Evolution of hybrid quantum–classical wavefunctions

Evolution of Hybrid Quantum-Classical Wavefunctions

“…The geometric structure of the quantum-classical model (3)-( 5) has been illustrated in [3] and here we will review the high points.…”

“…Then, after various manipulations we recover the system (3)-( 5). The purely quantum and classical cases are recovered by restricting to the cases X H = 0 and H = H1, respectively [3]. In addition, if one neglects the −terms in the Hamiltonian functional h, then the variational principle (7) recovers the Ehrenfest model, to which (3)-( 5) reduce.…”

“…Beyond the Ehrenfest model. Blending Koopman wavefunctions in classical mechanics with the geometry of prequantum theory, we recently formulated a quantum-classical model [3,4] which was developed in two stages. First, we provided an early quantum-classical model [1] that succeeded in satisfying only the properties 2)-5).…”

“…First, we provided an early quantum-classical model [1] that succeeded in satisfying only the properties 2)-5). Then, more recently, we upgraded this model in such a way that property 1) is also secured [3,4]. This upgrade was achieved by combining Lagrangian trajectories on the classical phase-space with a gauge principle which ensures that classical phases are unobservable, that is they do not contribute to measurable expectation values.…”

“…Nevertheless, the model is nonlinear and its explicit form is rather intricate due to the appearance of the non-Abelian gauge connection i[P, ∇P ], where P (x, x ′ ; q, p) = ψ(x; q, p)ψ * (x ′ ; q, p). In particular, this gauge connection emerges through the (Hermitian) operator-valued vector field Γ = i[P, X P ] in such a way that the model proposed in [3,4] reads…”