Part I: Vector Analysis of Spinors

TL;DR

This paper develops a geometric algebra framework for space and spinors, linking it to quantum mechanics and providing new insights into the structure of spinors and the Heisenberg uncertainty principle.

Contribution

It introduces a geometric algebra approach to space and spinors, offering a new proof of the Heisenberg uncertainty principle and a geometric interpretation of quantum spinors.

Findings

Geometric algebra of space is isomorphic to Pauli algebra.

Spinor algebra is represented on the Riemann sphere.

A new proof of the Heisenberg uncertainty principle is provided.

Abstract

Part I: The geometric algebra of space is derived by extending the real number system to include three mutually anticommuting square roots of plus one. The resulting geometric algebra is isomorphic to the algebra of complex 2x2 matrices, also known as the Pauli algebra. The so-called spinor algebra of C(2), the language of the quantum mechanics, is formulated in terms of the idempotents and nilpotents of the geometric algebra of space, including its beautiful representation on the Riemann sphere, and a new proof of the Heisenberg uncertainty principle. In "Part II: Spacetime Algebra of Dirac Spinors", the ideas are generalized to apply to 4-component Dirac spinors, and their geometric interpretation in spacetime.