Proof of the symmetry of the off - diagonal Hadamard / Seeley - deWitt ’ s coefficients in C ∞ Lorentzian mani - folds by a “ local Wick rotation ”

Abstract: Completing the results achieved in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth D-dimensional Lorentzian manifolds. To this end, it is shown that, in any Lorentzian manifold, a sort of “local Wick rotation” of the metric can be performed provided the metric is a locally analytic function of the coordinates and the coordinates are “physical”. No time-like Killing field is necessary. Such a local Wick rotation analytically continues the Lorentzian metric in a neighborhood of any point, or, more generally, in a neighborhood of a space-like (Cauchy) hypersurface, into a Riemannian metric. The continuation locally preserves geodesically convex neighborhoods. In order to make rigorous the procedure, the concept of a complex pseudo-Riemannian (not Hermitian or Kählerian) manifold is introduced and some features are analyzed. Using these tools, the symmetry of Hadamard/SeeleydeWitt off-diagonal coefficients is proven in Lorentzian analytical manifolds by analytical continuation of the (symmetric) Riemannian heat-kernel coefficients. This continuation is performed in geodesically convex neighborhoods in common with both the metrics. Then, the symmetry is generalized to C non analytic Lorentzian manifolds by approximating Lorentzian C metrics by analytic metrics in common geodesically convex neighborhoods. The symmetry requirement plays a central rôle in the point-splitting renormalization procedure of the one-loop stress-energy tensor in curved spacetimes for Hadamard quantum states.