Symmetry Invariance, Anticommutativity and Nilpotency in BRST Approach to QED: Superfield Formalism

Abstract

We provide the geometrical interpretation for the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry invariance of the Lagrangian density of a four $(3+1)$-dimensional ($4$D) interacting U(1) gauge theory within the framework of superfield approach to BRST formalism. This interacting theory, where there is an explicit coupling between the U(1) gauge field and matter (Dirac) fields, is considered on a $(4,2)$-dimensional supermanifold parametrized by the four spacetime variables $x^{\mu}(\mu=0,1,2,3)$ and a pair of Grassmannian variables $\theta$ and $\bar\theta$ (with $\theta^{2}=\bar\theta^{2}=0$, $\theta\bar\theta+\bar\theta\theta=0$). We express the Lagrangian density and (anti-)BRST charges in the language of the superfields and show that (i) the (anti-)BRST invariance of the $4$D Lagrangian density is equivalent to the translation of the super Lagrangian density along the Grassmannian direction(s) ($\theta$ and/or $\bar\theta$) of the $(4,2)$-dimensional supermanifold such that the outcome of the above translation(s) is zero, and (ii) the anticommutativity and nilpotency of the (anti-)BRST charges are the automatic consequences of our superfield formulation.