Conserved quantities of vectorial magnitudes within the material media

TL;DR

This paper uses octonion algebra to analyze conserved quantities of vector magnitudes in electromagnetic and gravitational fields, revealing which quantities can be conserved simultaneously within material media.

Contribution

It introduces a novel octonion-based framework to identify conserved vectorial quantities and their mutual relationships in electromagnetic and gravitational fields.

Findings

Identifies conserved quantities of vector magnitudes using octonion algebra.

Shows some conserved quantities cannot be simultaneously established.

Provides deeper understanding of conservation laws in field theories.

Abstract

By means of the rotational transformations of octonion coordinate systems, the paper aims to explore the physical properties of conserved quantities relevant to the vectorial magnitudes within the material media, revealing the simultaneity of some conserved quantities in the electromagnetic and gravitational fields. J. C. Maxwell first utilized the algebra of quaternions to describe the electromagnetic theory. The subsequent scholars studied the physical properties of electromagnetic and gravitational fields simultaneously, including the octonion linear momentum, angular momentum, torque and force within the material media. According to the algebra of octonions, the scalar parts of octonion physical quantities remain unchanged, while the vectorial parts may alter, in the rotational transformations of octonion coordinate systems. From the octonion characteristics, it is able to deduce a few conserved quantities related to the vectorial magnitudes in the octonion space $\mathbb{O}$, including the magnitudes of linear momentum, angular momentum, torque and force. Similarly, it is capable of inferring several conserved quantities relevant to the vectorial magnitudes in the transformed octonion space $\mathbb{O}_u$, including the magnitudes of electric current, magnetic moment and electric moment. Through the analysis and comparison, it is concluded that some conserved quantities, relevant to the vectorial magnitudes, are unable to be established simultaneously, from the point of view of the octonion spaces. This is helpful to deepen the further understanding of some conserved quantities related to the vectorial magnitudes.