The Van Hove limit in classical and quantum mechanics

C. Bartsch, Pedro Vidal

Rodrigo Vargas

David Taj

Marcel Reginatto, Andrés Darío Bermúdez Manjarres, Sebastian Ulbricht

Motivated by models of mixed quantum-classical hybrid theories, we consider the representation of classical observables as Hilbert-space operators based on van Hove's unitary representation of contact transformations. This leads to a reformulation of classical mechanics where the phase space density satisfies the Liouville equation and, more importantly, classical observables are represented by van Hove operators that satisfy a commutation algebra isomorphic to the classical Poisson algebra of functions in phase space. Thus, this Hilbert space approach captures all the essential features of the phase space formulation of classical mechanics in that it reproduces its algebraic structure -- something that is not achieved by other approaches such as the Koopman-von Neumann theory. As an important feature, the definition of appropriate physical states in this formalism imposes certain natural requirements on the phase of the classical wave function, which ensure correct expectation values for the observables. Furthermore, we discuss how to extend the approach to the hybrid van Hove theory of mixed classical-quantum systems and illustrate its application with the example of interacting classical and quantum oscillators. Finally, we focus on the question of how hybrid van Hove theory evades no-go theorems on mixed classical-quantum systems and point to the various assumptions of the no-go theorems that do not apply to our approach. This demonstrates that these no-go theorems are not universally applicable, contrary to their original claims.

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Remy Messadene

A recent formalism capturing the classical-quantum coupling in a Hamiltonian theory for probabilistic classical mechanics has been proposed: the Koopman-van Hove formulation. The aims of this report are twofolds. First, we rigourously expose this new framework for a non-expert audience and present a construction for the operatorial analogues to the classical angular and linear momentum operators. We then investigate the group actions generating their average as momentum maps as well as their associated dynamics. Finally, we illustrate the theory by deriving the Koopman-van Hove formulation to the Kepler problem, harmonic and anharmonic oscillators.