Charge quantization from a number operator

Abstract

We explain how an unexpected algebraic structure, the division algebras, can be seen to underlie a generation of quarks and leptons. From this new vantage point, electrons and quarks are simply excitations from the neutrino, which formally plays the role of a vacuum state. Using the ladder operators which exist within the system, we build a number operator in the usual way. It turns out that this number operator, divided by 3, mirrors the behaviour of electric charge. As a result, we see that electric charge is quantized because number operators can only take on integer values. Finally, we show that a simple hermitian form, built from these ladder operators, results uniquely in the nine generators of $SU(3)_c$ and $U(1)_{em}$. This gives a direct route to the two unbroken gauge symmetries of the standard model.