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Angular momentum and mutually unbiased bases. (English) Zbl 1093.81034

Summary: The Lie algebra of the group \(\text{SU}_{2}\) is constructed from two deformed oscillator algebras for which the deformation parameter is a root of unity. This leads to an unusual quantization scheme, the \(\{J^{2}, U_{r}\}\) scheme, an alternative to the familiar \(\{J^{2}, J_{z}\}\) quantization scheme corresponding to common eigenvectors of the Casimir operator \(J^{2}\) and the Cartan operator \(J_{z}\). A connection is established between the eigenvectors of the complete set of commuting operators \(\{J^{2}, U_{r}\}\) and mutually unbiased bases in spaces of constant angular momentum.

MSC:

81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R15 Operator algebra methods applied to problems in quantum theory
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