J.-B. Bru and W. de Siqueira Pedra
Classical Dynamics from Self-Consistency Equations in Quantum Mechanics
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ABSTRACT. During the last three decades, P. B\'{o}na has developed a non-linear 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 self-consistency. 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 many-body systems with long-range 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 well-known construction for finite-dimensional 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 real-valued 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 non-trivial to prove. New mathematical concepts are introduced in this paper: the convex weak$^{st }$ G\^{a}teaux derivative, state-dependent $C^{st }$-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.