Octonionic Reformulation of Vector Analysis

TL;DR

This paper extends three-dimensional vector analysis to seven dimensions using octonions, redefining core operators and exploring their properties, but faces limitations due to octonion non-associativity.

Contribution

It introduces a seven-dimensional vector analysis framework based on octonions, redefining gradient, divergence, and curl, and analyzes their properties and limitations.

Findings

Identity n(n-1)(n-3)(n-7)=0 holds only for 0, 1, 3, 7 dimensions.

Standard vector formulas lose meaning in 7D due to octonion non-associativity.

Vector formulas are valid under certain restrictions on octonions and split octonions.

Abstract

According to celebrated Hurwitz theorem, there exists four division algebras consisting of R (real numbers), C (complex numbers), H (quaternions) and O (octonions). Keeping in view the utility of octonion variable we have tried to extend the three dimensional vector analysis to seven dimensional one. Starting with the scalar and vector product in seven dimensions, we have redefined the gradient, divergence and curl in seven dimension. It is shown that the identity n(n-1)(n-3)(n-7)=0 is satisfied only for 0, 1, 3 and 7 dimensional vectors. We have tried to write all the vector inequalities and formulas in terms of seven dimensions and it is shown that same formulas loose their meaning in seven dimensions due to non-associativity of octonions. The vector formulas are retained only if we put certain restrictions on octonions and split octonions.