Lessons from Toy-Models for the Dynamics of Loop Quantum Gravity

TL;DR

This paper reviews various toy-model approaches to understanding the Hamiltonian dynamics in loop quantum gravity, highlighting insights into regularization, continuum limits, and the challenges faced in Thiemann's quantum Hamiltonian construction.

Contribution

It introduces toy models like parametrized field theories, BF models, and Regge lattice gravity to shed light on the complexities of loop quantum gravity's Hamiltonian formulation.

Findings

Toy models reveal ambiguities in Thiemann's Hamiltonian construction.

Insights into regularization and continuum limit issues in loop quantum gravity.

Comparison of different approaches enhances understanding of quantum gravity dynamics.

Abstract

We review some approaches to the Hamiltonian dynamics of (loop) quantum gravity, the main issues being the regularization of the Hamiltonian and the continuum limit. First, Thiemann's definition of the quantum Hamiltonian is presented, and then more recent approaches. They are based on toy models which provide new insights into the difficulties and ambiguities faced in Thiemann's construction. The models we use are parametrized field theories, the topological BF model of which a special case is three-dimensional gravity which describes quantum flat space, and Regge lattice gravity.