An Algebraic Theory of Non-Relativistic Spin

TL;DR

This paper introduces a purely algebraic, real-number based approach to understanding non-relativistic spin, emphasizing its geometric nature and identifying non-commutative multipole tensors as key observables.

Contribution

It provides a novel, elementary derivation of spin using real algebraic methods and reveals its geometric foundation without relying on complex numbers or dynamics.

Findings

Spin can be derived algebraically without complex numbers.

Non-commutative multipole tensors are key observables of spin.

Spin is fundamentally geometric in nature.

Abstract

In this paper we present a new, elementary derivation of non-relativistic spin using exclusively real algebraic methods. To do this, we formulate a novel method to decompose the domain of a real endomorphism according to its algebraic properties. We reveal non-commutative multipole tensors as the primary physically meaningful observables of spin, and indicate that spin is fundamentally geometric in nature. In so doing, we demonstrate that neither dynamics nor complex numbers are essential to the fundamental description of spin.