Proof of the Symmetry of the Off-Diagonal¶Hadamard/Seeley–deWitt's Coefficients in¶C ^∞ Lorentzian Manifolds 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. This result is relevant because it plays a central rôle in Physics, in particular in the theory of the stress-energy tensor renormalization procedure in quantum field theory in curved spacetime. 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 coordinate are appropriate. No time-like Killing field is necessary. Such a local Wick rotation analytically continues the Lorentzian metric in a neighborhood of any point (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/Seeley–deWitt 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.