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

TL;DR

This paper explores the mathematical structure of the quantum phase operator for photons, revealing an index theorem analogy with chiral anomaly, and discusses implications for quantum theory and gauge invariance.

Contribution

It establishes an index relation for the photon phase operator and draws a novel analogy with the Atiyah-Singer index theorem and chiral anomaly.

Findings

Index relation for harmonic oscillator operators

Analogy between phase operator problem and gauge theory anomaly

Implications for quantum phase operator formulation

Abstract

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.