# Four interactions in the sedenion curved spaces

## TL;DR

This paper extends the application of complex-sedenions to formulate and analyze the field equations of four fundamental interactions within curved spaces, bridging classical and quantum mechanics.

## Contribution

It introduces a method to extend field equations from flat to curved spaces using complex-sedenions, tangent-frames, and tensors, specifically for four fundamental interactions.

## Key findings

- Spatial parameters influence field operators in curved spaces.
- Field equations of four interactions are formulated in complex-sedenion curved spaces.
- Quantum-field equations are affected by the geometry of the curved space.

## Abstract

The paper aims to apply the complex-sedenions to explore the field equations of four fundamental interactions, which are relevant to the classical mechanics and quantum mechanics, in the curved spaces. J. C. Maxwell was the first to utilize the quaternions to describe the property of electromagnetic fields. Nowadays the scholars introduce the complex-octonions to depict the electromagnetic and gravitational fields. And the complex-sedenions can be applied to study the field equations of the four interactions in the classical mechanics and quantum mechanics. Further, it is able to extend the field equations from the flat space into the curved space described with the complex-sedenions, by means of the tangent-frames and tensors. The research states that a few physical quantities will make a contribution to certain spatial parameters of the curved spaces. These spatial parameters may exert an influence on some operators (such as, divergence, gradient, and curl), impacting the field equations in the curved spaces, especially the field equations of the four quantum-fields in the quantum mechanics. Apparently the paper and General Relativity both confirm and succeed to the Cartesian academic thought of `the space is the extension of substance'.