Suppose spinors have their own metric and affine connection that differs from tensors. Let us explore the possibilities. For spinors, we are going to use matrices instead of indices to stay in line with the typical semantics.

We will also make the following assumption about the metric matrix s in order for our lengths to be real numbers (and the metric will later become an operator):

We will also create a covariant derivative:

Using the assumption that if our spinor is “constant”, then its length will also be constant everywhere. We then can arrive at a…