Geometrical Applications of Split Octonions

TL;DR

This paper explores how split-octonion geometry and the automorphism group G2 can model fundamental physical properties like spacetime structure, particle behavior, and symmetries, revealing deep geometric origins of physical phenomena.

Contribution

It provides an explicit representation of G2 automorphisms on split-octonions, linking algebraic structures to physical properties such as Lorentz transformations, chirality, and particle null intervals.

Findings

Split-octonion geometry reproduces (3+1)-dimensional physics properties.

G2 automorphisms generate rotations and transformations analogous to Poincare symmetries.

Chirality of G2 algebra induces left-right asymmetry and parity violation.

Abstract

It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3+1)-theory (e.g. number of dimensions, existence of maximal velocities, Heisenberg uncertainty, particle generations, etc.). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie group G2. This group generates specific rotations of (3+4)-vector parts of split octonions with three extra time-like coordinates and in infinitesimal limit imitate standard Poincare transformations. In this picture translations are represented by non-compact Lorentz-type rotations towards the extra time-like coordinates. It is shown how the G2 algebra's chirality yields an intrinsic left-right asymmetry of a certain 3-vector (spin), as well as a parity violating effect on light emitted by a moving quantum system. Elementary particles are connected with the special elements of the algebra which nullify octonionic intervals. Then the zero-norm conditions lead to free particle Lagrangians, which allow virtual trajectories also and exhibit the appearance of spatial horizons governing by mass parameters.