Convergence of Dynamics on Inductive Systems of Banach Spaces

TL;DR

This paper introduces a flexible framework using soft inductive limits of Banach spaces to analyze the convergence of dynamics in physical theories, applicable to phenomena like phase transitions and quantum-classical transition.

Contribution

It develops general criteria for the convergence of dynamics within soft inductive limits, extending the applicability of inductive limits to diverse physical systems.

Findings

Criteria for convergence of dynamics established

Framework applies to phase transitions and quantum-classical limits

Generalization of inductive limits 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.