Algebraic Operators as Twistor Roots of the Levi-Civita Tensor: Towards a Unified Quantum Field Theory with Torsion

We present a novel interpretation of orthogonal imaginary operators I, J, K, L in geometricalgebraic unification theories as “twistor roots” of the Levi-Civita tensor ϵabcd. The operators are constructed explicitly as 4 × 4 matrices satisfying O^2 = −1, T r(O) = 0, and the orthogonality condition T r(OaOb) = 0 for a ̸= b. We demonstrate their natural emergence from twistor geometry, where they represent symplectic structures on twistor space. The connection to the Levi-Civita tensor is established via the relation ϵabcd ∝ T r(γ5[Pa, Pb][Pc, Pd]) , where Pa are projectors associated with the operators. This framework provides a mathematical foundation for the ultraviolet fixed point previously identified, explaining sector decoupling in quantum loops and offering new insights into confinement as a geometric phenomenon.