Abstract
We consider the global Hadamard condition and the notion of Hadamard parametrix whose use is pervasive in algebraic QFT in curved spacetime (see references 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 Mtimes M larger than Utimes U, for a normal convex neighbourhood 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 Kay and Wald and relies upon some non-trivial consequences of the paracompactness property. The proposed re-formulation is in agreement with Radzikowski’s microlocal version of the Hadamard condition.
Highlights
The use of Hadamard states is nowadays pervasive in algebraic QFT in curved spacetime
The original geometric definition of [19] of a global Hadamard parametrix has been exploited for instance to deal with rigorous interpretations of the Hawking radiation, see [29] and the recent interesting paper [22]
Just to mention some other applications of the Hadamard parametrix in algebraic QFT (aQFT), we can say that it plays a crucial role in the definition of locally covariant Wick powers [16,20], including the definition of the stress-energy tensor operator [27]
Summary
The use of Hadamard states is nowadays pervasive in algebraic QFT (aQFT) in curved spacetime (see, e.g. [1,2,7,8,9,10,11,12,16,17,27,28,35] and [3] for a recent survey on aQFT). To achieve our final goal, in the first part of the paper, we shall focus on the more abstract and mathematically minded problem of a well-posed definition of σ (and related geometric objects) in a neighbourhood of the diagonal of M × M This issue is the core of the problem with the Hadamard parametrix, but it may have other applications in mathematical physics, so that it deserves a separate study. Let x = (0, L/2) and x = (L/2, L) ≡ (L/2, −L) These points are causally related and J (x, x ) (a null line segment) can be thickened up to become a normal convex neighbourhood U. Thickening a timelike or a null geodesic from p to p0 to a convex normal neighbourhood leads to distinct singularity structures
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