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An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

M. Bożejko^a, E. W. Lytvynov^b, I. V. Rodionova<sup>b

a</sup> Institute of Mathematics, Wrocław University, Wrocław, Poland
^b Swansea University, Swansea, UK

Abstract: Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighbourhood of zero. An anyon Lévy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ on the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimes\nu)$, where $\varphi=\varphi(x)$ runs over a space of test functions on $\mathbb R^d$, while $\omega=\omega(x)$ is interpreted as an operator-valued distribution on $\mathbb R^d$. Let $L^2(\tau)$ be the noncommutative $L^2$-space generated by the algebra of polynomials in the variables $\langle \omega,\varphi\rangle$, where $\tau$ is the vacuum expectation state. Noncommutative orthogonal polynomials in $L^2(\tau)$ of the form $\langle P_n(\omega),f^{(n)}\rangle$ are constructed, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$, and are then used to derive a unitary isomorphism $U$ between $L^2(\tau)$ and an extended anyon Fock space $\mathbf F(L^2(\mathbb R^d,dx))$ over $L^2(\mathbb R^d,dx)$. The usual anyon Fock space $\mathscr F(L^2(\mathbb R^d,dx))$ over $L^2(\mathbb R^d,dx)$ is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Furthermore, the equality $\mathbf F(L^2(\mathbb R^d,dx))=\mathscr F(L^2(\mathbb R^d,dx))$ holds if and only if the measure $\nu$ is concentrated at a single point, that is, in the Gaussian or Poisson case. With use of the unitary isomorphism $U$, the operators $\langle \omega,\varphi\rangle$ are realized as a Jacobi (that is, tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. A Meixner-type class of anyon Lévy white noise is derived for which the corresponding Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure. Each anyon Lévy white noise of Meixner type is characterized by two parameters, $\lambda\in\mathbb R$ and $\eta\geqslant0$. In conclusion, the representation $\omega(x)=\partial_x^\dag+\lambda \partial_x^\dag\partial_x +\eta\partial_x^\dag\partial_x\partial_x+\partial_x$ is obtained, where $\partial_x$ and $\partial_x^\dag$ are the annihilation and creation operators at the point $x$.
Bibliography: 57 titles.

Keywords: anyon commutation relations, anyon Fock space, gamma process, Jacobi field, Lévy white noise, Meixner class of orthogonal polynomials.

UDC: 517.98

MSC: Primary 46L53, 60G51, 60H40; Secondary 33C45

Received: 01.12.2014

DOI: 10.4213/rm9668

 English version: Russian Mathematical Surveys, 2015, 70:5, 857–899

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