BRST and Related Superfield Approach to 1D and 2D Diffeomorphism Invariant Models of Particles and a Bosonic String: A Brief Review

In this brief review-cum-research article, we discuss the off-shell nilpotent (anti-)BRST symmetry transformations, (anti-)BRST invariant Curci-Ferrari (CF) type restriction(s), coupled Lagrangians/Lagrangian densities, etc., for the 1D diffeomorphism (i.e. reparameterization) invariant models of $(i)$ a non-relativistic and non-SUSY free particle, $(ii)$ a scalar (i.e. non-SUSY) relativistic free particle, $(iii)$ a spinning (i.e. SUSY) relativistic free particle, and $(iv)$ a 2D diffeomorphism invariant model of a specific bosonic string theory within the framework of BRST and related supervariable/superfield approach. We take up a new 1D diffeomorphism (i.e. reparameterization) invariant model of an interacting scalar relativistic particle with the electromagnetic field. The latter is treated as a constant background variable and is, therefore, independent of the evolution parameter of the trajectory of the above interacting scalar relativistic particle. We show that the universal nature of the CF-type restriction is maintained in the case of this new model, too. The universal nature of the CF-type restriction(s) is proven by using the modified Bonora-Tonin supervariable/superfield approach (MBTSA) to BRST formalism in the context of 1D and 2D diffeomorphism invariant theories. We further demonstrate that the 1D diffeomorphism invariant models of all kinds of particles are described by the singular Lagrangians. Thus, they also respect the infinitesimal gauge symmetry transformations that are generated by the first-class constraints that exist on them. The equivalence of the 1D diffeomorphism and gauge symmetries is explicitly clarified under specific conditions.