Tuesday, Apr 18th
**Bianca Dittrich, Perimeter Institute**
**Title: (3+1) LQG with homogeneously curved vacuum **
PDF of the talk (8M)
Audio+Slides [.mp4 17MB]

by Jorge Pullin, Louisiana State

The way geometries are studied mathematically one starts with a set of points that has a notion of proximity. One can say when points are close to each other. This is not the same as being able to measure distances in the set. That requires the introduction of an additional mathematical structure, a metric. The sets of points with notion of proximity are known as “manifolds”. General relativity is formulated on a manifold and is a theory about a metric to be imposed on that manifold. Ordinary quantum field theories, like quantum electrodynamics, require the introduction of a metric before they can be formulated, so they are of different nature than general relativity. Theories that do not require a metric in order to be formulated are known as “background independent”. Interestingly, although general relativity is a theory about a metric, it can be formulated without any prior metric. There exist quantum field theories that can be formulated without a metric. They are known as topological field theories and they typically, contrary to ordinary field theories, have only a finite number of degrees of freedom. This means that they are much easier to treat and to quantize.

An example of a topological field theory is general relativity in three space-time dimensions. In one dimension less than four, the Einstein equations just say that the metric is flat, except at a finite number of points. So space-time is flat everywhere with curvature concentrated at just a few points. An example of a space that is flat everywhere except at a point is a cone. The only place that is curved is the tip. One has to remember that the notion of curvature we are talking here is one that can be measured from inside the space-time (typically by going around a circle and seeing if a vector carried around returns parallel to itself). If you do that in a cone on any circle that does not thread the tip, the vector comes back parallel to itself. So space-times in three dimensional general relativity are said to have “conical singularities” at the points where the curvature is non-zero. As other topological field theories, general relativity in three space-time dimensions has a finite number of degrees of freedom. This explains why Witten was able to complete its quantization in the mid 1980’s whereas the quantization of four dimensional general relativity is still a big outstanding problem today.

In this talk a generalization of three dimensional general relativity to four dimensions was presented. The resulting theory in four space-time dimensions has curvature concentrated at edges (strings) –as opposed to points as we had in the three dimensional case- and elsewhere the metric is flat. This makes them much easier to quantize than general relativity. Among the results was the construction of four dimensional quantum geometries similar to those in a previous model by Crane and Yetter. Also a role for quantum groups, that had been conjectured to arise when one considers a cosmological constant was found providing more evidence to this assertion. The space of quantum states (Hilbert space) was rigorously constructed and leads to insights about how the continuum limit of the theory could emerge. The hope is that one could build on these theories to construct new representations for loop quantum gravity in four space-time dimensions and hopefully to implement on them the (quantum) dynamics of general relativity.

Also a notion of duality emerges. In this context, duality means a certain relationship between the metric and the curvature of the space-time at a classical level. Here it can be implemented at a quantum level and quantum space of states associated with the metric (areas) and curvatures can be introduced and are dual to each other. Similar spaces had been proposed for general relativity, but here there is much more mathematical control over them, so this provides a controlled arena to test ideas that are being put forward in the context of loop quantum gravity in four space-time dimensions.