  During the last three decades, P. B\'{o}na 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$^{\ast }$ continuous functionals on any
$C^{\ast }$-algebra. This is reminiscent of a well-known construction for
finite-dimensional Lie algebras. We then restrict this Poisson bracket to
states of this $C^{\ast }$-algebra, by taking quotients with respect to Poisson
ideals. This leads to densely defined symmetric derivations on the commutative
$C^{\ast }$-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 $C^{\ast }$-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 $^{\ast }$ G\^{a}teaux derivative, state-dependent
$C^{\ast }$-dynamical systems and the weak$^{\ast }$-Hausdorff hypertopology, a
new hypertopology used to prove, among other things, that convex weak$^{\ast
}$-compact sets generically have weak$^{\ast }$-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.
