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

TL;DR

This paper investigates the low density limit of multitime correlation functions of boson operators, revealing connections to free probability, Poisson statistics, and quantum white noise, with explicit constructions in special cases.

Contribution

It demonstrates that the limiting correlation functions can be computed using non-crossing pair partitions and establishes a link to free number operators and Poisson statistics.

Findings

Limiting correlation functions relate to non-crossing pair partitions.

Cumulants of certain elements match Poisson distribution cumulants.

Explicit quantum white noise representations are constructed in special cases.

Abstract

In the present paper the low density limit of the non-chronological multitime correlation functions of boson number type operators is investigated. We prove that the limiting truncated non-chronological correlation can be computed using only a sub-class of diagrams associated to non-crossing pair partitions and thus coincide with the non-truncated 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.