On the global Hadamard condition in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighborhoods

Valter Moretti
July 10, 2021
We consider the global Hadamard condition in algebraic QFT in curved spacetime, pointing out the existence of a technical problem in the literature concerning well-posedness of the global Hadamard parametrix in normal neighbourhoods of Cauchy surfaces. We discuss in particular the definition of the (signed) geodesic distance $\sigma$ and related structures in an open neighbourhood of the diagonal of $M\times M$ larger than $U\times U$, for a normal convex neighborhood $U$, where $(M,g)$ is a Riemannian or Lorentzian (smooth Hausdorff paracompact) manifold. We eventually propose a quite natural solution which slightly changes the original definition by B.S. Kay and R.M. Wald and relies upon some non-trivial consequences of paracompactness property. The proposed re-formulation is in agreement with M.J. Radzikowski's microlocal version of the Hadamard condition.

Hadamard sates, global Hadamard parametrix, signed geodesical distance