Nima who just returned from the Perimeter Institute was excited about very reasonable discussions with Laurent Freidel who is a loop quantum gravity person.
Laurent has shown that the so-called DSR (double or deformed special relativity) may arise naturally in 3 dimensions.
The Harvard interpretation is that 3 dimensions are special; there are no gravitons; and moreover, there is an invariant mass scale - the maximal mass you can have to avoid a closure of space (deficit angles exceeding 2 pi) - and these things won't hold in 4 dimensions or above four.
Nevertheless, the basic story of Laurent is quite interesting. Take 3D gravity coupled to a scalar field PHI with a cubic coupling, and integrate out the gravitational field. What you obtain is an action for PHI only; it differs from the original PHI-part of the action by having a new kind of a "star product" instead of the original one. However, it is not a Moyal product but rather a new kind of product relevant for addition of momenta in DSR.
The rule is
- exp(iPx) * exp (iQx) = exp(iRx)
- R = P sqrt(1-L^2.P^2) + Q sqrt(1-L^2.Q^2) + L P x Q