Computing volumes in spin foams

Tuesday, Sep 25th

Benjamin Bahr, DESY
Title: 4-volume in spin foam models from knotted boundary graphs 
PDF of the talk (3M)
Audio+Slides of the talk (15M)

by Jorge Pullin, LSU

There is an approach to quantum mechanics known as the path integral approach. In it, one considers all possible classical trajectories, not only the ones satisfying the equations of motion and assigns probabilities to each of them using a formula. The probabilities are summed and that gives the quantum probability to go from an initial state to a final state. In loop quantum gravity the initial and final states are given by spin networks, which are graphs with intersections and "colors" (a number) assigned to each edge. The trajectories connecting initial and final states therefore resemble a "foam" and are given the names of spin foams.

In this talk it was shown how to compute volumes of polytopes (regions of space-time bounded by flat sides, a generalization to higher dimensions of polyhedra of 3d) in spin foam quantum gravity. The calculation has nice connections with knot theory, the branch of math that studies how curves entangle with each other.

One of the central elements of spin foams is the formula that assigns the probabilities, known as a "vertex". The construction in this talk gives ideas for extending the current candidates for vertices, including the possibility of adding a cosmological constant and suggests possible connections with Chern-Simons theories (a special type of field theories) and also with quantum groups.