  Spinor representations of surfaces immersed into 4-dimensional
pseudo-riemannian manifolds are defined in terms of minimal left ideals and
tensor decompositions of Clifford algebras. The classification of spinor fields
and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes
spinor field on surfaces immersed into Lorentzian manifolds and on surfaces
conformally immersed into Minkowski spacetime is defined.
