Phase Operator for the Photon Field and an Index Theorem

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.