Boundary Terms for Massless Fermionic Fields

Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normal en A′ A to the boundary and a pair of independent spinor fields ψ and ψ̃ ′ . This paper studies the corresponding classical properties, i.e. the classical boundary-value problem and boundary terms in the variational problem. If √ 2 en A′ A ψ ∓ ψ̃A′ ≡ ΦA′ is set to zero on a 3-sphere bounding flat Euclidean 4-space, the modes of the massless spin1 2 field multiplying harmonics having positive eigenvalues for the intrinsic 3-dimensional Dirac operator on S should vanish on S. Remarkably, this coincides with the property of the classical boundary-value problem when spectral boundary conditions are imposed on S in the massless case. Moreover, the