Boundary terms for massless fermionic fields

Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normal _{e}n_{A}^{; ; A'} to the boundary and a pair of independent spinor fields psi^{A} and {widetilde psi}^{A'}. This paper studies the corresponding classical properties, i.e. the classical boundary-value problem and boundary terms in the variational problem. If sqrt{2} ; {_{e}n_{A}^{; ; A'}} ; psi^{A} mp {widetilde psi}^{A'} equiv Phi^{A'} is set to zero on a 3-sphere bounding flat Euclidean 4-space, the modes of the massless spin-1/2 field multiplying harmonics having positive eigenvalues for the intrinsic 3-dimensional Dirac operator on S^{3} should vanish on S^{3}. Remarkably, this coincides with the property of the classical boundary-value problem when spectral boundary conditions are imposed on S^3 in the massless case. Moreover, the boundary term in the action functional is proportional to the integral on the boundary of Phi^{A'} ; {_{e}n_{AA'}} ; psi^{A}.