Quantum Mechanics of Arc-Sine and Semi-Circle Distributions: A Unified Approach

TL;DR

This paper develops a unified quantum mechanical framework for arc-sine and semi-circle distributions, deriving operators, states, and algebraic structures, with applications in physical problems and differential equations.

Contribution

It introduces a unified approach to quantum mechanics for arc-sine and semi-circle distributions, including analytical operators, coherent states, and algebraic structures, extending previous semi-circle results.

Findings

Derived analytical expressions for momentum and kinetic energy operators.

Established a unified quantum framework for both distributions.

Identified perturbations of geometric distributions in photon statistics.

Abstract

This paper continues the program of applying beyond physics the technique of \textbf{probabilistic quantization} and extending to the quantum mechanics associated with the arc--sine distributions our previous results on the semi--circle distribution. We derive analytical expressions for the momentum and kinetic energy operators using the arc--sine weighted Hilbert transform and express corresponding evolutions as Neumann series of Bessel functions. These series are applicable in various physical problems and in solving certain mixed difference equations and differential equations. Moreover, exploiting the similarity between the Jacobi sequences of the semi-circle and arc-sine measures, we establish a unified formulation of their quantum mechanics. We introduce the semicircle and arc--sine exponential vectors and the corresponding coherent states and prove that, for both measures, the vacuum distributions of the number operator in these states (arc--sine photon statistics) are a \textit{perturbation} of the geometric distribution (Gibbs states in Boson physics: see the Introduction below for a discussion of the physical meaning of this perturbation). The $*$--Lie algebra generated by canonical creation and annihilation operators of both probability measures is isomorphic to the $*$--Lie algebra generated by all rank-one operators in corresponding $L^2$-spaces. The paper concludes with appendices that discuss the integral and Neumann series representations of the $1$--parameter unitary groups generated by momentum in semi-circle and arc-sine cases.