Representations and Applications of O(2, 1), SU(1, 1), and Sp(2)

The groups O(2, 1), SU(1, 1), SL(2, r), and Sp(2) are revisited. The importance of the group Sp(2) is well-known in the Hamiltonian formulations of both classical and quantum mechanics. It has also found a myriad of applications in relatively new research areas such as quantum information and computing. The correspondence of SU(1, 1) with SL(2, r) or Sp(2), consisting of real two-by-two matrices, is given. In this setting, the geometry of SL(2, r) or Sp(2) is made clear. Wigner, Bargmann, and Iwasawa decompositions are discussed within the context of little groups. It is shown that the study of O(2, 1) is that of the Legendre functions. Complex angular momentum is introduced, and its physical applications are demonstrated. We also outline the steps in examining the unitary irreducible representations of SU(1, 1).