The multitime correlation functions, free white noise, and the generalized Poisson statistics in the low density limit

In the present paper the low density limit of the nonchronological multitime correlation functions of boson number type operators is investigated. We prove that the limiting truncated nonchronological correlation functions can be computed using only a subclass of diagrams associated to noncrossing pair partitions and thus coincide with nontruncated correlation functions of suitable free number operators. The independent in the limit subalgebras are found and the limiting statistics is investigated. In particular, it is found that the cumulants of certain elements coincide in the limit with the cumulants of the Poisson distribution. An explicit representation of the limiting correlation functions and thus of the limiting algebra is constructed in a special case through suitably defined quantum white noise operators.