Internal Space-time Symmetries of Massive and Massless Particles and their Unification

TL;DR

This paper explores how the internal space-time symmetries of massive and massless particles are related through Lorentz transformations, showing that the E(2)-like symmetry of massless particles emerges as a limit of the O(3)-like symmetry of massive particles.

Contribution

It demonstrates that the gauge degrees of freedom of massless particles can be derived as a limit of the rotational symmetries of massive particles, unifying their internal symmetries.

Findings

Massless particle symmetry is an E(2)-like group.

Massive particle symmetry is an O(3)-like group.

Massless gauge degrees of freedom arise from the infinite-momentum limit.

Abstract

It is noted that the internal space-time symmetries of relativistic particles are dictated by Wigner's little groups. The symmetry of massive particles is like the three-dimensional rotation group, while the symmetry of massless particles is locally isomorphic to the two-dimensional Euclidean group. It is noted also that, while the rotational degree of freedom for a massless particle leads to its helicity, the two translational degrees of freedom correspond to its gauge degrees of freedom. It is shown that the E(2)-like symmetry of of massless particles can be obtained as an infinite-momentum and/or zero-mass limit of the O(3)-like symmetry of massive particles. This mechanism is illustrated in terms of a sphere elongating into a cylinder. In this way, the helicity degree of freedom remains invariant under the Lorentz boost, but the transverse rotational degrees of freedom become contracted into the gauge degree of freedom.