Statistical Assemblies of Particles with Spin

TL;DR

This paper develops a comprehensive framework using Lie group $SU(n)$ to describe all possible statistical assemblies of particles with various spins, extending the concept from simple oriented systems to more complex categories.

Contribution

It introduces a Lie group $SU(n)$ based approach to classify and describe all realizable statistical assemblies of particles with different spins, including non-oriented and aligned systems.

Findings

Probability domains form regular polyhedra in $ e^{n-1}$.

Higher spin systems contain multiple empirically determinable axes.

Only collinearity of axes yields simple oriented systems.

Abstract

Spin, $s$ in quantum theory can assume only half odd integer or integer values. For a given $s$, there exist $n=2s+1$ states $|s,m\rangle$, $m=s,s-1,........,-s$. A statistical assembly of particles (like a beam or target employed in experiments in physics) with the lowest value of spin $s=\frac {1}{2}$ can be described in terms of probabilities $p_m$ assigned to the two states $m=\pm \frac {1}{2}$. A generalization of this concept to higher spins $s>\frac {1}{2}$ leads only to a particularly simple category of statistical assemblies known as `Oriented systems'. To provide a comprehensive description of all realizable categories of statistical assemblies in experiments, it is advantageous to employ the generators of the Lie group $SU(n)$. The probability domain then gets identified to the interior of regular polyhedra in $\Re^{n-1},$ where the centre corresponds to an unpolarized assembly and the vertices represent `pure' states. All the other interior points correspond to `mixed' states. The higher spin system has embedded within itself a set of $s(2s+1)$ independent axes, which are determinable empirically. Only when all these axes turn out to be collinear, the simple category of `Oriented systems' is realized, where probabilities $p_m$ are assigned to the states $|s,m\rangle$. The simplest case of higher spin $s=1$ provides an illustrative example, where additional features of `aligned' and more general `non oriented' categories are displayed.