Global Boundary Conditions and ζ(0) Value for the Massless Spin-1/2 Field

Abstract

AbstractThe prefactor of the semiclassical approximation of the wave function is evaluated in quantum cosmological models where a massless Majorana spin-1/2 field is regarded as a perturbation around a flat Euclidean background bounded by a three-sphere of radius a. At first we outline the one-loop calculation in Schleich 1985 for pure gravity, and we discuss in some detail the global boundary conditions used in D’Eath and Halliwell 1987 and their relation to the spectral theory of elliptic operators. In performing the one-loop calculation, all untwiddled coefficients of the spin-1/2 field, multiplying unbarred harmonics having positive eigenvalues for the three-dimensional Dirac operator, are set to zero on S3. This means that half of the fermionic field, corresponding to harmonics of the intrinsic three-dimensional Dirac operator with positive eigenvalues, is required to vanish on the three-sphere boundary. The corresponding ζ(0) value is obtained studying the Laplace transform of the heat equation for the squared Dirac operator, and finally deriving the asymptotic expansion of the inverse Laplace transform, i.e. the heat kernel. This squared operator arises from the study of the coupled system of first-order eigenvalue equations for the perturbative modes, subject to the eigenvalue condition Jn+1(Ea) = 0, ∨ n ≥ 0, with degeneracy 2(n + 2)(n + 1). We compute in detail the infinite sums occurring in the asymptotic expansion of the heat kernel. The global boundary conditions previously described are shown to imply that the prefactor is still diverging in the limit of small three-geometry, because the generalized ζ-function is such that ζ(0) = 11/360 for massless Majorana fields.Finally, Hawking’s local boundary conditions involving the spin-1/2 field and the spinor version of the timelike future-pointing normal vector to S3 are introduced. The conditions under which one can get a classical boundary-value problem with a unique regular solution inside S3 are thus described.