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Gravitational path integral and group theory

**Pietro Dona, Penn State**
**Title: SU(2) graph invariants, Regge actions and polytopes **
PDF of the talk (10M)
Audio+Slides [.mp4 16MB]

By Jorge Pullin, LSU

In the loop quantum gravity approach to quantum gravity the quantum states are given by spin networks, graphs with intersections and "colors" associated with each link. The colors are a shorthand to characterize that each link in the graph is associated with a mathematical quantity known as an element of a group. A group is a type of mathematical set with a composition law that is associative, has a neutral element and has an opposite element. For instance, real numbers form a group under addition. Matrices of numbers also form groups under multiplication. When links of a spin network meet at an intersection, the respective group elements associated with them get multiplied into a mathematical entity known as "intertwiner". Such intertwiners are constructed with what are known as invariant tensors in the group.

One of the approaches to quantizations of field theories is the path integral approach. In it, one assigns probabilities to each physical trajectory and sums over all possible trajectories. When applied in the context of loop quantum gravity one gets trajectories in time of spin networks, which give rise to what are known as "spin foams". The probability of a given trajectory is quantified in terms of a number related to how the spin networks branch out into the future known as a "vertex". There are several proposals for such vertices to represent the dynamics of general relativity, at present it is not clear which one of the proposed ones represents nature more accurately. One of the most studied ones is the EPRL (Engle-Pereira-Rovelli-Livine) vertex. Other vertices that have simpler nature have also been proposed. This seminar deals with the evaluation of these vertices. This requires calculations in group theory. These calculations may have broader applicability than in just quantum gravity as these types of mathematical entities appear in many physical domains. Numerical calculations of the vertices have been carried out and asymptotic analyses performed for some of the more simplified vertices. The objective is to later extend the results to the EPRL vertex.
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