Definition Of Quantum Integrability Research Articles

TOWARDS A DEFINITION OF QUANTUM INTEGRABILITY

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 unitarilly 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.

Darboux Transformations of Bispectral Quantum Integrable Systems

We present an approach to higher-dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct nontrivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one dimensional systems) or by Darboux transformation of well-known bispectral systems such as quantum Calogero–Moser.

Quantum canonical transformations and integrability. Beyond unitary transformations

Quantum canonical transformations are defined in analogy to classical canonical transformations as changes of the phase space variables which preserve the Dirac bracket structure. In themselves, they are neither unitary nor non-unitary. A definition of quantum integrability in terms of canonical transformations is proposed which includes systems which have fewer commuting integrals of motion than degrees of freedom. The important role of non-unitary transformations in integrability is discussed.