Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt's coefficients in $C^{\infty}$ Lorentzian manifolds by a local Wick rotation

Details Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt's coefficients in $C^{\infty}$ Lorentzian manifolds by a local Wick rotation Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt's coefficients in $C^{\infty}$ Lorentzian manifolds by a local Wick rotation

1999-08-26

arxiv.org/abs/gr-qc/9908068v2

Commun.Math.Phys. 212 (2000) 165-189

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

30 pages, LaTeX, no figures, shortened version, minor errors corrected a note added. To appear in Commun. Math. Phys

Scientific paper

10.1007/s002200000202

Completing the results obtained 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 K\"ahlerian) 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 $C^\infty$ non analytic Lorentzian manifolds by approximating Lorentzian $C^{\infty}$ metrics by analytic metrics in common geodesically convex neighborhoods. The symmetry requirement plays a central r\^{o}le in the point-splitting renormalization procedure of the one-loop stress-energy tensor in curved spacetimes for Hadamard quantum states.

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