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 spin-1/2 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 alpha is found to be F(E) = [J(n + 1)(Ea)]2 - [J(n + 2)(Ea)]2 = 0, FOR ALL n greater-than-or-equal-to 0. Using the theory of canonical products, this function F may be expanded in terms of squared eigenvalues, in a way which has been used in other recent one-loop calculations involving eigenvalues of second-order operators. One can then study the generalized Riemann zeta function formed from these squared eigenvalues. The value of zeta(0) determines the scaling of the one-loop prefactor in the Hartle-Hawking amplitude in quantum cosmology. Suitable contour formulas, and the uniform asymptotic expansions of the Bessel functions J(m) and their derivatives J'm, yield, for a massless Majorana field, zeta(0) = 11/360. Combining this with zeta(0) values for other spins, one can then check whether the one-loop divergences in quantum cosmology cancel in a supersymmetric theory.