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Recently Kazama and Yokoi (arXiv:0801.1561 [hep-th]) have used a phase-spacemethod to study the Virasoro algebra of type IIB superstring theory in themaximally supersymmetric R-R plane wave background in a semi-light-cone gauge.Two types of normal ordering have been considered, namely "phase space normalordering" (PNO) and "massless normal ordering" (MNO). The second one, which isthe right one to choose in flat background, has been discarded with theargument that the Virasoro algebra closes only in the first case. To understandthis issue better with a completely covariant treatment we consider the easiestcase of bosonic strings propagating in an arbitrary pp-wave of the simplestkind. Using the phase-space method we show that MNO is in fact the right one tochoose because of the following reason. For both types of normal ordering theenergy-momentum tensor satisfies the desired Virasoro algebra up to anomalousterms proportional to the space-time equation of motion of the background.However, it is MNO which gives rise to the correct spectrum - we compute thequadratic space-time action by restricting the string field inside a transverseHilbert space. This turns out to be non-diagonal. Diagonalizing this actionreproduces the spectrum directly obtained in light-cone quantization. The samemethod with PNO gives rise to a spectrum with negative dimensions.

On the Conformal Field Theories for Bosonic Strings in PP-Waves