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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 $\theta \neq \pi$?

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  • $\begingroup$ 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. $\endgroup$ – FangXie Sep 26 at 14:09
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You bet. Heisenberg himself toyed with it (surprise!)-- see H. Rampacher, H. Stumpf, and F. Wagner, Fortschr.d.Physik 13 (1965) 385-480, (See Sec. III.9).

Quantum algebras cover this area, and the deformed Heisenberg algebra you wrote was pioneered by Cigler in 1979 and underlies nonstandard quantum statistics, anyonic physics, and a plethora of physical systems.

You might enjoy 4.g. et seq. of a brief 1990 review of mine where several such "q-mutators" and their properties are discussed and correlated. A review you might enjoy is by O W Greenberg 1993 , the father of parastatistics.

The book by F Wilczek (1990) Fractional Statistics and Anyon Superconductivity is a classic.

In fact, you may have several phases in your generalization to q-mutators, as explored by me and Fairlie in 1991.

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