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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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References:
[1] Faddeev L. D., Alg. Anal. 1 pp 178– (1989)
[2] DOI: 10.1142/S0217751X90000027 · Zbl 0709.17009
[3] Connes A., Publ. I.H.E.S. 62 pp 257– (1986)
[4] DOI: 10.1007/BF00420513 · Zbl 0771.17012
[5] DOI: 10.1016/0034-4877(89)90006-2 · Zbl 0707.47039
[6] DOI: 10.1016/0920-5632(91)90143-3
[7] DOI: 10.1063/1.530326 · Zbl 0807.16035
[8] DOI: 10.1088/0305-4470/22/21/020 · Zbl 0722.17009
[9] Biedenharn L., J. Phys. A 22 pp L– (1989)
[10] DOI: 10.1016/0370-2693(91)90640-C
[11] DOI: 10.1016/0370-2693(92)91044-A
[12] DOI: 10.1016/0370-2693(91)91227-M
[13] DOI: 10.1063/1.530204 · Zbl 0796.17016
[14] Koornwinder T. H., Trans. Am. Math. Soc. 333 pp 445– (1992)
[15] DOI: 10.1016/0370-2693(89)91366-X
[16] DOI: 10.1016/0370-2693(90)91927-4
[17] DOI: 10.1016/0370-2693(93)90785-G
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.