The Classical Origin of Spin: Vectors versus Bivectors

There are two ways of linearizing the Klein-Gordan equation: Dirac's choice, which introduces a matter–antimatter pair, and a second approach using a bivector, which Dirac did not consider. In this paper, we show that a bivector provides the classical origin of quantum spin. At high precessional frequencies, a symmetry transformation occurs in which classical reflection becomes quantum parity. We identify a classical spin-1 boson and demonstrate how bosons deliver energy, matter, and torque to a surface. The correspondence between classical and quantum domains allows spin to be identified as a quantum bivector, $i\sigma$, which is a spinning plane. Using geometric algebra, we show that a classical boson has two blades, corresponding to magnetic quantum number states $m=\pm 1$. We conclude that fermions are the blades of bosons, thereby unifying both into a single particle theory. We compare and contrast the Standard Model which uses chiral vectors as fundamental, with the Bivector Standard Model which uses bivectors, with two hands, as fundamental.