
Commutative properties of generalized number operators
 ZHOU Yulan, XUE Rui, CHENG Xiuqiang, CHEN Jia

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2021, 56(4):
94101.
doi:10.6040/j.issn.16719352.0.2020.494

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This paper considers the commutative ralations of the generalized number operator N_{h} and the quantum Bernoulli noise {ə_{σ},ə^{*}_{σ}:σ∈Γ} indexed by Γ, such as Lie bracket, the expressions of the composition of N_{h} and ə_{σ}(ə^{*}_{σ}), the commutative relation of N_{h} and ə_{σ}ə^{*}_{σ}(ə^{*}_{σ}ə_{σ}). The family of bounded linear operators {ə_{σ},ə^{*}_{σ}:σ∈Γ} on L^{}2(M) satisfies the canonical anticommutative relation, nilpotence and the composition are commutative if the intersection of the index is empty. Especially, {ə_{σ},ə^{*}_{σ}:σ∈Γ} satisfy “absorbing commutative relation”. In the following, the paper considers the commutative relations of N_{h} and {ə_{σ},ə^{*}_{σ}:σ∈Γ}. For any nonnegative function h on N, the Lie bracket of N_{h} and the σcreation ə^{*}_{σ}(σannihilation ə_{σ})are just #_{h}(σ)ə^{*}_{σ}(#_{h}(σ)ə_{σ}). Especially, if the support of h is not N, then N_{h} is commutative with some special kind of ə^{*}_{σ}(ə_{σ}). If the support of h is a finite subset of N, the composition of N_{h} and a special kind of ə^{*}_{σ}(ə_{σ}) are just the creation type(annihilation type)operators. Moreover, the paper obtains that N_{h} is commutative with {ə_{σ}ə^{*}_{σ},ə^{*}_{σ}ə_{σ}:σ∈Γ}.