Negative Binomial and Multinomial States: probability distributions and coherent states

Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the su(2) coherent states, we propose interpretation of su(1, 1) and su(r, 1) coherent states in terms of probability theory. They will be called the negative binomial (multinomial) states which correspond to the negative binomial (multinomial) distribution, the non-compact counterpart of the well known binomial (multinomial) distribution. Explicit forms of the negative binomial (multinomial) states are given in terms of various boson representations which are naturally related to the probability theory interpretation. Here we show fruitful interplay of probability theory, group theory and quantum theory. PACS: 03.65.-w, 05.30.ch, 42.50.Ar On leave of absence from Institute of Theoretical Physics, Northeast Normal University, Changchun 130024, P.R.China. E-mail: [email protected] 1