  We develop a finite temperature field theory formalism in any dimension that
has the filling fractions as the basic dynamical variables. The formalism
efficiently decouples zero temperature dynamics from the quantum statistical
sums. The zero temperature `data' is the scattering amplitudes. A saddle point
condition leads to an integral equation which is similar in spirit to the
thermodynamic Bethe ansatz for integrable models, and effectively resums
infinite classes of diagrams. We present both relativistic and non-relativistic
versions.
