Gauge fields and four interactions in the trigintaduonion spaces

TL;DR

This paper explores the use of trigintaduonion algebra to model and analyze four fundamental interactions simultaneously, revealing new insights into their properties and relationships with existing theories like electroweak and quark models.

Contribution

It introduces a novel application of trigintaduonion algebra to unify the description of four interactions and derives related gauge field equations, extending previous quaternion and octonion approaches.

Findings

Weak nuclear fields consist of three fundamental fields, aligning with electroweak theory.

Strong nuclear fields are composed of three fundamental fields, consistent with quark theory.

Yang-Mills equations can be deduced within the trigintaduonion framework.

Abstract

The paper aims to apply the trigintaduonion spaces to explore the physical properties of four interactions simultaneously, including the electromagnetic fields, gravitational fields, weak nuclear fields, and strong nuclear fields. J. C. Maxwell first applied the algebra of quaternions to study the physical properties of electromagnetic fields. It inspired some subsequent scholars to introduce the quaternions, octonions, sedenions, and trigintaduonions to research the electromagnetic fields, gravitational fields, weak nuclear fields, strong nuclear fields, quantum mechanics, gauge fields, and curved spaces and so forth. The algebra of trigintaduonions is able to discuss the physical quantities of four interactions, including the field potential, field strength, field source, linear momentum, angular momentum, torque, and force. In the field theories described with the algebra of trigintaduonions, the weak nuclear field is composed of three types of fundamental fields. These three fundamental fields, related to weak nuclear fields, can describe the physical properties of weak nuclear fields collectively. This is consistent with the conclusion of the electroweak theory. Meanwhile the strong nuclear field consists of three types of fundamental fields. These three fundamental fields relevant to strong nuclear fields may investigate the physical properties of strong nuclear fields mutually. It is coincident with the deduction of quark theory. According to the properties of trigintaduonions, one can deduce the Yang-Mills equation related to the gauge fields. It means that the electromagnetic field occupies a quaternion space. The gravitational field owns one different quaternion space. The weak nuclear fields occupy three mutually independent quaternion spaces. The properties of weak nuclear fields are different from those of electromagnetic fields or gravitational fields.