Sl(2)-modules bysl(2)-coherent states

Abstract

Irreducible sp(4)-module with highest weight, labeled by the azimuthal and magnetic quantum numbers l and m, is split into the direct sums of the irreducible su(2)- and su(1, 1)-submodules in four different ways: finite integer unitary irreducible subspaces corresponding to the orbital angular momentum algebra su(2), infinite positive discrete series of su(1, 1) with an arbitrary half-integer Bargmann index, and the positive and negative discrete series of su(1, 1) with both the Bargmann indices 1/4 and 3/4. Even and odd coherent states for the positive su(1, 1)-submodules with the Bargmann indices 1/4 and 3/4 are constructed and it is shown that they enjoy the property of completeness by two appropriate positive definite measures. We show that the even and odd coherent states themselves form the positive discrete series of su(1, 1) with the Bargmann indices 1/4 and 3/4, respectively. For these even and odd coherent states, we consider the uncertainty relations for the x- and y-components of the angular momentum as well as the generators of the negative discrete series of su(1, 1) with the Bargmann indices 1/4 and 3/4.