Local boundary conditions for the Dirac operator and one-loop quantum cosmology.

This paper studies local boundary conditions for fermionic fields in quantum cosmology, originally introduced by Breitenlohner, Freedman and Hawking for gauged supergravity theories in anti-de Sitter space. For a spin2 field the conditions involve the normal to the boundary and the undifferentiated field. A first-order differential operator for this Euclidean boundary-value problem exists which is symmetric and has self-adjoint extensions. The resulting eigenvalue equation in the case of a flat Euclidean background with a three-sphere boundary of radius a is found to be : F (E) = [Jn+1(Ea)] 2− [Jn+2(Ea)] = 0, ∀n ≥ 0. Using the theory of canonical products, this function F may be expanded