Local Boundary Conditions for the Dirac Operator and One-Loop Quantum Cosmology

TL;DR

This paper investigates local boundary conditions for fermionic fields in quantum cosmology, deriving eigenvalue equations and zeta-function values to analyze one-loop divergences and their potential cancellation in supersymmetric models.

Contribution

It introduces a specific boundary-value problem for fermions in quantum cosmology, derives eigenvalues, and computes zeta-function values relevant for one-loop amplitude scaling.

Findings

Eigenvalue equation for fermionic boundary conditions derived.

Zeta(0) for a massless Majorana field calculated as 11/360.

Framework established for checking divergence cancellations in supersymmetric theories.

Abstract

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.