  I adapt a numerical method, previously applied to investigate the Yang-Mills
vacuum wavefunctional, to the problem of extracting the effective Polyakov line
action from SU(N) lattice gauge theories, with or without matter fields. The
method can be used to find the variation of the effective Polyakov line action
along any trajectory in field configuration space; this information is
sufficient to determine the potential term in the action, and strongly
constrains the possible form of the kinetic term. The technique is illustrated
for both pure and gauge-Higgs SU(2) lattice gauge theory at finite temperature.
A surprise, in the pure gauge theory, is that the potential of the
corresponding Polyakov line action contains a non-analytic (yet
center-symmetric) term proportional to |P|^3, where P is the trace of the
Polyakov line at a given point, in addition to the expected analytic terms
proportional to even powers of P.
