Contractions of Wigner’s Little Groups as Limiting Procedures

Wigner’s little groups are the subgroups of the Poincaré group whose transformations leave the four-momentum of a relativistic particle invariant. The little group for a massive particle is SO(3)-like, whereas for a massless particle, it is E(2)-like. Multiple approaches to group contractions are discussed. It is shown that the Lie algebra of the E(2)-like little group for massless particles can be obtained from the SO(3) and from the SO(2, 1) group by boosting to the infinite-momentum limit. It is also shown that it is possible to obtain the generators of the E(2)-like and cylindrical groups from those of SO(3) as well as from those of SO(2, 1) by using the squeeze transformation. The contraction of the Lorentz group SO(3, 2) to the Poincaré group is revisited. As physical examples, two applications are chosen from classical optics. The first shows the contraction of a light ray from a spherical transparent surface to a straight line. The second shows that the focusing of the image in a camera can be formulated by the implementation of the focal condition to the ABCD matrix of paraxial optics, which can be regarded as a limiting procedure.