## Abstract

We prove that the singularity structure of all *n*-point distributions of a state of a generalised real free scalar field in curved spacetime can be estimated if the two-point distribution is of Hadamard form. In particular this applies to the free field and the result has applications in perturbative quantum field theory, showing that the class of all Hadamard states is the state space of interest. In our proof we assume that the field is a generalised free field, i.e. that it satisfies scalar (c-number) commutation relations, but it need not satisfy an equation of motion. The same arguments also work for anti-commutation relations and for vector-valued fields. To indicate the strengths and limitations of our assumption we also prove the analogues of a theorem by Borchers and Zimmermann on the self-adjointness of field operators and of a weak form of the Jost-Schroer theorem. The original proofs of these results make use of analytic continuation arguments. In our case no analyticity is assumed, but to some extent the scalar commutation relations can take its place.

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## Acknowledgements

I would like to thank Chris Fewster, Bernard Kay, Karl-Henning Rehren and Pedro Lauridsen Ribeiro for helpful suggestions and discussions. This research was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen and the Graduiertenkolleg 1493 “Mathematische Strukturen in der modernen Quantenphysik”. The results of Sect. 4 were obtained during the preparation of my PhD thesis at the University of York.

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Communicated by Y. Kawahigashi

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Sanders, K. Equivalence of the (Generalised) Hadamard and Microlocal Spectrum Condition for (Generalised) Free Fields in Curved Spacetime.
*Commun. Math. Phys.* **295**, 485–501 (2010). https://doi.org/10.1007/s00220-009-0900-7

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DOI: https://doi.org/10.1007/s00220-009-0900-7