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arXiv:2404.13718 · Infinite Dimensional Analysis Quantum Probability and Related Topics

Meixner random variables and their quantum operators

Nobuaki Obata, Aurel I. Stan, Hiroaki Yoshida

Published May 9, 2024• 0 citations
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Abstract

In this paper, we find the position–momentum decomposition of the quantum operators of the classic Meixner random variables. The position–momentum decomposition involves translation operators, which are used to give a new characterization of the Meixner random variables.
DOI
10.1142/s0219025724400034

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The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum-type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss-Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition.