BRST Cohomology and Its Application to QED

TL;DR

This paper develops the BRST cohomology framework with a positive definite inner product, establishing the Hodge decomposition and analyzing the quartet mechanism in QED to clarify the structure of the physical state space.

Contribution

It introduces a BRST cohomology construction with positive definite inner product and explicitly defines the co-BRST operator for QED.

Findings

Hodge decomposition theorem established for BRST cohomology

Harmonic states correspond to physical states with positive norm

Explicit analysis of the quartet mechanism in QED

Abstract

We construct the BRST cohomology under a positive definite inner product and obtain the Hodge decomposition theorem at a non-degenerate state vector space $V$. The harmonic states isomorphic with a BRST cohomology class correspond to the physical Hilbert space with positive norm as long as the completeness of $Q_{BRST}$ is satisfied. We explicitly define ``co-BRST'' operator and analyze the quartet mechanism in QED.