Spectral asymptotics of Euclidean quantum gravity with diffinvariant boundary conditions
Abstract
A general method is known to exist for studying Abelian and nonAbelian gauge theories, as well as Euclidean quantum gravity, at 1loop 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 boundaryvalue problem for the Laplacetype operator acting on h is known to be selfadjoint but not strongly elliptic. The latter is a technical condition ensuring that a unique smooth solution of the boundaryvalue problem exists, which implies, in turn, that the global heatkernel asymptotics yielding 1loop divergences and 1loop effective action actually exists. The present paper shows that, on the Euclidean 4ball, only the scalar part of perturbative modes for quantum gravity is affected by the lack of strong ellipticity. Further evidence for lack of strong ellipticity, from an analytic point of view, is therefore obtained. Interestingly, three sectors of the scalarperturbation problem remain elliptic, while lack of strong ellipticity is 'confined' to the remaining fourth sector. The integral representation of the resulting ζfunction asymptotics on the Euclidean 4ball is also obtained; this remains regular at the origin by virtue of a spectral identity here obtained for the first time.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 March 2005
 DOI:
 10.1088/02649381/22/6/005
 arXiv:
 arXiv:hepth/0412269
 Bibcode:
 2005CQGra..22..957E
 Keywords:

 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology
 EPrint:
 25 pages, Revtex4. Misprints in Eqs. (5.11), (5.14), (5.16) have been corrected