Conformal invariance on the light cone and canonical dimensions

Abstract

Covariance under infinitesimal transformations of the spinor group SU(2,2), the covering group of the conformal group, can place significant restrictions on the Wilson's type analysis of operator products on the light cone. For a discussion of the expansion it is relevant to analyze under such transformations the infinite set of local operators providing a basis for the expansion. An infinite ladder of irreducible representations provides such a basis. The existence of a scaling function places relations between the dimensions of tensors, belonging to inequivalent representations, which are all annihilated under Kλ, the generator of infinitesimal special conformal transformations. These relations are not deducible from conformal invariance alone. We establish a theorem which fixes the scale dimension of an irreducible symmetric local operator which is, together with its divergence, annihilated by Kλ. The theorem (or some possible extensions of it) may be useful to build up an algebraic scheme satisfying canonical dimensions. Finally, as an example of a mathematical mechanism providing for the required correlation of dimensions, we discuss an algebraic scheme based on enlarging the conformal algebra.