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

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.