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

Abstract

Let $\nu$ be a finite measure on $\mathbb{R}$ whose Laplace transform is analytic in a neighborhood of zero. An anyon Lévy white noise on $({\mathbb{R}}^d,dx)$ is a certain family of noncommuting operators $⟨\omega ,\phi ⟩$ in the anyon Fock space over $L^2({\mathbb{R}}^d\times \mathbb{R},dx\otimes \nu )$. Here $\phi =\phi (x)$ 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 variables $⟨\omega ,\phi ⟩$, where $\tau$ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in $L^2(\tau )$ of the form $⟨P_n(\omega ),f^{(n)}⟩$, where $f^{(n)}$ is a test function on $({\mathbb{R}}^d{)}^n$. Using these orthogonal polynomials, we derive a unitary isomorphism $U$ between $L^2(\tau )$ and an extended anyon Fock space over $L^2({\mathbb{R}}^d,dx)$, denoted by $\mathbf{F}(L^2({\mathbb{R}}^d,dx))$. The usual anyon Fock space over $L^2({\mathbb{R}}^d,dx)$, denoted by $\mathcal{F}(L^2({\mathbb{R}}^d,dx))$, is a subspace of $\mathbf{F}(L^2({\mathbb{R}}^d,dx))$. Furthermore, we have the equality $\mathbf{F}(L^2({\mathbb{R}}^d,dx))=\mathcal{F}(L^2({\mathbb{R}}^d,dx))$ if and only if the measure $\nu$ is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism $U$, we realize the operators $⟨\omega ,\phi ⟩$ as a Jacobi (i.e., tridiagonal) field in $\mathbf{F}(L^2({\mathbb{R}}^d,dx))$. We derive a Meixner-type class of anyon Lévy white noise for which the respective Jacobi field in $\mathbf{F}(L^2({\mathbb{R}}^d,dx))$ has a relatively simple structure.