we must guarantee that the Hamiltonian is self-adjoint in the region of with dimensions n + and n − , respectively, which are called deficiency indices of H. A necessary and sufficient condition for H to be self-adjoint is that n + = n − = 0. On the other hand, if n + = n − ≥ 1 then H has an infinite number of self-adjoint extensions parametrized by a unitary n × n matrix, where n = n + = n − .It is obvious that (n + , n − ) = (0, 0) for − ∂ 2 ∂z 2 on L 2 (R), the Hilbert space of square integrable functions on R. Therefore − ∂ 2 ∂z 2 is essentially self-adjoint. This being considered, we have also that translational invariance along the string axis (z-direction) reduces the problem to effectively two spacial dimensions.Since the typical size of a cosmic string radius is of the order of the GUT length scale (10 −30 cm) we can make the radius of the shielding cylinder a → 0 without loss of generality. Effectively...

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