Revisiting Takahashi's inversion theorem in discrete symmetry-based dual frameworks

Abstract

The so-called Takahashi's Inversion Theorem, the reconstruction of a given spinor based on its bilinear covariants [1], are re-examined, considering alternative dual structures. In contrast to the classical results, where the Dirac dual structure plays the central role, new duals are built using the discrete symmetries C,P,T. Their combinations are also taken into account. Furthermore, the imposition of a new adjoint structure led us also to re-examine the representation of the Clifford algebra basis elements, uncovering new bilinear structures and a new Fierz aggregate. Those results might help construct theories for new beyond standard model fields.