On the equivalence of two definitions of conformal primary fields in d > 2 dimensions

TL;DR

This paper clarifies and proves the equivalence of two common definitions of conformal primary fields in dimensions greater than two, serving as a tutorial for students and researchers in conformal field theory.

Contribution

It provides a clear, detailed proof of the equivalence of two definitions of conformal primary fields, filling a gap in the literature and aiding educational understanding.

Findings

Proof of equivalence of the two definitions

Simplified and detailed explanation suitable for tutorials

Uses minimal quantum field theory and conformal transformation properties

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.