Particle on a torus knot: Symplectic analysis

TL;DR

This paper applies symplectic analysis and BRST formalism to quantize a particle constrained on a torus knot, deriving the constraint spectrum, gauge invariance, and symmetry charges.

Contribution

It introduces a geometrically motivated symplectic approach to gauge invariance and BRST symmetry for a particle on a torus knot, avoiding additional variables.

Findings

Derived the constraint spectrum for the particle on a torus knot.

Reformulated the theory into a gauge-invariant form without extra variables.

Constructed conserved BRST charges satisfying physicality criteria.

Abstract

We quantize a particle confined to move on a torus knot satisfying constraint condition ($p \theta + q \phi) \approx 0$, within the context of a geometrically motivated approach - the Faddeev-Jackiw formalism. We also deduce the constraint spectrum and discern the basic brackets of the theory. We further reformulate the original gauge non-invariant theory into a physically equivalent gauge theory, which is free from any additional Wess-Zumino variables, by employing symplectic gauge invariant formalism. In addition, we analyze the reformulated gauge invariant theory within the framework of BRST formalism to establish the off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetries. Finally, we construct the conserved (anti-)BRST charges which satisfy the physicality criteria and turn out to be the generators of corresponding symmetries.