The Energy-Momentum Tensor in Field Theory II

In a previous paper, field theory in curved space was considered, and a formula that expresses the first order variation of correlation functions with respect to the external metric was postulated. The formula is given as an integral of the energy-momentum tensor over space, where the short distance singularities of the product of the energy-momentum tensor and an arbitrary composite field must be subtracted, and finite counterterms must be added. These finite counterterms have been interpreted geometrically as a connection for the linear space of composite fields over theory space. In this paper we will study a second order consistency condition for the variational formula and determine the torsion of the connection. A non-vanishing torsion results from the integrability of the variational formula, and it is related to the Bose symmetry of the product of two energy-momentum tensors. The massive Ising model on a curved two-dimensional surface is discussed as an example, and the short-distance singularities of the product of two energy-momentum tensors are calculated explicitly.