Gauge Independent Reduction of a Solvable Model with Gribov-Like Ambiguity

TL;DR

This paper introduces a gauge-independent method to analyze a solvable model with Gribov-like ambiguity, clarifying the impact of gauge fixing issues and connecting different Hamiltonian formalisms.

Contribution

It develops a gauge-independent Lagrangian approach to reduce the model's space, addressing gauge fixing complications and linking to previous solutions and formalisms.

Findings

Reduced space matches explicit solutions

Gauge fixing ambiguity causes nonuniqueness in transformations

Operator ordering issues are resolved with a new prescription

Abstract

We present a gauge independent Lagrangian method of abstracting the reduced space of a solvable model with Gribov-like ambiguity, recently proposed by Friedberg, Lee, Pang and Ren. The reduced space is found to agree with the explicit solutions obtained by these authors. Complications related to gauge fixing are analysed. The Gribov ambiguity manifests by a nonuniqueness in the canonical transformations mapping the hamiltonian in the afflicted gauge with that obtained gauge independently. The operator ordering problem in this gauge is investigated and a prescription is suggested so that the results coincide with the usual hamiltonian formalism using the Schr\"odinger representation. Finally, a Dirac analysis of the model is elaborated. In this treatment it is shown how the existence of a nontrivial canonical set in the ambiguity-ridden gauge yields the connection with the previous hamiltonian formalism.