Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Abstract

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le 1$, satisfying the Yang--Baxter equation. Bożejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal{F}(H)={⨁}_{n=0}^{\infty }{\mathcal{F}}_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces ${\mathcal{F}}_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal{F}(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:={\mathbb{R}}^d$ and $V:={\mathbb{C}}^m$ with $m\ge 2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(T{f}^{(2)})(x,y)=C_{x,y}{f}^{(2)}(y,x)$. Here, for a.a.\ $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le 1$ that satisfies $C_{x,y}^{\ast }=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

References in corpus (2)

• Anyons and the quantum Hall effect - a pedagogical review
• Non injectivity of the q-deformed von Neumann algebra

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