[Paper Review] Classical Dynamics from Self-Consistency Equations in Quantum Mechanics -- Extended Version

J. -B. Bru, W. de Siqueira Pedra|arXiv (Cornell University)|Sep 10, 2020

Advanced Operator Algebra Research92 references4 citations

TL;DR

This paper introduces a novel $C_0$-semigroup-based framework for Bóna's non-linear quantum mechanics, using self-consistency equations to derive classical dynamics from quantum processes. It establishes a Poisson bracket on polynomial functions of hermitian weak* continuous functionals on $C^*$-algebras, showing that symmetric derivations on state spaces generate $C_0$-groups of contractions, thereby embedding classical and quantum dynamics in a unified mathematical structure.

ABSTRACT

During the last three decades, P. Bóna 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 $^{\ast }$ Gâteaux 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.

Motivation & Objective

• To develop a mathematically rigorous framework for Bóna’s non-linear quantum mechanics beyond the original restrictive setting.
• To clarify the emergence of classical dynamics from quantum processes without relying on the $ackslash hbar \to 0$ limit.
• To establish the role of self-consistency in unifying classical and quantum dynamics within a single $C^*$-algebraic structure.
• To resolve the non-trivial issue of closability of unbounded symmetric derivations in commutative $C^*$-algebras of real-valued functions on state spaces.
• To introduce new topological tools—such as the weak*–Hausdorff hypertopology—to analyze convex weak* compact sets in infinite dimensions.

Proposed method

• Formalizing a Poisson bracket on polynomial functions over the space of hermitian weak* continuous functionals on a $C^*$-algebra using $C_0$-semigroup theory.
• Defining symmetric derivations on the commutative $C^*$-algebra of real-valued continuous functions on the state space via quotienting by Poisson ideals.
• Proving that these derivations generate $C_0$-groups of contractions, even in infinite-dimensional settings where closability is non-trivial.
• Introducing the convex weak* Gâteaux derivative as a tool to analyze state-dependent dynamics on $C^*$-algebras.
• Developing the weak*–Hausdorff hypertopology to study generic density of extreme points in weak* compact convex sets in infinite dimensions.
• Constructing state-dependent $C^*$-dynamical systems by associating quantum dynamics with evolving states, leading to classical evolution via reduction on invariant subspaces.

Experimental results

Research questions

• RQ1 How can self-consistency equations in quantum mechanics be formalized using $C_0$-semigroup theory to generate classical dynamics?
• RQ2 What is the role of Poisson brackets on functionals over $C^*$-algebras in connecting quantum and classical dynamics?
• RQ3 Under what conditions are unbounded symmetric derivations on commutative $C^*$-algebras of functions on state spaces closable, and when do they generate $C_0$-groups?
• RQ4 How does the weak*–Hausdorff hypertopology help characterize the generic density of extreme points in infinite-dimensional convex weak* compact sets?
• RQ5 In what way do state-dependent $C^*$-dynamical systems lead to consistent classical evolution laws?

Key findings

• The Poisson bracket on polynomial functions over weak* continuous functionals on a $C^*$-algebra is constructed via $C_0$-semigroup theory, extending Bóna’s framework to a broader class of $C^*$-algebras.
• Symmetric derivations on the commutative $C^*$-algebra of real-valued functions on the state space are shown to generate $C_0$-groups of contractions, even in infinite dimensions.
• The convex weak* Gâteaux derivative is introduced as a key tool to define state-dependent dynamics and analyze self-consistency in quantum systems.
• The weak*–Hausdorff hypertopology is proven to be a useful tool for analyzing convergence and genericity in hyperspaces, showing that convex weak* compact sets in infinite dimensions generically have weak* dense extreme boundaries.
• For antiliminal $C^*$-algebras—common in quantum spin and lattice fermion systems—the set of states has a weak* dense set of extreme points, supporting the emergence of classical dynamics from quantum mean-field systems.
• The framework confirms that macroscopic classical dynamics can emerge from microscopic quantum processes via self-consistent evolution, as demonstrated by recent results on Gross–Pitaevskii and Hartree hierarchies.