Hamiltonian Approach to 1+1 dimensional Yang-Mills theory in Coulomb gauge

TL;DR

This paper investigates the Hamiltonian formulation of 1+1 dimensional Yang-Mills theory in Coulomb gauge, analyzing gauge fixing, Gribov regions, and propagator behavior, revealing limitations of Dyson-Schwinger equations and implications for lattice gauge theory.

Contribution

It provides a detailed Hamiltonian analysis in different gauges, explores Gribov copy effects, and connects Dyson-Schwinger equations with gauge fixing issues in 1+1 dimensions.

Findings

Vacuum wave functional determined in pure Coulomb gauge

Propagators depend on the chosen Gribov region

Dyson-Schwinger equations do not fully specify the gauge theory

Abstract

We study the Hamiltonian approach to 1+1 dimensional Yang-Mills theory in Coulomb gauge, considering both the pure Coulomb gauge and the gauge where in addition the remaining constant gauge field is restricted to the Cartan algebra. We evaluate the corresponding Faddeev-Popov determinants, resolve Gauss' law and derive the Hamiltonians, which differ in both gauges due to additional zero modes of the Faddeev-Popov kernel in the pure Coulomb gauge. By Gauss' law the zero modes of the Faddeev-Popov kernel constrain the physical wave functionals to zero colour charge states. We solve the Schroedinger equation in the pure Coulomb gauge and determine the vacuum wave functional. The gluon and ghost propagators and the static colour Coulomb potential are calculated in the first Gribov region as well as in the fundamental modular region, and Gribov copy effects are studied. We explicitly demonstrate that the Dyson-Schwinger equations do not specify the Gribov region while the propagators and vertices do depend on the Gribov region chosen. In this sense, the Dyson-Schwinger equations alone do not provide the full non-abelian quantum gauge theory, but subsidiary conditions must be required. Implications of Gribov copy effects for lattice calculations of the infrared behaviour of gauge-fixed propagators are discussed. We compute the ghost-gluon vertex and provide a sensible truncation of Dyson-Schwinger equations. Approximations of the variational approach to the 3+1 dimensional theory are checked by comparison to the 1+1 dimensional case.