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 $q=exp(2\pi i\theta)$.
Categories: hep-th math.QA q-alg