Fractional Effective Quark-Antiquark Interaction in Symplectic Quantum Mechanics

TL;DR

This paper models a heavy quark-antiquark bound state using Symplectic Quantum Mechanics with fractional derivatives, solving the Schrödinger equation in phase space and analyzing the system's non-classical features via the Wigner function.

Contribution

It introduces a novel approach to quark-antiquark interactions using fractional derivatives within Symplectic Quantum Mechanics, providing explicit solutions and phase space analysis.

Findings

Ground state solution obtained and analyzed

Wigner function offers clearer insight into non-classicality

Comparison with Airy function for meson c̄c

Abstract

We investigate within the formalism of Symplectic Quantum Mechanics a two-dimensional non-relativistic strong interacting system that represents the bound heavy quark-antiquark state, where it was considered a linear potential in the context of generalized fractional derivatives. For this purpose, it was solved the Schr\"odinger equation in phase space with the linear potential. The solution (ground state) is obtained, analyzed through the Wigner function comparing with the original solution, the Airy function for the meson $c\overline{c}$. The identified eigenfunctions are connected to the Wigner function via the Weyl product and the Galilei group representation theory in phase space. In some ways, compared to the wave function, the Wigner function makes it simpler to see how the meson system is non-classical.