  The existing predictions of the energy density of gravitational waves
produced during preheating mostly rely on computer simulations in which the
matter field inhomogeneities essentially behave classically. In this article we
follow instead a full quantum treatment of the process within the $in$-$in$
formalism. We use this approach to determine the expected density of the
produced gravitational waves and numerically estimate its value in a simple
scalar field model, neglecting backreaction. Particular attention is devoted to
the regularization and renormalization of the divergences that appear in the
in-in formalism. We also address how our approach compares with the
conventional calculations used in the literature, and what elements could be
missed in the conventional analyses of gravitational wave production that rely
on numerical simulations. In the cases in which parametric resonance is
effective, our results agree with the predictions expected from numerical
simulations, as anticipated. At sufficiently low values of the resonance
parameter, however, numerical simulations fail, while our approach remains
applicable.
