Polyakov loop and its relation to static quark potentials and free energies

It appears well accepted in the literature that the correlator of Polyakov loops in a finite temperature system decays with the average free energy of the static quark-antiquark system and can be decomposed into singlet and adjoint (or octet for QCD) contributions. By fixing a gauge respecting the transfer matrix, attempts have been made to extract those contributions separately. In this paper we point out that the average and adjoint channels of Polyakov loop correlators are misconceptions. We show analytically that all channels receive contributions from singlet states only, and give a corrected definition of the singlet free energy. We verify this finding by simulations of the 3D SU(2) pure gauge theory in the zero-temperature limit, which allows one to cleanly extract the ground state exponents and the nontrivial matrix elements. The latter account for the difference between the channels observed in previous simulations.