What is the exact link between spinors and the Grassmann algebra? I'm pretty sure there's one, based on the following:
- The Berezin integral in path integrals is done over the Grassmann algebra of $\Bbb C$
- There's a mapping from the tangent bundle of a supermanifold (which is locally $\Bbb R^n \times $ the Grassmann algebra), and the spinor bundle of a spin manifold
- On the other hand, spinors are defined just as the vector space associated to a representation of the spin group.
I'm not quite sure what the relation is. If we deal with spinors larger than basic Dirac spinors $\Bbb C^2$ (like a product of two spinors), are they automatically described by a member of the Grassmann algebra? Can it be shown that only those transform properly under the spin group?