It's a good question and that's a helpful reply. Thanks to Curt and Tech! I've nothing important to add but will just amplify what Tech said.
We've heard so much in the popular media about "TOE" as a goal that there is a widepread impression that "final theory" is the overriding goal of today's physics.
TOE and Final Theory might actually be reasonable and relevant program goals or they might be buzzwords. Same with "unification of forces". I'll try to say something about that later.
The most important thing to get across, though, is that the Loop program's goal is different.
It is more like "First let's get the geometry right!"
For many years physics has been done on fixed geometry. Quantum field theory is usually constructed on flat 4D spacetime (of special relativity).
Another common statement of the program's goal is to learn how to construct a "background independent QFT". In Loop context, background independent means "no prior geometry". So the idea is to be able to do
Quantum Field Theory with no prior geometry specified.
One way that might be realized is to replace the spacetime continuum by a cell-complex or foam (the analogous thing to a graph in one higher dimension) and label the foam with both geometric information and matterfield quantum numbers. The graph or cell-complex is a way of representing the FINITE measurements we can imagine making which nail down the finite number of (geometry and matter) degrees of freedom which are in principle controllable, or that we can say something about, as in an experiment. Our information is finite so we truncate to a finite number of degrees of freedom in the way we represent it mathematically.
Such a thing can be thought of as a geometry+matter Feynman diagram. Where not only particles interact but also geometry interacts. Loop geometry can be formulated as a handful of Feynman rules for calculating amplitudes, and these rules then boil down to some integrals
(
http://arxiv.org/abs/1010.1939 )
So far the Feynman rules involve geometry only. The program is
incomplete in that sense. Last year a preliminary paper on putting fermions into the spinfoam picture appeared but there is still a ways to go in that direction.
The best overall presentation of the current status of Loop program is a February 2011 paper
http://arxiv.org/abs/1102.3660
It has an abbreviated treatment of the 1010.1939 stuff in an appendix---the interpretation of spinfoam as Feynman diagram.
Actually 1102.3660 is remarkably complete. It does a good job of describing and motivating and summarizing results so far in only about 20 pages. Plus a few pages of appendix.
The most interesting thing to me (as retired mathematician turned physics-watcher) is the potentially revolutionary impact of replacing the manifold with a different representation of spacetime. For roughly 200 years physics has been done on manifolds with fixed geometry.
Very often the geometry was flat. Or if not flat it was at least a prior fixed curvature setup.
But GR says that the geometry is live and interactive. It takes part in the action.
And QM says we can't say anything about geometry beyond what we can imagine measuring. Reality responds to measurement. And there are tradeoffs. Areas, angles and volumes may resist being once-for-all determined just like position momentum and spin do.
A manifold is uniformly the same down to infinitesimal scale and that might be wrong. If we cannot measure geometry past some point then we should not presume continuity past that point. The mathematical representation of spacetime should reflect what we measure or at least in principle infer and say something about.
So to me it seems that the Loop program is not even in the "TOE" ballpark. It is groping for a new idea of spacetime!
When you have a new idea of spacetime, you might build a TOE in it that turns out differently from any TOE you would construct in the old version of spacetime (the continuum with fixed prior geometry that we have been assuming for 200 years).
My personal feeling is I don't want to worry about "final theories". I'm interested in this new idea of the geometry that theories can be based on. Particularly since right now quantum geometry seems to be where the rapid progress is. The Loop program even though it involves only a couple of hundred people is evolving fast. Elsewhere it looks as if much of the theory-world is stalled.
Again that article 1102.3660 gives an idea of the rate of development, and the crucial new results just in the past 2 or 3 years.
One could say that the program's underlying attitude is that rather than unifying "forces" one should get a new idea of geometry and unify geometry with matter. It is a different idea of what one should unify.
The program's target areas for new understanding of nature are
A) the big bang singularity and ensuing early universe
B) black hole singularities
C) geometry at very small scale (or maybe it is geometry at high energy density--not sure which...)
And the program is still very incomplete! Again 1102.3660 is a good source. It has a list of open problems---some 17 problems if I remember right. It also deals frankly with unresolved issues and uncertainties faced by the Loop program.
