Classical vs. Quantum Groups as Symmetries of Quantized Systems

Metin Arik and Gökhan Ünel
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Metin Arik: Boğaziçi University, Department of Physics
Gökhan Ünel: Boğaziçi University, Department of Physics

A chapter in Symmetries in Science IX, 1997, pp 1-8 from Springer

Abstract: Abstract Quantization of nonlinear systems has been one of the fundamental problems of quantum mechanics. The idea that classical integrable nonlinear systems should be solvable after quantization, has lead to the introduction of quantum groups as “generalized symmetries” of a physical system. Quantum groups [1] can he obtained by applying the idea of q-deformation either to the Lie algebra or to a matrix representation of the corresponding classical group. However, as we will show a q-deformation of the algebra of quantum observables does not necessarily imply a q-deformation of the symmetry group. To clarify this point, we discuss the quantum group invariant and the classical group invariant q-oscillators separately in the second section. In the third section, we consider a specific quadratically constrained model [2] and show that the classical group invariant q-oscillator can be derived using Dirac bracket (diracket) quantization [3] with some modifications.

Keywords: Classical Group; Quantum Group; Poisson Bracket; Gauge Condition; Annihilation Operator (search for similar items in EconPapers)
Date: 1997
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4615-5921-4_1

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DOI: 10.1007/978-1-4615-5921-4_1

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