Wigner's little group and Berry's phase for massless particles

TL;DR

This paper explores the Lorentz group structure for massless particles, explicitly relates the Euclidean group E2 to Berry's phase, and discusses the physical implications of polarization density matrices.

Contribution

It provides an explicit method to obtain the rotation angle in E2 from Lorentz transformations and links this to Berry's topological phase for massless particles.

Findings

Explicit relation between Lorentz transformations and E2 rotation angle

Connection between Berry's phase and E2 rotation angle

Polarization density matrix is physically meaningless under Lorentz transformations

Abstract

The ``little group'' for massless particles (namely, the Lorentz transformations $\Lambda$ that leave a null vector invariant) is isomorphic to the Euclidean group E2: translations and rotations in a plane. We show how to obtain explicitly the rotation angle of E2 as a function of $\Lambda$ and we relate that angle to Berry's topological phase. Some particles admit both signs of helicity, and it is then possible to define a reduced density matrix for their polarization. However, that density matrix is physically meaningless, because it has no transformation law under the Lorentz group, even under ordinary rotations.