Effective constrained polymeric theories and their continuum limit

TL;DR

This paper develops a method to analyze the continuum limit of effective constrained polymer quantum theories, demonstrating its application to models like a constrained oscillator and loop quantum cosmology, and discusses implications for renormalization in quantum systems.

Contribution

It introduces a strategy for taking the continuum limit of constrained polymer theories using Dirac observables and applies it to specific models, advancing understanding of quantum constrained systems.

Findings

The continuum limit exists for the models studied.

A particular mbda-dependent internal time is crucial.

Results may inform renormalization approaches for quantum constrained systems.

Abstract

The classical limit of polymer quantum theories yields a one parameter family of `effective' theories labeled by \lambda. Here we consider such families for constrained theories and pose the problem of taking the `continuum limit', \lambda -> 0. We put forward criteria for such question to be well posed, and propose a concrete strategy based in the definition of appropriately constructed Dirac observables. We analyze two models in detail, namely a constrained oscillator and a cosmological model arising from loop quantum cosmology. For both these models we show that the program can indeed be completed, provided one makes a particular choice of \lambda-dependent internal time with respect to which the dynamics is described and compared. We show that the limiting theories exist and discuss the corresponding limit. These results might shed some light in the problem of defining a renormalization group approach, and its associated continuum limit, for quantum constrained systems.