Nature's natural numbers: relativistic quantum theory over the ring of complex quaternions

Abstract

The 2×2 complex matrix formulation of relativity and the two-component spin-1/2 formalism are merged with the complex quaternion algebra to yield a concise, manifestly covariant formalism of relativistic quantum mechanics. Along with reproducing all the old results of quantum theory, this complex quaternion formulation extends naturally the concept of scalar mass by adding to it orientation- and velocity-dependent parts giving a hyper-mass. The hyper-mass spin-1/2 equation, with the scalar part of the mass set equal to zero, gives a subtle variation on the two-component neutrino theory with very unsubtle consequences, such as an invariant mass parameter which could distinguishv eandv μ and elimination of the superposition principle.