158 research outputs found
Proof of the symmetry of the off-diagonal heat-kernel and Hadamard's expansion coefficients in general Riemannian manifolds
We consider the problem of the symmetry of the off-diagonal heat-kernel
coefficients as well as the coefficients corresponding to the
short-distance-divergent part of the Hadamard expansion in general smooth
(analytic or not) manifolds. The requirement of such a symmetry played a
central r\^{o}le in the theory of the point-splitting one-loop renormalization
of the stress tensor in either Riemannian or Lorentzian manifolds. Actually,
the symmetry of these coefficients has been assumed as a hypothesis in several
papers concerning these issues without an explicit proof. The difficulty of a
direct proof is related to the fact that the considered off-diagonal
heat-kernel expansion, also in the Riemannian case, in principle, may be not a
proper asymptotic expansion. On the other hand, direct computations of the
off-diagonal heat-kernel coefficients are impossibly difficult in nontrivial
cases and thus no case is known in the literature where the symmetry does not
hold. By approximating metrics with analytic metrics in common
(totally normal) geodesically convex neighborhoods, it is rigorously proven
that, in general Riemannian manifolds, any point admits a
geodesically convex neighborhood where the off-diagonal heat-kernel
coefficients, as well as the relevant Hadamard's expansion coefficients, are
symmetric functions of the two arguments.Comment: 30 pages, latex, no figures, minor errors corrected, English
improved, shortened version accepted for publication in Commun. Math. Phy
Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt's coefficients in Lorentzian manifolds by a local Wick rotation
Completing the results obtained in a previous paper, we prove the symmetry of
Hadamard/Seeley-deWitt off-diagonal coefficients in smooth -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 non analytic Lorentzian manifolds by
approximating Lorentzian 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.Comment: 30 pages, LaTeX, no figures, shortened version, minor errors
corrected a note added. To appear in Commun. Math. Phy
zeta-function regularization and one-loop renormalization of field fluctuations in curved space-times
A method to regularize and renormalize the fluctuations of a quantum field in
a curved background in the -function approach is presented. The method
produces finite quantities directly and finite scale-parametrized counterterms
at most. These finite couterterms are related to the presence of a particular
pole of the effective-action function as well as to the heat kernel
coefficients. The method is checked in several examples obtaining known or
reasonable results. Finally, comments are given for as it concerns the recent
proposal by Frolov et al. to get the finite Bekenstein-Hawking entropy from
Sakharov's induced gravity theory.Comment: 9 pages, standard LaTeX, no figure
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