  We canonically quantize closed string theory in the pp-wave background with a
non-zero flux of the three-form field strength by using the covariant BRST
operator formalism. In this canonical quantization, we completely construct new
covariant free-mode representations, for which it is particularly important to
take account of the commutation relations of the zero mode of the light-cone
string coordinate X^{-} with other modes. All covariant string coordinates are
composed of the free-modes. Moreover, employing these covariant string
coordinates for the energy-momentum tensor, we calculate the anomaly in the
Virasoro algebra and determine the number of dimensions of spacetime and the
ordering constant from the nilpotency condition of the BRST charge in the
pp-wave background.
