Geometrical derivation of Wigner's angle for arbitrary Lorentz transformations of massless particles

TL;DR

This paper provides a detailed geometric derivation of Wigner's angle for Lorentz transformations of massless particles, linking group transformations to spherical trigonometry.

Contribution

It offers a complete analytical derivation of Wigner's little group matrix and a closed-form formula for the Wigner angle for any Lorentz transformation.

Findings

Derived a closed formula for Wigner's angle.

Linked Lorentz transformations to classical spherical trigonometry.

Clarified the geometric structure of Wigner's little group.

Abstract

This note summarizes the physics and mathematics of Lorentz transformations for massless particles, specifically for photons. We provide a complete analytical derivation of Wigner's little group matrix and a closed formula for the calculation of Wigner's angle for arbitrary Lorentz transformations. Our derivation highlights the geometrical content of the sequence of little group transformations leading to Wigner's matrix and links it to classical theorems in spherical trigonometry.