QUANTIZATION OF A $q$-DEFORMED FREE RELATIVISTIC PARTICLE

TL;DR

This paper explores the BRST quantization of a $q$-deformed relativistic particle, revealing that the deformation parameter must be $ ext{±}1$ for consistency, thus linking deformation to symmetry preservation.

Contribution

It introduces a BRST formalism for a $q$-deformed relativistic particle and establishes the restriction $q= ext{±}1$ for algebraic and conservation consistency.

Findings

BRST charges are on-shell equivalent only for $q= ext{±}1$

Conservation of BRST charge requires $q= ext{±}1$

Solutions preserve $GL_q(2)$ invariance on-shell

Abstract

A $q$-deformed free scalar relativistic particle is discussed in the framework of the BRST formalism. The $q$-deformed local gauge symmetry and reparametrization invariance of the first-order Lagrangian have been exploited for the BRST quantization of this system on a $GL_{q}(2)$ invariant quantum world-line. The on-shell equivalence of these BRST charges requires the deformation parameter to be $ \pm 1 $ under certain identifications.The same restriction ($ q= \pm 1 $) emerges from the conservation of the $q$-deformed BRST charge on an arbitrary (unconstrained) manifold and the validity of the BRST algebra. The solutions for the equations of motion respect $GL_{q}(2)$ invariance on the mass-shell at any arbitrary value of the evolution parameter characterizing the quantum world-line.