Considerations on the hyperbolic complex Klein-Gordon equation

TL;DR

This paper explores a hyperbolic complex extension of the Klein-Gordon equation, emphasizing its geometric origin and potential for new interactions through hyperbolic complex gauge transformations.

Contribution

It introduces a novel hyperbolic complex framework for the Klein-Gordon equation, extending group and algebra structures with hyperbolic units for the first time.

Findings

Hyperbolic complex numbers provide a new perspective on the Klein-Gordon equation.

The geometric origin of the equation is highlighted through the Poincaré group.

Potential for new interactions via hyperbolic complex gauge transformations is discussed.

Abstract

The article summarizes and consolidates investigations on hyperbolic complex numbers with respect to the Klein-Gordon equation for fermions and bosons. The hyperbolic complex numbers are applied in the sense that complex extensions of groups and algebras are performed not with the complex unit, but with the product of complex and hyperbolic unit. The modified complexification is the key ingredient for the theory. The Klein-Gordon equation is represented in this framework in the form of the first invariant of the Poincar\'e group, the mass operator, in order to emphasize its geometric origin. The possibility of new interactions arising from hyperbolic complex gauge transformations is discussed.