From Quantum Hydrodynamics to Koopman Wavefunctions I

Gay–Balmaz, François; Tronci, Cesare

doi:10.1007/978-3-030-80209-7_34

Cited by 8 publications

(11 citation statements)

References 9 publications

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“…As discussed in [23,22], the Madelung transform of complex partial differential equations simply corresponds to expressing the equation in terms of amplitude and phase by writing Ψ in polar form. In the present section, this procedure will be applied to the KvH equation.…”

Section: Madelung Transform Of the Kvh Equationmentioning

confidence: 99%

“…Here we recall that the Lie algebra of the central extension Diff ω (T * Q) is isomorphic to the space of functions F (T * Q). See further details in [23,22]. Then, we are left with a relation between the momentum map (3) for the classical Liouville equation and the momentum map (17) associated to the KvH Madelung transform, that is…”

Section: Madelung Transform Of the Kvh Equationmentioning

confidence: 99%

“…In the present context, the hybrid density operator can be identified by rewriting the Hamiltonian functional (22) by using integration by parts as follows:…”

Section: The Hybrid Density Operatormentioning

confidence: 99%

“…Here, we have dropped the integration domain T * Q for convenience of notation and we have rewritten the original Hamiltonian functional (22) as…”

Section: Variational Structurementioning

confidence: 99%

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Koopman wavefunctions and classical states in hybrid quantum-classical dynamics

Gay-Balmaz,

Tronci

2021

Preprint

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We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum-classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive sign. In this way, the evolution allows identifying a classical and a quantum state in interaction at all times. After combining Koopman's Hilbert-space method in classical mechanics with van Hove's unitary representations in prequantum theory, the closure model is made available by the variational structure underlying a suitable wavefunction factorization. Also, we use Poisson reduction by symmetry to show that the hybrid model possesses a noncanonical Poisson structure that does not seem to have appeared before. As an example, this structure is specialized to the case of quantum two-level systems.

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Section: Madelung Transform Of the Kvh Equationmentioning

confidence: 99%

Section: Madelung Transform Of the Kvh Equationmentioning

confidence: 99%

“…In the present context, the hybrid density operator can be identified by rewriting the Hamiltonian functional (22) by using integration by parts as follows:…”

Section: The Hybrid Density Operatormentioning

confidence: 99%

“…Here, we have dropped the integration domain T * Q for convenience of notation and we have rewritten the original Hamiltonian functional (22) as…”

Section: Variational Structurementioning

confidence: 99%

See 2 more Smart Citations

Koopman wavefunctions and classical states in hybrid quantum-classical dynamics

Gay-Balmaz,

Tronci

2021

Preprint

Self Cite

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“…For example, the Lagrange-to-Euler map (3.3) now reads D a (z, t) = ´Da,0 (z)δ(z − η(z 0 , t)) d 2 z 0 , or equivalently in terms of the Jacobian J η = det ∇η. We note that, since the χ a 's all obey the same KvH equation, the Lagrangian path η is the same for each a = 1, ..., N. Upon defining In the limit case Σ → 0, the von Neumann operator associated to the singular Wigner function W = Dδ(λ − A) was presented explicitly in [31].…”

Section: A Von Neumann Operators In Koopman Mechanicsmentioning

confidence: 99%

Evolution of hybrid quantum-classical wavefunctions

Gay-Balmaz,

Tronci

2021

Preprint

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A gauge-invariant wave equation for the dynamics of hybrid quantum-classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge invariance in the variational principle underlying a hybrid wave equation previously proposed by the authors. While the original construction ensures a positive-definite quantum density matrix, the present model also guarantees the same property for the classical Liouville density. After a suitable wavefunction factorization, gauge invariance is achieved by resorting to the classical Lagrangian paths made available by the Madelung transform of Koopman wavefunctions. Due to the appearance of a phase-space analogue of the Berry connection, the new hybrid wave equation is highly nonlinear and it is proposed here as a platform for further developments in quantum-classical dynamics. Indeed, the associated model is Hamiltonian and appears to be the first to ensure a series of consistency properties beyond positivity of quantum and classical densities. For example, the model possesses a quantum-classical Poincaré integral invariant and its special cases include both the mean-field model and the Ehrenfest model from chemical physics.

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Evolution of hybrid quantum–classical wavefunctions

Gay–Balmaz

Tronci

2022

Physica D: Nonlinear Phenomena

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From Quantum Hydrodynamics to Koopman Wavefunctions I

Cited by 8 publications

References 9 publications

Koopman wavefunctions and classical states in hybrid quantum-classical dynamics

Koopman wavefunctions and classical states in hybrid quantum-classical dynamics

Evolution of hybrid quantum-classical wavefunctions

Evolution of hybrid quantum–classical wavefunctions

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