Fermions in one-loop quantum cosmology. II. The problem of correspondence between covariant and noncovariant formalisms.

Abstract

In our preceding paper we calculated the contribution of fermions to the one-loop prefactor of the Hartle-Hawking wave function of the Universe. It was shown that the values of scaling factors obtained by using the $\ensuremath{\zeta}$-function technique differ from those obtained by using covariant calculations of the Schwinger-DeWitt coefficient ${A}_{2}$. It is known 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 inappropriate use of the 3+1 decomposition during the application of the $\ensuremath{\zeta}$-function technique on the manifolds where such a decomposition could not be done consistently. To check this hypothesis we calculate the $\ensuremath{\zeta}(0)$ value for massive Dirac fermions on the flat manifold bounded by two concentric three-spheres. The result coincides with the one obtained by using covariant calculations. It is also shown that different expressions for $\ensuremath{\zeta}(0)$ obtained in the preceding paper for Majorana and Dirac fermions on the part of a de Sitter sphere bounded by a three-sphere at local and spectral boundary conditions have the same limiting value in the case of a full sphere. This value coincides with the covariant one. In addition, all these expressions give the same results in the case of a hemisphere. We discuss briefly also the problem of the discrepancy for other higher-spin fields.