Spectral Boundary Conditions in One-Loop Quantum Cosmology

TL;DR

This paper investigates spectral boundary conditions for fermionic fields in quantum cosmology, showing they yield the same one-loop amplitude as local conditions in specific flat background cases, with implications for boundary value problems.

Contribution

It demonstrates that spectral boundary conditions produce identical one-loop quantum cosmology results as local boundary conditions in certain flat backgrounds, clarifying boundary condition choices.

Findings

Spectral boundary conditions match local boundary condition results for fermions in flat backgrounds.

Calculated zeta(0) values are 11/360 for spin-1/2 and -289/360 for spin-3/2 fields.

Spectral conditions involve setting half of the fermionic modes to zero on the boundary.

Abstract

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.