Choice of Boundary Conditions in One-Loop Quantum Cosmology

Abstract

The problem of boundary conditions in a supersymmetric theory of quantum cosmology is studied. For fermionic fields one has a choice of local or nonlocal boundary conditions. N = 1 supergravity is the simplest supersymmetric model, which we study in a background cosmological model which is flat Euclidean space bounded by a three-sphere. We pick out the physical degrees of freedom by imposing the supersymmetry constraints and choosing a gauge condition. A set of naturally occurring nonlocal boundary conditions for fermions is such that the coefficients of the physical degrees of freedom are regular at the origin, and half of them are set equal to zero on S 3. However, supersymmetry does not relate them to Dirichlet or other local boundary conditions for the gravitational field. Studying N =1 supergravity at 1 loop about flat space, this approach is shown to yield a nonvanishing result for the PDF ζ(0). Namely, the PDF contribution to the prefactor due to the spin-3/2 field is proportional to a 289/360 (a being the three-sphere radius), which does not cancel a −278/45 due to the gravitational field subject to Dirichlet boundary conditions for the perturbed three-metric.