Group Contractions

Although the mathematics of group contraction is not widely discussed in the established physics curriculum, we very often think of E(2) as a limiting case of O(3). For instance, when we commute from home to school, we are making transformations within a two-dimensional Euclidean space. However, when we travel on the surface of the earth, we know that we are performing rotations. We then wonder how the motions on the E(2) plane can be reconciled with those on the spherical surface. We explain first how the squeeze transformation can be used and that it is possible a straight line can be produced from a function on the xy-plane. We give a summary of Wigner’s E(2) little group and the cylindrical group, and contract the O(3) into both the E(2) and cylindrical groups. We then consider group contractions and unitary representations of E(2). The Lorentz and the Galilei transformations are discussed as well as the O(2, 1) Lorentz group. Additionally, we show how we can use the squeeze transformation to contract O(2, 1) into the E(2) and cylindrical groups and the Lorentz group O(3, 1) into the Galilean group.