Abstract
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.
Highlights
During the last few years, there has been an upsurge of interest in the study of diffeomorphism invariant theories because one of the key and decisive features of the gravitational andstring theories is the observation that they respect the classical diffeomorphism symmetry transformations
The latter symmetry transformations can be exploited within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism [1,2,3,4] where the classical diffeomorphism symmetry transformation is elevated to the quantumBRST symmetry transformations
We have purposely taken a reparameterization invariant NR and NSUSY system so that we could discuss theoretical aspects that are different from our earlier works on the NSUSY relativistic scalar and SUSY relativistic spinning particles [23, 24]
Summary
During the last few years, there has been an upsurge of interest in the study of diffeomorphism invariant theories because one of the key and decisive features of the gravitational and (super)string theories is the observation that they respect the classical diffeomorphism symmetry transformations. In a recent couple of papers [23, 24], we have applied the theoretical beauty of the MBTSA as well as ACSA (i.e., (anti)chiral superfield/supervariable approach) to BRST formalism [25,26,27,28,29] in the context of the 1D diffeomorphism (i.e., reparameterization) invariant theories of the non-SUSY (i.e., scalar) as well as SUSY (i.e., spinning) relativistic free particles. We have applied the beautiful blend of theoretical ideas from MBTSA and ACSA to derive the proper (anti-)BRST symmetries and CF-type restriction for this NR system This model is interesting in its own right as it is a NR system (unlike our earlier discussions [23, 24] on the relativistic systems), and “time” itself has been treated as a physical observable that depends on the evolution parameter τ.
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