Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System

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 space $\left(x\right)$ and time $\left(t\right)$ variables are a function of an evolution parameter $\left(\tau \right)$ . The infinitesimal reparameterization (i.e., 1D diffeomorphism) symmetry transformation of our theory is defined w.r.t. this evolution parameter $\left(\tau \right)$ . 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 by $\tau$ ) is generalized onto a $\left(1,2\right)$ -dimensional supermanifold which is characterized by the superspace coordinates $Z^M=\left(\tau ,\theta ,\overline{\theta }\right)$ where a pair of the Grassmannian variables satisfy the fermionic relationships: ${\theta }^2={\overline{\theta }}^2=0$ , $\theta \overline{\theta }+\overline{\theta }\theta =0$ , and $\tau$ is the bosonic evolution parameter. In the context of ACSA, we take into account only the $\left(1,1\right)$ -dimensional (anti)chiral super submanifolds of the general $\left(1,2\right)$ -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.