  An index relation $dim\ ker\ a^{\dagger}a - dim\ ker\ aa^{\dagger} = 1$ is
satisfied by the creation and annihilation operators $a^{\dagger}$ and $a$ of a
harmonic oscillator. A hermitian phase operator, which inevitably leads to
$dim\ ker\ a^{\dagger}a - dim\ ker\ aa^{\dagger} = 0$, cannot be consistently
defined. If one considers an $s+1$ dimensional truncated theory, a hermitian
phase operator of Pegg and Barnett which carries a vanishing index can be
defined. However, for arbitrarily large $s$, we show that the vanishing index
of the hermitian phase operator of Pegg and Barnett causes a substantial
deviation from minimum uncertainty in a characteristically quantum domain with
small average photon numbers. We also mention an interesting analogy between
the present problem and the chiral anomaly in gauge theory which is related to
the Atiyah-Singer index theorem. It is suggested that the phase operator
problem related to the above analytic index may be regarded as a new class of
quantum anomaly. From an anomaly view point ,it is not surprising that the
phase operator of Susskind and Glogower, which carries a unit index, leads to
an anomalous identity and an anomalous commutator.
