The Standard Model in noncommutative geometry and Morita equivalence

We discuss some properties of the spectral triple (AF,HF,DF,JF,γF) describing the internal space in the noncommutative geometry approach to the Standard Model, with AF=C⊕H⊕M3(C). We show that, if we want HF to be a Morita equivalence bimodule between AF 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 BF=C⊕M2(C)⊕M3(C).