Foldy–Wouthuysen transformation of Dirac equation in the context of conformable fractional derivative

In this paper, we investigate the non-relativistic limit of the Dirac equation for relativistic spin-1/2 particles within the framework of the conformable fractional derivative (CFD) using the Foldy–Wouthuysen (FW) transformation. This approach leads to the derivation of a conformable fractional Schrödinger–Pauli equation. We propose and employ a conformable fractional version of the FW transformation, thoroughly examining its efficacy and behavior in the non-relativistic limit. Additionally, based on perturbation theory, we compute the energy shifts within the context of CFD and derive a conformable fractional fine structure of the hydrogen spectrum.