Introduction

Abstract

Clifford (geometric) algebras extend the real number system to include vectors u, v, w,…, and their products uv, uvw,…. They are useful for modeling geometry, and their vector products, representing surfaces and higher-dimensional objects, allow simple but rigorous descriptions of rotations, reflections, and other geometric transformations. The name Clifford algebra honors the English mathematician William Kingdon Clifford (1845–79), who recognized the importance of ideas set forth by the German high-school mathematics teacher Hermann Günther Grassmann (1809–77). Clifford developed Grassmann’s ideas into what he called geometric algebras. Complex numbers and quaternions (“hypercomplex numbers”), form two particularly simple geometric algebras. The aim of this introduction is to present a brief overview of Clifford algebras and their applications in physics.