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.
Keywords:Hadamard sates, global Hadamard parametrix, signed geodesical distance