  The notion of index is applied to analyze the phase operator problem
associated with the photon. We clarify the absence of the hermitian phase
operator on the basis of an index consideration. We point out an interesting
analogy between the phase operator problem and the chiral anomaly in gauge
theory, and an appearance of a new class of quantum anomaly is noted. The
notion of index, which is invariant under a wide class of continuous
deformation, is also shown to be useful to characterize the representations of
Q-deformed oscillator algebra.
