We study the spontaneous emission in high-gain free-electron lasers operating in the quantum regime
and its detrimental effect on coherent emission. A quantum model describing the coherent and spontaneous
emission in free electron lasers has been recently proposed and investigated [G. R. M. Robb
and R. Bonifacio, Phys. Plasmas 19, 073101 (2012)]. The model is based on a Wigner distribution
describing the electron beam dynamics, coupled to Maxwell equations for the emitted radiation field.
Here, we rephrase the model in a more rigorous way, considering a discrete Wigner distribution
defined for a periodic space coordinate for which the electron momentum is discrete. From its numerical
solution, we find good agreement with the approximate continuous model. In the quantum
regime of the free-electron laser, we obtain a simple density matrix equation for two momentum
states, where the role of the spontaneous emission has a clear interpretation in terms of coherence
decay and population transfer.