Group Contractions: Inonu, Wigner, and Einstein

TL;DR

This paper explores how the internal symmetry groups of massive and massless particles are related through group contraction, revealing that rotational degrees of freedom for massive particles become gauge degrees of freedom for massless particles.

Contribution

It demonstrates the contraction of the $O(3)$ symmetry group for massive particles to the $E(2)$ group for massless particles, linking internal symmetries to gauge degrees of freedom.

Findings

$O(3)$ symmetry contracts to $E(2)$ in the large-radius limit.

Transverse rotational degrees of freedom become gauge degrees of freedom.

Group contraction explains the symmetry transition from massive to massless particles.

Abstract

Einstein's $E = mc^{2}$ unifies the momentum-energy relations for massive and massless particles. According to Wigner, the internal space-time symmetries of massive and massless particles are isomorphic to $O(3)$ and $E(2)$ respectively. According to Inonu and Wigner, $O(3)$ can be contracted to $E(2)$ in the large-radius limit. It is noted that the $O(3)$-like little group for massive particles can be contracted to the $E(2)$-like little group for massless particles in the limit of large momentum and/or small mass. It is thus shown that transverse rotational degrees of freedom for massive particles become contracted to gauge degrees of freedom for massless particles.