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.
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