Solutions to Dirac equation
are usually obtained by acting from the left with complex conjugate Dirac operator
and realizing that this is a wave equation. I wonder if the additional multiplication by $(i\gamma^\mu\partial_\mu+m)$ above just restricts the full set of solutions to plane waves only, and so whether the original first order equation might have other types of solutions as well? Is it proven that plane waves are the complete set of solutions to the Dirac equation?