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arXiv:1512.08862 · Journal of Mathematical Physics

Radial Bargmann representation for the Fock space of type B

Nobuhiro Asai, Marek Bożejko, Takahiro Hasebe

Published Jan 13, 2016• 5 citations
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Abstract

Let να,q be the probability and orthogonality measure for the q-Meixner-Pollaczek orthogonal polynomials, which has appeared in the work of Bożejko, Ejsmont, and Hasebe [J. Funct. Anal. 269, 1769–1795 (2015)] as the distribution of the (α, q)-Gaussian process (the Gaussian process of type B) over the (α, q)-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of να,q. Our main results cover not only the representation of q-Gaussian distribution by van Leeuwen and Maassen [J. Math. Phys. 36, 4743–4756 (1995)] but also of q2-Gaussian and symmetric free Meixner distributions on ℝ. In addition, non-trivial commutation relations satisfied by (α, q)-operators are presented.

DOI
10.1063/1.4939748

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Let να,q be the probability and orthogonality measure for the q-Meixner-Pollaczek orthogonal polynomials, which has appeared in the work of Bożejko, Ejsmont, and Hasebe [J. Funct. Anal. 269, 1769–1795 (2015)] as the distribution of the (α, q)-Gaussian process (the Gaussian process of type B) over the (α, q)-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of να,q. Our main results cover not only the representation of q-Gaussian distribution by van Leeuwen and Maassen [J. Math. Phys. 36, 4743–4756 (1995)] but also of q2-Gaussian and symmetric free Meixner distributions on ℝ. In addition, non-trivial commutation relations satisfied by (α, q)-operators are presented.
On (alpha,q)(alpha, q)(alpha,q)-Poisson Operators and Their Distributions (Mathematical aspects of quantum fields and related topics)
The q-deformed Poisson distribution has been introduced as the orthogonalizing probability measure for a certain q-deformation of Charlier polynomials, which is reduced to the usual Poisson distribution in the limit q→1 and takes the free Poisson distribution in case of q=0. We give the operator on the q-Fock space, which has the q-deformed Poisson distribution with respect to the vacuum state. It is a linear combination of a q-number operator, a q-Gaussian random variable, and a scalar operator. This representation is of the same form as of Hudson and Parthasarathy on the symmetric Fock space.
We consider Fock representations of the Q-deformed commutation relations ∂s∂t†=Q(s,t)∂t†∂s+δ(s,t) for s,t∈T. Here T:=Rd (or more generally T is a locally compact Polish space), the function Q:T2→C satisfies |Q(s,t)|≤1 and Q(s,t)=Q(t,s)¯, and ∫T2h(s)g(t)δ(s,t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt), σ being a fixed reference measure on T. In the case, where |Q(s,t)|≡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=R2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H:=L2(T→C,σ) is constructed. An explicit form of the orthogonal projection of H⊗n onto the n-particle space Fn(H) is derived. A scalar product in Fn(H) is given by an operator Pn≥0 in H⊗n which is strictly positive on Fn(H). We realize the smeared operators ∂t† and ∂t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂s†∂t†=Q(t,s)∂t†∂s†, ∂s∂t=Q(t,s)∂t∂s, valid for those s,t∈T for which |Q(s, t)| = 1.