# Color Confinement and Spatial Dimensions in the Complex-sedenion Space

## TL;DR

This paper explores the use of complex-sedenions to model non-Abelian gauge fields, revealing how spatial dimensions in complex-quaternion wavefunctions can naturally represent color confinement in quantum chromodynamics.

## Contribution

It introduces a novel approach using complex-sedenions to describe gauge fields and demonstrates how spatial dimensions can represent color degrees of freedom, providing new insights into color confinement.

## Key findings

- Complex-sedenions can depict classical and quantum field equations.
- Three spatial dimensions in complex-quaternion wavefunctions correspond to color degrees of freedom.
- Color confinement emerges from the spatial structure of wavefunctions, not physical properties.

## Abstract

The paper aims to apply the complex-sedenions to explore the wavefunctions and field equations of non-Abelian gauge fields, considering the spatial dimensions of a unit vector as the color degrees of freedom in the complex-quaternion wavefunctions, exploring the physical properties of the color confinement essentially. J. C. Maxwell was the first to employ the quaternions to study the physical properties of electromagnetic fields. His method inspires some subsequent scholars to introduce the quaternions, octonions, and sedenions to research the electromagnetic field, gravitational field, nuclear field, quantum mechanics, and gauge field. The application of complex-sedenions is capable of depicting not only the field equations of the classical mechanics on the macroscopic scale, but also the field equations of the quantum mechanics on the microscopic scale. The latter can be degenerated into the Dirac equation and Yang-Mills equation. In contrast to the complex-number wavefunction, the wavefunction in the complex-quaternion space possesses three new degrees of freedom, that is, three color degrees of freedom. One complex-quaternion wavefunction is equivalent to three conventional wavefunctions with the complex-numbers. It means that the three spatial dimensions of unit vector in the complex-quaternion wavefunction can be considered as the `three colors', naturally the color confinement will be effective. In other words, in the complex-quaternion space, the `three colors' are only the spatial dimensions, rather than any property of physical substance. The existing `three colors' can be merged into the wavefunction, described with the complex-quaternions.