Fock representations of $Q$-deformed commutation relations

Abstract

We consider Fock representations of the $Q$-deformed commutation relations $$\partial_s\partial^\dag_t=Q(s,t)\partial_t^\dag\partial_s+\delta(s,t), \quad s,t\in T.$$ Here $T:=\mathbb R^d$ (or more generally $T$ is a locally compact Polish space), the function $Q:T^2\to \mathbb C$ satisfies $|Q(s,t)|\le1$ 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 $\mathcal P_n\ge0$ in $\mathcal H^{\otimes n}$ which is strictly positive on $\mathcal F_n(\mathcal H)$. We realize the smeared operators $\partial_t^\dag$ 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 $\partial^\dag_s\partial^\dag_t=Q(t,s)\partial^\dag_t\partial^\dag_s$, $\partial_s\partial_t=Q(t,s)\partial_t\partial_s$, valid for those $s,t\in T$ for which $|Q(s,t)|=1$.