Weyl gauging and conformal invariance

Abstract

Scale-invariant actions in arbitrary dimensions are investigated in curved space to clarify the relation between scale, Weyl and conformal invariance on the classical level. The global Weyl group is gauged. Then the class of actions is determined for which Weyl gauging may be replaced by a suitable coupling to the curvature (Ricci gauging). It is shown that this class is exactly the class of actions which are conformally invariant in flat space. The procedure yields a simple algebraic criterion for conformal invariance and produces the improved energy-momentum tensor in conformally invariant theories in a systematic way. It also provides a simple and fundamental connection between Weyl anomalies and central extensions in two dimensions. In particular, the subset of scale-invariant Lagrangians for fields of arbitrary spin, in any dimension, which are conformally invariant is given. An example of a quadratic action for which scale invariance does not imply conformal invariance is constructed.