A Unified Understanding of Spin and Orbital Angular Momentum in the Complex Plane

TL;DR

This paper explores the complex plane representation of angular momentum in quantum mechanics, revealing how spin may originate from non-orbital contributions constrained by Cauchy-Riemann equations.

Contribution

It introduces a complex plane framework for angular momentum, linking spin to non-orbital terms constrained by C-R equations in quantum particles.

Findings

Decomposition of angular momentum into orbital and non-orbital parts.

Identification of C-R equations as constraints defining complex domains.

Proposal that spin arises from non-orbital contributions in the complex plane.

Abstract

The quantum mechanical operator for angular momentum is transformed from the real plane into the complex plane. In doing so, the Cauchy-Riemann (C-R) equations are interpreted as constraint conditions defining two distinct domains where complex differentiation is permitted. It is shown each of these domains contains an orbital angular momentum contribution plus an non-orbital term that cancels out between them. It is further shown the field equations for spinning quantum particles include C-R equations that restrict the particles to a single complex constraint space. It is therefore proposed the non-orbital term in the constraint space angular momentum is the source of the spin.