# Real Clifford algebras and their spinors for relativistic fermions

## TL;DR

This paper reviews real Clifford algebras across various dimensions, their spinor representations, and their application to relativistic fermions, emphasizing classification, algebraic structures, and physical relevance.

## Contribution

It provides a comprehensive classification of real Clifford algebras and detailed constructions of spinor representations for describing relativistic fermions.

## Key findings

- Classified Clifford algebras as matrix algebras over real, complex, or quaternionic fields.
- Defined spinors as elements of minimal ideals within Clifford algebras.
- Introduced two types of Dirac adjoint spinors and discussed their physical applications.

## Abstract

Real Clifford algebras for arbitrary number of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real, complex or quaternionic type. Spinors are defined as elements of minimal or quasi-minimal left ideals within the Clifford algebra and as representations of the pin and spin groups. Two types of Dirac adjoint spinors are introduced carefully. The relation between mathematical structures and applications to describe relativistic fermions is emphasized throughout.