A New Spectral Cancellation in Quantum Gravity
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A general method exists for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at one-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry. In the de Donder gauge, however, the resulting boundary-value problem for the Laplace type operator acting on h is known to be self-adjoint but not strongly elliptic. The present paper shows that, on the Euclidean four-ball, only the scalar part of perturbative modes for quantum gravity is affected by the lack of strong ellipticity. Interestingly, three sectors of the scalar-perturbation problem remain elliptic, while lack of strong ellipticity is confined to the remaining fourth sector. The integral representation of the resulting zeta-function asymptotics on the Euclidean four-ball is also obtained; this remains regular at the origin by virtue of a peculiar spectral identity obtained by the authors. There is therefore encouraging evidence in favour of the zeta(0) value with fully diff-invariant boundary conditions remaining well defined, at least on the four-ball, although severe technical obstructions remain in general.
26 pages, Latex file with Latex macros. In the final version, style of References has been further improved
26 pages, Latex file with Latex macros. In the final version, style of References has been further improved