4-Spinors and 5D Spacetime

TL;DR

This paper explores the mathematical relationship between 4-spinors and 5D spacetime, showing how mappings that commute with Lorentz transformations can be expressed using quaternionic structures, linking spinor geometry to higher-dimensional spacetime.

Contribution

It demonstrates that mappings from 4-spinors to 3+1 spacetime can be embedded in 5D spacetime using quaternionic algebra, revealing a geometric interpretation of spinors as square roots of 5D spacetime.

Findings

Mappings can be expressed as products of quaternions or split-quaternions.

A point in 5D spacetime corresponds to a uniquely determined 4-spinor up to quaternionic phase.

The geometry of 4-spinors relates to the square root of 5D spacetime.

Abstract

We revisit the subject exploring maps from the space of 4-spinors to 3+1 space-time that commute with the Lorentz transformation. All known mappings have a natural embedding in a higher five dimensional spacetime, and can be succinctly expressed as products of quaternions, or split-quaternions, depending on the signature of the embedding 5D spacetime. It is in this sense that we may view the geometry of 4-spinors as being related to the `square root' of five dimensional spacetime. In particular, a point in 5D spacetime may be identified with a corresponding 4-spinor that is uniquely determined up to a quaternionic - or split-quaternionic - phase.