Dirac's Equation in Different Numerical Rings

TL;DR

This paper explores Dirac's equations formulated within different numerical rings (real, complex, quaternions), revealing unique symmetries and conserved currents, and discusses the implications of the underlying numerical field.

Contribution

It introduces a consistent formulation of Dirac's equations in R, C, and H, highlighting the distinct symmetries and conserved currents arising from quaternionic conjugation.

Findings

Quaternions yield three conserved currents linked to color symmetry.

Formulations in R, C, and H exhibit distinct symmetry properties.

Discussion on the role of the numerical field in physical theories.

Abstract

Dirac's equations are formulated in a consistent way by using only elements from each of R, C, and H. In H, the quaternions, the symmetry resulting from a "single" conjugation (i, j, or k) results in three conserved currents - possibly associated with "color." The role that the numerical field plays is discussed and speculated on.