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Journal of Physics A Mathematical and General

Aharonov–Bohm quantum systems on a punctured 2-torus

Toshihiro Iwai, Yoshiro Yabu

Published Jan 11, 2006• 3 citations

Abstract

This paper deals with Aharonov–Bohm (A-B) quantum systems on a punctured two-dimensional torus from the geometric and the operator theoretic point of view. First, flat connections on the U (1)-bundles over the punctured 2-torus are studied, which serve as vector potentials for A-B effect magnetic fields. It is proved that the moduli space of flat connections is identified with the ( N + 1)-dimensional torus T N +1 , if the punctured torus has N > 0 pinholes at which solenoids are assumed to penetrate the 2-torus. For a given point of T N +1 , an associated flat connection is constructed in terms of the Weierstrass zeta function on C together with an inhomogeneous linear function on R 2 . A-B quantum systems are defined in terms of position operators and momentum operators coupled with the A-B potentials. Necessary and sufficient conditions are given for two A-B quantum systems to be unitarily equivalent. Further, the A-B Hamiltonian is defined and analysed from the viewpoint of operator theory. The deficiency indices of the A-B Hamiltonian are determined to be ( N + M , N + M ), where M is the number of solenoids whose fluxes are not quantized. Finally, the eigenvalue problem is studied for the A-B Hamiltonian with all fluxes quantized to obtain eigenvalues together with eigenfunctions which are described in terms of the Weierstrass sigma functions.

DOI: 10.1088/0305-4470/39/4/003

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This paper deals with Aharonov–Bohm (A-B) quantum systems on a punctured two-dimensional torus from the geometric and the operator theoretic point of view. First, flat connections on the U (1)-bundles over the punctured 2-torus are studied, which serve as vector potentials for A-B effect magnetic fields. It is proved that the moduli space of flat connections is identified with the ( N + 1)-dimensional torus T N +1 , if the punctured torus has N > 0 pinholes at which solenoids are assumed to penetrate the 2-torus. For a given point of T N +1 , an associated flat connection is constructed in terms of the Weierstrass zeta function on C together with an inhomogeneous linear function on R 2 . A-B quantum systems are defined in terms of position operators and momentum operators coupled with the A-B potentials. Necessary and sufficient conditions are given for two A-B quantum systems to be unitarily equivalent. Further, the A-B Hamiltonian is defined and analysed from the viewpoint of operator theory. The deficiency indices of the A-B Hamiltonian are determined to be ( N + M , N + M ), where M is the number of solenoids whose fluxes are not quantized. Finally, the eigenvalue problem is studied for the A-B Hamiltonian with all fluxes quantized to obtain eigenvalues together with eigenfunctions which are described in terms of the Weierstrass sigma functions.

In this paper we study the stationary scattering problem of the Aharonov–Bohm (AB) effect. To achieve this goal we construct a Hamiltonian from the most general representations of the Heisenberg algebra. Such representations are defined on a non-simply-connected manifold which we set as the flat circular annulus. By means of the von Neumann's self-adjoint extensions formalism, the scattering data are then provided. No solenoid is considered in this paper. The corresponding Hamiltonian is based on a topological quantum degree of freedom inherent in such representations. This variable stands for the magnetic vector gauge potential at quantum level. Our outcomes confirm the topological nature of this effect.

We discuss the Aharonov-Bohm (

A-B

) effect and the Dirac (

D

) monopole of magnetic charge

g={{1}\over{2}}

in the context of bundle theory, exhibiting a purely geometric relation between them. If

ξ_{A-B}

and

ξ_D

are the respective

U(1)

-bundles, we show that

ξ_{A-B}

is isomorphic to the pull-back of

ξ_D

induced by the inclusion of the corresponding base spaces

ι:(D_0^2)^*\to S^2

}. The fact that the

A-B

effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux

Φ_0={{2π}\over{\vert e\vert}}

associated with the electric charge

\vert e\vert

, reflects here as a consequence of the pull-back by

ι

of the Dirac connection in

ξ_D

to

ξ_{A-B}

, and the Dirac quantization condition. We also show the necessary vanishing in

ξ_{A-B}

of the pull-back of the Chern class

c_1

in

ξ_D

.

We discuss the Aharonov-Bohm (A -B) effect and the Dirac (D) monopole of magnetic charge g = 1 2 in the context of bundle theory, which allows to exhibit a deep geometric relation between them.If ξA-B and ξD are the respective U (1)-bundles, we show that ξA-B is isomorphic to the pull-back of ξD induced by the inclusion of the corresponding base spaces.The fact that the A -B effect disappears when the magnetic flux in the solenoid equals an integer number of times the quantum of flux associated with the unit of electric charge, reflects here as a consequence of the pull-back of the Dirac connection in ξD to ξA-B, and the Dirac quantization condition.

The celebrated Aharonov-Bohm effect is perhaps the first example in which the the interplay between classical topology and quantum theory was explored. This connection has continued to shed light on diverse areas of physics like quantum statistics, anomalies, condensed matter physics, and gauge theories. Several attempts were made to generalize the Aharonov-Bohm effect by modifying the simple solenoidal current distribution used by them to the case of multiple solenoids, and a toroidal solenoid, for example. A particularly ambitious task is to confine the magnetic flux to the interior of a knotted solenoid. While it is to be expected that a non-trivial phase factor will be picked up by the wave function of a charged particle travelling in the complement of the knot in three-dimensional space, the lack of symmetry defied attempts to explicitly solve the associated scattering problem. In this paper we report on a way to make progress towards this problem based on multipole expansions. The vector potential produced by a knot is obtained by making a multipole expansion, which is then used to calculate the S-matrix for the scattering of the charged particle, in the Born approximation. It is found that the S-matrix carries an imprint of the knottedness at the octopole order. For the case of a torus knot, a curious factorization property is seen to hold.

We present an application of the affine covariant integral quantization (ACIQ) (Adv. Oper. Theory, 5, 2020; Adv. Oper. Theory, 7, 2022) to quantum mechanics on the punctured plane. The associated four-dimensional phase space is identified with the similitude group SIM(2), which comprises translations, rotations, and dilations of the plane. Due to the topology of the punctured plane, our quantization procedure gives rise to an affine vector potential. This potential can be interpreted as the Aharonov-Bohm (AB) gauge field produced by an infinite solenoid. This observation supports a reinterpretation of the AB effect: it emerges from the topological constraint imposed by the impenetrable coil rather than from an externally applied classical gauge field. In addition to this gauge structure, ACIQ also generates a repulsive, centrifugal-like scalar potential, a feature already encountered when applying ACIQ to motion on the half-line, whose phase space is the open half-plane. These results provide a new perspective on the AB effect, highlighting the central roles of topology and symmetry in quantum mechanics.

In this paper, we apply covariant affine integral quantization to study motion in the 2D punctured plane. The associated four-dimensional phase space is identified with the similitude group SIM(2), representing transformations of the plane composed of translations, rotations, and dilations. Near the punctured point, we observe that the topology influencing quantum fluctuations gives rise to an affine vector potential, which can be interpreted as the Aharonov-Bohm (AB) gauge potential generated by an infinite coil. This finding suggests that the AB effect originates from the topology enforced by the impenetrable coil, rather than from a classical gauge potential. Our results offer a novel perspective on the AB effect, emphasizing the fundamental role of topology in quantum mechanics.