This paper establishes a grand unified theory of particle physics based on the hierarchical structure of normed division algebras: the real numbers R, complex numbers C, quaternions H, and octonions O. Hurwitz’s theorem guarantees that these are the only four normed division algebras over the real numbers, with dimensions 1, 2, 4, and 8, which correspond precisely to the numbers of degrees of freedom of the four fundamental interactions. In the particle rest frame, the intrinsic properties are represented by an octonion O = m01 + ql + sxi + syj + szk + gril + ggjl + gbkl, whose norm gives the Lorentz invariant. The weak interaction emerges as a quantum relativistic effect of the color interaction in the presence of electric charge, realized through the chiral operator W+ = (kl ◦ jl ◦ il) = +1 (left-handed fermions) and W− = (il ◦ jl ◦kl) = −1 (right-handed antifermions). We derive the exact relation gwgs= 2π√α, which unifies the strong, weak, and electromagnetic coupling constants. The Higgs boson is identified as the scalar weak boson, parallel to the vector weak bosons W and Z, analogous to the relation between pseudoscalar and vector mesons in strong interactions. Physical constants remain energy-independent, while processes evolve with energy. We further establish the U(1, 3) grand unification framework, which is mathematically isomorphic to the general linear group over quaternions GL(2, H), the Clifford algebra group Pin(1, 3), and a subgroup of the octonion automorphism group Aut(O); physically, it corresponds to the four perspectives of four-dimensional spacetime, four-dimensional energy-momentum, intrinsic space, and gauge symmetry. Within this framework, we introduce dark quarks— lementary particles carrying color charge but zero electric charge—as dark matter candidates. Their weak mixing angle retains the bare value 30◦, and they couple to weak bosons only through weak moments (analogous to magnetic moments) rather than weak charges. This weak moment originates from the projection of the color magnetic moment under the chiral projection. Through the classification of electromagnetic moments, color magnetic moments, and color-electric mixed moments, we reveal that the essence of the weak interaction is a color-electric mixed moment interaction, thereby unifying electromagnetic, strong, and weak interactions within a self-consistent framework of normed division algebras. The theoretical predictions are in precise agreement with all precision electroweak data and provide testable dark matter signal
shifa liu (Wed,) studied this question.