Spinors in Spacetime Algebra and Euclidean 4-Space

TL;DR

This paper investigates the geometric algebra structures of Minkowski spacetime and Euclidean 4-space, revealing their algebraic isomorphism to quaternion-based matrix algebra and unifying different algebraic frameworks used in physics.

Contribution

It demonstrates the algebraic isomorphism between Minkowski spacetime and Euclidean 4-space algebras and unifies Hestenes' Space-Time Algebra with Baylis' Algebra of Physical Space.

Findings

Both algebras are isomorphic to 2x2 quaternion matrices

Provides geometric insights into spinors and hyperbolic geometry

Unifies different algebraic approaches in spacetime physics

Abstract

This article explores the geometric algebra of Minkowski spacetime, and its relationship to the geometric algebra of Euclidean 4-space. Both of these geometric algebras are algebraically isomorphic to the 2x2 matrix algebra over Hamilton's famous quaternions, and provide a rich geometric framework for various important topics in mathematics and physics, including stereographic projection and spinors, and both spherical and hyperbolic geometry. In addition, by identifying the time-like Minkowski unit vector with the extra dimension of Euclidean 4-space, David Hestenes' Space-Time Algebra of Minkowski spacetime is unified with William Baylis' Algebra of Physical Space.