The Standard Model in noncommutative geometry and Morita equivalence

We discuss some properties of the spectral triple (A(F), H-F,H- D-F, J(F), gamma(F)) describing the internal space in the noncommutative geometry approach to the Standard Model, with A(F) = C circle plus H circle plus M-3. (C). We show that, if we want H-F to be a Morita equivalence bimodule between A(F) and the associated Clifford algebra, two terms must be added to the Dirac operator; we then study its relation with the orientability condition for a spectral triple. We also illustrate what changes if one considers a spectral triple with a degenerate representation, based on the complex algebra B-F = C circle plus M-2 (C) circle plus M-3 (C).