Spectral boundary conditions in one-loop quantum cosmology.

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 ζ-function formed from the squared eigenvalues of the four-dimensional fermionic operators. For a massless Majorana spin2 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 ζ(0) = 11 360 as was found previously by the authors using local boundary conditions. A similar calculation for a spin-32 field, working only with physical degrees 1 Spectral Boundary Conditions in One-Loop ... of freedom (and, hence, excluding gauge and ghost modes, which contribute to the full Becchi-Rouet-Stora-Tyutin-invariant amplitude), again gives a value ζ(0) = − 360 equal to that for the natural local boundary conditions. Phys. Rev. D 44, 1713 (1991).