  In this paper we study in detail the connection between the solutions to the
Dirac and Weyl equation and the associated electromagnetic 4-potentials. First,
it is proven that all solutions to the Weyl equations are degenerate, in the
sense that they correspond to an infinite number of electromagnetic
4-potentials. As far as the solutions to the Dirac equation are concerned, it
is shown that they can be classified into two classes. The elements of the
first class correspond to one and only one 4-potential, and are called
non-degenerate Dirac solutions. On the other hand, the elements of the second
class correspond to an infinite number of 4-potentials, and are called
degenerate Dirac solutions. Further, it is proven that at least two of these
4-potentials are gauge-inequivalent, corresponding to different electromagnetic
fields. In order to illustrate this particularly important result we have
studied the denerate solutions to the force-free Dirac equation and shown that
they correspond to massless particles. We have also provided explicit examples
regarding solutions to the force-free Weyl equation and the Weyl equation for a
constant magnetic field. In all cases we have calculated the infinite number of
different electromagnetic fields corresponding to these solutions.
