Wigner rotations in laser cavities

The Wigner rotation is important in many branches of physics, chemistry, and engineering sciences. It is a group theoretical effect resulting from two Lorentz boosts. The net effect is one boost followed or preceded by a rotation. While the term “Wigner rotation” is derived from Wigner’s little group whose transformations leave the four-momentum of a given particle invariant, it is shown that the Wigner rotation is different from the rotations in the little group. This difference is clearly spelled out, and it is shown to be possible to construct the corresponding Wigner rotation from the little-group rotation. It is shown also that the $\mathrm{ABCD}$ matrix for light beams in a laser cavity shares the same mathematics as the little-group rotation, from which the Wigner rotation can be constructed.