Hadamard--Tetrahedral Symmetric Spacetime and Clifford-Like Factorizations of Dirac and Maxwell Operators
Abstract
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 qa 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 κaκa, 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 Boxq, 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.
Related
- Almgren frequency and Agmon convexity on rotationally symmetric Cartan-Hadamard manifolds
- Classification of Symmetric Hadamard Matrices Up to Order 32
- On approximate support versions of Donoho--Stark uncertainty principles for k-Hadamard operators
- Non-Elliptic Quadratic 𝒫𝒯-Symmetric Operators and Similarity to Self-Adjoint Operators
- On the Finite-Zone PT-symmetric Dirac Operators
BibTeX
@misc{aghili2026hadamardtetrahedral,
title = {{Hadamard--Tetrahedral Symmetric Spacetime and Clifford-Like Factorizations of Dirac and Maxwell Operators}},
author = {Farhad Aghili},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09617},
eprint = {2609.09617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09617},
}
RIS
TY - UNPB
TI - Hadamard--Tetrahedral Symmetric Spacetime and Clifford-Like Factorizations of Dirac and Maxwell Operators
AU - Farhad Aghili
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09617
UR - https://arxiv.org/abs/2609.09617
AB - 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.
ER -