Interaction of a spin-1/2 particle with an electromagnetic field

An equation for the action of a spin-1/2 particle is proposed. The proposed action results in a second-order differential equation for the bispinor wavefunction of the particle. The obtained equation is compared with the Dirac equation. It is shown that all of the solutions of the Dirac equation satisfy the obtained equation. However, among the solutions of the second-order differential equation are solutions not satisfying the Dirac equation. The equation for the current density four-vector is derived. It is shown that the CPT transformation properties of the obtained current density four-vector are qualitatively different from that derived from those of the Dirac equation. The solution of the proposed equation for the problem of electron motion in the Coulomb field is found in analytically tractable form. The calculated spectrum is different from that described by the well-known Sommerfeld formula of the Dirac theory. The predicted spectrum includes the hyperfine structure which is doublet for the levels with the zero angular momentum and triplet for the levels with the non-zero orbital momentum. Analytically tractable solutions of the proposed equation are also found for the problems of electron motion in a uniform magnetic field, and in a superposition of Coulomb and uniform magnetic fields.