Proof of the symmetry of the off-diagonal heat-kernel and Hadamard's expansion coefficients in general $C^{\infty}$ Riemannian manifolds

TL;DR

This paper rigorously proves that in smooth Riemannian manifolds, the off-diagonal heat-kernel and Hadamard expansion coefficients are symmetric functions within geodesically convex neighborhoods, confirming a key assumption used in quantum field theory renormalization.

Contribution

It provides the first rigorous proof of the symmetry of off-diagonal heat-kernel and Hadamard coefficients in general smooth Riemannian manifolds, previously assumed without proof.

Findings

Symmetry holds in geodesically convex neighborhoods.

Proof applies to both analytic and non-analytic smooth metrics.

Supports assumptions used in quantum field theory renormalization.

Abstract

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 $C^\infty$ metrics with analytic metrics in common (totally normal) geodesically convex neighborhoods, it is rigorously proven that, in general $C^\infty$ 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.