Spin-1/2 Fields in Riemannian Geometries

Part of the Fundamental Theories of Physics book series (FTPH, volume 69)


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 ψ A and \( \widetilde\psi ^{A'} \) . This chapter studies the corresponding classical properties, i.e. the classical boundary-value problem and boundary terms in the variational problem. If \( \sqrt 2 _e n_A^{A'} \psi ^{\rm A} \mp \widetilde\psi ^{A'} \equiv \Phi ^{A'} \) is set to zero on a three-sphere bounding flat Euclidean four-space, the modes of the massless spin-1/2 field multiplying harmonics having positive eigenvalues for the intrinsic three-dimensional Dirac operator on S3 should vanish on S3. Remarkably, this coincides with the property of the classical boundary-value problem when spectral. boundary conditions are imposed on S3 in the massless case. Moreover, the boundary term in the action functional is proportional to the integral on the boundary of Φ A’ e n AA’ ψ A . The existence of self-adjoint extensions of the Dirac operator subject to supersymmetric boundary conditions is then proved. The global theory of the Dirac operator in compact Riemannian manifolds is finally described.


Dirac Operator Boundary Term Riemannian Geometry Symmetric Operator Spinor Field 
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© Kluwer Academic Publishers 2002

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