# Generalized commutator/anticommutator via phase factor

We know that the commutator between two operators $$A$$ and $$B$$ reads $$[A,B]_{-}=AB - BA$$, while the anticommutator reads $$[A,B]_+=AB + BA$$.

I am wondering if someone has ever used a generalized commutator $$[A,B]_{\theta}=AB + e^{-i\theta} BA$$

where with $$\theta=\pi$$ one has the commutator, while for $$\theta=0$$ one has the anticommutator. If this exists, in what areas of physics has ever been used, and to do what? Are there operators that commute with $$\theta\neq 0$$ and $$\theta \neq \pi$$?

• The Abelian anyons appeared in level m Lauphlin states (FQHE) changes its phase by $e^{i2\pi/m}$ under particle permutation. Also, on the edge of FQHE, the excitations are described by level m chiral boson, whose vertex operators satisfy $[\psi(x),\psi(x')]_{2\pi/3}$ = 0. – FangXie Sep 26 '19 at 14:09