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 a is found to be: $F(E)=[J_{n+1}(Ea)]^{2}-[J_{n+2}(Ea)]^{2}=0 , \forall n \geq 0$. 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 formulae, and the uniform asymptotic expansions of the Bessel functions and their first derivatives, 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.