The Dirac equation vs. the Dirac type tensor equation

Approximate -state Solutions to the Dirac Mobius Square-Yukawa and Mobius Square-quasi Yukawa Problems under Pseudospin and Spin Symmetry limits with Coulomb-like Tensor Interaction

In this paper, we present the Dirac equation for the Mobius-square-Yukawa potentials including the tensor interaction term within the framework of pseudospin and spin symmetry limit with arbitrary spin-orbit quantum number . We obtained the energy eigenvalues and the corresponding wave functions using the supersymmetry method. The limiting cases of this potential model reduce to the Deng-Fan, Yukawa and Coulomb potentials, respectively

On Dirac equation for a Coulomb scalar, vector, and tensor interaction

In their recent paper (Inter. J. Mod. Phys. A 26 (2011) 1011), Zarrinkamar and coauthors have considered the radial Dirac equation for a Coulomb scalar, vector and tensor interaction. The exact solutions for the energy eigenvalues they have reported for spin-symmetry and pseudo-spin symmetry were mishandled and are incorrect eigenvalues, therefore.

Approximate analytical solutions of the Dirac equation for Yukawa potential plus Tensor Interaction with any $κ$-value

Approximate analytical solutions of the Dirac equation are obtained for the Yukawa potential plus a tensor interaction with any $κ$-value for the cases having the Dirac equation pseudospin and spin symmetry. The potential describing tensor interaction has a Yukawa-like form. Closed forms of the energy eigenvalue equations and the spinor wave functions are computed by using the Nikiforov-Uvarov method. It is observed that the energy eigenvalue equations are consistent with the ones obtained before. Our numerical results are also listed to see the effect of the tensor interaction on the bound st…

Maxwell-Dirac stress-energy tensor in terms of Fierz bilinear currents

We analyze the stress-energy tensor for the self-coupled Maxwell-Dirac system in the bilinear current formalism, using two independent approaches. The first method used is that attributed to Belinfante: starting from the spinor form of the action, the well-known canonical stress-energy tensor is augmented, by extending the Noether symmetry current to include contributions from the Lorentz group, to a manifestly symmetric form. This form admits a transcription to bilinear current form. The second method used is the variational derivation based on the covariant coupling to general relativity. Th…

Pure Coulomb tensor interaction in the Dirac equation

In this work we compute bound solutions for particles and anti-particles of the Dirac equation for a pure tensor radial Coulomb potential plus a constant. We find that the binding depends on the sign the tensor constant potential, and allows only bound solutions for a certain sign and magnitude of the $κ$ quantum number, which is related to the spin-orbit coupling in the Dirac equation. This relation is reversed for anti-particle solutions. On the other hand, the Coulomb tensor potential, although not biding, changes the range of $κ$ values for which there are bound solutions.

Dirac-Kahler equation in curved space-time, relation between spinor and tensor formulations

A common view is that generalization of a wave equation on Riemannian space-time is substantially determined by what a particle is - boson or fermion. As a rule, they say that tensor equations for bosons are extended in a simpler way then spinor equations for fermions. In that context, a very interesting problem is of extension a wave equation for Dirac--Kähler field (Ivanenko--Landau field was historically first term, also the term a vector field of general type was used). The article relates a generally covariant tensor formalism to a spinor one when these both are applied to description of …

Effect of Tensor Interaction in the Dirac-Attractive Radial Problem under Pseudospin Symmetry limit

We approximately investigated pseudospin symmetric solutions of the Dirac equation for attractive radial potential including a Coulomb-like tensor interaction under pseudospin symmetry limit for any spin-orbit quantum number . By using the parametric generalization of the Nikiforov-Uvarov method, the energy eigenvalues equation and the corresponding wave functions have been obtained in closed forms. Some numerical results are also given. We presented tensor interaction removes degeneracy between two states in pseudospin doublets.

Approximate bound state solutions of Dirac equation with Hulthén potential including Coulomb-like tensor potential

We solve the Dirac equation approximately for the attractive scalar $S(r)$ and repulsive vector $V(r)$ Hulthén potentials including a Coulomb-like tensor potential with arbitrary spin-orbit coupling quantum number $κ.$ In the framework of the spin and pseudospin symmetric concept, we obtain the analytic energy spectrum and the corresponding two-component upper- and lower-spinors of the two Dirac particles by means of the Nikiforov-Uvarov method in closed form. The limit of zero tensor coupling and the non-relativistic solution are obtained. The energy spectrum for various levels is presented f…

Superalgebras of Dirac operators on manifolds with special Killing-Yano tensors

We present the properties of new Dirac-type operators generated by real or complex-valued special Killing-Yano tensors that are covariantly constant and represent roots of the metric tensor. In the real case these are just the so called complex or hyper-complex structures of the K\" ahlerian manifolds. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. In this way the Dirac operators are related among themselves through continuous transformations associated with specific d…

一种适用于非线性狄拉克系统的旋量适配几何方法及其在闵可夫斯基时空附近张量波 - 狄拉克系统中的应用

A spinor-adapted geometric approach for nonlinear Dirac systems and its application to a tensorial wave-Dirac system near Minkowski spacetime

研究(1 + 3)维渐近平坦时空上的非线性张量波 - 狄拉克系统，通过保留狄拉克方程一阶性质，结合多种能量估计建立耦合能量方法，排除奇异非线性相互作用，得到小数据解全局存在及稳定性，为研究更一般非线性狄拉克系统提供思路。

Equations of Motion and Energy-Momentum 1-Forms for the Coupled Gravitational, Maxwell and Dirac Fields

A theory where the gravitational, Maxwell and Dirac fields (mathematically represented as particular sections of a convenient Clifford bundle) are supposed fields in Faraday's sense living in Minkowski spacetime is presented. In our theory there exist a genuine energy-momentum tensor for the gravitational field and a genuine energy-momentum conservation law for the system of the interacting gravitational, Maxwell and Dirac fields. Moreover, the energy-mometum tensors of the Maxwell and Dirac fields are symmetric, and it is shown that the equations of motion for the gravitational potentials is …

Exact solutions of a spatially-dependent mass Dirac equation for Coulomb field plus tensor interaction via Laplace transformation method

The spatially-dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction potential under the spin and pseudospin (p-spin) symmetric limits by using the Laplace transformation method (LTM). Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublet…

A tensor interpretation of the 2D Dirac equation

We consider the Dirac equation in flat Minkowski 3-space and rewrite it as the Maxwell equation in Minkowski 4-space with torsion. The torsion tensor is defined as the dual of the electromagnetic vector potential. Our model clearly distinguishes the electron and the positron without resorting to "negative frequencies": we produce a real scalar invariant (charge) which indicates whether we are looking at an electron or a positron. Another interesting feature of our model is that the free electron and positron are identified with gradient type solutions of the standard (torsion free) Maxwell equ…

Symmetries of the Dirac operators associated with covariantly constant Killing-Yano tensors

The continuous and discrete symmetries of the Dirac-type operators produced by particular Killing-Yano tensors are studied in manifolds of arbitrary dimensions. The Killing-Yano tensors considered are covariantly constant and realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. The Dirac operators are related among themselves through continuous or discrete transformations. It is shown that the groups of the continuous symmet…

Approximate Spin and Pseudospin Solutions of the Dirac equation with Rosen-Morse Potential including a Coulomb Tensor Interaction

By applying the Pekeris-type approximation to deal with the (pseudo or) centrifugal term, the spin and pseudospin symmetry solutions of the Dirac equation for the Rosen-Morse potential including a Coulomb-like tensor potential with arbitrary spin-orbit coupling quantum number $κ$ are obtained by standard method. It has been shown from the numerical results that the degeneracies between spin and pseudospin state doublets are removed by the tensor interaction. Special case of this potential, that is, the spin and pseudospin solutions of the Dirac equation with the modified Pöschl-Teller potentia…

Dirac Notation, Fock Space and Riemann Metric Tensor in Information Retrieval Models

Using Dirac Notation as a powerful tool, we investigate the three classical Information Retrieval (IR) models and some their extensions. We show that almost all such models can be described by vectors in Occupation Number Representations (ONR) of Fock spaces with various specifications on, e.g., occupation number, inner product or term-term interactions. As important cases of study, Concept Fock Space (CFS) is introduced for Boolean model; the basic formulas for Singular Value Decomposition (SVD) of Latent Semantic Indexing (LSI) Model are manipulated in terms of Dirac notation. And, based on …

The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor

We discuss from scratch the classical structure of Dirac spinors on an arbitrary globally hyperbolic, Lorentzian spacetime, their formulation as a locally covariant quantum field theory, and the associated notion of a Hadamard state. Eventually, we develop the notion of Wick polynomials for spinor fields, and we employ the latter to construct a covariantly conserved stress-energy tensor suited for back-reaction computations. We shall explicitly calculate its trace anomaly in particular.

Bound states of spatially dependent mass Dirac equation with the Eckart potential including Coulomb tensor interaction

We investigate the approximate solutions of the Dirac equation with the position-dependent mass particle in the Eckart potential field including the Coulomb tensor interaction by using the parametric Nikiforov-Uvarov method. Taking an appropriate approximation to deal with the centrifugal term, the Dirac energy states and the corresponding normalized two-spinor components of the wave function are obtained in closed form. Some special cases of our solution are investigated. Furthermore, we present the correct solutions obtained via the asymptotic iteration method which are in agreement with the…

Exact Spin and Pseudo-Spin Symmetric Solutions of the Dirac-Kratzer Problem with a tensor potential via Laplace Transform Approach

Exact bound state solutions of the Dirac equation for the Kratzer potential in the presence of a tensor potential are studied by using the Laplace transform approach for the cases of spin- and pseudo-spin symmetry. The energy spectra is obtained in the closed form for the relativistic as well as non-relativistic cases including the Coulomb potential. It is seen that our analytical results are in agrement with the ones given in literature. The numerical results are also given in a table for different parameter values.

洛伦兹不变性的多维狄拉克-海斯坦方程

Lorentz Invariance of the Multidimensional Dirac-Hestenes Equation

本文研究多维狄拉克-海斯坦方程的洛伦兹不变性，通过张量和旋量两种方法比较其在不同变换下的形式不变性及协变性。

A Study of the Dirac-Sidharth Equation

The Dirac-Siddharth Equation has been constructed from the Siddharth hamiltonian by quantization of the energy and momentum in Pauli algebra. We have solved this equation by using tensor product of matrices.

Some Consequences from the Dirac-Kahler Theory: on Intrinsic Spinor Sub-structure of the Different Boson Wave Functions

Properties of tensors equivalent to the direct product of two different 4-spinors are investigated. It is shown that the tensors obey additional 8 nonlinear restrictions, those are presented in Lorentz covariant form. In the context of the Dirac-Kähler theory, such a property can be interpreted as follows: if one wishes to consider the Dirac-Kähler field as consisting of two 4-spinor fields, one must impose additional restrictions on tensors of the Dirac-Kähler field, which leads to a non-linear wave equation for a complex boson field (composed on the base of two 4-spinor fields). Instead, the…

Interplay between Dynamic Systems Described by the Klein-Gordon and Dirac Equations

It is shown that in the two-dimensional space-time the dynamic system, described by the free Klein-Gordon equation, turns to the dynamic system, described by the free Dirac equation, provided the current and the energy-momentum tensor are redefined in a proper way.

The dihedral Dunkl--Dirac symmetry algebra with negative Clifford signature

The Dunkl--Dirac symmetry algebra is an associative subalgebra of the tensor product of a Clifford algebra and the faithful polynomial representation of a rational Cherednik algebra. In previous work, the finite-dimensional representations of the Dunkl--Dirac symmetry algebra in three dimensions linked with a dihedral group were given. We give here the necessary results to proceed to the same construction when the Clifford algebra in the tensor product has negative signature.

Separability of the massive Dirac's equation in 5-dimensional Myers-Perry black hole geometry and its relation to a rank-three Killing-Yano tensor

The Dirac equation for the electron around a five-dimensional rotating black hole with two different angular momenta is separated into purely radial and purely angular equations. The general solution is expressed as a superposition of solutions derived from these two decoupled ordinary differential equations. By separating variables for the massive Klein-Gordon equation in the same space-time background, I derive a simple and elegant form for the Stackel-Killing tensor, which can be easily written as the square of a rank-three Killing-Yano tensor. I have also explicitly constructed a symmetry …

Symmetry operators for Dirac's equation on two-dimensional spin manifolds

It is shown that the second order symmetry operators for the Dirac equation on a general two-dimensional spin manifold may be expressed in terms of Killing vectors and valence two Killing tensors. The role of these operators in the theory of separation of variables for the Dirac equation is studied.

Analytical bound state solutions of the Dirac equation with the Hulthén plus a class of Yukawa potential including a Coulomb-like tensor interaction

We examine the bound state solutions of the Dirac equation under the spin and pseudospin symmetries for a new suggested combined potential, Hulten plus a class of Yukawa potential including a Coulomb-like tensor interaction. An improved scheme is employed to deal with the centrifugal (pseudo-centrifugal) term. Using the Nikiforov-Uvarov and SUSYQM methods, we analytically develop the relativistic energy eigenvalues and associated Dirac spinor components of wave functions. We find that both methods give entirely the same results. Modifiable of our results into some particular potential cases, u…

Comparison of two formulas for metric connections in the bundle of Dirac spinors

Two explicit formulas for metric connections in the bundle of Dirac spinors are studied. Their equivalence is proved. The explicit formula relating the spinor curvature tensor with the Riemann curvature tensor is rederived.

A coordinateless form of the Dirac equation

We present a so called Dirac-type tensor equation (DTTE). This equation is written in coordinateless form with the aid of differential operators $d$ and $δ$. A wave function of DTTE belongs to a minimal left ideal of the algebra of exterior forms with respect to the Clifford product. We show that a coordinate form of DTTE is identical to the Dirac equation in a fixed coordinate system.

Hidden symmetries and Killing tensors on curved spaces

Higher order symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing-Yano tensors and non-standard supersymmetries is pointed out. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The general results are applied to space-times which appear in modern studies. One presents the infinite dimensional superalgebra of Dirac type operators on the 4-dimensional Euclidean Taub-NUT space that can be s…

Dirac-Kähler field, spinor technique, and 2-potential approach to electrodynamics with two charges

From the 16-component Dirac-Kähler field theory, spinor equations for two types of massless vector photon fields with different parities have been derived. Their equivalent tensor equations in terms of the strength tensor $F_{ab}$ and respective 4-vector $A_{b}$ and 4-pseudovector $\tilde{A}_{b}$ depending on intrinsic photon parity are derived; they include additional sources, electric 4-vector $j_{b}$ and magnetic 4-pseudovector $\tilde{j}_{b}$. The theories of two types of photon fields are explicitly uncoupled, their linear combination through summing or subtracting results in Maxwell elec…

Symmetries and supersymmetries of the Dirac operators in curved spacetimes

It is shown that the main geometrical objects involved in all the symmetries or supersymmetries of the Dirac operators in curved manifolds of arbitrary dimensions are the Killing vectors and the Killing-Yano tensors of any ranks. The general theory of external symmetry transformations associated to the usual isometries is presented, pointing out that these leave the standard Dirac equation invariant providing the correct spin parts of the group generators. Furthermore, one analyzes the new type of symmetries generated by the covariantly constant Killing-Yano tensors that realize certain square…

关于 $Spin(3,1)$ 张量、Dirac 场和 $GL(k, \mathbb{R})$ 对称性的注记

Note on $Spin(3,1)$ tensors, the Dirac field and $GL(k, \mathbb{R})$ symmetry

本文研究了 $Spin(3,1)$ 张量实矩阵的秩分解，导出受 $GL(k, \mathbb{R})$ 对称性约束的 Majorana 双旋量与 Dirac 场，并详细分析了 $k=1$ 时的对称性及守恒流，同时澄清了关于矩阵秩约束的代数条件问题。

非对称非交换 torus 的 Einstein 张量消失

Asymmetric noncommutative torus has vanishing Einstein tensor

研究通过部分共形缩放计算非交换 torus 上谱三元组的谱度量，发现其 torsion 和 Einstein 张量均消失。

A nonlinear evolution equation with 2 + 1 dimensions related to nonstationary Dirac-type system

In this paper the inverse scattering problem for the nonstationary Dirac-type system on the whole plane was considered. A nonlinear evolution sytem of equation related to nonstationary Dirac-type system is introduced and the solviblity of this sytem using the IST method is studied.

Spin and Pseudo-Spin Symmetries in Radial Dirac Equation and Exceptional Hermite Polynomials

We have generalized the solutions of the radial Dirac equation with a tensor potential under spin and pseudospin symmetry limits to exceptional orthogonal Hermite polynomials family. We have obtained new general rational potential models which are the generalization of the nonlinear isotonic potential families and also energy dependent.

Relativistic new Yukawa-like potential and tensor coupling

We approximately solve the Dirac equation for a new suggested generalized inversely quadratic Yukawa (GIQY) potential including a Coulomb-like tensor interaction with arbitrary spin-orbit coupling quantum number In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions, in closed form, by using the parametric Nikiforov-Uvarov (NU) method. The numerical results show that the Coulomb-like tensor interaction, removes degeneracies between spin and p-spin state doublets. The Dirac solutions in the presence of exact s…

Quantum Dirac fermions in half space and their interaction with electromagnetic field

We study the polarization tensor of a Dirac field in $(3+1)$ dimensions confined to a half space -- a problem motivated by applications to the condensed matter physics, and to Topological Insulators in particular. Although the Pauli-Villars regularization scheme has a number of advantages, like explicit gauge invariance and decoupling of heavy modes, it is not applicable on manifolds with boundaries. Here, we modify this scheme by giving an axial mass to the regulators and to the physical field. We compute the renormalized polarization tensor in coordinate representation. We discuss then the i…

Semiclassical Description of Relativistic Spin without use of Grassmann variables and the Dirac equation

We propose a relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation. Both $Γ$\,-matrices and the relativistic spin tensor are produced through the canonical quantization of the classical variables which parametrize the properly constructed relativistic spin surface. Although there is no mass-shell constraint in our model, our particle's speed cannot exceed the speed of light. The classical dynamics of the model is in correspondence with the dynamics of mean values of the corresponding operators in the Dirac theory. In particular,…

The Dirac equation in curved spacetimes using coordinate-free notation

The Dirac equation in curved spacetimes is formulated using coordinate-free notation. A Lagrangean density which corresponds to the subject equation is presented. The subject equation is invariant under a local rotation of the coframe. The current is independent of the local orientation of the coframe and it is conserved. The subject equation has an equivalent formulation which uses the Christoffel gamma. The stress-energy tensor is calculated.

Symmetry operators and separation of variables for Dirac's equation on two-dimensional spin manifolds with external fields

The second order symmetry operators that commute with the Dirac operator with external vector, scalar and pseudo-scalar potentials are computed on a general two-dimensional spin-manifold. It is shown that the operator is defined in terms of Killing vectors, valence two Killing tensors and scalar fields defined on the background manifold. The commuting operator that arises from a non-trivial Killing tensor is determined with respect to the associated system of Liouville coordinates and compared to the the second order operator that arises from that obtained from the unique separation scheme ass…

Pseudo-Hermitian Dirac operator on the torus for massless fermions under the action of external fields

The Dirac equation in $(2+1)$ dimensions on the toroidal surface is studied for a massless fermion particle under the action of external fields. Using the covariant approach based on general relativity, the Dirac operator stemming from a metric related to the strain tensor is discussed within the Pseudo-Hermitian operator theory. Furthermore, analytical solutions are obtained for two cases, namely, constant and position-dependent Fermi velocity.

Symmetry operators and separability of massive Klein-Gordon and Dirac equations in the general 5-dimensional Kerr-(anti-)de Sitter black hole background

It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr-(anti-)de Sitter black hole with two independent angular momenta. A first order symmetry operator that commutes with the Dirac operator is constructed in terms of a rank-three Killing-Yano tensor whose square is a second order symmetric Stackel-Killing tensor admitted by the five-dimensional Kerr-(anti-)de Sitter spacetime. We highlight the construction procedure of such a symmetry operator. In addition, the first law of black hole thermodynamics has been extended to the case that the cosmological…

The Index of Some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fields

We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed with the help of explicitly describing the dimension of the cohomology groups of generalised (`harmonic') Dirichlet and Neumann tensor fields. The main results of this contribution are the computation of the indices of Dirac-type operators associated to the elasticity complex and the newly found biharmonic complex, relevant for the biharmonic equation, elasticity, and fo…

Dirac operator and its cohomology for the quantum group $U_q(\mathfrak{sl}_2)$

We introduce a Dirac operator $D$ for the quantum group $U_q(\mathfrak{sl}_2)$, as an element of the tensor product of $U_q(\mathfrak{sl}_2)$ with the Clifford algebra on two generators. We study the properties of $D$, including an analogue of Vogan's conjecture. We compute the cohomology of $D$ acting on various $U_q(\mathfrak{sl}_2)$-modules.

Separability of a modified Dirac equation in a five-dimensional rotating, charged black hole in string theory

The aim of this paper is to investigate the separability of a spin-1/2 spinor field in a five-dimensional rotating, charged black hole constructed by Cvetic and Youm in string theory, in the case when three U(1) charges are set equal. This black hole solution represents a natural generalization of the famous four-dimensional Kerr-Newman solution to five dimensions with the inclusion of a Chern-Simons term to the Maxwell equation. It is shown that the usual Dirac equation can not be separated by variables in this general spacetime with two independent angular momenta. However if one supplements…

Energy methods for Dirac-type equations in two-dimensional Minkowski space

In this article we develop energy methods for a large class of linear and nonlinear Dirac-type equations in two-dimensional Minkowski space. We will derive existence results for several Dirac-type equations originating in quantum field theory, in particular for Dirac-wave maps to compact Riemannian manifolds.

The Dirac equation without spinors

In the first part of the paper we give a tensor version of the Dirac equation. In the second part we formulate and analyse a simple model equation which for weak external fields appears to have properties similar to those of the 2--dimensional Dirac equation.

关于几何谱函数的探讨

On Geometric Spectral Functionals

本文研究了与Dirac和拉普拉斯型微分算子相关的谱函数，扩展了经典结果至含torsion的几何结构，揭示了谱函数与体积形式、标量曲率等几何张量的关系，并引入了chiral谱函数。