  We discuss the construction of low-energy tight-binding Hamiltonians for
condensed matter systems with a strong coupling to the quantum electromagnetic
field. Such Hamiltonians can be obtained by projecting the continuum theory on
a given set of Wannier orbitals. However, different representations of the
continuum theory lead to different low-energy formulations, because different
representations may entangle light and matter, transforming orbitals into
light-matter hybrid states before the projection. In particular, a multi-center
Power-Zienau-Woolley transformation yields a dipolar Hamiltonian which
incorporates the light-matter coupling via both Peierls phases and a
polarization density. We compare this dipolar gauge Hamiltonian and the
straightforward Coulomb gauge Hamiltonian for a one-dimensional solid, to
describe sub-cycle light-driven electronic motion in the semiclassical limit,
and a coupling of the solid to a quantized cavity mode which renormalizes the
band-structure into electron-polariton bands. Both descriptions yield the same
result when many bands are taken into account, but the dipolar Hamiltonian is
more accurate when the model is restricted to few electronic bands, while the
Coulomb Hamiltonian requires fewer electromagnetic modes.
