Twinor Theory: Field Equations from $(A)dS_4^{\mathbb C}$ Spin-Tractors: A Twin to Twistor Theory Modelled on $(A)dS_4^{\mathbb C}$

Abstract

Twinor theory is formulated as a spacetime-based counterpart to twistor theory using holomorphic reductive Cartan geometry modelled on complexified $(A)dS_4$. The fundamental spin-tractor bundle defines Dirac twinors, its symmetric powers define ladder twinors, and their extremal Lorentz components define chiral twinors. The general \textit{twinor field equation} is $D\mathbf Y=\mathbf J$. Its adjacent contractions and mixed divergence are constrained by source relations; no additional equation is imposed on its completely symmetric projection. On a flat torsion-free Lorentzian background, the adjacent equations with the mass-normalization source and vanishing additional contracted sources, together with vanishing mixed divergence, reproduce the massive \textit{Dirac--Fierz--Pauli system}; the massless chiral realization gives the zero-rest-mass equations. Within the fixed mass normalization, the reality condition yielding a positive real mass squared is the \textit{anti-de Sitter reality condition} on the particle scale, independently of the cosmological scale and background curvature. The mass-coupling phase is therefore present in the classical connection before quantization. Nonlinear gravity is formulated through the full adjoint curvature, with energy--momentum and spin currents specifying Einstein--Cartan dynamics without requiring an action. The adjoint Bianchi identity constrains source compatibility, while a BGG-type sequence organizes torsion-free infinitesimal deformations. Matter interactions enter through the coupled connection and contracted source relations, with curvature compatibility recorded by $D\mathbf J=\mathcal F_D\mathbf Y$. Both chiralities are retained, and the additional consistency requirements for interacting higher-spin fields are distinguished from the underlying bundle construction.