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The quantum deformation of the oscillator algebra and its implications on thephase operator are studied from a view point of an index theorem by using anexplicit matrix representation. For a positive deformation parameter $q$ or$q=exp(2\pi i\theta)$ with an irrational $\theta$, one obtains an indexcondition $\dml a - \dml a^{\dagger} = 1$ which allows only a non-hermitianphase 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 thenon-hermitian one with $\dml \expon^{i \varphi} - \dml(\expon^{i\varphi})^{\dagger} = 1$. Implications of this interpretation of thequantum deformation are discussed. We also show how to overcome the problem ofnegative norm for $q=exp(2\pi i\theta)$.Comment: 13 pages. A rather substantial revision has been made by employing an  explicit matrix representation of q-deformed oscillator algebra. In  particular, it is shown how to overcome the problem of negative norm for the  deformation parameter at a primitive root of unity. The revised version is in  press for Mod. Phys. Lett.

Q-DEFORMED OSCILLATOR ALGEBRA AND AN INDEX THEOREM FOR THE PHOTON PHASE OPERATOR