Constraints, Conserved Charges and Extended BRST Algebra for a 3D Field-Theoretic Example for Hodge Theory

TL;DR

This paper analyzes a 3D field theory exemplifying Hodge theory, exploring its classical and quantum symmetries, conserved charges, and constraints within the BRST formalism, revealing novel aspects of charge nilpotency and physical state conditions.

Contribution

It provides a detailed classical and quantum constraint analysis of a 3D Hodge theory model, highlighting the derivation of non-nilpotent charges and their nilpotent counterparts, and emphasizes the role of a unique CF-type restriction.

Findings

Noether charges associated with symmetries are derived.

Non-nilpotent BRST charges are converted into nilpotent forms.

Physical states are shown to be annihilated by operator constraints.

Abstract

We perform the constraint analysis of a three (2 + 1)-dimensional (3D) field-theoretic example for Hodge theory $(i)$ at the classical level within the ambit of Lagrangian formulation, and $(ii)$ at the quantum level within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We derive the conserved charges corresponding to the six continuous symmetries of our present theory. These six continuous summery transformations are the nilpotent (anti-)BRST and (anti-)co-BRST symmetries, a unique bosonic symmetry and the ghost-scale symmetry. It turns out that the Noether conserved (anti-)BRST charges are found to be non-nilpotent even though they are derived from the off-shell nilpotent versions of the continuous and infinitesimal (anti-)BRST symmetry transformations. We obtain the nilpotent versions of the (anti-)BRST charges from the non-nilpotent Noether (anti-)BRST charges and discuss the physicality criteria w.r.t. the latter to demonstrate that the operator forms of the first-class constraints (of the classical gauge theory) annihilate the physical states at the quantum level. This observation is consistent with Dirac's quantization conditions for the systems that are endowed with the constraints. We lay emphasis on the existence of a single (anti-)BRST invariant Curci-Ferrari (CF) type restriction in our theory and derive it from various theoretical angles.