Spin and Clifford algebras, an introduction

TL;DR

This paper provides an introductory overview of spin groups, spinors, and Clifford algebras, linking their constructions and extending concepts to general spacetimes through bundles, aimed at pedagogical understanding.

Contribution

It offers a unified, group-theoretic and algebraic introduction to spin and Clifford algebras, including their generalizations to arbitrary spacetimes.

Findings

Clarifies the relationship between spin groups and Clifford algebras.

Constructs low-dimensional Clifford algebras independently.

Introduces the concept of spin and spinor bundles for general spacetimes.

Abstract

In this short pedagogical presentation, we introduce the spin groups and the spinors from the point of view of group theory. We also present, independently, the construction of the low dimensional Clifford algebras. And we establish the link between the two approaches. Finally, we give some notions of the generalisations to arbitrary spacetimes, by the introduction of the spin and spinor bundles.