Aharonov-Bohm Scattering From Knots

TL;DR

This paper explores the quantum scattering effects of particles around knotted magnetic fluxes, using multipole expansions to analyze the Aharonov-Bohm effect in complex topologies like knots, revealing knottedness signatures in the scattering matrix.

Contribution

It introduces a method to analyze Aharonov-Bohm scattering from knotted solenoids using multipole expansions and calculates the S-matrix, highlighting the influence of knot topology.

Findings

The S-matrix reflects knottedness at the octopole order.

A multipole expansion approach enables progress in complex topological scattering problems.

A factorization property is observed for torus knots.

Abstract

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.