Loop Representation of Wigner's Little Groups

TL;DR

This paper presents a unified graphical representation of Wigner's little groups, capturing internal symmetries of relativistic particles across different spins and masses, including parity, time reversal, and charge conjugation.

Contribution

It introduces a graphical approach to unify the mathematical descriptions of Wigner's little groups for all particle types and spins using two-by-two and four-by-four matrix representations.

Findings

Unified graphical description of internal symmetries

Representation for various spins and masses

Dirac matrices as two-by-two representations

Abstract

Wigner's little groups are the subgroups of the Lorentz group whose transformations leave the momentum of a given particle invariant. They thus define the internal space-time symmetries of relativistic particles. These symmetries take different mathematical forms for massive and for massless particles. However, it is shown possible to construct one unified representation using a graphical description. This graphical approach allows us to describe vividly parity, time reversal, and charge conjugation of the internal symmetry groups. As for the language of group theory, the two-by-two representation is used throughout the paper. While this two-by-two representation is for spin-1/2 particles, it is shown possible to construct the representations for spin-0 particles, spin-1 particles, as well as for higher-spin particles, for both massive and massless cases. It is shown also that the four-by-four Dirac matrices constitute a two-by-two representation of Wigner's little group.