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arXiv 1603.03075math-phmath.MPmath.OA
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$Q$变形对易关系的Fock表示

Fock representations of $Q$-deformed commutation relations

Marek Bożejko, Eugene Lytvynov, Janusz Wysoczański

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摘要：

我们考虑Fock表示的$Q$变形的交换关系$$\partial_s\partial^†_t=Q(s,t)\partial_t^†\partial_s+δ(s,t), \quad s,t\in T.$$在这里$T:=\mathbb R^d$（或更一般地$T$是一个局部紧波兰空间），功能$Q:T^2\to \mathbb C$满足$|Q(s,t)|\le1$和$Q(s,t)=\overline{Q(t,s)}$，和$$\int_{T^2}h(s)g(t)δ(s,t)\,σ(ds)σ(dt):=\int_T h(t)g(t)\,σ(dt),$$ $σ$是一个固定的参考措施$T$。在$|Q(s,t)|\equiv 1$的情况下，$Q$变形的对易关系描述了Liguori和Mintchev（1995）研究的广义统计。这些广义统计包含任意子统计作为特殊情况（$T=\mathbb R^2$和$Q$函数的特殊选择）。构造了$\mathcal H:=L^2(T\to\mathbb C,σ)$上相应的$Q$变形Fock空间$\mathcal F(\mathcal H)$.导出了$\mathcal H^{\otimes n}$在$n$ -粒子空间$\mathcal F_n(\mathcal H)$上的正交投影的一种显式形式. $\mathcal F_n(\mathcal H)$中的标量积由$\mathcal H^{\otimes n}$中的算子$\mathcal P_n\ge0$给出，该算子在$\mathcal F_n(\mathcal H)$上是严格正的。在$\mathcal F(\mathcal H)$中分别实现了模糊算子$\partial_t^†$和$\partial_t$作为产生算子和湮灭算子。额外的$Q$ -交换关系之间的创建运营商和湮灭运营商。它们的形式为$\partial^†_s\partial^†_t=Q(t,s)\partial^†_t\partial^†_s$、$\partial_s\partial_t=Q(t,s)\partial_t\partial_s$，对$|Q(s,t)|=1$适用的$s,t\in T$有效。

英文摘要：

We consider Fock representations of the $Q$-deformed commutation relations $$\partial_s\partial^†_t=Q(s,t)\partial_t^†\partial_s+δ(s,t), \quad s,t\in T.$$ 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 $$\int_{T^2}h(s)g(t)δ(s,t)\,σ(ds)σ(dt):=\int_T h(t)g(t)\,σ(dt),$$ $σ$ 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 $\mathcal H:=L^2(T\to\mathbb C,σ)$ 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 $\partial_t^†$ 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 $|Q(s,t)|=1$.

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