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A \(q\) deformation of the Gauss distribution. (English) Zbl 0841.60089

Summary: The \(q\) deformed commutation relation \(aa^* - qa^* a = 1\) for the harmonic oscillator is considered with \(q \in [-1,1]\). An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of \(a + a^*\) in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural \(q\) deformation of the Gaussian.

MSC:

60K40 Other physical applications of random processes
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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