On $q$-Deformed Spinning Relativistic Particle

TL;DR

This paper explores a $q$-deformed spinning relativistic particle, revealing how its gauge symmetries and invariances depend on the deformation parameter $q$, with special cases at $q= \\pm 1$.

Contribution

It introduces three equivalent $q$-deformed Lagrangians for a spinning relativistic particle and analyzes their gauge symmetries and invariances, highlighting the special role of $q= \\pm 1$.

Findings

Gauge symmetries are equivalent only at $q= \\pm 1$.

$q$-commutator of supersymmetric transformations generates reparametrization at $q= \\pm 1$.

Solutions respect $GL_{\\surd q}(1|1)$ and $GL_q(2)$ invariances.

Abstract

A $q$-deformed free spinning relativistic particle is discussed in the framework of the Lagrangian formalism. Three equivalent Lagrangians are obtained for this system which are endowed with $q$-deformed local (super)gauge symmetries and reparametrization invariance. It is demonstrated that these symmetries are on-shell equivalent only for $ q = \pm1 $ under particular identification of the transformation parameters. The same condition ($ q=\pm1 $) emerges due to the requirement that the $q$-commutator of two supersymmetric gauge transformations should generate a reparametrization plus a supersymmetric gauge transformation. For a specific gauge choice, the solutions for equations of motion respect $GL_{\surd q}(1|1)$ and $GL_{q}(2)$ invariances for any arbitrary value of the evolution parameter characterizing the quantum super world-line.