  We present a theory of the cavity quantum electrodynamics of the graphene
cyclotron resonance. By employing a canonical transformation, we derive an
effective Hamiltonian for the system comprised of two neighboring Landau levels
dressed by the cavity electromagnetic field (integer quantum Hall polaritons).
This generalized Dicke Hamiltonian, which contains terms that are quadratic in
the electromagnetic field and respects gauge invariance, is then used to
calculate thermodynamic properties of the quantum Hall polariton system.
Finally, we demonstrate that the generalized Dicke description fails when the
graphene sheet is heavily doped, i.e. when the Landau level spectrum of 2D
massless Dirac fermions is approximately harmonic. In this case we `integrate
out' the Landau levels in valence band and obtain an effective Hamiltonian for
the entire stack of Landau levels in conduction band, as dressed by strong
light-matter interactions.
