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dc.contributor.authorBru, J.-B. 
dc.contributor.authorde Siqueira Pedra, W. 
dc.description.abstractDuring the last three decades, P. Bóna has developed a non-linear generalization of quantum mechanics, based on symplectic structures for normal states and offering a general setting which is convenient to study the emergence of macroscopic classical dynamics from microscopic quantum processes. We propose here a new mathematical approach to Bona's one, with much brother domain of applicability. It highlights the central role of self-consistency. This leads to a mathematical framework in which the classical and quantum worlds are naturally entangled. We build a Poisson bracket for the polynomial functions on the hermitian weak∗ continuous functionals on any C∗-algebra. This is reminiscent of a well-known construction for finite-dimensional Lie algebras. We then restrict this Poisson bracket to states of this C∗-algebra, by taking quotients with respect to Poisson ideals. This leads to densely defined symmetric derivations on the commutative C∗-algebras of real-valued functions on the set of states. Up to a closure, these are proven to generate C0-groups of contractions. As a matter of fact, in general commutative C∗-algebras, even the closableness of unbounded symmetric derivations is a non-trivial issue. Some new mathematical concepts are introduced, which are possibly interesting by themselves: the convex weak ∗ Gâteaux derivative, state-dependent C∗-dynamical systems and the weak∗-Hausdorff hypertopology, a new hypertopology used to prove, among other things, that convex weak∗-compact sets generically have weak∗-dense extreme boundary in infinite dimension. Our recent results on macroscopic dynamical properties of lattice-fermion and quantum-spin systems with long-range, or mean-field, interactions corroborate the relevance of the general approach we present here. Note that the present paper is an extended version of the published one.en_US
dc.description.sponsorshipThis work was supported by CNPq (Grant No. 309723/2020-5); FAPESP (Grant No. 2017/22340-9); the Basque Government through the BERC 2022-2025 program; the Spanish Ministry of Science, Innovation and Universities (BCAM Severo Ochoa accreditation Grant Nos. SEV-2017-0718 and PID2020-112948GB-I00); and the COST Action CA18232 financed by the European Cooperation in Science and Technology (COST).en_US
dc.rightsReconocimiento-NoComercial-CompartirIgual 3.0 Españaen_US
dc.subjectPoisson algebrasen_US
dc.subjectquantum mechanicsen_US
dc.subjectclassical mechanicsen_US
dc.titleClassical dynamics from self-consistency equations in quantum mechanicsen_US
dc.journal.titleJournal of Mathematical Physicsen_US

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Reconocimiento-NoComercial-CompartirIgual 3.0 España
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