  In previous articles we have developed a theory of down conversion in
nonlinear crystals, based on the Wigner representation of the radiation field.
Taking advantage of the fact that the Wigner function is always positive in
parametric down conversion experiments, we construct a local hidden variables
model where the amplitudes of the field modes are taken as random variables
whose probability distribution is the Wigner function. In order to achieve our
goal we give a model of detection which is fully local but departs from quantum
theory. In our model the zeropoint (vacuum) level of radiation lies below a
threshold of the detectors and only signals above the threshold are detectable.
The predictions of the model agree with those of quantum mechanics if the
signal intensities surpase some level and the efficiency is low. This is
consistent with the known fact that quantum mechanics is compatible with local
realism in that case (a fact called the ``efficiency loophole''). Our model
gives a number of constraints which do not follow from the quantum theory of
detection and are experimentally testable.
