We exploit the theoretical strength of the supervariable and Becchi-Rouet-Stora-Tyutin (BRST) formalisms to derive the proper (i.e., off-shell nilpotent and absolutely anticommuting) (anti-)BRST symmetry transformations for the reparameterization invariant model of a nonrelativistic (NR) free particle whose spacexand timetvariables are a function of an evolution parameterτ. The infinitesimal reparameterization (i.e., 1D diffeomorphism) symmetry transformation of our theory is defined w.r.t. this evolution parameterτ. We apply the modified Bonora-Tonin (BT) supervariable approach (MBTSA) as well as the (anti)chiral supervariable approach (ACSA) to BRST formalism to discuss various aspects of our present system. For this purpose, our 1D ordinary theory (parameterized byτ) is generalized onto a1,2-dimensional supermanifold which is characterized by the superspace coordinatesZM=τ,θ,θ¯where a pair of the Grassmannian variables satisfy the fermionic relationships:θ2=θ¯2=0,θ θ¯+θ¯ θ=0, andτis the bosonic evolution parameter. In the context of ACSA, we take into account only the1,1-dimensional (anti)chiral super submanifolds of the general1,2-dimensional supermanifold. The derivation of the universal Curci-Ferrari- (CF-) type restriction, from various underlying theoretical methods, is a novel observation in our present endeavor. Furthermore, we note that the form of the gauge-fixing and Faddeev-Popov ghost terms for our NR and non-SUSY system is exactly the same as that of the reparameterization invariant SUSY (i.e., spinning) and non-SUSY (i.e., scalar) relativistic particles. This is a novel observation, too.
Read full abstract