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Annals of Physics

Calculation of non-relativistic scattering of charged scalar particles from a classical Aharonov-Bohm solenoid without an explicit gauge choice

A. Y. Shiekh. Blackett Laboratory

Published Jan 1, 1986• 0 citations
Publisher
DOI
10.1016/0003-4916(86)90060-6

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Relativistic corrections to the Aharonov-Bohm scattering

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Aharonov-Bohm Scattering From Knots

Kaustav Giri, V. V. Sreedhar

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.

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Scattering matrix for the two-solenoid Aharonov-Bohm effect

Pavel Šťovíček

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Is the Aharonov-Bohm phase shift for a non-closed path a measurable quantity ?

M. Wakamatsu

There recently appear some interesting attempts to explain the AB-effect through the interaction between the charged particle and the solenoid current mediated by the exchange of a virtual photon. A vital assumption of this approach is that AB-phase shift is proportional to the change of the interaction energy between the charged particle and solenoid along the path of the moving charge. Accordingly, they insist that the AB-phase change along a path does not depend on the gauge choice so that the AB-phase shift for a non-closed path is in principle measurable. We however notice the existence of two fairly different discussions on the interaction energy between the solenoid and a charge particle, the one is due to Boyer and the other is due to Saldanha and others. In the present paper, based on a self-contained quantum mechanical treatment of the combined system of a solenoid, a charged particle, and the quantized electromagnetic fields, we show that both interaction energies of Boyer and of Saldanha are in fact gauge invariant at least for non-singular gauge transformations but they are destined to cancel each other. Our analysis rather shows that the origin of the AB-phase can be traced back to other part of our effective Hamiltonian. Furthermore, based on the path-integral formalism with our effective Lagrangian, we explicitly demonstrate that the AB-phase shift for a non-closed path is not a gauge-variant quantity, which means that it would not correspond to direct experimental observables.

arXiv:2305.15658•May 25, 2023•1 citation

Quantum effects in the scattering by the magnetic field of a solenoid

Gabriela Murguía, Moreno, Matias

We present a relativistic quantum calculation at first order in perturbation theory of the differential cross section for a Dirac particle scattered by the magnetic field of a solenoid. The resulting cross section is symmetric in the scattering angle as those obtained by Aharonov and Bohm (AB) in the string limit and by Landau and Lifshitz (LL) for the non relativistic case. We show that taking pr_0 sin(theta/2) << 1 in our expression of the differential cross section it reduces to that one reported by AB, and if additionally we assume theta << 1 our result becomes the one obtained by LL. However, these limits are explicitly singular in hbar as opposed to our initial result. We analyse the singular behavior in hbar and show that the perturbative Planck limit (hbar -> 0) is consistent, contrarily to those of the AB and LL expressions.

arXiv:physics/0202020•Feb 7, 2002•0 citations
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.
There recently appear some interesting attempts to explain the AB-effect through the interaction between the charged particle and the solenoid current mediated by the exchange of a virtual photon. A vital assumption of this approach is that AB-phase shift is proportional to the change of the interaction energy between the charged particle and solenoid along the path of the moving charge. Accordingly, they insist that the AB-phase change along a path does not depend on the gauge choice so that the AB-phase shift for a non-closed path is in principle measurable. We however notice the existence of two fairly different discussions on the interaction energy between the solenoid and a charge particle, the one is due to Boyer and the other is due to Saldanha and others. In the present paper, based on a self-contained quantum mechanical treatment of the combined system of a solenoid, a charged particle, and the quantized electromagnetic fields, we show that both interaction energies of Boyer and of Saldanha are in fact gauge invariant at least for non-singular gauge transformations but they are destined to cancel each other. Our analysis rather shows that the origin of the AB-phase can be traced back to other part of our effective Hamiltonian. Furthermore, based on the path-integral formalism with our effective Lagrangian, we explicitly demonstrate that the AB-phase shift for a non-closed path is not a gauge-variant quantity, which means that it would not correspond to direct experimental observables.
We present a relativistic quantum calculation at first order in perturbation theory of the differential cross section for a Dirac particle scattered by the magnetic field of a solenoid. The resulting cross section is symmetric in the scattering angle as those obtained by Aharonov and Bohm (AB) in the string limit and by Landau and Lifshitz (LL) for the non relativistic case. We show that taking pr_0 sin(theta/2) << 1 in our expression of the differential cross section it reduces to that one reported by AB, and if additionally we assume theta << 1 our result becomes the one obtained by LL. However, these limits are explicitly singular in hbar as opposed to our initial result. We analyse the singular behavior in hbar and show that the perturbative Planck limit (hbar -> 0) is consistent, contrarily to those of the AB and LL expressions.