  A system evolving under the driven Jaynes--Cummings model will undergo a
phase transition at a critical driving field amplitude. This transition is
foreshadowed by a collapse of the quasienergy level spectra of the system and
remains present as the model is extended to include a counter-rotating
interaction. We study this critical response and obtain the eigenvalues and
eigenstates of the extended model by presenting a correspondence between the
Jaynes--Cummings model and a charged Dirac particle subject to an external
electromagnetic field. Under this correspondence, the field and two-level
system that compose the Jaynes--Cummings model map to the external and internal
degrees of freedom describing the Dirac particle, respectively. The phases of
the system below (above) the critical drive are then characterized by discrete
(continuous) solutions, with the manipulations required to obtain these
solutions appearing naturally as Lorentz transformations.
