The Construction of Spinors in Geometric Algebra

TL;DR

This paper develops a unified, real geometric algebra framework for spinors, clarifying their relationships across different physical theories and simplifying their mathematical construction compared to complex-based methods.

Contribution

It introduces a consistent approach defining spinors as elements of the even subalgebra of real geometric algebra, unifying various types of spinors and their physical interpretations.

Findings

Explicit relationships between Dirac, Lorentz, Weyl, and Majorana spinors.

Simpler construction of spinors using real geometric algebra.

Enhanced conceptual and theoretical clarity in spinor theory.

Abstract

The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap between algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of $\text{U}(1)$, $\SU(2)$, and $\text{SL}(2,\mathbb{C})$ spinors. The physical observables in Schr\"{o}dinger-Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches.