Global Conformal Invariance

Abstract

This relatively short chapter provides a general introduction to conformal symmetry in arbitrary dimension. Conformal transformations are introduced in Sect. 4.1, with their generators and commutation relations. The conformal group in dimension d is identified with the noncompact group SO(d + 1,1). In Sect. 4.2 we study the action of a conformal transformation on fields, at the classical level. The notion of a quasi-primary field is defined. We relate scale invariance, conformal invariance, and the tracelessness of the energy momentum tensor. In Sect. 4.3 we look at the consequences of conformal invariance at the quantum level on the structure of correlation functions. The form of the two- and three-point functions is given, and the Ward identities implied by conformal invariance are derived. Aspects of conformal invariance that are specific to two dimensions, including local (not globally defined) conformal transformations, are studied in the next chapter. However, the proof that the trace T^μ_μvanishes for a two-dimensional theory with translation, rotation, and dilation invariance is given at the end of the present chapter.