分裂八元数中的 Dirac 与 Maxwell 系统

Octonion and Split Octonion Representation of SO(8) Symmetry

The 8 $\times$ 8 matrix representation of SO(8) Symmetry has been defined by using the direct product of Pauli matrices and Gamma matrices. These 8 $\times$ 8 matrices are being used to describe the rotations in SO(8) symmetry. The comparison of 8$\times$8 matrices with octonions has also been shown. The transformations of SO(8) symmetry are represented with the help of Octonions and split Octonions spinors.

Geometrical Applications of Split Octonions

It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3+1)-theory (e.g. number of dimensions, existence of maximal velocities, Heisenberg uncertainty, particle generations, etc.). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie group G2. This group generates specific rotations of (3+4)-vector parts of split octonions with three extra time-like coordinates and in infinitesimal limit imitate standard …

八元数与分裂八元数平面的直射群

Collineation groups of octonionic and split-octonionic planes

本文用 Veronese 向量刻画八元数与分裂八元数射影、双曲平面及其直射群实形式，并借助 Okubo 代数和副八元数代数给出最小构造。

（分裂）八元数中的狄拉克方程：起源、变体与现代背景

The Dirac equation in (split-)octonions: origins, variants, and modern context

综述八元数和分裂八元数表达狄拉克方程的方式，分四组介绍，阐述二因子方法起源，研究‘分裂八元数狄拉克方程’，经显式保结构旋转表明其与直积表示相同，肯定其真正贡献。

Non-associative Gauge Theory

We present a construction of gauge theory which its structure group is not a Lie group, but a Moufang loop which is essentially non-associative. As an example of non-associative algebra, we take octonions with norm one as a Moufang loop, with which we can produce an octonionic gauge theory. Our octonionic gauge theory is a natural generalization of Maxwell U(1)= S^1 gauge theory and Yang-Mills SU(2)= S^3 gauge theory. We also give the BPST like instanton solution of our octonionic gauge theory in 8 dimension.

Octonionic Planes and Real Forms of G2, F4 and E6

In this work we present a useful way to introduce the octonionic projective and hyperbolic plane through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups of motions over the octonionic plane recovering all real forms of G2, F4and E6 groups and finally give a classification of all octonionic and split-octonionic planes as symmetric spaces.

Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra J. The automorphism group F_4 of J and its maximal Borel - de Siebenthal subgroups are studied in detail and applied to the classification of fundamental fermions and gauge bosons. Their intersection in F_4 is demonstrated to coincide with the gauge group of the Standard Model of particle p…

G2 and the Rolling Ball

Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2: it acts as the symmetries of a 'spinorial ball rolling on a projective plane', again when the ratio of radii is 1:3. We explain this ratio in simple terms, use the dot product and cross p…

A magic approach to octonionic Rosenfeld spaces

In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfe…

基于物理信息神经网络的时域麦克斯韦方程组与分裂场完美匹配层

Physics-Informed Neural Networks for the Time-Domain Maxwell Equations with Split-Field Perfectly Matched Layers

提出一种结合分裂场完美匹配层的物理信息神经网络（PINN）方法求解时域麦克斯韦方程组，通过一维和二维高斯脉冲问题验证，结果与解析解和FDTD参考解吻合良好。

Maxwell-Dirac系统的时间分裂谱方法

A time-splitting spectral scheme for the Maxwell-Dirac system

提出一种针对Maxwell-Dirac系统的时间分裂谱方法，在半经典和非相对论区域中保持洛伦兹规范条件，具有无条件稳定性和高效性，并数值展示了正电子模式产生和电子-正电子相互作用。

Fierz bilinear formulation of the Maxwell-Dirac equations and symmetry reductions

We study the Maxwell-Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwel…