Spacetime Grand Unified Theory

Abstract

The gauge symmetries and fermionic sector of the Standard Model are derived from the free Dirac Lagrangian in 8-dimensional spacetime. Motivated by second quantization, we embed the 4-dimensional Clifford algebra of the Dirac Lagrangian into that of 8-dimensional spacetime, showing this embedding carries a redundancy described by a complexified Pati-Salam group acting on field multiplets represented as Clifford algebra ideals, externally and internally via left and right multiplication. Invariance of the Dirac Lagrangian selects different real slices of the group: internal transformations follow $SU(4)_C \times SU(2)_L \times SU(2)_R$, external ones $SU(4)_F \times Spin(1,3)$, with $SU(4)_F$ acting on four particle families. Although $SU(4)_C$ and $SU(4)_F$ can be broken spontaneously and independently, we prove that fixing a triality in $C\ell_{8,0}$ simultaneously gives $SU(4)_C \to SU(3)_C \times U(1)_{B-L}$ and $SU(4)_F \to SU(3)_F \times U(1)_{F}$, originating a leptonic sector and a non-mixing fourth family whose stable neutrino is a potential dark matter candidate. Breaking $U(1)_{F}$ yields a hierarchy governed by a generalized Koide formula, with mass scales showing a modular nature. Most importantly, every quantity entering the Standard Model acquires a direct algebraic interpretation in higher dimensional spacetime.