  We study the low-energy quantum electrodynamics of electrons and holes, in a
thin graphene wire. We develop an effective field theory (EFT) based on an
expansion in p/p_T, where p_T is the typical momentum of electrons and holes in
the transverse direction, while p are the momenta in the longitudinal
direction. We show that, to the lowest-order in (p/p_T), our EFT theory is
formally equivalent to the exactly solvable Schwinger model. By exploiting such
an analogy, we find that the ground state of the quantum wire contains a
condensate of electron-hole pairs. The excitation spectrum is saturated by
electron-hole collective bound-states, and we calculate the dispersion law of
such modes. We also compute the DC conductivity per unit length at zero
chemical potential and find g_s =e^2/h, where g_s=4 is the degeneracy factor.
