$2$-Spinors via Linear Algebra

TL;DR

This paper presents a simplified linear algebra approach to 2-spinors and the Dirac equation, demonstrating bundle isomorphisms and the relationship between solutions and conjugate spinor fields.

Contribution

It offers a streamlined linear algebra-based framework for understanding 2-spinors and the Dirac equation, including bundle isomorphisms and solution structures.

Findings

Proves Dirac bundle is isomorphic to associated bundles involving SL_2(C) and SU_2.

Shows solutions of the Dirac equation correspond to conjugate 2-spinor fields.

Provides a linear algebra perspective simplifying the understanding of spinor geometry.

Abstract

We give a streamlined account of $2$-spinors, up to and including the Dirac equation, using little more than the resources of linear algebra. We prove that the Dirac bundle is isomorphic to the associated bundles $\mathrm{SL}_2(\mathbb{C}) \times_{\mathrm{SU}_2} S$ and $\mathrm{SL}_2(\mathbb{C}) \times_{\mathrm{SU}_2} \bar{S}$. A solution of the Dirac equation determines a pair of conjugate $2$-spinor fields over the mass shell $X_m$.