Particle on a Torus Knot: Constrained Dynamics and Semi-Classical Quantization in a Magnetic Field

TL;DR

This paper investigates the constrained dynamics and semi-classical quantization of a particle on a torus knot in a magnetic field, revealing non-planar effects and fractional angular momentum through advanced symplectic and quantization analyses.

Contribution

It derives the symplectic structure and explores semi-classical quantization of a particle on a torus knot, highlighting non-planar features and fractional spin effects.

Findings

Derived Dirac brackets in different coordinate systems.

Identified non-planar corrections to fractional angular momentum.

Compared EBK and Bohr-Sommerfeld quantization schemes.

Abstract

Kinematics and dynamics of a particle moving on a torus knot poses an interesting problem as a constrained system. In the first part of the paper we have derived the modified symplectic structure or Dirac brackets of the above model in Dirac's Hamiltonian framework, both in toroidal and Cartesian coordinate systems. This algebra has been used to study the dynamics, in particular small fluctuations in motion around a specific torus. The spatial symmetries of the system have also been studied. In the second part of the paper we have considered the quantum theory of a charge moving in a torus knot in the presence of a uniform magnetic field along the axis of the torus in a semiclassical quantization framework. We exploit the Einstein - Brillouin - Keller (EBK) scheme of quantization that is appropriate for multidimensional systems. Embedding of the knot on a specific torus is inherently two dimensional that gives rise to two quantization conditions. This shows that although the system, after imposing the knot condition reduces to a one dimensional system, even then it has manifest non-planar features which shows up again in the study of fractional angular momentum. Finally we compare the results obtained from EBK (multi-dimensional) and Bohr-Sommerfeld (single dimensional) schemes. The energy levels and fractional spin depend on the torus knot parameters that specifies its non-planar features. Interestingly, we show that there can be non-planar corrections to the planar anyon-like fractional spin.