Fock representations of $Q$-deformed commutation relations

Abstract

We consider Fock representations of the $Q$-deformed commutation relations $${\partial }_s{\partial }_t^{†}=Q(s,t){\partial }_t^{†}{\partial }_s+\delta (s,t),s,t\in T.$$ Here $T:={\mathbb{R}}^d$ (or more generally $T$ is a locally compact Polish space), the function $Q:T^2\to \mathbb{C}$ satisfies $|Q(s,t)|\le 1$ and $Q(s,t)=\overline{Q(t,s)}$, and $${\int }_{T^2}h(s)g(t)\delta (s,t)\sigma (ds)\sigma (dt):={\int }_{T}h(t)g(t)\sigma (dt),$$ $\sigma$ being a fixed reference measure on $T$. In the case where $|Q(s,t)|\equiv 1$, the $Q$-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev (1995). These generalized statistics contain anyon statistics as a special case (with $T={\mathbb{R}}^2$ and a special choice of the function $Q$). The related $Q$-deformed Fock space $\mathcal{F}(\mathcal{H})$ over $\mathcal{H}:=L^2(T\to \mathbb{C},\sigma )$ is constructed. An explicit form of the orthogonal projection of ${\mathcal{H}}^{\otimes n}$ onto the $n$-particle space ${\mathcal{F}}_n(\mathcal{H})$ is derived. A scalar product in ${\mathcal{F}}_n(\mathcal{H})$ is given by an operator ${\mathcal{P}}_n\ge 0$ in ${\mathcal{H}}^{\otimes n}$ which is strictly positive on ${\mathcal{F}}_n(\mathcal{H})$. We realize the smeared operators ${\partial }_t^{†}$ and ${\partial }_t$ as creation and annihilation operators in $\mathcal{F}(\mathcal{H})$, respectively. Additional $Q$-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ${\partial }_s^{†}{\partial }_t^{†}=Q(t,s){\partial }_t^{†}{\partial }_s^{†}$, ${\partial }_s{\partial }_t=Q(t,s){\partial }_t{\partial }_s$, valid for those $s,t\in T$ for which $|Q(s,t)|=1$.