QuantumGeometry 2009-01-07 08:40:14 2009-01-07 08:49:31 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. \subsection{Lattice methods and spin foams} \subsection{Quantum $6j$-symbols related to spherical tetrahedra (rather than flat tetrahedra)} Several quantum observables whose expectation values generally give topological information about the nature of quantized spacetime have been already considered but-- with very few exceptions-- the results in this area has remain largely mathematical in nature; thus, surprisingly little is understood about the physics of such observables, although some are most likely to be related to length and perhaps volumes, whereas other observables are connected to scattering amplitudes.