Proposed equation for the action of a particle with spin 1 / 2 interacting with the electromagnetic field . Proposed relativistic and calibration invariant action. The main difference between the proposed action and the previously known relativistic equations is that it is a quadratic form relative to the 4-momentum operator and contains three material constants characterizing the properties of the particle: rest mass, charge and magnitude of the magnetic moment. Wave equations for particle and electromagnetic field. Analytical solution of equations for the case of electron motion in the Coulomb field and in a constant magnetic field. The obtained solutions show the presence of an ultra-thin structure of levels of hydrogen-like atoms and an electron in a magnetic field due to the difference between the magnetic moment of the electron and the Bohr magneton .. It is shown that the obtained equations describe the interaction between nucleons. Comparison with the results of experiments on precision laser spectroscopy of hydrogen atom.