Boundary terms for massless fermionic fields

Abstract

Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normalenA/A′ to the boundary and a pair of independent spinor fieldsψA and\(\tilde \psi ^{{\rm A}'} \). This paper studies the corresponding classical properties, i.e., the classical boundary-value problem and boundary terms in the variational problem. If\(\sqrt 2 _e n_{\rm A}^{{\rm A}'} \tilde \psi ^{{\rm A}'} \mp \tilde \psi ^{{\rm A}'} \equiv \Phi ^{{\rm A}'} \) 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 onS3 should vanish onS3. Remarkably, this coincides with the property of the classical boundary-value problem when spectral boundary conditions are imposed onS3 in the massless case. Moreover, the boundary term in the action functional is proportional to the integral on the boundary of ΦA′enAA′ψA.