On the equivalence of two definitions of conformal primary fields in d > 2 dimensions
arXiv:2112.01837 · doi:10.1140/epjp/s13360-022-03228-y
Abstract
Conformal primary fields are of central importance in a conformal field theory with d > 2 spacetime dimensions. They can be defined in two ways. A first definition involves commutators between the field and the generators of the conformal group; a second definition characterizes a primary field according to its behavior under a finite conformal transformation. In the existing literature, the proof of the equivalence of the definitions is either omitted or carried out with little details. In this paper we present a clear and concise review of the two definitions and provide a simple and detailed proof for their equivalence, using some minimal results from quantum field theory and basic properties of conformal transformations. The paper is intended as a tutorial for an introductory lecture course in conformal field theory.
23 pages. v2: final version accepted for publication in Eur.Phys.J.Plus