Ehrenfest principle and unitary dynamics of quantum-classical systems with general potential interaction

Abstract

Representation of classical dynamics by unitary transformations has been used to develop a unified description of hybrid classical-quantum systems with a particular type of interaction, and to formulate abstract systems interpolating between classical and quantum ones. We solved the problem of a unitary description of two interpolating systems with general potential interaction. The general solution is used to show that with arbitrary potential interaction between the two interpolating systems the evolution of the so-called unobservable variables is decoupled from that of the observable ones if and only if the interpolation parameters in the two interpolating systems are equal.

Projective representation of the Galilei group for classical and quantum–classical systems*

“…Besides the purely classical results, the KvN and KvH formalism are of interest to contrast/compare quantum and classical mechanics and the study of quantization/dequantization rules [13][14][15][16][17]. The KvN and KvH theories are also of interest for the formulation of some quantum-classical hybrid theories [9,10,[18][19][20][21][22][23][24][25]. In this last regard, the knowledge of the non-relativistic space-time symmetries of the KvN theory was used to study the viability of quantum-classical hybrids of the Sudarshan type [26].…”

Wigner–Lindblad Equations for Quantum Friction

“…As an outcome, ODM guarantees that the resulting equations of motion have the desired physical structure to reproduce the supplied dynamical observations. This formalism has provided new interpretation of the Wigner function, unveiled conceptual inconstancies in finite-dimensional quantum mechanics, formulated dynamical models in topologically nontrivial spaces, advanced the study of quantum-classical hybrids, and led to development of efficient numerical techniques. , …”

Galilean covariance of quantum-classical hybrid systems of the Sudarshan type

“…It is worth noting that in the literature there can be found interaction terms that do not respect the commutation relations from the Galilei algebra for the total system (see, for example, equation (13) in Ref [1] or equation (18) in Ref [8]. See also [21]). It was noted in [8] that a quantum-classical coupling need not to conserve the total energy, here a more general result was given since conditions for the conservation of the total momentum were obtained.…”