The Generalized Cauchy-Riemann-Fueter Equation and Handedness

Abstract

A generalization of the Cauchy-Riemann condition in complex analysis is described for complex numbers, quaternions and complex quaternions. The generalization called here generalized Cauchy-Riemann-Fueter analycity encompasses not just the left and right-handed versions of quaternion analysis but also generates other variants for complex quaternions. These multiple variants are shown to satisfy an analogue of Cauchy's Theorem and to have similarities with the generalized Cauchy-Riemann conditions that define monogenic functions on R n + 1; they are also similar to Fueter-type operators and the Moisil-Theodoresco operator. The multiple variants are shown to have an interpretation that unifies analycity into a single definition. Thus left and right-handedness in quaternions are shown to be two sides of the same concept, and likewise for complex quaternions. This is then shown to have possible physical interpretation for example in understanding the nature of chirality and the 'arrow of time'.