We prove in this paper that the vacuum expectation of any product of field operator of an interacting Fock space with the interactions λn's is driven by pair partitions if and only if each interaction λn is a linear combination of permutation operators.
We study some probabilistic aspects of the (q,2)-Fock space over a given Hilbert space. Specifically, our focus is on determining the vacuum distribution of the field operator. We ascertain the precise expression for the vacuum expectation of any arbitrary product of creation and annihilation operators, with particular emphasis on the vacuum moments of the field operator. Utilizing these results, we calculate the moment-generating function of the field operator and subsequently determine the corresponding distribution.
Cabana-Duvillard and lonescu11have proved that any symmetric probability measure with moments of any order can be obtained as central limit theorem of self-adjoint, weakly independent and symmetrically distributed (in a quantum souse) random variables. Results of this type will be called "universal central limit theorem". Using Interacting Fock Space (IFS) techniques we extend this result in two directions: (i) we prove that the random variables can be taken to be generalized Gaussian in the sense of Accardi and Bożejko3and we give a realization of such random variables as sums of creation, annihilation and preservation operators acting on an appropriate IFS; (ii) we extend the above-mentioned result to the nonsymmetric case. The nontrivial difference between the symmetric and the nonsymmetric case is explained at the end of the introduction below.