# Dirac Spinors as Representation of $SL(2,\mathbb{C})$ over Grassmann algebra

Recently, I've learned that the Clifford algebra can be regarded as the quantization of Grassmann algebra. This is shown from the following two papers by Berezin.

'Classical spin and Grassmann algebra'

http://www.jetpletters.ac.ru/ps/1476/article_22521.shtml

'Particle spin dynamics as the Grassmann variant of classical mechanics'

http://www.sciencedirect.com/science/article/pii/0003491677903359

I've also noticed that when doing classical Dirac fields, sometimes they are treated as complex-valued spinors but sometimes they are treated as Grassmann-valued spinors. They are treated as complex-valued spinors because they are the representation of the group $SL(2,\mathbb{C})$. But when we are dealing with the canonical and path-integral quantization of Dirac fields, we have to treat them as Grassmann-valued spinors.

If there were such a representation, does it make any sense to construct Grassmann-valued spin up and spin down states in non-relativistic quantum mechanics?

• The bilinears (and higher order $Spin$-symmetric combinations) are objects of usual representation theory. There's no need to think of the inputs as anything other than spinors over $\mathbb{C}$. – Ryan Thorngren Aug 31 '18 at 16:49