  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 to cleanly extract the ground state exponents and the non-trivial
matrix elements. The latter account for the difference between the channels
observed in previous simulations.
