Dirac–Kratzer problem with spin and pseudo-spin symmetries in curved space–time

In this work, the Dirac–Kratzer problem with spin and pseudo-spin symmetries in a deformed nucleus is analyzed. Thus, the Dirac equation in curved space–time was considered, with a line element given by [Formula: see text], where [Formula: see text] is a scalar potential, coupled to vector [Formula: see text] and tensor [Formula: see text] potentials. Defining the vector and scalar potentials of the Kratzer type and the tensor potential given by a term centrifugal-type term plus a term cubic singular at the origin, we obtain the Dirac spinor in a quasi-exact way and the eigenenergies numerically for the spin and pseudo-spin symmetries, so that these symmetries are removed due to the coupling of an Coulomb-type effective tensor potential coming from the curvature of space, however, when such potential is null the symmetries return. The probability densities were analyzed using graphs to compare the behavior of the system with and without spin and pseudo-spin symmetries.

Curvature tensors of higher-spin gauge theories derived from general Lagrangian densities

Curvature tensors of higher-spin gauge theories have been known for some time. In the past, they were postulated using a generalization of the symmetry properties of the Riemann tensor (curl on each index of a totally symmetric rank-n field for each spin-n). For this reason they are sometimes referred to as the generalized 'Riemann' tensors. In this article, a method for deriving these curvature tensors from first principles is presented; the derivation is completed without any a priori knowledge of the existence of the Riemann tensors or the curvature tensors of higher-spin gauge theories. To perform this derivation, a recently developed procedure for deriving exactly gauge invariant Lagrangian densities from quadratic combinations of N order of derivatives and M rank of tensor potential is applied to the N = M = n case under the spin-n gauge transformations. This procedure uniquely yields the Lagrangian for classical electrodynamics in the N = M = 1 case and the Lagrangian for higher derivative gravity (`Riemann' and `Ricci' squared terms) in the N = M = 2 case. It is proven here by direct calculation for the N = M = 3 case that the unique solution to this procedure is the spin-3 curvature tensor and its contractions. The spin-4 curvature tensor is also uniquely derived for the N = M = 4 case. In other words, it is proven here that, for the most general linear combination of scalars built from N derivatives and M rank of tensor potential, up to N=M=4, there exists a unique solution to the resulting system of linear equations as the contracted spin-n curvature tensors. Conjectures regarding the solutions to the higher spin-n N = M = n are discussed.

Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances

The use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex variable method utilising Airy’s stress function, which can be generalised to a first integral methodology based on the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Basic questions relating to the existence and gauge freedoms of the potential fields and the satisfaction of the boundary conditions required for closure are addressed; with respect to (i), the properties of self-adjointness and Galilean invariance are of particular interest. The application and use of both approaches is explored through the solution of four purposely selected problems; three of which are tractable analytically, the fourth requiring a numerical solution. In all cases, the results obtained are found to be in excellent agreement with corresponding solutions available in the open literature.

A connection between linearized Gauss–Bonnet gravity and classical electrodynamics

A connection between linearized Gauss–Bonnet gravity and classical electrodynamics is found by developing a procedure which can be used to derive completely gauge-invariant models. The procedure involves building the most general Lagrangian for a particular order of derivatives ([Formula: see text]) and a rank of tensor potential ([Formula: see text]), then solving such that the model is completely gauge-invariant (the Lagrangian density, equation of motion and energy–momentum tensor are all gauge-invariant). In the case of [Formula: see text] order of derivatives and [Formula: see text] rank of tensor potential, electrodynamics is uniquely derived from the procedure. In the case of [Formula: see text] order of derivatives and [Formula: see text] rank of symmetric tensor potential, linearized Gauss–Bonnet gravity is uniquely derived from the procedure. The natural outcome of the models for classical electrodynamics and linearized Gauss–Bonnet gravity from a common set of rules provides an interesting connection between two well-explored physical models.

Solutions to the Dirac Equation for Manning-Rosen Plus Shifted Deng-Fan Potential and Coulomb-Like Tensor Interaction Using Nikiforov-Uvarov Method

We solve the Dirac equation for the Manning-Rosen plus shifted Deng-Fan potential including a Coulomb-like tensor potential with arbitrary spin–orbit coupling quantum number κ. In the framework of the spin and pseudospin (pspin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions in closed form by using the Nikiforov–Uvarov method. Also Special cases of the potential as been considered and their energy eigen values as well as their corresponding eigen functions are obtained for both relativistic and non-relativistic scope.