Spatial Wavefunctions of Spin

TL;DR

This paper introduces a new spatial wavefunction formulation for quantum angular momentum using Wigner D-functions, linking these to fundamental particles' spin and internal quantum numbers, with implications for particle physics and dark matter.

Contribution

It proposes that Wigner D-functions with specific internal quantum numbers serve as spatial wavefunctions for particle spin, offering new insights into Standard Model particles and dark matter candidates.

Findings

Elementary particles with spin have wavefunctions with specific internal quantum numbers.

Standard Model particles fit into categories based on these quantum numbers.

The approach suggests new predictions for neutrino nature and dark matter candidates.

Abstract

We present an alternative formulation of quantum mechanical angular momentum, based on spatial wavefunctions that depend on the Euler angles $\phi, \theta, \chi$, and have an additional internal projection $n$. The wavefunctions are Wigner D-functions, $D_{n m}^s (\phi, \theta, \chi)$, for which the body-fixed projection quantum number $n$ has the unusual value $n=|s|=\sqrt{s(s+1)}$, or $n=0$. We show that the states $D_{\sqrt{s(s+1)},m}^s (\phi, \theta, \chi)$ of elementary particles with spin $\sqrt{s(s+1)}$ give a gyromagnetic ratio of $g=2$ for $s>0$, and we identify these as the spatial angular-momentum wavefunctions of known fundamental charged particles with spin. All known Standard-Model particles can be categorized with either value $n=\sqrt{s(s+1)}$ or $n=0$, and all known particle reactions are consistent with the conservation of its projection in the internal frame, and with internal-frame Clebsch-Gordan coefficients of unity. Therefore, we make the case that the $D_{n m}^s (\phi, \theta, \chi)$ are useful as spatial wavefunctions for angular momentum. Some implications and new predictions related to the quantum number $n$ for fundamental particles are discussed, such as the proposed Dirac-fermion nature of the neutrino, the explanation of some Standard-Model structure, and some proposed dark-matter candidates.