Classical dynamics from self-consistency equations in quantum mechanics

Abstract

During the last three decades, Pavel Bóna 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 a general setting that is very convenient to study the emergence of macroscopic classical dynamics from microscopic quantum processes. We propose here a new mathematical approach to Bóna’s non-linear quantum mechanics. It is based on C0-semigroup theory and has a domain of applicability that is much broader than Bóna’s original one. It highlights the central role of self-consistency. This leads to a mathematical framework in which the classical and quantum worlds are naturally entangled. In this new mathematical approach, 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 generic 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 and the state-dependent C*-dynamical systems. 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.