A discrete symmetry and the ladder operators for the dynamical group of monopole

✍ Scribed by Hou Bo-Yu

• Publisher: Elsevier Science
• Year: 1981
• Tongue: English
• Weight: 149 KB
• Volume: 134
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(81)90214-1

✦ Synopsis

The algebraic regularity of monopole eigenfunctions in various dynamical 0(2, 1) bases shows the symmetry in radial phase space. The multipole tensors serve as ladder operators between different 0(2, 1) @ O(3) quasi-bound states.

Recently Jackiw [l] has shown the existence of a time reparametrization invariance for the interaction between a monopole and a a charge and has investigated in detail the survived 0(2, 1) 0 O(3) symmetry for nonrelativistic scalar charged particles. Some noteworthy regularities are exhibited by the special functions in this problem. In this paper we try to explain it by a discrete symmetry in 0(2, 1). Moreover, we introduce the multipole operators as the transition operators between Jackiw's various quasi-harmonic bound states.

The nonrelativistic Hamiltonian of a charged e scalar particle with mass m around an infinitely heavy monopole with strength g is (1) where: A,=O, A,=O, A@=-g/( r sin B)(cos 8 + dy/d#) (y = kd, Dirac-Wu-Yang gauge, y = 0 Schwinger gauge), ,U z ge, p,=-i+zr, L=rx (p-eA)--r/r.

The invariant group for angular part is O(3) generated by Li (i = 1,2, 3) with eigenfunctions 11, 13), (0, 4 1 I, 13) = I$,(#, 6, y) ~PTi]4~r.

The radial part possesses a dynamical group 0(2, 1) generated by Jo, J, , J, .

The irreducible representation of 0(2, 1) is labeled by the eigenvalue of J2, which equals ;[I(I + 1) -,u2 -i] = J(J + 1). ( [Ji, L,] = 0). One may use either the elliptic 313