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

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 $q=exp(2πiθ)$ with an irrational $θ$, one obtains an index condition $\dml a - \dml a^{\dagger} = 1$ which allows only a non-hermitian phase operator with $\dml \expon^{i φ} - \dml (\expon^{iφ})^{\dagger} = 1$. For $q=exp(2πiθ)$ with a rational $θ$ , one formally obtains the singular situation $\dml a =\infty$ and $ \dml a^{\dagger} = \infty$, which allows a hermitian phase operator with $\dml \expon^{i Φ} - \dml (\expon^{iΦ})^{\dagger} = 0$ as well as the non-hermitian one with $\dml \expon^{i φ} - \dml (\expon^{iφ})^{\dagger} = 1$. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for $q=exp(2πiθ)$.