Classical Dynamics from Self-Consistency Equations in Quantum Mechanics -- Extended Version

• URL: http://arxiv.org/abs/2009.04969v2
• Date: Thu, 7 Jan 2021 15:18:43 GMT
• Title: Classical Dynamics from Self-Consistency Equations in Quantum Mechanics -- Extended Version
• Authors: J.-B. Bru and W. de Siqueira Pedra
• Abstract summary: We propose a new mathematical approach to Bona's non-linear generalization of quantum mechanics. It highlights the central role of self-consistency. Some new mathematical concepts are introduced, which are possibly interesting by themselves.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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