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

TL;DR

This paper addresses a technical issue in defining the global Hadamard parametrix in algebraic quantum field theory on curved spacetime, proposing a natural reformulation that extends the signed geodesic distance beyond convex neighborhoods.

Contribution

It identifies a problem with existing definitions of the Hadamard parametrix and introduces a new approach based on paracompactness to extend the signed geodesic distance.

Findings

Resolved the well-definedness issue of the Hadamard parametrix in larger neighborhoods.

Proposed a reformulation compatible with Radzikowski's microlocal condition.

Ensured consistency with existing theoretical frameworks.

Abstract

We consider the global Hadamard condition and the notion of Hadamard parametrix whose use is pervasive in algebraic QFT in curved spacetime (see refences in the main text). We point out the existence of a technical problem in the literature concerning well-definedness 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 the paracompactness property. The proposed re-formulation is in agreement with M.J. Radzikowski's microlocal version of the Hadamard condition.