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Communications in Theoretical Physics

Relationship Between Negative Binomial State and SU(1,1) Coherent State and Its q -Deformed Generalization

Hong-Yi Fan, Sicong Jing

Published Oct 30, 1995• 12 citations
Publisher

Abstract

By choosing a suitable single-mode boson realization of SU(1,1) algebra we show that the associated SU(1,1) coherent state can be directly reformed as a negative binomial state. The q -deformed negative binomial state is introduced, and the relationship between this state and the SU q (1,1) coherent state is also demonstrated.
DOI
10.1088/0253-6102/24/3/377

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Using the q analog of the Holstein-Primakoff boson realization of the su(2) generators, we show that a type of q-deformed binomial state that corresponds to the Heine distribution can be identified as an su(2)q{)}_{\mathit{q}})q​ coherent state. This fact is a q extension of the fact that the ordinary binomial state is a particular su(2) coherent state when the Holstein-Primakoff transformation is employed.
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.
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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)su(2)su(2) coherent states, we propose ``interpretation'' of su(1,1)su(1,1)su(1,1) and su(r,1)su(r,1)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.