Conserved Currents and the Energy Momentum Tensor in Conformally Invariant Theories for General Dimensions

The implications of conformal invariance, as relevant in quantum field theories at a renormalisation group fixed point, are analysed with particular reference to results for correlation functions involving conserved currents and the energy momentum tensor. Ward identities resulting from conformal invariance are discussed. Explicit expressions for two and three point functions, which are essentially determined by conformal invariance, are obtained. As special cases we consider the three point functions for two vector and an axial current in four dimensions, which realises the usual anomaly simply and unambiguously, and also for the energy momentum tensor in general dimension $d$. The latter is shown to have two linearly independent forms in which the Ward identities are realised trivially, except if $d = 4$, when the two forms become degenerate. This is necessary in order to accommodate the two independent forms present in the trace of the energy momentum tensor on curved space backgrounds for conformal field theories in four dimensions. The coefficients of the two trace anomaly terms are related to the three parameters describing the general energy momentum tensor three point function. The connections with gravitational effective actions depending on a background metric are described. A particular form due to Riegert is shown to be unacceptable. Conformally invariant expressions for the effective action in four dimensions are obtained using the Green function for a differential operator which has simple properties under local rescalings of the metric.