2026/07/20 by N. Furey · 1 voice
#hep-ph #hep-th
Where do the Standard Model's internal symmetries come from? Treating \mathbbO⊕ℍ⊕ℂ⊕ℝ as a module for its own multiplication algebra enables a particular origin story for the Standard Model's pre-Higgs, \mathfrakgSM:=\mathfraksu(3)C ⊕ \mathfraksu(2)L ⊕ \mathfraku(1)Y, and post-Higgs, \mathfrakgLE:=\mathfraksu(3)C ⊕ \mathfraku(1)Q, symmetries. We recognize both these endomorphisms and their modules alike as ℤ2n-graded algebras. Then, annihilating certain highest grade (volume) elements, and imposing an equal-trace condition on anti-hermitian operators leads precisely to \mathfrakgSM and \mathfrakgLE. Weak hypercharge and electric charge operators, Y and Q, take on a remarkably simple form: ∑ (1)/(n)\mathbbIn× n. With the help of auxiliary imaginary units, this 15 ℝ dimensional \mathbbO⊕ℍ⊕ℂ⊕ℝ embeds naturally as a vector space into several well-studied 16 ℝ dimensional algebras, which we generically refer to as \mathbbV. With this embedding, the Standard Model's internal symmetries may then be seen to arise in part from the sequence of nested inclusions: ℝ⊂ℂ⊂ℍ⊂ \mathbbO⊂\mathbbV. In the sedenionic case of \mathbbV = \mathbbS, the full sequence becomes a Cayley-Dickson tower. We define the notion of endomorphic models of particle physics, and connect End_ℝ(\mathbbV)≃ Cl(0,8) to the earlier ideas of Bott Periodic Particle Physics. We comment on a possible connection between the existence of multiple complex structures and the baryon asymmetry problem.