Quantum Field Theory on Curved Backgrounds. II. Spacetime Symmetries

We study space-time symmetries in scalar quantum field theory on an arbitrary static space-time. We first consider Euclidean quantum field theory, and show that the isometry group is generated by one-parameter subgroups which have either selfadjoint or unitary quantizations. We then show that the self-adjoint semigroups thus constructed can be analytically continued to one-parameter unitary groups, and using this analytic continuation we construct a unitary representation of the isometry group of the Lorentz-signature metric. We illustrate our method for the explicit example of hyperbolic space, whose Lorentzian continuation is Anti-de Sitter space.