Yulan Zhou, Cuicui Liu, Qingqing Yang, Wanying Wei, Zhouning Wang
In this paper, we define a family of bounded linear operators on the square integrable Bernoulli functional space , with the finite power set of as the index set, which includes QBNs, preserving some properties of QBNs, which is called -QBNs; the discussion shows that -QBNs has some new properties, such as the quasi-exchangeability, the quasi-nilpotent property, the absorbed anti-commutation relation, the canonical binomial anti-commutation relation and the multi-indicator absorbed anti-commutation relation, which is a multi-index generalization of the canonical anti-commutation relation of QBNs; in particular, -QBNs is ``quantum generators" of the full space . Also, its isochronous mixture product is an ``orthogonal" projection operator on , and -QBNs can be represented by the Dirac operator generated by the orthogonal basis of .