Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime
Ko Sanders
TL;DR
The paper proves that, for generalised real free fields in curved spacetime, a Hadamard form for the two-point distribution suffices to bound the singularities of all higher-point distributions via microlocal analysis, ensuring the microlocal spectrum condition for the entire state. By working with generalised commutation relations (scalar, or anti-commuting), it avoids analyticity assumptions and shows that higher truncated n-point distributions become smooth (for n≠2) when the two-point data satisfy the Hadamard condition, and that the Hadamard class is closed under algebraic operations. It also extends two cornerstone results from Wightman theory to curved spacetimes—Borchers–Zimmermann self-adjointness and a weak form of Jost–Schroer—under the framework of generalised free fields. Collectively, these results reinforce the Hadamard framework as the natural foundation for perturbative QFT in curved spacetime, enabling robust construction of Wick polynomials and time-ordered products and suggesting avenues for perturbation theory around generalised free fields. The work has practical implications for the mathematical consistency and physical applicability of QFT in curved backgrounds.
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.