Mixing quantum and classical mechanics and uniqueness of Planck's constant

Abstract

Observables of quantum or classical mechanics form algebras called quantum or classical Hamilton algebras respectively (Grgin E and Petersen A (1974

The gravity-related decoherence master equation from hybrid dynamics

“…Perhaps the deepest reason is that the notion of deterministic classical trajectories of one subsystem becomes lost under the influence of the other subsystem which is subject to quantum uncertainties. I proposed a possible remedy long time ago [3], another approach was shown together with Gisin and Strunz [6]; further mathematical structures of hybrid dynamics appear from time to time [7,8,9,10,11,12]. A comparative analysis is missing.…”

“…Hybrid dynamics has obtained certain theoretical importance in foundations, in cosmology, in measurement problem. A very incomplete list of related works [1,2,3,4,5,6,7,8,9,10,11,12] shows the diversity of motivations.…”

Hybrid classical-quantum formulations ask for hybrid notions

“…For instance, the dynamics is dictated by a bracket failing to satisfy Jacobi's identity and Leibniz's rule. Under reasonable hypotheses it has been shown that there is no Lie bracket for a semiquantized theory of this form [21,[25][26][27][28]. If one permits the bracket to be of non-Lie type, it can be shown that the theory will lose the positivity of its density matrix [2,21] (positive operators could have nonpositive expectation values).…”

Classical-Quantum Limits

“…In the literature these are known as classicalquantum hybrid schemes, and a wide range of distinct schemes have been developed for various purposes including modeling of chemical reactions, analyzing decoherence, and studying measurement theory [2,4,5,7,8,[13][14][15][17][18][19][20][21][22][23][24][25]. Additionally, a number of studies of the general structure and properties of classical-quantum hybrid schemes have been carried out [1,6,[9][10][11][26][27][28].…”