  For fermionic fields on a compact Riemannian manifold with boundary one has a
choice between local and non-local (spectral) boundary conditions. The one-loop
prefactor in the Hartle-Hawking amplitude in quantum cosmology can then be
studied using the generalized Riemann zeta-function formed from the squared
eigenvalues of the four-dimensional fermionic operators. For a massless
Majorana spin-1/2 field, the spectral conditions involve setting to zero half
of the fermionic field on the boundary, corresponding to harmonics of the
intrinsic three-dimensional Dirac operator on the boundary with positive
eigenvalues. Remarkably, a detailed calculation for the case of a flat
background bounded by a three-sphere yields the same value zeta(0)=11/360 as
was found previously by the authors using local boundary conditions. A similar
calculation for a spin-3/2 field, working only with physical degrees of freedom
(and, hence, excluding gauge and ghost modes, which contribute to the full
Becchi-Rouet-Stora-Tyutin-invariant amplitude), again gives a value
zeta(0)=-289/360 equal to that for the natural local boundary conditions.
