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Convergence of Dynamics on Inductive Systems of Banach Spaces

Lauritz van Luijk, Alexander Stottmeister, Reinhard F. Werner

Abstract

Many features of physical systems, both qualitative and quantitative, become sharply defined or tractable only in some limiting situation. Examples are phase transitions in the thermodynamic limit, the emergence of classical mechanics from quantum theory at large action, and continuum quantum field theory arising from renormalization group fixed points. It would seem that few methods can be useful in such diverse applications. However, we here present a flexible modeling tool for the limit of theories: soft inductive limits constituting a generalization of inductive limits of Banach spaces. In this context, general criteria for the convergence of dynamics will be formulated, and these criteria will be shown to apply in the situations mentioned and more.

Convergence of Dynamics on Inductive Systems of Banach Spaces

Abstract

Many features of physical systems, both qualitative and quantitative, become sharply defined or tractable only in some limiting situation. Examples are phase transitions in the thermodynamic limit, the emergence of classical mechanics from quantum theory at large action, and continuum quantum field theory arising from renormalization group fixed points. It would seem that few methods can be useful in such diverse applications. However, we here present a flexible modeling tool for the limit of theories: soft inductive limits constituting a generalization of inductive limits of Banach spaces. In this context, general criteria for the convergence of dynamics will be formulated, and these criteria will be shown to apply in the situations mentioned and more.
Paper Structure (15 sections, 40 theorems, 119 equations, 2 figures)

This paper contains 15 sections, 40 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Convergent nets in inductive systems
  3. Soft inductive limits
  4. Nets of operations on inductive systems
  5. Dynamics on inductive systems
  6. Inductive systems of C*-algebras and completely positive dynamics
  7. Examples and Applications
  8. Quantum dynamics in the classical limit
  9. Mean field limit
  10. Spin systems, dynamics and the thermodynamic limit
  11. Quantum scaling limits
  12. Recovering symmetries: Thompson's group actions à la Jones
  13. Comparison with the literature
  14. Convergence of implementors for dynamics in representations
  15. Interchanging Lie-Trotter limits and inductive limits

Key Result

Theorem A

Given a (soft) inductive system $(E,j_{})$ along with approximating dynamics $T_{n}(t)$ admitting generators $A_{n}$. Then we have the following equivalent characterizations of the dynamics on the limit space $E_{\infty}$: We conclude that the limit dynamics $T_\infty(t)$ is a strongly continuous one-parameter semigroup with generator $A_\infty$. The latter's domain is obtained by acting with the

Figures (2)

  • Figure 1: Streamlines of the vector field $X = (-\alpha q-p)\partial_q + (q-\alpha p)\partial_p$ on phase space describing the (classical) damped harmonic oscillator with damping constant $\alpha>0$.
  • Figure 2: An example of a Thompson's group element $f\in F$ as a map between two incomplete dyadic partitions.

Theorems & Definitions (84)

  • Theorem A: The evolution theorem, cf. \ref{['thm:evolution']}
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • Example 4: Constant inductive systems
  • Proposition 5
  • Definition 6
  • Lemma 7
  • ...and 74 more