MICRO-LOCAL APPROACH TO THE HADAMARD CONDITION IN QUANTUM-FIELD THEORY ON CURVED SPACE-TIME

For the two-point distribution of a quasi-free Klein-Gordon neutral sc alar quantum field on an arbitrary four dimensional globally hyperboli c curved space-time we prove the equivalence of (1) the global Hadamar d condition, (2) the property that the Feynman propagator is a disting uished parametrix in the sense of Duistermaat and Hormander, and (3) a new property referred to as the wave front set spectral condition (WF SSC), because it is reminiscent of the spectral condition in axiomatic quantum field theory on Minkowski space. Results in micro-local analy sis such as the propagation of singularities theorem and the uniquenes s up to C-infinity of distinguished parametrices are employed in the p roof. We include a review of Kay and Wald's rigorous definition of the global Hadamard condition and the theory of distinguished parametrice s, specializing to the case of the Klein-Gordon operator on a globally hyperbolic space-time. As an alternative to a recent computation of t he wave front set of a globally Hadamard two-point distribution on a g lobally hyperbolic curved spacetime, given elsewhere by Kohler (to cor rect an incomplete computation in [32]), we present a version of this computation that does not use a deformation argument such as that used in Fulling, Narcowich and Wald and is independent of the Cauchy evolu tion argument of Fulling, Sweeny and Wald (both of which are relied up on in Kohler's proof). This leads to a simple micro-local proof of the preservation of Hadamard form under Cauchy evolution (first shown by Fulling, Sweeny and Wald) relying only on the propagation of singulari ties theorem. In another paper [33], the equivalence theorem is used t o prove a conjecture by Kay that a locally Hadamard quasi-free Klein-G ordon state on any globally hyperbolic curved space-time must be globa lly Hadamard.