Moyal Quantization and Levi-Civita Transformation of Effective Quark-Antiquark Hamiltonian: Application to Heavy Quarkonia and Heavy-Light Mesons

Using the framework of non-commutative geometry, specifically the concepts of symplectic structure and the Weyl product, unitary representations of the Galilei group are constructed with the operators [Formula: see text] and [Formula: see text]. This formulation allows the Schrödinger equation to be expressed in phase space, where the quark-antiquark interaction is modeled using the Hulthen potential plus a linear term. However, direct application of this potential to meson spectroscopy presents certain challenges in the context of Moyal quantization. To address these difficulties, a methodological approach based on the Levi-Civita transformation is introduced to derive the mass spectrum of bound states. The results of these calculations are then applied to the study of several heavy mesons, including charmonium [Formula: see text], bottomonium [Formula: see text], and the mixed-flavor meson [Formula: see text], as well as heavy-light mesons such as [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The analysis is restricted to [Formula: see text]-wave states, as our theoretical model is found to be applicable primarily to such configurations. A key insight of our approach is that the Levi-Civita transformation naturally leads to quark confinement within mesons. Furthermore, the calculated meson masses exhibit good agreement with experimental data, reinforcing the validity of the method. The Wigner functions of the studied mesons are also analyzed. It is observed that they exhibit negative values in the first excited state, whereas they remain strictly positive in the ground state.