Dirac-K\"ahler approach connected to quantum mechanics in Grassmann space

TL;DR

This paper compares a Grassmann space approach to deriving spinors with the Dirac-K"ahler formulation, highlighting simplicity and potential for generalizations in describing internal degrees of freedom and charges.

Contribution

It introduces a simplified Grassmann formulation for spinors, clarifies Lorentz transformation issues, and discusses generalizations and discrete symmetries.

Findings

Grassmann formulation simplifies spinor description

Two types of time-reversal operators identified

Four families of spinors are shown to be unavoidable

Abstract

We compare the way one of us got spinors out of fields, which are a priori antisymmetric tensor fields, to the Dirac-K\"ahler rewriting. Since using our Grassmann formulation is simple it may be useful in describing the Dirac-K\"ahler formulation of spinors and in generalizing it to vector internal degrees of freedom and to charges. The ``cheat'' concerning the Lorentz transformations for spinors is the same in both cases and is put clearly forward in the Grassmann formulation. Also the generalizations are clearly pointed out. The discrete symmetries are discussed, in particular the appearance of two kinds of the time-reversal operators as well as the unavoidability of four families.