# Mathematics Colloquia and Seminars

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### Introduction to Observables in Three Dimensional Quantum Gravity

**QMAP Seminar**

Speaker: | Catherine Meusburger, Hamburg |

Location: | 2112 MSB |

Start time: | Mon, Jun 8 2009, 2:10PM |

Part I of a joint Quantum Geometry + Geometric Topology RFG Activity Flat three-dimensional Lorentzian manifolds arise as the solutions of the vacuum Einstein equations in three-dimensional general relativity. They exhibit a rich mathematical structure and are of interest as toy models for (quantum) gravity in four-dimensions. However, this requires relating the variables used in the classical description and quantisation of the theory to concrete geometrical quantities measured by observers. This has been difficult so far even in the classical theory. In this talk, we relate the geometry of such spacetimes to the fundamental physical observables (Wilson loops) of the theory and show that they correspond to realistic measurements by observers. By considering an observer who probes the geometry of the spacetime with returning lightrays, we obtain three realistic physical quantities that could be measured by observers - the eigentime elapsed between the emission and the return of the lightray, the angles between the directions associated with returning lightrays and the relative frequency shift of the emitted and returning lightray. We give explicit expressions for these quantities as functions on the phase space and discuss their physical interpretation. We demonstrate how an observer can reconstruct the geometry of the spacetime from these measurements and that the fundamental observables of the theory (Wilson loops) arise naturally in the description. Reference: arXiv:0811.4155