Explanation from the Kaehler Calculus of Why Positrons Spuriously Emerge with Negative Energy in Dirac's Theory
This poster is the second complement to the talk by the present author "A unifying calculus for physics: a demonstration in relativistic quantum mechanics".
Though the mother equation obtained in the first poster has large (or even) and small (odd) components, both of them belong to the electron, if one has picked its ideal. We now choose the positron's ideal and obtain a very similar mother system and the same Hamiltonian, except that the large components are now odd and the small components are now even.
We return to the mother system for the electron, which we transform in such a way that the role of the large and small components is reversed. This is achieved by time and (sign of) charge reversal, but the role of the potential energy term is now played by the potential energy minus twice the rest mass. This amounts to having the total energy decreased by 2mc2, or as if the rest energy were the negative of the usual one. This is a consequence of incorrectly treating the small components as if they constituted the wave function of the antiparticle.