Hadamard--四面体对称时空与Dirac和Maxwell算子的类Clifford分解

中文摘要 AI 辅助

通过将归一化的四阶Hadamard变换与虚时间坐标相结合，构造了Minkowski时空的一种对称表示。所得复坐标$q^a$将时间和空间均等地分配到四个分量中，并将Minkowski度规变换为欧几里得形式的复双线性度规。四个Hadamard空间符号向量构成一个正四面体，并同时出现在变换后的Dirac和Maxwell算子中。Dirac矩阵满足欧几里得形式的Clifford代数，并包含Pauli矩阵的四面体组合，而Riemann--Silberstein Maxwell矩阵实现了相应的自旋-1结构。在相同坐标下，构造了规范不变的QED拉格朗日量。在傅里叶空间中，Dirac和Maxwell系统由共同的不变量$\kappa_a\kappa^a$支配，而完整的Maxwell算子允许精确的矩形类Clifford分解。该表述进一步推广到弱场Einstein--Dirac--Maxwell系统。在谐和规范下，线性化引力场由相同的标量波动算子$\Box_q$支配，而Dirac和Maxwell方程的引力修正通过微扰tetrad、自旋联络和曲率引入。这种共同的主算子结构暗示了耦合场求解器共享谱核和标量波预处理。

英文摘要

A symmetric representation of Minkowski spacetime is constructed by combining a normalized fourth-order Hadamard transformation with an imaginary temporal coordinate. The resulting complex coordinates $q^a$ distribute time and space equally among four components and transform the Minkowski metric into a Euclidean-form complex bilinear metric. The four Hadamard spatial sign vectors form a regular tetrahedron and appear simultaneously in the transformed Dirac and Maxwell operators. The Dirac matrices satisfy a Euclidean-form Clifford algebra and contain tetrahedral combinations of Pauli matrices, whereas the Riemann--Silberstein Maxwell matrices realize the corresponding spin-1 structure. A gauge-invariant QED Lagrangian is formulated in the same coordinates. In Fourier space, the Dirac and Maxwell systems are governed by the common invariant $κ_aκ^a$, while the complete Maxwell operator admits an exact rectangular Clifford-like factorization. The formulation is further extended to the weak-field Einstein--Dirac--Maxwell system. In harmonic gauge, the linearized gravitational field is governed by the same scalar wave operator $\Box_q$, while gravitational corrections to the Dirac and Maxwell equations enter through the perturbed tetrad, spin connection, and curvature. This common principal-operator structure suggests shared spectral kernels and scalar-wave preconditioning for coupled field solvers.

发表机构

• Concordia University（康考迪亚大学）

机构由 AI 辅助整理，请以论文原文为准。