Towards a definition of quantum integrability

Abstract

We briefly review the most relevant aspects of complete integrability for classical systems and identify those aspects which should be present in a definition of quantum integrability. We show that a naive extension of classical concepts to the quantum framework would not work because all infinite dimensional Hilbert spaces are unitarily isomorphic and, as a consequence, it would not be easy to define degrees of freedom. We argue that a geometrical formulation of quantum mechanics might provide a way out.

37 pages, AmsLatex, 1 figure

References in corpus (5)

• Geometry of quantum systems: density states and entanglement
• Geometrization of Quantum Mechanics
• Alternative Structures and Bihamiltonian Systems
• Alternative structures and bi-Hamiltonian systems on a Hilbert space
• Alternative linear structures for classical and quantum systems

Cited by in corpus (7)

• Remarks on the notion of quantum integrability
• Integrability vs Quantum Thermalization
• Dynamical aspects in the Quantizer-Dequantizer formalism
• Reduction of Lie-Jordan Banach algebras and quantum states
• Tensorial dynamics on the space of quantum states
• Integrability in time-dependent systems with one degree of freedom
• Tomography on f-oscillators