  We report on deep BVR-imaging of the field of the nearby millisecond pulsar
PSR J0030+0451 obtained with the ESO/VLT/FORS2. We do not detect any optical
counterpart down to B>27.3, V>27.0 and R>27.0 in the immediate vicinity of the
radio pulsar position. The closest detected sources are offset by >3'', and
they are excluded as counterpart candidates by our astrometry. Using our upper
limits in the optical, and including recent XMM-Newton X-ray data we show that
any nonthermal power-law spectral component of neutron star magnetospheric
origin, as suggested by the interpretation of X-ray data, must be suppressed by
at least a factor of ~500 in the optical range. This either rules out the
nonthermal interpretation or suggests a dramatic spectral break in the
0.003-0.1 keV range of the power-law spectrum. Such a situation has never been
observed in the optical/X-ray spectral region of ordinary pulsars, and the
origin of such a break is unclear. An alternative interpretation with a purely
thermal X-ray spectrum is consistent with our optical upper limits. In this
case the X-ray emission is dominated by hot polar caps of the pulsar.
