On classical and quantum integrability

Michael J. Davis, Ellen B. Stechel, Eric J. Heller

Andrii Nikolaevich Liashyk, Nicolai Reshetikhin, Ivan Andreevich Sechin

Jonathan Oppenheim, Zachary Weller-Davies

Consistent dynamics which couples classical and quantum degrees of freedom exists. This dynamics is linear in the hybrid state, completely positive and trace preserving. Starting from completely positive classical-quantum master equations, we derive a general path integral representation for such dynamics in terms of a classical-quantum action, which includes the necessary and sufficient conditions for complete positivity and trace preservation. The path integral we study is a generalization of the Feynman path integral for quantum systems, and the stochastic path integral used to study classical stochastic processes, allowing for interaction between the classical and quantum systems. When the classical-quantum Hamiltonian is at most quadratic in the momenta we are able to derive a configuration space path integral, providing a map between master equations and covariant classical-quantum path integrals.

Stefan Weigert

R. Bullough, N. M. Bogoliubov, JUSSI TIMONEN

Ruguang Zhou, Wen‐Xiu Ma

A new finite-dimensional classical integrable system and a new quantum integrable system are generated from a spectral problem for the MKdV hierarchy through the nonlinearization technique of Lax systems. Our classical integrable system is an example of Gaudin magnet with boundary and relates to the finite band solutions of the MKdV hierarchy. Its Lax representation and r -matrix is given, and its separation of variables is performed. Based on a direct link between r -matrix formulas for classical systems and quantum problems, a quantum integrable system with separated variables is presented.