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 SUc(3) and Uem(1). This gives a direct route to the two unbroken gauge symmetries of the standard model.
Highlights
We explain how an unexpected algebraic structure, the division algebras, can be seen to underlie a generation of quarks and leptons
The real numbers are used almost universally in physics; the complex numbers are central to quantum theory; the quaternions lead to the Pauli matrices, and are tightly entwined with the Lorentz algebra
What is to be said for the octonions, O, the fourth, and final division algebra? With R, C, and H each undeniably etched into fundamental physics, it is hard not to wonder: is it really the case that O has been omitted in nature?
Summary
One finds a system of ladder operators within the complex octonions. For all f in C ⊗ O, and assuming right-to-left multiplication, these three lowering operators obey the anticommutation relations. We will be concerned only with operators, such as the αi , as opposed to the object f. This being the case, it will be understood that all equations will hold over all f in C ⊗ O, even though f will not be mentioned explicitly. {αi, α j} = αiα j + α jαi = 0 for all i, j = 1, 2, 3 These operators acting on f may be viewed as 8 × 8 complex matrices acting on f , an eight-complex-dimensional column vector. Taking into account the above paragraph, our equations from here on in can be considered as relations only between the matrices
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