Topics in Supersymmetric and Noncommutative Quantum Cosmology

In the present article we review the work carried out by us and collaborators on supersymmetric quantum cosmology, noncommutative quantum cosmology and the application of GUPs to quantum cosmology and black holes. The review represents our personal view on these subjects and it is presented in chronological order.

A theoretical study of the modified equal scalar and vector Manning-Rosen potential within the deformed Klein-Gordon and Schrödinger in RNCQM and NRNCQM symmetries

In this research work, within the framework of relativistic and nonrelativistic noncommutative quantum mechanics, the deformed Klein–Gordon and Schrödinger equations were solved with the modified equal vector scalar Manning-Rosen potential that has been of significance interest in recent years using Bopp's shift method and standard perturbation theory in the first-order in the noncommutativity parameters in 3-dimensions noncommutative quantum mechanics. By employing the improved approximation of the centrifugal term, the relativistic and nonrelativistic bound state energies were obtained for some diatomic molecules such as (HCl, CH, LiH, CO, NO, O2, I2, N2, H2, and Ar2). The obtained energy eigenvalues appear as a function of the generalized Gamma function, the parameters of noncommutativity, and the parameters of studied potential, in addition to the atomic quantum numbers . In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result is consistent with what is obtained in the literature. We have seen that the improved approximation of the centrifugal term is better than the other approximations in finding the approximate analytical solutions of the Klein-Gordon and Schrödinger equations for the modified Manning–Rosen potential in RNCQM and NRNCQM.

The Relativistic and Nonrelativistic Solutions for the Modified Unequal Mixture of Scalar and Time-Like Vector Cornell Potentials in the Symmetries of Noncommutative Quantum Mechanics

In this work, we have obtained analytically the bound state solution for both the relativistic modified Klein-Gordon equation MKG and non-relativistic modified Schrödinger equation for the modified unequal mixture of scalar and time-like vector Cornell (MUSVC) potentials in the relativistic noncommutative three-dimensional real space (RNC: 3D-RS) symmetries. The unequal mixture of scalar and time-like vector Cornell potentials is extended by including new radial terms. Also, MUSVC potentials are proposed as a quark-antiquark interaction potential for studying the masses of heavy and heavy-light mesons in (RNC: 3D-RSP) symmetries. The ordinary Bopp’s shift method and perturbation theory are surveyed to get generalized excited states’ energy as a function of shift energy and the energy of USVC potentials in the relativistic quantum mechanics RQM and NRQM. Furthermore, the obtained preservative solutions of discrete spectrum depended on the parabolic cylinder function, the gamma function, the ordinary discrete atomic quantum numbers, as well as the potential parameters and the two infinitesimal parameters (θ and σ) which are generated with the effect of (space-space) noncommutativity properties. We have also applied our obtained results for bosonic particles, like the charmonium cc ¯ and bottomonium bb ¯ mesons (that have quark and antiquark flavour) and cs ¯ mesons with spin-(0 and 1) and shown that MKG equation under MUSVC potentials becomes similar to the Duffin–Kemmer equation. We have shown that the degeneracy of the initial spectral under USVC potentials in RQM is changed radically and replaced by the newly triplet degeneracy of energy levels under the MUSVC potentials; this gives more precision in measurement and better results compared to the results of ordinary RQM under USVC potentials. Keywords: Klein-Gordon equation, Schrödinger equation, Unequal mixture of scalar and time-like vector Cornell potentials, Noncommutative quantum mechanics, Star product, Bopp’s shift method, Heavy–light mesons. PACS Nos.: 03.65.Ta; 03.65.Ca; 03.65.Ge.

⋆-cohomology, third type Chern character and anomalies in general translation-invariant noncommutative Yang–Mills

A representation of general translation-invariant star products ⋆ in the algebra of [Formula: see text] is introduced which results in the Moyal–Weyl–Wigner quantization. It provides a matrix model for general translation-invariant noncommutative quantum field theories in terms of the noncommutative calculus on differential graded algebras. Upon this machinery a cohomology theory, the so-called ⋆-cohomology, with groups [Formula: see text], [Formula: see text], is worked out which provides a cohomological framework to formulate general translation-invariant noncommutative quantum field theories based on the achievements for the commutative fields, and is comparable to the Seiberg–Witten map for the Moyal case. Employing the Chern–Weil theory via the integral classes of [Formula: see text] a noncommutative version of the Chern character is defined as an equivariant form which contains topological information about the corresponding translation-invariant noncommutative Yang–Mills theory. Thereby, we study the mentioned Yang–Mills theories with three types of actions of the gauge fields on the spinors, the ordinary, the inverse, and the adjoint action, and then some exact solutions for their anomalous behaviors are worked out via employing the homotopic correlation on the integral classes of ⋆-cohomology. Finally, the corresponding consistent anomalies are also derived from this topological Chern character in the ⋆-cohomology.