Nonstandard vector connections given by nonstandard spinor connections

A unique two-component spinor connection, which we call the standard connection, is determined by the requirement that it be compatible with the spinor inner product and that it give rise to the standard 4-vector connection. Here we take the most general spinor connection, presuming that the conjugate spinor connection is uniquely determined by it, and examine which 4-vector connections are thereby determined. We classify such nonstandard spinor connections and the resulting 4-vector connections and show that the most general torsion tensor can be so generated. However, it is not possible to generate in this way the most general tensor describing incompatibility of the 4-vector connection with the 4-vector inner product. These results illuminate the relationship which must exist between nonstandard theories for spinors and for vectors.