A LOCAL-TO-GLOBAL SINGULARITY THEOREM FOR QUANTUM-FIELD THEORY ON CURVED SPACE-TIME

We prove that if a reference two-point distribution of positive type o n a time orientable curved space-time (CST) satisfies a certain condit ion on its wave front set (the ''class P-M,P-g condition'') and if any other two-point distribution (i) is of positive type, (ii) has the sa me antisymmetric part as the reference module smooth function and (iii ) has the same local singularity structure, then it has the same globa l singularity structure. In the proof we use a smoothing, positivity-p reserving pseudo-differential operator the support of whose symbol is restricted to a certain conic region which depends on the wave front s et of the reference state. This local-to-global theorem, together with results published elsewhere, leads to a verification of a conjecture by Kay that for quasi-free states of the Klein-Gordon quantum field on a globally hyperbolic CST, the local Hadamard condition implies the g lobal Hadamard condition. A counterexample to the local-to-global theo rem on a strip in Minkowski space is given when the class P-M,P-g cond ition is not assumed.