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

J. -B. Bru, W. de Siqueira Pedra

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.

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

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 continuous functionals on any -algebra. This is reminiscent of a well-known construction for finite-dimensional Lie algebras. We then restrict this Poisson bracket to states of this -algebra, by taking quotients with respect to Poisson ideals. This leads to densely defined symmetric derivations on the commutative -algebras of real-valued functions on the set of states. Up to a closure, these are proven to generate -groups of contractions. As a matter of fact, in general commutative -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, state-dependent -dynamical systems and the weak-Hausdorff hypertopology, a new hypertopology used to prove, among other things, that convex weak-compact sets generically have weak-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.

Paper Structure

This paper contains 33 sections, 49 theorems, 339 equations.

Table of Contents

  1. Introduction
  2. Classical View on Quantum Systems
  3. State Space of $C^{\ast }$-Algebras
  4. Phase Space of $C^{\ast }$-Algebras
  5. Generic Weak$^{\ast }$-Compact Convex Sets in Infinite Dimension
  6. Classical $C^{\ast }$-Algebra of Continuous Functions on the State Space
  7. Classical $C^{\ast }$-Algebra of Continuous Functions on the Phase Space
  8. Poisson Structures in Quantum Mechanics
  9. Bóna's Poisson Structures
  10. Poisson Algebra of Polynomial Functions on the Continuous Self-Adjoint Functionals on a $C^{\ast }$-Algebra
  11. Poisson Ideals Associated with State and Phase Spaces
  12. Convex Weak$^{\ast }$ Gateaux Derivative
  13. Explicit Construction of Poisson Brackets for Functions on the State Space
  14. Poissonian Symmetric Derivations
  15. Hamiltonian Flows for States from Self-Consistent Quantum Dynamics
  16. ...and 18 more sections

Key Result

Theorem \oldthetheorem

Let $\mathcal{X}$ be an infinite-dimensional separable Banach space. Then, the set $\mathcal{D}$ of all nonempty convex weak$^{\ast }$-compact sets $K$ with a weak$^{\ast }$-dense set $\mathcal{E}(K)$ of extreme points is a weak$^{\ast }$-Hausdorff-dense $G_{{ \if@compatibility \mathchar"010E {}

Theorems & Definitions (68)

  • Definition 1: State space
  • Definition 2: Phase space
  • Definition 3: Weak$^{\ast }$-Hausdorff hypertopology for convex sets
  • Theorem \oldthetheorem: Generic convex weak$^{\ast }$-compact sets
  • Theorem \oldthetheorem: Generic weak$^{\ast }$-compact convex subset of the state space
  • Remark 1
  • Definition 4: Poisson bracket
  • Proposition \oldthetheorem: Usual properties of Poisson brackets
  • Corollary \oldthetheorem: Poisson algebra
  • Proposition \oldthetheorem: Folia of states and Poisson ideals
  • ...and 58 more