Gauge-invariant fields in the temporal gauge, Coulomb-gauge fields, and the Gribov ambiguity

TL;DR

This paper explores the relationship between gauge-invariant and gauge-dependent fields in two-color QCD, highlighting how the Gribov ambiguity manifests in the multiplicities of gauge-invariant fields in the temporal gauge.

Contribution

It demonstrates the non-uniqueness of Coulomb-gauge fields and links gauge-invariant field multiplicities to the Gribov ambiguity, providing new insights into gauge fixing issues.

Findings

Gauge-invariant fields exhibit multiplicities linked to Gribov copies.

The differential equations for gauge-invariant fields mirror the Gribov equation nonlinearities.

Gauge-invariant field multiplicities only appear with Gauss's law and gauge invariance imposed.

Abstract

We examine the relation between Coulomb-gauge fields and the gauge-invariant fields constructed in the temporal gauge for two-color QCD by comparing a variety of properties, including their equal-time commutation rules and those of their conjugate chromoelectric fields. We also express the temporal-gauge Hamiltonian in terms of gauge-invariant fields and show that it can be interpreted as a sum of the Coulomb-gauge Hamiltonian and another part that is important for determining the equations of motion of temporal-gauge fields, but that can never affect the time evolution of ``physical'' state vectors. We also discuss multiplicities of gauge-invariant temporal-gauge fields that belong to different topological sectors and that, in previous work, were shown to be based on the same underlying gauge-dependent temporal-gauge fields. We argue that these multiplicities of gauge-invariant fields are manifestations of the Gribov ambiguity. We show that the differential equation that bases the multiplicities of gauge-invariant fields on their underlying gauge-dependent temporal-gauge fields has nonlinearities identical to those of the ``Gribov'' equation, which demonstrates the non-uniqueness of Coulomb-gauge fields. These multiplicities of gauge-invariant fields --- and, hence, Gribov copies --- appear in the temporal gauge, but only with the imposition of Gauss's law and the implementation of gauge invariance; they do not arise when the theory is represented in terms of gauge-dependent fields and Gauss's law is left unimplemented.