  We use state-of-the-art, three-dimensional non-local thermodynamic
equilibrium (non-LTE) radiative magnetohydrodynamic simulations of the quiet
solar atmosphere to carry out detailed tests of chromospheric magnetic field
diagnostics from free-free radiation at millimeter and submillimeter
wavelengths (mm/submm). The vertical component of the magnetic field was
deduced from the mm/submm brightness spectra and the degree of circular
polarization synthesized at millimeter frequencies. We used the frequency bands
observed by the Atacama Large Millimeter/Submillimeter Array (ALMA) as a
convenient reference. The magnetic field maps obtained describe the
longitudinal magnetic field at the effective formation heights of the relevant
wavelengths in the solar chromosphere. The comparison of the deduced and model
chromospheric magnetic fields at the spatial resolution of both the model and
current observations demonstrates a good correlation, but has a tendency to
underestimate the model field. The systematic discrepancy of about 10 percent
is probably due to averaging of the restored field over the heights
contributing to the radiation, weighted by the strength of the contribution. On
the whole, the method of probing the longitudinal component of the magnetic
field with free-free emission at mm/submm wavelengths is found to be applicable
to measurements of the weak quiet-Sun magnetic fields. However, successful
exploitation of this technique requires very accurate measurements of the
polarization properties (primary beam and receiver polarization response) of
the antennas, which will be the principal factor that determines the level to
which chromospheric magnetic fields can be measured. Consequently,
high-resolution and high-precision observations of circularly polarized
radiation at millimeter wavelengths can be a powerful tool for producing
chromospheric longitudinal magnetograms.
