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States of a generic quantum field theory on a curved spacetime are consideredwhich satisfy the KMS condition with respect to an evolution associated with acomplete (Killing) vector field. It is shown that at any point where the vectorfield is spacelike, such states cannot satisfy a certain microlocal conditionwhich is weaker than the microlocal spectrum condition in the case ofasymptotically free fields.

letters in mathematical physics/letters in mathematical physics

Local incompatibility of the microlocal spectrum condition with the KMS  property along spacelike directions in quantum field theory on curved  spacetime