Euclidean Group as a Dynamical Symmetry of Surface Fluctuations: The Planar Interface and Critical Behavior

The statistical mechanics of the interface between two discrete thermodynamic phases is studied in terms of a field $f$ which describes the deviation of the interface from planar. Exploiting the dynamical Euclidean invariance of the Hamiltonian of the field $f$, we construct $\epsilon$ expansions for Ising-like critical behavior in $1+\epsilon$ dimensions, with a critical temperature of order $\epsilon$, Scaling functions for the interface profile and width in a pinning potential are obtained.