United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS GRAVITATIONAL WARD IDENTITIES

The gravitational field of general relativity constitutes an example of a gauge field. That is, the gravitational action is invariant with respect to an infinite-dimensional pseudogroup-the group of general co-ordinate transformations. Consequently the quantized version of this theory must possess a feature which is characteristic of al l gauge theories: a set of conserved currents which are intimately connected with the Bource currents of the gauge quanta. In this case the gauge quanta are of course gravitons. However, in contrast with gauge theories of the internal kind such as eleotrodynamics or Yang-Mills theory, the connection between the two kinds of current- the conserved ones and the source currents-is more subtle. In electrodynamics, for example, these currents are identical: the photon source is precisely the. electromagnetic con-served1 current, i.e., the photon couples to the electric charge which is conserved. In Yang-Mills theories, on the other hand, the source ourrent differs from the conserved one by an additive term which, though generally different from zero, is a gauge-dependent quantity. This term vanishes in some gauges with the result that the Yang-Mills particle couples to the conserved isospin only in some gauges. The graviton source, however, never coincides with the conserved current- in this case the energy-momentum tensor. We give now*a demonstration of thist Consider the classical action functional ^,(g»(p) which governs the motion of the gravitational field, gnV, and a set of ("boson) matter fields, (p. Since this functional is invariant under in-finitesimal co-ordinate transformations