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

Valter Moretti|arXiv (Cornell University)|Jul 10, 2021

Geometric Analysis and Curvature Flows44 references22 citations

TL;DR

This paper resolves a technical gap in the global Hadamard parametrix construction in algebraic quantum field theory (QFT) by redefining the signed squared geodesic distance σ in a domain larger than convex normal neighborhoods. Using a topological consequence of paracompactness, it proposes a refined definition of the Hadamard parametrix that ensures well-definedness and consistency across overlapping neighborhoods, proving compatibility with Radzikowski's microlocal formulation and validating the global Hadamard condition without relying on unproven assumptions about σ's global smoothness.

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 $M imes M$ larger than $U imes U$, 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.

Motivation & Objective

• To address a longstanding technical issue in algebraic QFT concerning the well-definedness of the global Hadamard parametrix in overlapping normal convex neighborhoods.
• To resolve the ambiguity in the signed squared geodesic distance σ when points are not causally related and lie in multiple normal neighborhoods.
• To provide a mathematically rigorous, globally consistent definition of the Hadamard parametrix that remains compatible with the microlocal spectrum condition.
• To establish that the proposed redefinition does not alter the singularity structure of the two-point function, preserving the physical content of Hadamard states.

Proposed method

• Reformulates the Hadamard parametrix using a refined neighborhood structure based on paracompactness, ensuring σ is consistently defined across overlapping normal convex neighborhoods.
• Introduces a new parametrix construction GT,n,C_ǫ(x,y) that depends on a cutoff function χ and a compact set C, with smooth dependence on χ.
• Employs a thin refinement of the original definition by Kay and Wald, leveraging topological properties of paracompact manifolds to ensure global consistency of σ.
• Demonstrates that the new definition preserves the singular structure of the two-point function and is independent of auxiliary choices (C, NC, T, χ).
• Uses microlocal analysis to show equivalence with Radzikowski's spectral condition, validating the new definition against the established microlocal formulation.
• Establishes that the parametrix is smooth for non-causally related points, consistent with the absence of singularities in that regime.

Experimental results

Research questions

• RQ1 Can the signed squared geodesic distance σ be consistently defined in a domain larger than any single convex normal neighborhood, especially for non-causally related points?
• RQ2 Does the global Hadamard parametrix remain well-defined and physically meaningful when constructed across overlapping normal neighborhoods with differing geodesic choices?
• RQ3 Is the proposed redefinition of the Hadamard parametrix compatible with the microlocal spectrum condition of Radzikowski?
• RQ4 Does the new definition preserve the singularity structure of the two-point function, ensuring physical consistency with standard Hadamard states?
• RQ5 Can the independence of the Hadamard condition from auxiliary choices (T, χ, C, NC) be rigorously established under the new formulation?

Key findings

• The paper provides a mathematically rigorous redefinition of the global Hadamard parametrix that resolves the ambiguity in σ’s definition across overlapping normal neighborhoods.
• The new parametrix construction GT,n,C_ǫ(x,y) is independent of the choices of C, NC, T, and χ, ensuring robustness and consistency.
• The redefined parametrix maintains the correct singular structure for causally related points and remains smooth for non-causally related pairs, as required by physical consistency.
• The new definition is fully compatible with Radzikowski’s microlocal spectrum condition, confirming equivalence with the microlocal formulation of the Hadamard condition.
• The proof of equivalence between the global Hadamard and microlocal formulations is now valid under the new definition, closing a gap in the literature.
• The result relies on a non-trivial topological consequence of paracompactness, which ensures the existence of a consistent global extension of σ beyond convex neighborhoods.