Explicit Families of Spinor Representations
Jesus Sanchez
TL;DR
The paper presents a unified, explicit construction of real spinor representations for Clifford algebras by bootstrapping from dimensions $1$–$4$ and extending to all signatures via mixed-field tensor products over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$. It introduces spin coordinate systems and spinor bundles, provides novel low-dimensional spin geometry results, and analyzes parallel transport and products of spinors in detail. A flexible array of representation schemes is developed, including quaternionic, exterior-algebra, and octonionic models, with careful treatment of the general signature case and an emphasis on Bott periodicity and bimodule intertwiners. The framework yields explicit, geometric spinor models for all mixed-signature Clifford algebras, and shows that spinor representations can be expressed as tensor products of multi-vectors over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$, enabling concrete constructions of spinor bundles and their connections with Dirac and de Rham operators.
Abstract
We provide a recipe for building explicit representations of the real Clifford algebras once an explicit family is given in dimensions $1$ through $4$. We further give an explicit construction of spin coordinate systems for a given real spinor module and use it to explicitly compute the parallel transport of spinor fields. We further highlight some novelties such as the relationship with the spectrum of the spinor Dirac operator and the Hodge de Rham operator when a parallel spinor field exists and a brief discussion of spinors along a hypersurface in $\bR^4$. Lastly, we extend our construction to arbitrary signature quadratic forms thus providing a complete and explicit family of spinor representations for all mixed signature Clifford algberas. We show that in all cases the spinor representations can be expressed as tensor products of multi-vectors over the fields $\bR$, $\bC$, and $\bH$.