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 L\'evy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\omega,\varphi\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimes\nu)$. Here $\varphi=\varphi(x)$ 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 variables $\langle \omega,\varphi\rangle$, where $\tau$ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in $L^2(\tau)$ of the form $\langle P_n(\omega),f^{(n)}\rangle$, 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 $\langle \omega,\varphi\rangle$ as a Jacobi (i.e., tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. We derive a Meixner-type class of anyon L\'evy white noise for which the respective Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure.