Q-Deformed Oscillator Algebra and an Index Theorem for the Photon Phase Operator

TL;DR

This paper explores the quantum deformation of oscillator algebra and its impact on the photon phase operator, revealing conditions for hermitian and non-hermitian phase operators through an index theorem approach.

Contribution

It introduces an index theorem framework to analyze the quantum deformation effects on the photon phase operator, including cases with rational and irrational deformation parameters.

Findings

For irrational q, only a non-hermitian phase operator exists.

For rational q, both hermitian and non-hermitian phase operators are possible.

The paper discusses overcoming negative norm issues for rational q.

Abstract

The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a view point of an index theorem by using an explicit matrix representation. For a positive deformation parameter $q$ or $q=exp(2\pi i\theta)$ with an irrational $\theta$, one obtains an index condition $\dml a - \dml a^{\dagger} = 1$ which allows only a non-hermitian phase operator with $\dml \expon^{i \varphi} - \dml (\expon^{i\varphi})^{\dagger} = 1$. For $q=exp(2\pi i\theta)$ with a rational $\theta$ , one formally obtains the singular situation $\dml a =\infty$ and $ \dml a^{\dagger} = \infty$, which allows a hermitian phase operator with $\dml \expon^{i \Phi} - \dml (\expon^{i\Phi})^{\dagger} = 0$ as well as the non-hermitian one with $\dml \expon^{i \varphi} - \dml (\expon^{i\varphi})^{\dagger} = 1$. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for $q=exp(2\pi i\theta)$.