Deformation Quantization of the Isotropic Rotator

We perform a deformation quantization of the classical isotropic rigid rotator. The resulting quantum system is not invariant under the usual SU(2)×SU(2) chiral symmetry, but instead SUq−1(2)× SUq(2). The classical isotropic rotator is known to be invariant under SU(2) × SU(2) chiral transformations. In ref. [1] a new Hamiltonian formulation of the isotropic rotator was found where the left and right SU(2) transformations are not canonical symmetries but rather Poisson Lie group symmetries.[2-8] The treatment given in ref. [1] further differs from the standard one because the classical Hamiltonian can not be expressed as the square of the angular momentum Ji, nor does Ji satisfy an SU(2) algebra. On the other hand, from this formulation one obtains the usual equations of motion for the isotropic rotator. They state that an SU(2) matrix-valued degree of freedom g denoting the orientation of the rigid body undergoes a uniform precession. This can be expressed as follows: ġg = i 2 Jiσi , J̇i = 0 , (1) where the dot dentoes a time derivative, σi are Pauli matrices and we have set the moment of inertia equal to one. In the usual formalism a general chiral transformation is given by g → wgv , Jiσi → w Jiσiw , w, v ∈ SU(2) , which leaves (1) invariant. In this article we quantize the system of ref. [1] using the method of deformation quantization [9]. We show that the resulting system is invariant under SUq−1(2)×SUq(2), and this is the quantum analogue of the classical SU(2) × SU(2) Poisson Lie group symmetry. The quantum mechanical observables for the system are associated with a pair of Hopf algebras[10] or equivalently a quantum double. We obtain dynamics on the quantum double which reduces to (1) when h̄ → 0. Furthermore in analogy to (1) the quantum dynamics is such that the quantum operator corresponding to g (now taking values in SUq(2)) undergoes a “uniform precession”. We first review the classical Hamiltonian formalism of ref. [1]. There it was shown that the six dimensional phase space describing a rigid body can be taken to be the group