Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1995-04-26
Mod.Phys.Lett. A10 (1995) 2543-2552
Physics
High Energy Physics
High Energy Physics - Theory
13 pages. A rather substantial revision has been made by employing an explicit matrix representation of q-deformed oscillator
Scientific paper
10.1142/S0217732395002684
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 $q=exp(2\pi i\theta)$.
Fujikawa Kazuo
Kwek Leong Chuan
Oh Chang-Heon
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