Spinors and the quaternionic Poincaré group

TL;DR

The paper investigates how the symmetry of a quaternionic ambient manifold ${\mathcal M}_{\mathbb{H}}$ constrains spin and internal quantum numbers in four-dimensional spacetime ${\mathcal R}$. It adopts a nonlinear realization framework, employing coset bundles ${G_{\mathbb{H}}}/{H}$ and Whitney sums to couple spin representations on the base ${\rm Sp}(3)/{\rm SO}(3)$ (massive) and ${\rm Sp}(2)/{\rm SO}(2)$ (massless) to the real spacetime, thereby generating new internal-like quantum numbers such as ${\rm SU}(2)$ charges and scalar components. The main results show that linear spin representations do not appear for the ambient complex/quaternionic structures; spin arises only after Whitney-sum couplings, with a three-fold degeneracy in the massless base and an emergent color-like structure on the base, suggesting a geometric route to generation-like features. The framework provides a possible bridge to standard-model-like structures, where gauge fields would correspond to connections on the coset bases and the masses depend on topological/measures on the manifolds.

Abstract

When four dimensional spacetime R is considered as locally embedded on a larger manifold M, labelled by higher division algebra coordinates, a natural question to ask is how much of the symmetry properties of the larger space are inherited by R. Here this question is studied when M is a quaternion manifold. Of particular relevance is the absence of spinors in the linear representations of the symmetry group of the larger manifold and the emergence of new quantum numbers when, by Whitney sums, spinors are implemented on the vector bundles associated to the coset manifolds of the symmetry groups of M. A possible relation to the structures of the standard model is briefly discussed.