A unified approach to spinor duals via Clifford algebras and $G_Ω$ groups
R. T. Cavalcanti, J. M. Hoff da Silva
TL;DR
The paper develops a unified, group-theoretic framework for spinor duals within the Clifford algebra $\mathcal{C}\ell_{1,3}$, extending Dirac's dual via covariant generalized duals and the $\Delta$/$\Omega$ formalisms. It introduces $G_\Omega$ as a group of dual mappings, derives conditions for group structure (Abelian), and defines $\Omega$-equivalence classes to classify dual spinors. It then situates these constructions in the full Clifford algebra by showing the largest admissible group of duals is $GL(2,\mathbb{H})$ (with reductions to $GL(2,\mathbb{C})$ for the even subalgebra), and maps the corresponding subgroups (Spin$^+$, Pin, etc.) into a hierarchical diagram. The results provide a principled, algebraic scheme to organize and distinguish generalized dual spinors, with potential implications for extended fermion theories and beyond.
Abstract
Recent developments in the construction of generalized Dirac duals have revealed, within the structure of the Clifford algebra $\mathbb{C}\otimes\mathcal{C}\ell_{1,3},$ the existence of distinct algebraic formulations of spinors duals with potential applications in quantum field theoretic models. In this work, after reviewing the matrix formulation, we employ the recent covariant formulation of the generalized spinor dual and establish its interplay with the algebra $\mathcal{C}\ell_{1,3}$. We construct dual mappings governed by groups denoted by $G_Ω$ and introduce the notion of $Ω$-equivalence classes as a tool to classify dual spinors from a group-theoretic perspective.