On q-Deformed Spinning Relativistic Particle

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.