Minimum uncertainty states for Dirac's number-phase pair

With an analytic Hardy space representation, the number-phase statistical p roperties are investigated from the viewpoint of uncertainty relation in th e original Dirac model and operator index theory. The minimum uncertainty s tates are identified. They are the shifted Barut-Girardello coherent states (or the so called philophase states). The fact that Wigner's semi-circle l aw arises in the probability distribution of the cosine operators is indica ted. (C) 2000 Elsevier Science B.V. All rights reserved.