Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime

TL;DR

This paper demonstrates that the singularity structure of n-point distributions in generalised free scalar fields in curved spacetime can be controlled by the two-point Hadamard form, with implications for quantum field theory.

Contribution

It establishes the equivalence of Hadamard and microlocal spectrum conditions for generalised free fields without requiring equations of motion.

Findings

All n-point distributions' singularities are estimable from the two-point Hadamard form.

The class of Hadamard states characterizes the physically relevant state space.

Analogues of key theorems are proven without relying on analyticity.

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 real free scalar 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 satisies scalar (c-number) commutation relations, but it need not satisfy an equation of motion. The same argument also works for anti-commutation relations and it can be generalised to 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 very weak form of the Jost-Schroer theorem. The original proofs of these results in the Wightman framework 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.