On Wick algebras with additional twisted commutation relations

Remarks on quantum statistics

“…Note that the existence of nontrivial kernel of operator P 2 ≡ R 1 ≡ id E⊗E + T is essential for the nondegeneracy of the scalar product [31]. One can see that if this kernel is trivial, then we obtain well-defined system with generalized statistics [33,34].…”

“…It is interesting that all commutation relations for particles equipped with arbitrary statistics can be described as representations of the so-called quantum Weyl or Wick algebra W. Such algebraic formalism has been considered by Jörgensen, Schmith and Werner [31] and further developed by the author in the series of papers [22,23,24,25,26,27,28,29,30]. and also by Ralowski [31,32,33,34]. Similar approach has been also considered by others authors, see [35,36,37] and [38,39].…”

“…then one can prove that there is well defined action of annihilation operators and scalar product. In this case we need two operators T and B satisfying the above consistency conditions for the model with generalized statistics [32,33,34].…”

“…for s = P m (s) and t = P n ( t). One can define here the action of annihilation operators in such a way that we obtain well-defined system with generalized statistics [34].…”