Narrow Results

Localized noncommutative structures for manifolds with connection are constructed based on the use of vertical star products. The model's main feature is that two points that are far away from each other will not be subjected to a deviation from classical geometry while space-time becomes noncommutative for pairs of points that are close to one another.

We introduce the notion of generalized Weyl system, and use it to define *-products which generalize the commutation relations of Lie algebras. In particular we study in a comparative way various *-products which generalize the κ-Minkowski commutation relation.

We consider the regularization of a gauge quantum field theory following a modification of the Pochinski proof based on the introduction of a cutoff function. We work with a Poincaré invariant deformation of the ordinary pointwise product of fields introduced by Ardalan, Arfaei, Ghasemkhani and Sadooghi, and show that it yields, through a limiting procedure of the cutoff functions, to a regularized theory, preserving all symmetries at every stage. The new gauge symmetry yields a new Hopf algebra with deformed costructures, which is inequivalent to the standard one.

Under the influence of the deformation space-space symmetries, the improved Mobius square plus generalized Yukawa potentials (IMSGYPs) have been employed to solve the deformed Klien–Gordon equation in three-dimensional noncommutative relativistic quantum space (3D-RNCQS) symmetries. Combined with the approximation approach suggested by Greene and Aldrich, we also employ the parametric Bopp’s shift approach and standard perturbation theory to derive novel relativistic energy eigenvalues. The new relativistic energy eigenvalues of (N₂, K₂, NI, ScI, and RbH) diatomic molecules under the IMSGYPs were shown to be sensitive to the atomic quantum numbers ($j,l,s,m$), the mixed potential depths ($V_0,A,B,C,D,V_1$), the screening parameter’s inverse $\alpha$ and non-commutativity parameters ($\mathrm{\Theta }$, $\beta$, $\zeta$). In addition, we analyzed the new non-relativistic energy values in three-dimensional noncommutative non-relativistic quantum space (3D-NRNCQS) symmetries, by applying the well-known mapping in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the newly generalized Mobius square potential, the newly generalized Yukawa potential, and the newly generalized Deng-Fan potential. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system, such as $c\overline{c}$, bottomonium $b\overline{b}$, $\overline{b}c$ and light mesons $c\overline{s}$ and $b\overline{s}$, that have the quark and antiquark flavors within the framework of the IMSGYPs model in 3D-NRNCQS symmetries.

In this paper, the modified Dirac equation (MDE) has been solved approximately for improved Manning–Rosen plus quasi-Hellman potentials (IMRQHPs), including the improved Yukawa tensor interaction under new spin symmetry and pseudospin symmetry for arbitrary spin–orbit number $k$, an appropriate approximation employed on the centrifugal terms in the symmetries of three-dimensional extended relativistic quantum mechanics (3D-ERQM). This potential is a superposition of Manning–Rosen plus quasi-Hellman potentials and some other exponential terms. Through the application of both new approximations to deal with the centrifugal term, Bopp’s shift method and the independent-time perturbation theory method, the recent modified approximate energy corrections under IMRQHPs are obtained for nuclei ${}^{17}$O and ${}^{17}$F and (O₂, I₂, N₂, H₂, LiH, CO and NO) diatomic molecules in 3D-ERQM and three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries. The recent values that we get appear sensitive to $(j,k,l∕\tilde{l},s∕\tilde{s},m∕\tilde{m})$, which are known as the discrete atomic quantum numbers, the mixed potential depths ($C∕A$, $D∕B$) and $(V_0∕a,V_1∕b)$, the range of the potential $\alpha$ and non-commutativity parameters $(\mathrm{\Theta },\sigma ,\chi )$. In both 3D-ERQM and 3D-ENRQM symmetries, we show that the recent corrections obtained on the new bound states under IMRQHPs are infinitesimal to the principal part of energy in the ordinary cases of 3D-RQM and 3D-NRQM symmetries. In the new symmetries of 3D-ERQM symmetries, it is not possible to get the exact analytical solutions $k=1$ and $k\ne -1$, only the approximate solutions are possible. In addition, we discussed in detail the non-relativistic study of the studied mixed potentials. Four special cases, i.e. we investigated some special cases in the context of modified Schrödinger theories. In the framework of known relativistic quantum mechanics, we have clearly demonstrated that the modified Schrödinger equation can be represented as the Dirac equation under the influence of IMRQHPs.

Perturbative deformations of symmetry structures on noncommutative spaces are studied in view of noncommutative quantum field theories. The rigidity of enveloping algebras of semisimple Lie algebras with respect to formal deformations is reviewed in the context of star products. It is shown that rigidity of symmetry algebras extends to rigidity of the action of the symmetry on the space. This implies that the noncommutative spaces considered can be realized as star products by particular ordering prescriptions which are compatible with the symmetry. These symmetry preserving ordering prescriptions are calculated for the quantum plane and four-dimensional quantum Euclidean space. The result can be used to construct invariant Lagrangians for quantum field theory on noncommutative spaces with a deformed symmetry.

The problem of finding a generic star product on ℝⁿ is reduced to the computation of a skew-symmetric biderivation.

We have obtained the approximate analytical solutions of the nonrelativistic Hydrogen-like atoms such as $({\mathrm{\text{He}}}^{+},{\mathrm{\text{Li}}}^{2+}$ and ${\mathrm{\text{Be}}}^{+})$ and neutral atoms such as (${}^{22}\mathrm{\text{Na}}{,}^{\mathrm{\text{12}}}\mathrm{\text{C}}$ and ${}^{\mathrm{\text{158}}}\mathrm{\text{Au}}$) atoms with a newly proposed generalized perturbed Yukawa potential with centrifugal barrier (GPYPCB) model using the generalized Bopp’s shift method and standard perturbation theory in the symmetries of noncommutative three-dimensional real space phase (NC: 3D-RSP). By approximating the centrifugal term through the Greene–Aldrich approximation scheme, we have obtained the energy eigenvalues and generalized Hamiltonian operator for all orbital quantum numbers $\ell$ in the symmetries of NC: 3D-RSP. The potential is a superposition of the perturbed Yukawa potential and new terms proportional with $(\frac{exp(-2ar)}{r^4},\frac{exp(-2ar)}{r^3},\frac{exp(-ar)}{r^3}\text{ and }\frac{exp(-ar)}{r^2}$) appear as a result of the effects of noncommutativity properties of space and phase on the perturbed Yukawa potential model. The obtained energy eigenvalues appear as functions of the generalized Gamma function, the discreet atomic quantum numbers $(j,n,l,s\text{ and }m)$, two infinitesimal parameters $(\mathrm{\Theta },\overline{𝜃})$, which are induced by (position–position and phase–phase). In addition, the dimensional parameters $(a,V_0,V_1,\alpha )$ of perturbed Yukawa potential with centrifugal barrier model in NC: 3D-RSP. Furthermore, we have shown that the corresponding Hamiltonian operator in (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of perturbed Yukawa potential model and the two operators are modified spin–orbit interaction and the modified Zeeman operator for the previous Hydrogenic and neutral atoms.

In this study, the analytical solutions of the Klein–Gordon equation for any $l$ states of the modified effective mass potential under the modified unequal scalar and vector Coulomb–Hulthén potential (MUSVCH-P) are derived by using an approximation method to the centrifugal potential term in the symmetries of relativistic noncommutative three-dimensional real space (RNC: 3D-RS). The new analytical expressions for eigenvalues of the energy spectrum and the new mass of mesons, such as charmonium and bottomonium that have the quark and antiquark flavor, have been estimated by using Bopp’s shift method, and perturbation theory. The energy state equation depends on the global parameters characterizing the noncommutativity space and the potential parameter $(v_0,s_0,v_1,s_1,m_0,m_1,\delta )$ in addition to the Gamma function and the discreet atomic quantum numbers $(j,l,s,m)$. The expression for the new energy spectra is applied to obtain the new mass spectra of heavy quarkonium systems (charmonium and bottomonium) in the symmetries of (RNC: 3D-RS). The comparisons show that our theoretical results are in very good agreement with the reported works.

The deformed Klein–Gordon equation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the newly improved screened Kratzer potential (NISKP) model under the influence of the deformation space–space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H₂) molecules under the NISKP model were shown to be sensitive to the atomic quantum numbers ($j,l,s,m$), mixed potential depths ($D_e,r_e$), the screening parameter’s inverse $(\alpha ,\delta )$, control parameter $\tau$ and non-commutativity parameters ($\mathrm{\Theta }$, $\beta$, $\zeta$). In addition, we analyzed the non-relativistic energy values by applying the well-known transmission rules known in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the new screened cosine Kratzer potential, the new screened Kratzer potential (SKP), the new Kratzer potential, the new coshine Yukawa potential, the new coshine Yukawa potential, the new shifted improved SKP and the new shifted SKP. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system such as charmonium $c\overline{c}$, bottomonium $b\overline{b}$, $\overline{b}c$ and light mesons $c\overline{s}$ and $b\overline{s}$ that have the quark and antiquark flavor within the framework of the NISKP model in 3D-ENRQM symmetries.