Gribov vs BRST

TL;DR

This paper explores how the Gribov problem affects BRST quantization in simple quantum models, revealing conditions under which the BRST formalism avoids or suffers from the Gribov issue, impacting unitarity and gauge dependence.

Contribution

It demonstrates that supplementary conditions on gauge fixing fermions determine whether the BRST physical states align with Dirac states, clarifying the impact of the Gribov problem on BRST quantization.

Findings

BRST physical states can be isomorphic to Dirac states if conditions are met

Conventional gauge fixing fermions fail to satisfy supplementary conditions due to Gribov problem

Violations of the Batalin-Vilkovisky theorem lead to gauge dependence and unitarity issues

Abstract

We investigate the way in which the Gribov problem is manifested in the BRST quantization of simple quantum mechanical models by comparing models with and without a Gribov problem. We show that the hermiticity and nilpotency of the BRST charge together with the Batalin-Vilkovisky theorem yield non-trivial supplementary conditions on gauge fixing fermions. If the gauge fixing fermion satisfies the supplementary conditions, the BRST physical states form a space isomorphic to the Dirac space, and the BRST formal path integral does not suffer from the Gribov problem. The conventional gauge fixing fermion, that gives rise to the Faddeev-Popov integral, fails to satisfy the supplementary conditions due to the Gribov problem. Alternatively, enforcing the conventional gauge fixing fermion, these supplementary conditions imply restrictions on the BRST physical states for which the Batalin-Vilkovisky theorem holds. We find that these BRST physical states are not isomorphic the Dirac states. This can be interpreted as a violation of the Batalin-Vilkovisky theorem on the space of Dirac states and implies a breakdown of unitarity and a general dependence of physical quantities on the gauge condition.