Local approach to dilatation invariance

Scale invariance is analyzed locally by coupling the energy-momentum tensor to a source which is the metric field of curved space-time. The resulting theory at the classical level has no mass parameters only if the general coordinate transformation group can be represented in Weyl's scheme. We further discuss the quantum extension of the theory; the Ward identities become anomalous under radiative corrections and the anomaly is shown to be connected with the instability of the classical metric field representation. The anomalies, recognized as the well-known trace anomalies for the energy-momentum tensor, are then reabsorbed by a perturbative alteration of the original metric field transformation law and we prove the modified Ward identities to be renormalizable in the flat limit. Finally we show that our approach is equivalent to the well-known parametric equations of the Callan-Symanzik type only if the dilatation invariance is not spontaneously broken. In the presence of spontaneous scale breaking we derive a functional equation which will be applied to cases of physical interest in a forthcoming paper.