Spacetime Grand Unified Theory

Abstract

The Standard Model of particle physics is derived from first principles from the free Dirac Lagrangian in 8-dimensional spacetime. Motivated by second quantization arguments, we embed the 4-dimensional Clifford algebra of the Dirac Lagrangian into the Clifford algebra of 8-dimensional spacetime. We show this process carries a natural redundancy described by the SM gauge group and an additional $U(1)_{B-L}$ symmetry. All known fermionic particle representations, with additional right handed neutrinos, arise as Dirac spinors transforming under this symmetry. Four particle families are predicted with mixing intrinsically restricted to the first three, while avoiding common challenges related to a fourth family. The strong force arises from Spin(8) triality, with chirality emerging as the property of rotations left invariant by this automorphism. The symmetry group acts internally and externally, via right and left multiplications on Dirac spinors, respectively. The external counterpart results in a $U(3)_F$ family interaction and a $U(2)_L$ symmetry acting on spinor indexes whose gauging yields a 4-dimensional left-handed spin connection. The proposed breaking of $U(3)_F$ results in a hierarchy governed by a generalized Koide formula, with mass scales displaying a modular nature. Internal and external transformations carry a direct algebraic interpretation in 8-dimensional spacetime while avoiding the Coleman-Mandula theorem. Weak interactions are encoded in the Clifford algebra of the observed 4-dimensional spacetime, while strong interactions live in the Clifford algebra of the four extra dimensions. The theory is anomaly free and devoid of proton decay.