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-½ 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 $F\left(E\right)={\left[J_{n+1\mathrm{}}\mathrm{}\right(\mathrm{Ea}\left)\right]}^2-{\left[J_{n+2\mathrm{}}\mathrm{}\right(\mathrm{Ea}\left)\right]}^2=0\mathrm{}$, $\forall n\ge 0$. 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 $\zeta \mathrm{}\left(0\right)\mathrm{}$ 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 $J_m^{\prime }$, yield, for a massless Majorana field, $\zeta \mathrm{}\left(0\right)=\frac{11}{360}$. Combining this with $\zeta \mathrm{}\left(0\right)\mathrm{}$ values for other spins, one can then check whether the one-loop divergences in quantum cosmology cancel in a supersymmetric theory.