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 $q=exp(2\pi i\theta )$ with an irrational $\theta$, one obtains an index condition $\dml a - \dml a^{\dagger} = 1$ which allows only a non-hermitian phase operator with $\dml \expon^{i φ} - \dml (\expon^{iφ})^{\dagger} = 1$. 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 $\dml \expon^{i Φ} - \dml (\expon^{iΦ})^{\dagger} = 0$ 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 $q=exp(2\pi i\theta )$.

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. A

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