Singularity structure of the two-point function in quantum field theory in curved spacetime

In the point-splitting for renormalizing the stress-energy tensor of a scalar field in curved spacetime, it is assumed that the anticommutator expectation value G(chi, chi')= has a singularity of the Hadamard form as chi → chi'. We prove here that if G(chi, chi') has the Hadamard singularity structure in an open neighborhood of a Cauchy surface, then it does so everywhere, i.e., Cauchy evolution preserves the Hadamard singularity structure. In particular, in a spacetime which is flat below a Cauchy surface, for the 'in' vacuum state G(chi, chi') is of the Hadamard form everywhere, and thus the point-splitting prescription in this case has been rigorously shown to give meaningful, finite answers. (orig.)