Degeneracy of Relativistic Cyclotron Motion

The quantum-mechanical problem of the relativistic cyclotron motion of a charged particle in a uniform magnetic field is solved by consideration of the symmetry which the system obeys. It is shown that its symmetry is isomorphic to the Lie group called G(0,1) or G(1,0), and doubly degenerate infinite series of wave functions with a constant energy eigenvalue are labeled by the eigenvalues of the operators ${\mathcal{H}}^2$, $L_z+S_z$, and $S_z$. Here $\mathcal{H}$ is the relativistic Hamiltonian referred to in the present problem, and $L_z$ and $S_z$ are the usual orbital and spin angular momentum operators, respectively.