The quantum mechanics canonically associated to free probability Part I: Free momentum and associated kinetic energy

TL;DR

This paper explores the quantum mechanics associated with the semi-circle distribution, deriving explicit operator actions and connecting free probability with quantum operators using orthogonal polynomials and special functions.

Contribution

It introduces a novel quantum framework linked to the semi-circle law, explicitly characterizing operators and their actions via orthogonal polynomials and special functions.

Findings

Explicit expressions for semi-circle translation and free evolution operators.

Connection between quantum operators and the Hilbert transform in free probability.

Application of special functions like Bessel and hypergeometric series in operator analysis.

Abstract

After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the \textbf{standard semi--circle random variable} $X$, characterized by the fact that its probability distribution is the semi--circle law $\mu$ on $[-2,2]$. We prove that, in the identification of $L^2([-2,2],\mu)$ with the $1$--mode interacting Fock space $\Gamma_{\mu}$, defined by the orthogonal polynomial gradation of $\mu$, $X$ is mapped into position operator and its canonically associated momentum operator $P$ into $i$ times the $\mu$--Hilbert transform $H_{\mu}$ on $L^2([-2,2],\mu)$. In the first part of the present paper, after briefly describing the simpler case of the $\mu$--harmonic oscillator, we find an explicit expression for the action, on the $\mu$--orthogonal polynomials, of the semi--circle analogue of the translation group $e^{itP}$ and of the semi--circle analogue of the free evolution $e^{itP^2/2}$ respectively in terms of Bessel functions of the first kind and of confluent hyper--geometric series. These results require the solution of the \textit{inverse normal order problem} on the quantum algebra canonically associated to the classical semi--circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of $e^{-tH_{\mu}}$ and $e^{-itH_{\mu}^2/2}$ on the $\mu$--orthogonal polynomials is difficult, % aaa ask T if it is solved the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials.