The GUT of the light: On the Abelian Complexifications of the Euclidean R^n spaces

TL;DR

This paper explores the mathematical framework of ternary complexifications of Euclidean spaces, proposing new geometries, symmetries, and algebraic structures with potential implications for fundamental physics and the Standard Model.

Contribution

It introduces the concept of C_n complexification of R^n spaces and investigates their geometric, algebraic, and symmetry properties, linking them to physical theories.

Findings

C_n complexification of R^n spaces and their invariant surfaces

C_n holomorphicity and harmonicity properties

Connection between C_n holomorphicity and spin 1/n origin

Abstract

The new great development in Physics could be related to the excited progress of a new mathematics: ternary theory of numbers, ternary Pithagor theorem and ternary complex analysis, ternary algebras and symmetries, ternary Clifford algebras,ternary differential geometry, theory of the differential wave equations of the higher degree n>2 and etc. Especially, we expect the powerful influence of this progress into the Standard Model (SM) and beyond, into high energy neutrino physics, Gravity and Cosmology. This can give the further development in the understanding of the Lorentz symmetry and matter-antimatter symmetry, the geometrical origin of the gauge symmetries of the Standard Model, of the 3-quark-lepton family and neutrino problems, dark matter and dark energy problems in Cosmology. The new ambient geometry can be related to a new space-time symmetry leading at high energies to generalization of the Special Theory of Relativity. We related the future of this development with C_n numbers, n-algebras (n>2) and corresponding geometrical objects. We will discuss the following results - C_n complexification of R^n spaces - C_n structure and the invariant surfaces - C_n holomorphicity and harmonicity - The link between C_n holomorphicity and the origin of spin 1/n - New geometry and N-ary algebras/symmetries - Root system of a new ternary TU(3) algebra - N-ary Clifford algebras - Ternary 9-plet and 27-plet number surfaces