Conformal Basis for Scaling Operators

Abstract

A preliminary attempt to identify the vertex with scaling operators in c < 1 minimal models, as described in the previous Chapter, turned out to be unsuccessful, since the results seemed incompatible at the level of 1- and 2-point functions already. Here we shall see how one can define alternative sets of scaling operators, related to the original ones by analytical transformations in the space of coupling constants. One of these sets, referred to as the conformal basis of scaling operators, happens to exhibit the orthogonality properties of vertex operators correlation functions, within a suitable scaling regime, and thus shall be taken as the appropriate set for comparisons with the Liouville formulation of the theory. The conformal basis was originally found in ref. [31] using the minisuperspace approximation of the Wheeler-DeWitt equation. Physical wave functions are expected to obey such equation and section 4.4. is dedicated to its discussion.