  An index relation $dim\ ker\ a - dim\ ker\ a^{\dagger} = 1$ is satisfied by
the creation and annihilation operators $a^{\dagger}$ and $a$ of a harmonic
oscillator. Implications of this analytic index on the possible form of the
phase operator are discussed. A close analogy between the present phase
operator problem and chiral anomaly in gauge theory, which is associated with
Atiyah-Singer index theorem, is emphasized.
