The Lorentz group in the oscillator realization. I. The group SO(2,1) and the transformation matrices connecting the SO(2) and SO(1,1) bases

Abstract

The unitary transformation connecting the SO(2) and SO(1,1) bases for the principal and discrete series of representations of the three-dimensional Lorentz group is determined by using the oscillator representation technique. The Hilbert space and the SO(1,1) basis, in this realization, have a simple appearance while the compact basis appears as the solution of an ordinary differential equation reducible to the confluent hypergeometric equation by a simple substitution. The Taylor expansion of this solution obtained by the use of certain functional identities yields the continuous spectrum of the SO(1,1) representations and the unitary transformation from the compact to the noncompact basis after the Sommerfeld–Watson transformation.