Hadamard renormalization in curved space-time (Q794006)

In quantum field theory of a free scalar field in curved background, for each choice of a basis of solutions of the Klein-Gordon equation, the kernel \(G_ 1(x,x')\) is the vacuum expectation value of the product of field operators at two different points x and x'; the kernel is crucial for the construction of the vacuum state of the theory as well as the renormalized vacuum expectation value of the energy-momentum tensor in terms of the point-splitting method. Motivated by the facts that the flat space kernel is a Hadamard elementary solution as a function of geodesic distance and that the renormalized vacuum expectation value of the energy-momentum tensor is covariantly conserved, when the Hadamard solution is symmetric, the authors construct all possible kernels for a scalar field theory the kernels of which have the structure of a symmetrical Hadamard elementary solution involving only geometrical quantities. It is found that in the massless limit no such kernel generally exists. The relationship between the appearance of a trace anomaly in the renormalized vacuum expectation value of the energy-momentum tensor and the renormalization scheme employed is also discussed.