Koopman wavefunctions and classical–quantum correlation dynamics

Abstract

Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman–von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical–quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect—the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical–quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.

Momentum maps for mixed states in quantum and classical mechanics

“…where A = −p · dq is the symplectic potential such that the canonical symplectic form on R 6 is given as ω can = dA. In turn, as shown in [11], this action produces the Clebsch representation momentum map…”

“…For example, Koopman's wavefunction description of classical dynamics [42] has been attracting increasing attention (see e.g. [10,11,53]) due to its analogies to quantum mechanics. However, other types of Clebsch representations also appeared in the context of density matrix evolution.…”

“…The latter differs from the physical total energy, which instead would read´H|ψ| 2 by following the KvN prescription f = |ψ| 2 . This apparent inconsistency, was recently overcome in [11] by considering an alternative action of Diff Ham (R 6 ) × S 1 on classical wavefunctions ψ ∈ L 2 (R 6 ). As reported in van Hove's thesis, this action is given by…”

“…Partly inspired by Kirillov [38], this equation (51) has been called Koopman-van Hove (KvH) equation and the self-adjoint operator L H is called prequantum operator in prequantization theory [29]. As mentioned earlier, the KvH equation (51) first appeared in [28,41], although the relation (50) between the classical wavefunction ψ(z) and the Liouville density function was discovered only recently in [11]. The main relation between the KvN and KvH constructions is that KvH reproduces the KvN equation for the modulus D = |ψ| 2 , while it also carries the evolution for the phase.…”

“…(up to a global phase factor), as it emerges by formally integrating the KvH equation (51). The KvH construction was recently used in [11] to formulate a classical-quantum wave equation for the Hamiltonian dynamics of hybrid classical-quantum systems. Such a formulation has been an open question for over 40 years, since Sudarshan's first proposal [57] of using KvN for modeling hybrid systems.…”