Phase operator for the photon, an index theorem, and quantum anomaly

Abstract

An index relation $dimkera-dimker{a}^{†}=1$ is satisfied by the creation and annihilation operators $a^{†}$ 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, 1995