The Hadamard Condition and Kay's Conjecture in (axiomatic) Quantum Field Theory on Curved Space-Time

We interpret the global Hadamard condition for a two-point distribution of a Klein-Gordon neutral scalar quantum field model on an arbitrary globally hyperbolic curved space-time in terms of distinguished parametrices (of Duistermaat and Hörmander) and a wave front set spectrum condition. Microlocal results by Duistermaat and Hörmander such as the propagation of singularities theorem and the uniqueness of distinguished parametrices are employed in the proof. Using a smoothing, positivity-preserving pseudo-differential operator, one obtains a local-to-global singularity theorem, generalizing a conjecture by Kay that for quasi-free Klein-Gordon states, local Hadamard implies global Hadamard. This theorem relies on a general wave front set spectrum condition for the two-point distribution; a counterexample is given on Minkowski space when this condition is violated. We postulate a wave front set condition for any m-point distribution on a space-time and show consistency up to C with the usual spectrum condition on Minkowski space and exact correspondence with this condition in the scaling limit. Axioms implying a spin-statistics theorem are suggested for quantum field models on curved space-time.