 18111 J.B. Bru and W. de Siqueira Pedra
 Classical Dynamics from SelfConsistency Equations in Quantum Mechanics
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Nov 14, 18

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Abstract. During the last three decades, P. B\'{o}na has developed a nonlinear generalization of quantum mechanics, which is based on symplectic structures for normal states. One important application of such a generalization is to offer a general setting to understand the emergence of macroscopic dynamics from microscopic quantum processes. We propose a more general approach based on $C_{0}$semigroup theory, highlighting the central role of selfconsistency. This leads to a new mathematical framework for which the classical and quantum worlds are entangled. Such a feature is generally imperative to describe the dynamics of macroscopic quantum manybody systems with longrange interactions, as shown in subsequent papers. In this new mathematical approach, we build a Poisson bracket for the polynomial functions on the hermitian weak$^{st }$ continuous functionals on any $C^{st }$algebra. This is reminiscent of a wellknown construction for finitedimensional Lie groups. We then restrict this Poisson bracket to states of this $C^{st }$algebra by taking quotients with respect to Poisson ideals. This leads to densely defined symmetric derivations on the commutative $C^{st }$algebras of realvalued functions on the set of states. Up to a closure, these are proven to generate $C_{0}$groups of contractions. As a matter of fact, in general commutative $C^{st }$algebras, even the closeability of unbounded symmetric derivations is nontrivial to prove. New mathematical concepts are introduced in this paper: the convex weak$^{st }$ G\^{a}teaux derivative, statedependent $C^{st }$dynamical systems and the weak$^{st }$Hausdorff hypertopology, a new hypertopology used to prove, among other things, that convex weak$^{st }$compact sets generically have weak$^{st}$dense extreme boundary in infinite dimension.
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