MULTISCALE REPRESENTATION OF GENERATING AND CORRELATION-FUNCTIONS FORSOME MODELS OF STATISTICAL-MECHANICS AND QUANTUM-FIELD THEORY

We consider models of statistical mechanics and quantum field theory ( in the Euclidean formulation) which are treated using renormalization group methods and where the action is a small perturbation of a quadra tic action. We obtain multiscale formulas for the generating and corre lation functions after n renormalization group transformations which b ring out the relation with the nth effective action. We derive and com pare the formulas for different RGs. The formulas for correlation func tions involve (1) two propagators which are determined by a sequence o f approximate wave function renormalization constants and renormalizat ion group operators associated with the decomposition into scales of t he quadratic form and (2) field derivatives of the nth effective actio n. For the case of the block field ''delta-function'' RG the formulas are especially simple and for asymptotic free theories only the deriva tives at zero field are needed; the formulas have been previously used directly to obtain bounds on correlation functions using information obtained from the analysis of effective actions. The simplicity can be traced to an ''orthogonality-of-scales' property which follows from a n implicit wavelet structure. Other commonly used RGs do not have the ''orthogonality of scales'' property.