Phase operator problem and an index theorem for Q-deformed oscillator

TL;DR

This paper uses index theory to analyze the photon phase operator problem, clarifying the non-existence of a Hermitian phase operator and revealing connections to quantum anomalies and Q-deformed oscillator representations.

Contribution

It introduces an index-based approach to the phase operator problem and establishes a novel analogy with chiral anomalies, advancing understanding of Q-deformed oscillators.

Findings

Clarifies the absence of a Hermitian phase operator using index theory

Identifies an analogy between phase operator problem and chiral anomaly

Characterizes Q-deformed oscillator representations via index invariance

Abstract

The notion of index is applied to analyze the phase operator problem associated with the photon. We clarify the absence of the hermitian phase operator on the basis of an index consideration. We point out an interesting analogy between the phase operator problem and the chiral anomaly in gauge theory, and an appearance of a new class of quantum anomaly is noted. The notion of index, which is invariant under a wide class of continuous deformation, is also shown to be useful to characterize the representations of Q-deformed oscillator algebra.