From Dirac’s Aether to the Dirac Equation

In 1951, Dirac proposed a formalism for a Lorentz invariant Aether with a vacuum state that contains all possible velocity states at each space-time point. Dirac showed no explicit path from the Aether towards the Quantum Mechanics. In this paper, we demonstrate that Dirac’s proposed Aether can be described by a lattice of possible events in space-time built in the local Lorentz frame. The idealised case of single velocity state leads to the famous Dirac equation for a plane wave state and is compatible with quantum statistics. On the lattice, possible space-time events are connected by the Dirac spinors which provide the probability of observing an event. The inertial mass of a particle is shown to be equivalent to the density of possible events on the lattice. Variation of the lattice density of events modifies the metric and provides a space-time curvature leading to the Hilbert action associated with general relativity. In classical limit, the perturbation in the density of possible events of the Aether is proportional to the Newtonian gravitational potential.

紧Spin流形上Dirac-Laplace方程解的存在性

本研究利用对称形式的山路定理阐释了紧Spin流形上Dirac-Laplace方程解的存在性,改进了文献[2]中的条件,得到了Dirac-Laplace方程具有无穷多对不同的解,这为高阶Dirac方程解的存在性研究提供了重要的参考.

一种基于Dirac半金属的可调控极化偏转器件的仿真分析

本文提出了一种基于体Dirac半金属(Bulk Dirac Semimetal,BDS)的动态超材料结构极化偏转器.与传统的极化偏转器不同,该偏转器可以通过改变费米能级来动态控制偏振转换率和不对称传输.模拟结果表明,当BDS的费米能级调整为90 meV时,偏转器的偏振转换率在1.085~1.872 THz频率范围内都保持在90%以上(带宽为0.787 THz),在1.159 THz处偏振转换率达到峰值98%.该偏振转换器的不对称传输参数在1.229~1.831 THz大带宽范围内大于60%.本文还研究了偏振旋转角、椭圆度角、表面电流和电场分布情况以阐明该器件的极化转换特性和物理机制;通过施加栅极电压控制BDS费米能级,实现了对该极化偏转器的偏振转换率以及不对称传输的动态调谐.该极化偏转器的可调谐不对称传输特性,为基于动态不对称传输所设计的多路复用器、太赫兹二极管等设备的制造提供了新的思路.

含有Dirac位势的非线性S-P方程的约束极小元

该文研究了一类含有Dirac位势的非线性Schrödinger-Poisson方程的约束变分问题,在一定的参数和指标假设条件下,证明了约束极小元的存在性,推广了文献[2]中的相关结论.

具有手性边界条件的耦合Dirac系统

本文研究具有手性边界条件的Dirac系统解的存在性.通过在恰当的分数阶索伯列夫乘积空间上建立解析框架,给出了具有超二次增长的非线性项的Dirac系统解的存在性,把GONG和LU(2017)研究的结果推广到手性边界条件下的情形.

共形扭化Dirac算子与非交换留数

考虑了一类共形扭化Dirac算子,给出了其Lichnerowicz公式.利用Laplace算子的局部表示,在法坐标系下计算出了无边流形与共形扭化Dirac算子相关的非交换留数.

一类边界条件中含谱参数的Dirac算子的谱性质

本文考虑了一类内部具有两个不连续点且边界条件依赖谱参数的Dirac算子的谱性质。首先通过引入适当的Hilbert空间并在其上定义新的自伴算子,使得所考虑问题的特征值与该算子的特征值一致。然后通过构造基本解得到了特征值的一些性质。最后给出了问题的Green函数和预解算子。In this paper, we consider the spectral properties of a class of Dirac operators with two internal discontinuities and spectral parameter-dependent boundary conditions. First, the eigenvalues of the problem under consideration are made to coincide with the eigenvalues of the operator by introducing a suitable Hilbert space and defining a new self-adjoint operator on it. Then some properties of the eigenvalues are obtained by constructing the basic solution. Finally, Green’s function and the resolvent operator of the problem are given.

一类具有复权函数和两个不连续点的Dirac算子的逆问题

本文考虑了一类具有转移条件和复值权函数的Dirac算子的逆问题,采用了谱映射的方法得到了具有两个不连续点的Dirac算子的势函数在整个区间能被两组谱或Weyl函数唯一确定。

Dirac method for nonlinear and non-homogenous boundary value problems of plates

The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.

Photoinduced Floquet higher-order Weyl semimetal in C_(6) symmetric Dirac semimetals

Topological Dirac semimetals are a parent state from which other exotic topological phases of matter, such as Weyl semimetals and topological insulators, can emerge. In this study, we investigate a Dirac semimetal possessing sixfold rotational symmetry and hosting higher-order topological hinge Fermi arc states, which is irradiated by circularly polarized light. Our findings reveal that circularly polarized light splits each Dirac node into a pair of Weyl nodes due to the breaking of time-reversal symmetry, resulting in the realization of the Weyl semimetal phase. This Weyl semimetal phase exhibits rich boundary states, including two-dimensional surface Fermi arc states and hinge Fermi arc states confined to six hinges.Furthermore, by adjusting the incident direction of the circularly polarized light, we can control the degree of tilt of the resulting Weyl cones, enabling the realization of different types of Weyl semimetals.

Magnetic Topological Dirac Semimetal Transition Driven by SOC in EuMg_(2)Bi_(2)

Magnetic topological semimetals have been at the forefront of condensed matter physics due to their ability to exhibit exotic transport phenomena.Investigating the interplay between magnetic and topological orders in systems with broken time-reversal symmetry is crucial for realizing non-trivial quantum effects.We delve into the electronic structure of the rare-earth-based antiferromagnetic Dirac semimetal EuMg_(2)Bi_(2) using first-principles calculations and angle-resolved photoemission spectroscopy.Our calculations reveal that the spin-orbit coupling(SOC)in EuMg_(2)Bi_(2) prompts an insulator to topological semimetal transition,with the Dirac bands protected by crystal symmetries.The linearly dispersive states near the Fermi level,primarily originating from Bi 6p orbitals,are observed on both the(001)and(100)surfaces,confirming that EuMg_(2)Bi_(2) is a three-dimensional topological Dirac semimetal.This research offers pivotal insights into the interplay between magnetism,SOC and topological phase transitions in spintronics applications.

Massive Dirac particles based on gapped graphene with Rosen-Morse potential in a uniform magnetic field

We explore the gapped graphene structure in the two-dimensional plane in the presence of the Rosen-Morse potential and an external uniform magnetic field.In order to describe the corresponding structure,we consider the propagation of electrons in graphene as relativistic fermion quasi-particles,and analyze it by the wave functions of two-component spinors with pseudo-spin symmetry using the Dirac equation.Next,to solve and analyze the Dirac equation,we obtain the eigenvalues and eigenvectors using the Legendre differential equation.After that,we obtain the bounded states of energy depending on the coefficients of Rosen-Morse and magnetic potentials in terms of quantum numbers of principal n and spin-orbit k.Then,the values of the energy spectrum for the ground state and the first excited state are calculated,and the wave functions and the corresponding probabilities are plotted in terms of coordinates r.In what follows,we explore the band structure of gapped graphene by the modified dispersion relation and write it in terms of the two-dimensional wave vectors K_(x) and K_(y).Finally,the energy bands are plotted in terms of the wave vectors K_(x) and K_(y) with and without the magnetic term.

Coexistence of Dirac and Weyl points in non-centrosymmetric semimetal NbIrTe_(4)

Using angle-resolved photoemission spectroscopy and density functional theory calculations methods,we investigate the electronic structures and topological properties of ternary tellurides NbIrTe_(4),a candidate for type-II Weyl semimetal.We demonstrate the presence of several Fermi arcs connecting their corresponding Weyl points on both termination surfaces of the topological material.Our analysis reveals the existence of Dirac points,in addition to Weyl points,giving both theoretical and experimental evidences of the coexistence of Dirac and Weyl points in a single material.These findings not only confirm NbIrTe_(4) as a unique topological semimetal but also open avenues for exploring novel electronic devices based on its coexisting Dirac and Weyl fermions.

BUBBLING ANALYSIS FOR A NONLINEAR DIRAC EQUATION ON SURFACES

In this paper,we apply the three circle type method and a Hardy type inequality to a nonlinear Dirac type equation on surfaces,and provide alternative proofs to the energy quantization results.

Single pair of charge-2 Dirac and charge-2Weyl phonons in GeO_(2)

The presence of a pair of Weyl and Dirac points(WP-DP)in topological semimetal states is intriguing and sought after due to the effects associated with chiral topological charges.However,identifying these states in real materials poses a significant challenge.In this study,by means of first-principles calculations we predict the coexistence of charge-2 Dirac and charge-2 Weyl phonons at high-symmetry points within a noncentrosymmetric P4_(1)2_(1)2 space group.Furthermore,we propose GeO_(2)as an ideal candidate for realizing these states.Notably,we observe two distinct surface arcs that connect the Dirac and Weyl points across the entire Brillouin zone,which could facilitate their detection in future experimental investigations.This study not only presents a tangible material for experimentalists to explore the topological properties of WP-DP states but also opens up new avenues in the quest for ideal platforms to study chiral particles.

Controllable optical bistability in a Fabry-Pérot cavity with a nonlinear three-dimensional Dirac semimetal

Optical bistability(OB)is capable of rapidly and reversibly transforming a parameter of an optical signal from one state to another,and homologous nonlinear optical bistable devices are core components of high-speed all-optical communication and all-optical networks.In this paper,we theoretically investigated the controllable OB from a Fabry-Pérot(FP)cavity with a nonlinear three-dimensional Dirac semimetal(3D DSM)in the terahertz band.The OB stems from the third-order nonlinear bulk conductivity of the 3D DSM and the resonance mode has a positive effect on the generation of OB.This FP cavity structure is able to tune the OB because the transmittance and the reflectance can be modulated by the Fermi energy of the 3D DSM.We believe that this FP cavity configuration could provide a reference concept for realizing tunable bistable devices.

CONVERGENCE OF A QUANTUM LATTICE BOLTZMANN SCHEME TO THE NONLINEAR DIRAC EQUATION FOR GROSS-NEVEU MODEL IN 1+1 DIMENSIONS

This paper studies the strong convergence of the quantum lattice Boltzmann(QLB)scheme for the nonlinear Dirac equations for Gross-Neveu model in 1+1 dimensions.The initial data for the scheme are assumed to be convergent in L^(2).Then for any T≥0 the corresponding solutions for the quantum lattice Boltzmann scheme are shown to be convergent in C([0,T];L^(2)(R^(1)))to the strong solution to the nonlinear Dirac equations as the mesh sizes converge to zero.In the proof,at first a Glimm type functional is introduced to establish the stability estimates for the difference between two solutions for the corresponding quantum lattice Boltzmann scheme,which leads to the compactness of the set of the solutions for the quantum lattice Boltzmann scheme.Finally the limit of any convergent subsequence of the solutions for the quantum lattice Boltzmann scheme is shown to coincide with the strong solution to a Cauchy problem for the nonlinear Dirac equations.

Equivalence of the Aharonov-Bohm and Dirac Monopole Potentials

The Aharonov-Bohm effect (experimentally verified) constitutes an undubitable proof of the non local nature of quantum mechanics and of the gauge character of the electromagnetic interaction. On the other hand, the existence of a Dirac monopole (not yet experimentally confirmed) leads to the quantization of the electric charge. Both phenomena can be mathematically described in the context of fiber bundle theory. Using this approach, we briefly review the mutual determination of the corresponding connections ωA−B, ωDand potentials AA−B±, AD±. This mathematical result gives an additional theoretical support to present day active search of the magnetic charge.

Dirac粒子的正-反粒子自由度和正-反粒子量子数

对Dirac粒子引进了正 反粒子自由度和相应的内部τ空间的算子,把γ矩阵分解成自旋σ算子和正 反粒子τ算子;Dirac方程的解出现了正 反粒子量子数;正 反粒子变换是Dirac粒子的哈密顿量的反对称变换,Dirac粒子负能态能量的负值来自正 反粒子量子数的负值;γ矩阵这种分解是处理物理相互作用的需要.

常型Dirac算子的谱分解

借助于 Green函数 ,利用留数方法讨论了 Dirac特征值问题的基本问题 ,证明了向量函数 f (x)分别在空间D和 L2 (a,b)上 Dirac特征值问题按特征向量函数展开的定理 ,给出了定义域 D上产生的 Dirac算子的谱分解 .