Spectral boundary conditions in one-loop quantum cosmology

For fermionic fields on a compact Riemannian manifold with a boundary, one has a choice between local and nonlocal (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-½ 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 $\zeta \mathrm{}\left(0\right)=\frac{11}{360}$ as was found previously by the authors using local boundary conditions. A similar calculation for a spin-$\frac{3}{2}$ field, working only with physical degrees of freedom (and, hence, excluding gauge and ghost modes, which contribute to the full Becchi-Rouet-Stora-Tyutininvariant amplitude), again gives a value $\zeta \mathrm{}\left(0\right)=-\frac{289}{360}$ equal to that for the natural local boundary conditions.