# Why if 2 operators commute they have a common set of eigenvectors and what's the relation to 2 fold degeneracy?

I have the following sentence in my lecture notes "Dirac hamiltonian and helicity have a common set of eigenvectors, this is also the reason for the two fold degeneracy found for every energy eigenstate of the Dirac hamiltonian" can you explain why this is the case?

• – AccidentalFourierTransform Jan 31 '18 at 19:25
• $\hat h = \vec\Sigma \cdot \hat p$ so it has eigenvalues $\pm 1/2$ and commutes with the Dirac hamiltonian. Going to the frame where momenta are in the z direction, for the sake of argument, the same $\sigma_z$ solution will have an Energy-"twin" moving in the opposite direction, so with the opposite helicity. – Cosmas Zachos Jan 31 '18 at 20:56