Noncommutative Integration of the Dirac Equation in a Flat Space and in the de Sitter Space

Noncommutative integration of the Dirac massive and massless equations is performed in a four-dimensional flat space and in the de Sitter space of arbitrary signature. A new class of rigorous solutions of the Dirac equation is constructed in these spaces. The properties of the solutions obtained are examined.     

Massless Fields on Dirac Six-Cone and De Sitter Ambient Space

We have proceeded to obtain manifestly conformally invariant (CI) equations for thinkable graviton fields in de Sitter (dS) space-time. The tensor fields are originally considered in 4+2 dimensional conformal space or Dirac’s six-cone and then project to dS space which is embedded in 4+1 dimensional ambient space. It will be shown that, by projecting these tensor fields there exists a correspondence between the massless fields on the cone and dS space. Also, we have shown that for rank-2 tensor field the divergenceless condition, which is necessary when we attempt to correspond the tensor field with the unitary irreducible representations (UIRs) of dS group, is not really a condition at all, it is a consequence of ambient space property. Due to the combined occurrences of corresponding fields and divergenceless property, the appropriate CI field equations have obtained in a fairly simple way and without imposing any extra condition.     

Symmetry and Conservation Laws for a Dirac Equation in a Riemannian Space

The linkage of the linear symmetry operators of the Dirac–Fock equation for electrons and massless particles with the differential conservation laws and the symmetry of equations in conformal spaces are studied.     

Semi-Analytical Solution of Dirac Equation in Schwarzschild-de Sitter Spacetime

In the present paper, we solve the radial parts of Dirac equation between the inner and the outer horizon in the Schwarzschild-de Sitter (SdS for short) geometry. Complete physical parameter space is divided into two regions depending on the height of the potential barrier and the energy of the incoming particle. In each region, we concentrate on two limiting cases. The first case is when the two horizons are close to each other and the second case is when the horizons are far apart. In each case, we give the semi-analytical solution by using WKB (Wentzel-Krames-Brillouin) approximation and show the instantaneous reflection and transmission coefficients as well as the radial wave functions graphically. PACS: 04.20.-q, 04.70.-s, 04.70.Dy, 95.30.Sf     

Dirac Equation in the Spatially Flat Friedmann Model

The unitary transformation which diagonalizesthe field-free Dirac Hamiltonian in the spatially flatFriedmann-Robertson-Walker metric is analyzed, and apair of simultaneous first-order nonlinear differential equations is derived for the two parameters(two angles) that characterize the transformation. Theequations are solved approximately for a test particlewhose kinetic energy is small compared to its mass energy, and minimum-uncertainty wave packetsare constructcd from the solutions. It is found thatgeneral relativity limits the quantum mechanical spreadof the wave packets, but forces then to expand with the expanding space, as if they were embeddedin it. The massless Dirac equation is solved exactly forthe two-component neutrino spinor, and yieldsgeneralized nonspreading wave packets which display no quantum mechanical spread at all, but areconstrained to expand with the expanding space as theyfollow null geodesics.     

Quantum Minkowski Space and the q-Analogue of Dirac Equation

The covariant differential calculus on the quantum Minkowski space is presented with the help of the generalized Wess-Zumino method and the quantum Pauli matrices. The quantum Dirac matrices are constructed parallel to those in the classical case. Combining these two aspects a q-analogue of Dirac equation follows directly.     

Dirac Equation in Space–Time with Torsion

The Dirac equation in a curved space–time endowed with compatible affine connection is reconsidered. After a detailed decomposition of the total action, the equation is obtained by varying with respect to the Dirac spinor and the torsion field. The result is a known Dirac-like equation with constraints that can be interpreted as the equation of a self-interacting spin 1/2 particle in curved space–time. The scheme is then translated into the language of the 2-spinor formalism of curved space–time based on the choice of a null tetrad frame. The spinorial equation so obtained coincides with the standard one in case of no torsion, while in general it remains a nonlinear equation describing a self-interacting spin 1/2 particle. The nonlinearity is produced by the interaction of the particle with its own current that remains conserved as in the free torsion case.     

Solutions of Dirac Equation for Makarov Potential with Pseudospin Symmetry

The pseudospin symmetry in the Makarov potential is investigated systematically by solving the Dirac equation. The analytical solution for the Makarov potential with pseudospin symmetry is obtained by Nikiforov-Uvarov （N-U） method. The eigenfunctions and eigenenergies are presented with equal mixture of vector and scalar potentials in opposite signs, for which is exact.     

Solutions of Dirac Equation for Makarov Potential with Pseudospin Symmetry

The pseudospin symmetry in the Makarov potential is investigated systematically by solving the Dirac equation. The analytical solution forthe Makarov potential with pseudospin symmetry is obtained byNikiforov-Uvarov (N-U) method. The eigenfunctions and eigenenergies arepresented with equal mixture of vector and scalar potentials in oppositesigns, for which is exact.     

Woods-Saxon势下的赝自旋对称性

通过求解具有Woods-Saxon径向标量势与矢量势的Dirac方程, 分别分析了原子核中赝自旋双重态能级劈裂和波函数下分量劈裂随着Woods-Saxon势参数的变化关系, 其中Wood-Saxon势中的参数a以及R对于赝自旋能级劈裂和波函数劈裂的影响最大, 而势阱深度V₀只产生微小的影响. 随着平均场的变化, 赝自旋双重态会出现能级反转现象. 具有较大n或l的赝自旋双重态能级劈裂对于参数的变化更敏感. 赝自旋波函数下分量劈裂随着参数a和R的增大向核表面扩散, 并且在核表面附近达到最大. 赝自旋波函数劈裂随着参数a的增大而增大, 但是随着参数R或V₀的增大它却在减小的. 由于在特定的同位素链中, 参数R和V₀都与核子数有关系, 而参数a又是描述核的表面性质的, 所以以这些参数为变量对于赝自旋劈裂的研究是有意义的, 研究的结果至少在定性上可以应用到大部分原子核中.     

Some Problems of the Existence of Symmetry Spinor Operators for the Dirac Equation

Conditions necessary for the existence of a class of fields that can be used to construct the spinor symmetry operators for the Dirac equation in Riemannian space are specified in the present paper. The metrics of spaces with four-dimensional groups of motions in which these fields exist are indicated. A class of spaces is identified in which the Dirac equation admits no separation of variables within the framework of the definition adopted, but the algebra of symmetry of the Dirac equation satisfies the conditions of theorems of the noncommutative intergrability.     

Approximate Analytical Solutions of the Dirac Equation with the Hyperbolic Potential in the Presence of the Spin Symmetry and Pseudo-spin Symmetry

By employing a new improved approximation scheme to deal with the centrifugal term and pseudo-centrifugal term, we solve approximately the Dirac equation with the hyperbolic potential for the arbitrary spin-orbit quantum number κ. Under the condition of the spin and pseudospin symmetry, the bound state energy eigenvalues and the associated two-component spinors of the Dirac particle are obtained approximately by using the basic concept of the supersymmetric shape invariance formalism and the function analysis method.     

A Static Solution of Yang-Mills Equation on Anti-de Sitter Space

Since product metric on AdS space has played a very important role in Lorentz version of AdS/CFT correspondence, the Yang Mills equation on AdS space with this metric is considered and a static solution is obtained in this paper, which helps to understand the AdS/CFT correspondence of Yang Mills fields.     

Spontaneous Edge Currents for the Dirac Equation in Two Space Dimensions

Spontaneous edge currents are known to occur in systems of two space dimensions in a strong magnetic field. The latter creates chirality and determines the direction of the currents. Here we show that an analogous effect occurs in a field-free situation when time reversal symmetry is broken by the mass term of the Dirac equation in two space dimensions. On a half plane, one sees explicitly that the strength of the edge current is proportional to the difference between the chemical potentials at the edge and in the bulk, so that the effect is analogous to the Hall effect, but with an internal potential. The edge conductivity differs from the bulk (Hall) conductivity on the whole plane. This results from the dependence of the edge conductivity on the choice of a selfadjoint extension of the Dirac Hamiltonian. The invariance of the edge conductivity with respect to small perturbations is studied in this example by topological techniques Mathematics Subject Classification (2000). 81Q10, 58J32     

On Symmetry of the Generalized Breit Equation

Abstract

In this paper we find the complete set of symmetry operators for the two-particle Breit equation in the class of first-order differential operators with matrix coefficients. A new integral of motion is obtained.     

Dirac wave equation in the de Sitter universe

We present and discuss the Dirac wave equation in the de Sitter universe. This equation is obtained by factoring the second-order Casimir invariant operator associated to the Fantappié-de Sitter group.