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

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 $μ$ on $[-2,2]$. We prove that, in the identification of $L^2([-2,2],μ)$ with the $1$--mode interacting Fock space $Γ_μ$, defined by the orthogonal polynomial gradation of $μ$, $X$ is mapped into position operator and its canonically associated momentum operator $P$ into $i$ times the $μ$--Hilbert transform $H_μ$ on $L^2([-2,2],μ)$. In the first part of the present paper, after briefly describing the simpler case of the $μ$--harmonic oscillator, we find an explicit expression for the action, on the $μ$--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 $e^{-tH_μ}$ and $e^{-itH_μ^2/2}$ on the $μ$--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.