Application of the Doubly Special Relativity to the Dirac equation formalism

In this letter we generalize the Dirac equation in the context of the Magueijo-Smolin (MS) [1] approach to the Doubly Special Relativity. This recent theory introduced by Amelino-Camelia [2] is based on the addition to the speed of light of a second relativistic invariant κ related to the Planck energy that implies a noncommutative space-time structure, the κ–Minkowski space-time. In a recent paper [3] we deduced the "Dirac equation" in momentum space in MS base (see also [4]). Interesting studies with this equation have also been conducted, in particular in [5] where it was applied to a calculation of the hydrogen atom spectrum, or in [6] where the discreteness of the "generalized uncertainty principle" was used. Here we will establish the Dirac equation in position space.