EXACTLY SOLVABLE CONFORMALLY INVARIANT QUANTUM FIELD MODELS IN D DIMENSIONS

Abstract

A class of exactly solvable models of the conformally invariant quantum field theory in D dimensions is proposed. It is shown that in any conformal theory of the field φ(x) with the scale dimension d there exists an infinite collection of the tensor fields Ps of the ranks and the dimensions ds=d+s independently of the type of interaction. These tensor fields appear in the product Tµν(x1)φ(x2) operator expansion where Tµν is the energy-momentum tensor. The fields Ps are analogues to the certain superpositions of the secondary fields of D=2 models. The existence of the fields Ps follows from the structure of the Ward identities for the energy-momentum tensor conformally invariant Green functions. Each model of the above-mentioned class is defined by the operator equation Ps(x)=0. The method of solving these models is proposed. Some of the models coincide with the certain lagrangian models. The method allows us to obtain closed differential equations for each Green function of fundamental and composite fields, and also algebraic equations for scale dimensions of the fields. The derivation of all these equations is based on the energy momentum conformal Ward identities’ specific property. Detailed analysis of the Ward identities is given in the paper. It is shown that each D>2-model involves a special scalar field PD−2 with the scale dimension dP=D−2. This field is an analogue of the central charge of D=2 models and becomes the constant coinciding with the central charge when D=2. A new class of D=2 models with broken infinite parametric symmetry is obtained. In each model the closed differential equations for the highest Green functions are derived and the central charge is calculated. The general method of solving the D≥2 models is illustrated on examples of Thirring and Wess-Zumino-Witten models and on a trivial model in four-dimension space.