Approaches to Quantum Gravity

Questions and answers 579would allow small finite dimensional vector spaces, which is what is involvedin the representation theory of SU(2). Indeed, q-bits are elementary reps ofSU(2). So I could imagine being pushed to go far enough to believe in one ora few q-bits per Planck volume, evolving in a way that requires one q-gate perPlanck time. But this does not allow the representation theory of non-compactgroups.As for your question, the answer to it is actually pretty straightforward,one has to compute the propagator for such states, under the evolution givenby the local moves. The mass matrix is then the inverse propagator at zeromomentum. To derive the propagator there are three steps. (1) Show that thebraids do propagate on spin networks by local moves. (This is shown for thethree valent moves in a paper in preparation by Jonathan Hackett and forthe four valent case in another paper we have in preparation with Wan.) (2)Show that if the spin network has an approximate translation symmetry thereare noise free subsystems spanned by identical braids in different positions,so that momentum is an approximate conserved quantity. (This is done inprinciple as it is a consequence of the Kribs and Markopoulous paper.) (3)In a given spin foam model, which gives amplitudes to the local moves, onethen computes the propagator analogously to how the graviton propagatorwas recently computed.I can also report that the extension of the results to the 4-valent case hasbeen accomplished, thanks mainly to some insights of Yidun Wan and is nowbeing written up. This is relevant for the Barrett–Crane and similar spin foammodels. We show that the braid preon states both propagate and interact witheach other in the 4-valent case.