Anharmonic analysis of lattice field theories

A new calculational procedure for polynomial lattice field theories is discussed that utilizes an anharmonic basis and a general orthogonal transformation of coordinates. The standard blocking procedure is shown to correspond to a discrete Haar transform of the field coordinates. Some generalizations of the Haar transform are given which allow one to block an arbitrary number of sites. This is applied to both the energy density and to the correlation functions. In this paper only an "unperturbed" problem will be discussed, but the unperturbed Hamiltonian will be chosen using a variational principle—it includes couplings and nonharmonic effects in a very nontrivial way. Numerical results will be given for certain critical indices for a ${\phi }^4$ theory in one space dimension.