On the temporal Wilson loop in the Hamiltonian approach in Coulomb gauge

TL;DR

This paper explores the calculation of the temporal Wilson loop in Yang-Mills theory using the Hamiltonian approach in Coulomb gauge, comparing known results in simple cases and approximations in the complex (3+1) dimensional case.

Contribution

It demonstrates how Coulomb gauge can be used to compute the Wilson loop with an approximate vacuum wave functional in (3+1) dimensions, showing agreement between Wilson and Coulomb string tensions.

Findings

Wilson and Coulomb string tensions agree within approximation

Unitary transformations reproduce known results in simple cases

Approximate vacuum wave functional yields consistent Wilson loop results

Abstract

We investigate the temporal Wilson loop using the Hamiltonian approach to Yang-Mills theory. In simple cases such as the Abelian theory or the non-Abelian theory in (1+1) dimensions, the known results can be reproduced using unitary transformations to take care of time evolution. We show how Coulomb gauge can be used for an alternative solution if the exact ground state wave functional is known. In the most interesting case of Yang-Mills theory in (3+1) dimensions, the vacuum wave functional is not known, but recent variational approaches in Coulomb gauge give a decent approximation. We use this formulation to compute the temporal Wilson loop and find that the Wilson and Coulomb string tension agree within our approximation scheme. Possible improvements of these findings are briefly discussed.