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Uncertainty relation in quantum mechanics with quantum group symmetry. (English) Zbl 0877.17017
Summary: The commutation relations, uncertainty relations, and spectra of position and momentum operators are studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of nonzero minimal uncertainties in the positions and momenta. The usual quantum mechanical behavior is recovered as a limit case for not too small and not too large distances and momenta.

##### MSC:
 17B81 Applications of Lie (super)algebras to physics, etc. 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 81S05 Commutation relations and statistics as related to quantum mechanics (general) 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory
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