2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/54336An 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.To be published in the Proceedings of International Symposium on Foundations of Quantum Mechanics, Tokyo, 1995High Energy Physics - Theorytext