The Quantum Mechanics Canonically Associated to Free Probability I: Free Momentum and Associated Kinetic Energy

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 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 L2([-2, 2],μ) with the 1-mode interacting Fock space Γμ, 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 Hμ on L2([-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 eitP and of the semi-circle analogue of the free evolution eitP2/2, 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 e-tHμ and e-itHμ2/2 on the μ-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.