atyy said:
I don't think LQG has moved toward GFT. Rovelli seems cool towards it, he's pushing TQFT.
...
That's a nice sketch of where those various approaches are at present! I will explain what I meant about LQG moving in the direction of GFT.
The LQG Hilbertspace---the space of quantum states of geometry---is very basic to LQG. It used to be defined using a spacetime manifold.
Now Rovelli defines it essentially as a Lebesgue space of complex valued square integrable functions defined on a Cartesian product of SU(2).
So to me that looks like Group Field Theory, because the theory's basic Hilbertspace is functions defined on a group manifold. (You actually have to take quotient by an equivalence relation to get rid of some gauge redundancy but that's the idea.)
To me when I first saw it, it was a very refreshing and encouraging change. You eliminate the spacetime continuum from the picture and deal with pure information about geometric relationships embodied in your group manifold (simply the cartesian product of as many copies as you need of the underlying symmetry group).
And the Hilbertspace of squareintegrable functions, say on the unit interval, L
2[0,1] is one of the first nontrivial examples of a Hilbertspace you meet in undergraduate mathematics. It is a familiar and loved thing, like a teddybear is to a child.
So take a compact group manifold G, with Haar measure, and do the same thing.
L
2(G).
Haar measure on a compact group manifold is another familiar thing. It is the natural uniform probability measure that you get by using the group itself to spread out the measure evenly.
That is what the new LQG formulation's move toward GFT means to me. The group manifold with an old familiar measure and an old familiar Hilbertspace. The inner product is just the integral of the two functions multiplied together (one is complex conjugated). Probably what Hilbert himself had in mind when he axiomatized Hilberspaces.
It is not the FULL GFT, in your sense probably, but it puts a large part of the new LQG formulation conceptually within reach of an undergraduate math major. And probably a physics major as well. I like that, for sure.
The key to doing spacetime geometry that way, of course, is the
graph. Each finite graph (with its links and nodes) has such a Hilbertspace associated with it.
Because each link of the graph can have an element of the Lie group SU(2) living on it! That's why you need the cartesian product---of as many copies of the group as there are links in the graph.