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 = ±1 under a particular identification of the transformation parameters. The same condition (q = ±1) 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 the equations of motion respect GLνq(11) and GLq(2) invariances for any arbitrary value of the evolution parameter characterizing the quantum super world-line.