# Polymer Quantum Mechanics and its Continuum Limit

## TL;DR

This paper investigates the polymer quantum mechanics framework, deriving it from standard quantum mechanics and analyzing its continuum limit to understand its relation to conventional theory and potential implications for quantum gravity.

## Contribution

It provides a detailed derivation of polymer quantum mechanics from Schrödinger theory and explores the continuum limit to connect discrete polymer models with standard quantum mechanics.

## Key findings

- Polymer quantum mechanics can be derived as a limit of Schrödinger quantum mechanics.
- The continuum limit of polymer theory recovers standard quantum mechanics for key models.
- Examples include harmonic oscillator, free particle, and simple cosmological models.

## Abstract

A rather non-standard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schroedinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schroedinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schroedinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle and a simple cosmological model.