Quantum integrability and action operators in spin dynamics

A new formulation of the quantum integrability condition for spin systems is proposed. It eliminates the ambiguities inherent in formulations derived from a direct transcription of the classical integrability criterion. In the new formulation, quantum integrability of an N-spin system depends on the existence of a unitary transformation which expresses the Hamiltonian as a function of N action operators. All operators are understood to be algebraic expressions of the spin components with no restriction to any finite-dimensional matrix representation. The consequences of quantum (non)integrability on the structure of quantum invariants are discussed in comparison with the consequences of classical (non)integrability on the corresponding classical invariants. Our results indicate that quantum integrability is universal for systems with N = 1 and contingent for systems with N 2.