Born-Infeld Lagrangian using Cayley-Dickson algebras

TL;DR

This paper reformulates the Born-Infeld Lagrangian using Cayley-Dickson algebras, expressing it as a determinant of larger matrices distinguished by a single parameter, potentially simplifying its analysis.

Contribution

It introduces a novel representation of the Born-Infeld Lagrangian via Cayley-Dickson algebra-based matrices, highlighting a unique parameter that characterizes these matrices.

Findings

Reformulation of the Lagrangian as a larger determinant matrix

Identification of a single parameter distinguishing matrices

Proposed condition to fix the parameter

Abstract

We rewrite the Born-Infeld Lagrangian, which is originally given by the determinant of a $4 \times 4$ matrix composed of the metric tensor $g$ and the field strength tensor $F$, using the determinant of a $(4 \cdot 2^n) \times (4 \cdot 2^n)$ matrix $H_{4 \cdot 2^{n}}$. If the elements of $H_{4 \cdot 2^{n}}$ are given by the linear combination of $g$ and $F$, it is found, based on the representation matrix for the multiplication operator of the Cayley-Dickson algebras, that $H_{4 \cdot 2^{n}}$ is distinguished by a single parameter, where distinguished matrices are not similar matrices. We also give a reasonable condition to fix the paramete