Dirac equation on hyperbolic octonions

摘要

When extending complex number algebra using nonreal square roots of +1, the resulting arithmetic has long exhibited signs for potential applicability in physics. This article provides proof to a statement by Musès [C. Musès, Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries, Appl. Math. Comput. 6 (1980) 63–94] that the Dirac equation in physics can be found in conic sedenions (or 16-dimensional M-algebra). Hyperbolic octonions (or counteroctonions), a subalgebra of conic sedenions, are used to describe the Dirac equation sufficiently in a simple form. In the example of conic sedenions, a method is then outlined on how hypernumbers could potentially further aid mathematical description of physical law, by transitioning between different geometries through genuine hypernumber rotation.