Spinor metrics, spin connection compatibility and spacetime geometry from spin geometry

We show first that it is possible to consider the charge conjugation matrix as a metric (inner product) on the spin space. This metric is complementary to the usual Dirac spinor metric in that the Dirac metric defines the inner product of a spinor with a conjugate spinor, whereas the charge conjugation metric defines the inner product of a spinor with another spinor. The invariance group of the Dirac metric, U(2, 2), is distinct from that of the charge metric, Sp(4; C), but their joint subgroup, Sp(4; R), contains the cover of the Lorentz group, Sl(2; C). It is possible to find a spin connection that is metric compatible with both spin metrics, and also compatible with covariant constancy of the Dirac matrices, and this condition also then determines the spacetime curvature as the spin curvature. However, we show that if the condition of covariant constancy of the Dirac matrices is relaxed, it is possible to maintain metricity for both spin metrics, and to obtain both spacetime curvature and torsion from the spin curvature.