Geometric Number Systems and Spinors

TL;DR

This paper extends real numbers with new anticommuting roots to form a geometric algebra that models space, spinors, and Lorentz transformations, providing a unified geometric framework for quantum and relativistic physics.

Contribution

It introduces a geometric number system that unifies algebraic and geometric representations of space, spinors, and Lorentz transformations, enhancing understanding of space-time and quantum objects.

Findings

Geometric algebra models Pauli matrices and spinors.

Provides a geometric interpretation of Lorentz boosts.

Unifies algebraic and geometric descriptions of space-time.

Abstract

The real number system is geometrically extended to include three new anticommuting square roots of plus one, each such root representing the direction of a unit vector along the orthonormal coordinate axes of Euclidean 3-space. The resulting geometric (Clifford) algebra provides a geometric basis for the famous Pauli matrices which, in turn, proves the consistency of the rules of geometric algebra. The flexibility of the concept of geometric numbers opens the door to new understanding of the nature of space-time, and of Pauli and Dirac spinors as points on the Riemann sphere, including Lorentz boosts.