  We examine the equilibrium properties of hot, dilute, non-relativistic
plasmas. The partition function and density correlation functions of a
classical plasma with several species are expressed in terms of a functional
integral over electrostatic potential distributions. The leading order,
field-theoretic tree approximation automatically includes the effects of Debye
screening. Subleading, one-loop corrections are easily evaluated. The two-loop
corrections, however, have ultraviolet divergences. These correspond to the
short-distance, logarithmic divergence which is encountered in the spatial
integral of the Boltzmann exponential when it is expanded to third order in the
Coulomb potential. Such divergences do not appear in the underlying quantum
theory --- they are rendered finite by quantum fluctuations. We show how such
divergences may be removed and the correct finite theory obtained by
introducing additional local interactions in the manner of modern effective
quantum field theories. We obtain explicit results for density-density
correlation functions through two-loop order and thermodynamic quantities
through three-loop order. The induced couplings are shown to obey
renormalization group equations, and these equations are used to characterize
all leading logarithmic contributions in the theory. A linear combination of
pressure and energy and number densities is shown to be described by a
field-theoretic anomaly. The effective theory allows us to evaluate very easily
the algebraic long-distance decay of density correlation functions.
