Minimal Uncertainty States for Quantum Groups

The problem of how to obtain quasi-classical states for quantum groups is examined. A measure of quantum indeterminacy is proposed, which involves expectation values of some natural quantum group operators. It is shown that within any finite dimensional irreducible representation, the highest weight vector and those unitarily related to it are the quasi-classical states. Quantum groups [1] have been intensively studied in recent years. Their applications have already led to significant progress in statistical mechanics and low dimensional topology. It is also widely believed that quantum groups play some important role in quantum physics as well. In this letter, we will investigate the problem of how to obtain quasi classical states for quantum groups. We will propose a measure of quantum indeterminacy, which involves expectation values of some combinations of Drinfeld’s v operator and the universal R matrix. A quasi-classical state is characterized as having minimal indeterminacy. It will be shown that for any finite dimensional irreducible representation, the highest weight vector and those unitarily related to it are the states having this property. Our study here is an extension to quantum groups of the investigation carried out in [2] some twenty years ago, where the corresponding problem for compact simple Lie groups was resolved by one of us. In the limit q → 1, we recover the results of that publication. Given a compact simple Lie group G, we denote its Lie algebra by g. Now there exists a basis {ei}, which is self dual with respect to the Killing form, in which the quadratic Casimir operator can be expressed as C = ∑ i ei ei. It was shown in [2] that the following dispersion