{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T16:47:50Z","timestamp":1762015670147,"version":"3.41.2"},"reference-count":13,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:p>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 standard semi-circle random variable [Formula: see text], characterized by the fact that its probability distribution is the semi-circle law [Formula: see text] on [Formula: see text]. We prove that, in the identification of [Formula: see text] with the [Formula: see text]-mode interacting Fock space [Formula: see text], defined by the orthogonal polynomial gradation of [Formula: see text], [Formula: see text] is mapped into position operator and its canonically associated momentum operator [Formula: see text] into [Formula: see text] times the [Formula: see text]-Hilbert transform [Formula: see text] on [Formula: see text]. In the first part of the present paper, after briefly describing the simpler case of the [Formula: see text]-harmonic oscillator, we find an explicit expression for the action, on the [Formula: see text]-orthogonal polynomials, of the semi-circle analogue of the translation group [Formula: see text] and of the semi-circle analogue of the free evolution [Formula: see text], respectively, in terms of Bessel functions of the first kind and of confluent hyper-geometric series. These results require the solution of the 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 [Formula: see text] and [Formula: see text] on the [Formula: see text]-orthogonal polynomials is difficult, 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.<\/jats:p>","DOI":"10.1142\/s1230161222500172","type":"journal-article","created":{"date-parts":[[2023,2,2]],"date-time":"2023-02-02T03:26:24Z","timestamp":1675308384000},"source":"Crossref","is-referenced-by-count":3,"title":["The Quantum Mechanics Canonically Associated to Free Probability I: Free Momentum and Associated Kinetic Energy"],"prefix":"10.1142","volume":"29","author":[{"given":"Luigi","family":"Accardi","sequence":"first","affiliation":[{"name":"Centro Vito Volterra, Universit\u00e0 di Roma \u201cTor Vergata\u201d Via Columbia 2, 00133 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tarek","family":"Hamdi","sequence":"additional","affiliation":[{"name":"Department of Management Information Systems, College of Business Management, Qassim University, Ar Rass, Saudi Arabia"},{"name":"Laboratoire d\u2019Analyse Math\u00e9matiques et Applications LR11ES11, Universit\u00e9 de Tunis El-Manar, Tunisie"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yun Gang","family":"Lu","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 di Bari, via Orabona 4, 70125 Bari, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2023,2,1]]},"reference":[{"key":"S1230161222500172BIB001","unstructured":"L. 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