A relationship between the discrete Dirac-Kähler equation and discrete analogues of some Dirac type equations in the geometric spacetime algebra is discussed. We show that a solution of the discrete Dirac-Kähler equation can be represented as the sum and difference of solutions of discrete Dirac type equations with the corresponding sign of the mass term.
In this work we study and exactly solve the Dirac oscillator with three different topological defects, namely the cosmic string spacetime ($Λ_\mp$), the magnetic cosmic string spacetime ($Θ_\mp$) and the cosmic dislocation spacetime ($Π_\mp$). Moreover, we show that the radial part of this problem possess an $SU(1,1)$ symmetry. From this, we obtain the wave functions and their respective energy spectrum by means of the Schrödinger factorization. Also, we compute the radial coherent states of the eigenfunctions of each problem and their time evolution.
In this article we study two-dimensional Dirac Hamiltonians with non-commutativity both in coordinates and momenta from an algebraic perspective. In order to do so, we consider the graded Lie algebra $\mathfrak{sl}(2|1)$ generated by Hermitian bilinear forms in the non-commutative dynamical variables and the Dirac matrices in $2+1$ dimensions. By further defining a total angular momentum operator, we are able to express a class of Dirac Hamiltonians completely in terms of these operators. In this way, we analyze the energy spectrum of some simple models by constructing and studying the represe…
The Bundles of Algebraic and Dirac-Hestenes Spinor Fields
The main objective of this paper is to clarify the ontology of Dirac-Hestenes spinor fields (DHSF) and its relationship with sum of even multivector fields, on a general Riemann-Cartan spacetime admitting a spin structure and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE) when spacetime is a Lorentzian. To this aim we introduce the Clifford bundle of multivector fields (Cl(M,g)) and the left (Cl_{ Spin_{1,3} ^{e}}^{l}(M)) and right (Cl_{ Spin_{1,3} ^{e}}^{r}(M)) spin-Clifford bundles on the spin manifold (M,g) The relation between left ideal algebra…
The massless Dirac particle in the $C$-metric, representing the exterior gravitational field of a uniformly accelerating black hole, is studied. Classes of (quasi-)polynomial solutions to the radial and the polar parts of the Dirac equation, each of which is equivalent to the general Heun equation~(GHE), are obtained exploiting the underlying $su(1,1)$ algebraic structures of the GHE.
In this article we present the algebraic rearrangement, or matrix inversion of the Dirac equation in a curved Riemann-Cartan spacetime with torsion, the presence of non-vanishing torsion is implied by the intrinsic spin-1/2 of the Dirac field. We then demonstrate how the inversion leads to a reformulation of the fully non-linear and self-interactive Einstein-Cartan-Dirac field equations in terms of Dirac bilinears. It has been known for some decades that the Dirac equation for charged fermions interacting with an electromagnetic field can be algebraically inverted, so as to obtain an explicit …
Basically (2 + 1) dimensional Dirac equation with real deformed Lorentz scalar potential is investi gated in this study. The position dependent Fermi velocity function transforms Dirac Hamiltonian into a Klein-Gordon-like effective Hamiltonian system. The complex Hamiltonian and its real energy spectrum and eigenvectors are obtained analytically. Moreover, the Lie algebraic analysis is performed.
General conditions for vanishing the current J^{z} for Dirac field on boundaries of the domain between two planes
在有外部磁场存在的旋量场的卡西米尔效应中，特别令人感兴趣的是狄拉克方程在两个平面限制的区域中的解，其中守恒电流J^{z}在两个边界上的第三投影为零。给出了流为零的一般条件，它们归结为线性齐次代数系统，只有当线性系统的行列式为零时才有解，即四阶代数方程关于变量e^{2 ik a}的根，其中a为平面间的半距离，k为Dirac粒子动量的第三投影.方程的所有解都已被明确地找到，它们中的每一个原则上都为我们提供了一种特殊的可能性，以获得参数k的量子化规则;大多数产生的根的表达式只能关于参数k数值求解。通常，解e^{2 ik a}取决于4个任意相位参数，这些参数影响具有消失电流的适当波函数：J^{z}（z=-z，+a）=0。
I review the various algebraic foundations of quantum mechanics. They have been suggested since the birth of this theory till up to last year. They are the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930), von Neumann (1936), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi (2009) and Buchholz and Fregenhagen (2019). Three cases are stressed: 1) the misinterpretation of Dirac foundation; 2) von Neumann conversion from the analytic approach of Hilbert space to the algebraic approach of the rings of operators; 3) the recent foundation of quantum mechanics upon the a…
Elementary commutator method for the Dirac equation with long-range perturbations
Graduate School of Mathematical Sciences, The University of Tokyo（东京大学大学院数理科学研究科）；Department of Mathematics, Graduate School of Science, Kobe University（神户大学理学研究科数学系）
