Maxwell's Octonion Equations

Abstract

Vadim Sovetov

Analytically, Maxwell's equations for the octonion are obtained. It is shown that when using shortened notations for the scalar and vector products of the Hamiltonian operator on the electric and magnetic field vectors, the equations have the same form as Maxwell's equations for a quaternion. This means that the equations describe the same physical phenomena, only for spaces of different dimensions. According to the Cauchy-Riemann conditions (CRC) and, accordingly, the law of conservation of energy, an electron cannot transition to an arbitrary state, but only makes quantum jumps to those locations in space where it conserves energy. Consequently, an electron moves in space along an orbit. The resulting Maxwell equations contain Gauss's law as a scalar part, which is related to the imaginary parts by CRC and, therefore, a change in the scalar part determines the value of the imaginary parts of the equations. A quaternion has 4D dimensions, while an octonion has 8D dimensions. It is shown that the octonion matrix can be decomposed into two non-intersecting matrices in 8D space. A quaternion can be either single-frequency or three-frequency, an octonion can be single-frequency or seven-frequency. Using the formula for representing the octonion matrix through two quaternion matrices, the corresponding images of the change in the scalar part of the seven-frequency octonion for 4 quaternions are shown.

PDF