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Physical Review A

Quantum group realized in a symmetric-top system

Zhe Chang

Published Apr 1, 1992• 8 citations
Publisher

Abstract

We show that the quantum group ${\mathrm{SU}}_{\mathit{q}}$(2) can be realized in a classical symmetric-top system as long as a suitable Hamiltonian and symplectic structure are chosen. The usual Lie group can be recovered when the deformation parameter q equals 1. The Hamiltonian system in which the quantum group is realized obeys the same equations of motion with the standard Hamiltonian system, but these two Hamiltonian systems have different constants of motion that reflect that the symmetries are different.

DOI
10.1103/physreva.45.4303

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We show that the quantum group SUq{\mathrm{SU}}_{\mathit{q}}SUq​(2) can be realized in a classical symmetric-top system as long as a suitable Hamiltonian and symplectic structure are chosen. The usual Lie group can be recovered when the deformation parameter q equals 1. The Hamiltonian system in which the quantum group is realized obeys the same equations of motion with the standard Hamiltonian system, but these two Hamiltonian systems have different constants of motion that reflect that the symmetries are different.
Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, with the standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining the symplectic structure, any of these three constants of motion can be used as a Hamiltonian, and that the remaining two, together with the total energy, generate a dynamical symmetry group isomorphic to SU(1,1). We also show that the standard energy levels of the quantum two-dimensional isotropic harmonic oscillator and their degeneracies are obtained making use of the appropriate representations of SU(1,1), provided that the canonical commutation relations are modified according to the new symplectic structure. Whereas in classical mechanics the different symplectic structures lead to equivalent formulations of the equations of motion, in quantum mechanics the modifications of the commutation relations should be accompanied by modifications in the interpretation of the formalism in order to obtain results equivalent to those found with the common relations.
Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part of. While such transformations were shown to be symmetries of the system's Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work, we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations.
Starting from the symplectic Lie groups Sp (2 n , ) of an n -mode system in classical and quantum mechanics we discuss, in particular, Sp (2, )~ SO (2,1) for a single mode and Sp (4, )~ SO (3,2) with some of its important subgroups such as SU (1,1)~ Sp (2, ) and SU (2). We then look at their applications in quantum optics (squeezing, phase states, beamsplitter and polarization). We explicitly construct the root scheme of the Lie algebra sp (4, ) in two different bases as it is an important means for the investigation of the group properties and the construction of the irreducible representations. Apart from the well known realizations of Sp (2, ) by single-mode and two-mode systems, we discuss some other realizations of this group by fractal combinations of boson operators.
It is proposed that an unconstrained quantum system is completely specified by its configuration space CCC and its group of symmetries GGG. It is then shown that the theory of projective unitary representations of GGG on a certain Hilbert space determined by CCC and GGG leads to a definitive resolution of supposed ambiguities relating to superselection rules, choice of wave functions, topological properties of GGG and CCC, anomalies, etc.
By means of geometric quantization, we discuss the quantum-deformed-symmetric-top model. We show that the quantum-deformed symmetric top and the quantum-standard symmetric top obey the same Heisenberg equations of motion. The quantum-deformed-symmetric-top model possesses quantum group symmetry; thus the quantum group can be dynamical symmetries of the symmetric-top system. The energy levels and wave functions of stationary states of the quantum-deformed-symmetric-top model are obtained exactly by a quantum-group-theoretic approach. Using the exactly solved model, we investigate the rotational spectra of symmetric-top molecules and diatomic molecules when the moment of inertia about the line joining the nuclei is not exactly zero.