QuantumGeometry 2009-01-07 08:40:14 2009-01-07 08:40:14 Topic quantum operator algebras quantum geometry % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here This is a contributed topic on: \section{Quantum Geometry} In 4 dimensions, one of the attractive programs of spacetime quantization is “quantum geometry”, often represented as “loop quantum gravity” . Loop quantum gravity starts with a Hamiltonian formulation of the first order formalism, with constraints, written in analogy to the (3+1)-dimensional case that take the form: $$D_i E^{ia} =0$$, $$ E^i_a R^a_{ij} =0,$$ and $$\epsilon_{abc}E^{ib}E^{jc}R^a_{ij}=0,$$ where the indices $i,j$ and $k$ are the spatial indices on a surface of constant time, $$E^{ia}= \epsilon^{ij}e^a_j$$, $D_i$ is the $SO(2,1)$ gauge-covariant derivative for the connection $\omega$, and the $R^a_{ij}$ are the spatial components of the curvature two-form.