GAUGE-INVARIANT FIELDS IN THE TEMPORAL GAUGE, COULOMB-GAUGE FIELDS, AND THE GRIBOV AMBIGUITY

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 a 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 nonuniqueness 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' law and the implementation of gauge invariance; they do not arise when the theory is represented in terms of gauge-dependent fields and Gauss' law is left unimplemented.