FERMIONS IN ONE-LOOP QUANTUM COSMOLOGY .2. THE PROBLEM OF CORRESPONDENCE BETWEEN COVARIANT AND NONCOVARIANT FORMALISMS

In our preceding paper we calculated the contribution of fermions to t he one-loop prefactor of the Hartle-Hawking wave function of the Unive rse. It was shown that the values of scaling factors obtained by using the zeta-function technique differ from those obtained by using covar iant calculations of the Sehwinger-DeWitt coefficient A(2). It is know n that an analogous discrepancy appears in the calculations of scaling factors of other fields of higher spins too. Here we put forward the hypothesis that the reason for this discrepancy consists in the inappr opriate use of the 3+1 decomposition during the application of the zet a-function technique on the manifolds where such a decomposition could not be done consistently To check this hypothesis we calculate the ze ta(0) value for massive Dirac fermions on the flat manifold bounded by two concentric three-spheres. The result coincides with the one obtai ned by using covariant calculations. It is also shown that different e xpressions for zeta(0) obtained in the preceding paper for Majorana an d Dirac fermions on the part of a de Sitter sphere bounded by a three- sphere at local and spectral boundary conditions have the same limitin g value in the case of a full sphere. This value coincides with the co variant one. In addition, all these expressions give the same results in the case of a hemisphere. We discuss briefly also the problem of th e discrepancy for other higher-spin fields.