Loop Quantization of a 3D AbelianBFModel withσ-Model Matter

Diego C M Mendonça,Olivier Piguet

doi:10.1155/2015/307132

Abstract

The main goal of this work is to explore the symmetries and develop the dynamics associated with a 3D AbelianBFmodel coupled to scalar fields submitted to a sigma model like constraint, at the classical and quantum levels. Background independence, on which the model is founded, strongly constrains its nature. We adapt to the present model the techniques of Loop Quantum Gravity in order to construct its physical Hilbert space and its observables.

Highlights

The quasi-hundred-year-old General Relativity as a theory of gravitation, despite its tremendous successes in accounting for predicting phenomena, still lacks a quantum version
Very important progresses have been made in the last decades, especially in the framework of Loop Quantum Gravity (LQG) [6,7,8,9], based on the canonical Hamiltonian approach of Dirac [10, 11] applied to the Ashtekar-Barbero [12,13,14] parametrization of the theory
The LQG program entails the difficult task of implementing the constraints of the theory as quantum operators in some predefined kinematic Hilbert space and to solve them, leaving as a subspace the physical Hilbert space in which act the selfadjoint operators representing the observables of the theory

Summary

Introduction

The quasi-hundred-year-old General Relativity as a theory of gravitation, despite its tremendous successes in accounting for predicting phenomena, still lacks a quantum version. General Relativity, as a background independent theory—in the sense that no background geometry is given a priori, geometry being dynamical—is a fully constrained theory, its Hamiltonian being merely a sum of constraints generating the gauge invariances of the theory. The lower-dimensional gravitation theories are much more easy to handle, since they can be described as topological gauge theories, when not coupled to matter [17,18,19,20,21,22,23]. The purpose of this paper is to present the loop quantization of a background independent theory of the BF type [26, 27] with the Abelian group U(1) as a gauge group. Provided the topology of 2-dimensional space is nontrivial, global degrees of freedom are present in the theory.

Formulation of the Model

Hamiltonian Analysis and Constraints

Quantization

Conclusions