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Fock representations of \(Q\)-deformed commutation relations. (English) Zbl 1456.81266

Summary: We consider Fock representations of the \(Q\)-deformed commutation relations \(\partial_s \partial_t^{\dagger} = Q(s, t) \partial_t^{\dagger} \partial_s + \delta(s, t)\) for \(s, t \in T\). Here \(T : = \mathbb{R}^d\) (or more generally \(T\) is a locally compact Polish space), the function \(Q : T^2 \rightarrow \mathbb{C}\) satisfies \(| Q(s, t) | \leq 1\) and \(Q(s, t) = \overline{Q(t, s)}\), and \(\int_{T^2} h(s) g(t) \delta(s, t) \sigma(d s) \sigma(d t) : = \int_T h(t) g(t) \sigma(d t)\), \(\sigma\) 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. 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 \(\mathcal{H} : = L^2(T \rightarrow \mathbb{C}, \sigma)\) 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 \(\mathcal{P}_n \geq 0\) in \(\mathcal{H}^{\otimes n}\) which is strictly positive on \(\mathcal{F}_n(\mathcal{H})\). We realize the smeared operators \(\partial_t^{\dagger}\) 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 \(\partial_s^{\dagger} \partial_t^{\dagger} = Q(t, s) \partial_t^{\dagger} \partial_s^{\dagger}\), \(\partial_s \partial_t = Q(t, s) \partial_t \partial_s\), valid for those \(s, t \in T\) for which \(|Q(s,t)| = 1\).{
©2017 American Institute of Physics}

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81S05 Commutation relations and statistics as related to quantum mechanics (general)
30H20 Bergman spaces and Fock spaces
14D15 Formal methods and deformations in algebraic geometry
81V27 Anyons
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