First a word of advice. My impression is that condensed matter physics might have better career prospects. Also there is something of a boom going on in astrophysics. New instruments both ground-based and in orbit. (Auger, GLAST, MAGIC, Planck). Astrophysics offers information both about the universe and about fundamental physics gathered by instruments which exploit a new technologies and cost less than accelerators and colliders.
I would say that QG is something to know about, to keep in one's periferal vision. But not something to delve into as a US undergraduate. As far as I know, string theory has been something of a disappointment---output and citations began to dwindle as early as 2003. String faculty jobs are being cut back (according to a 2007 HEPAP report.) Admittedly, there are newer non-string approaches to QG which are on the upswing. (They have obscure names like CDT, QEG, LQC, NCG. Sorry about all the acronyms.) But a lot of the non-string QG scene is outside the US----Canada, UK, Europe. Loll's group, for instance, is mainly at Utrecht in Holland. Rovelli's group is at Marseille and several other French universities. The QEG people are at Mainz and Trieste. And the numbers of people involved are very small. Nonstring QG has at most several hundred people worldwide, counting all the different approaches.
I'm glad you express an interest in QG and give me a chance to talk about it, but don't want to divert you from the main business of a solid undergrad physics major, hopefully with specialization in some line like astrophysics and condensed matter or wherever the jobs are! Other people here at PF can provide knowledge and guidance on that score.
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Anyway here's why I like Jan Ambjorn and Renate Loll's CDT approach.
It is totally minimal. They don't make up a lot of extra dimensions and garbage. And they get it to run in the computer.
They get little universes to pop into existence, grow, shrink, and disappear. The universes are in a sense random because they arise and evolve according to a quantized General Relativity rule, and so they are different each time. The researchers use a Monte Carlo method which means they produce thousands of little universes at random, study and measure them, and take averages.
An interesting aspect of these simulated universes, their dimensionality, can be measured by generating one and chosing a point in it, and seeing how radius and volume are related around that point. The dimensionality is not predetermined. It is a quantum observable (like position and momentum of a particle). The dimensionality will normally depend on the scale at which you look. At larger scale the spatial dimension will be 3D and the spacetime dimension will be 4D, just as one would expect. At smaller scale the geometry is more like a foam or a fractal, rather than a smooth continuum. Indeed at smaller scale the dimensionality may not be an integer and may be considerably less than 3D.
Another way the dimensionality of these little universes can be measured is by stopping one and running a diffusion experiment or random walk in it, starting from some arbitrarily chosen point.
The method used in CDT (Loll's approach) is essentially the same as the Feynman path integral way of studying the motion of a particle. With the path integral, you approximate using piecewise linear paths----line segments----and make a weighted sum of all possible jagged paths. Then you let the length of the segments go to zero.
With CDT the triangulated spacetime geometry is analogous to the jagged piecewise linear path-----the zigzag chain of line segments.
Again there is a weighted sum. Again you let the size go to zero.
You may have heard that in the Feynman path integral picture you think of the particle as exploring all possible ways of getting from A to B. (the amplitude weighting is such that when it's all added up the really bad ways tend to cancel out).
what happens here is closely analogous. A spacetime is like a path through possible 3D geometries, so you can think of the universe as exploring all possible ways of getting from spatial geometry A to spatial geometry B. There is an amplitude weighting attached to each way. The initial and final geometry states A and B can be minimal---thats how it is in their Monte Carlo runs.
I hope i haven't made things more confusing! If I have, i apologize. Please just overlook the blunder and don't worry about what is poorly described or incomprehensible.
The main thing is that the Utrecht people have something working that begins to look like quantum gravity. That is, a quantum geometry which evolves according to a quantized version of General Relativity----(to speak in gibberish: a Feynmanesque path integral using the discrete Regge form of the Einstein Hilbert action
). The reality is simpler than it sounds: they are beginning to have a quantum gravity (in other words, quantum spacetime geometry) that you can run in a computer and study what happens with, at various scales. It begins to look like quantum gravity ought to look-----in my humble view.
Especially impressive to me is the foamy or fractally structure at Planck scale----smoothing out to a conventional continuum appearance at larger scale.
I think this is how, some 40 years ago, John Archibald Wheeler (Feynman's thesis advisor at Princeton) thought it ought to look.
That's good. And the minimality is good: the fact that their approach is barebones, no extra junk, no extra dimensions----just the absolute most simple straightforward path integral quantization of spacetime geometry.
Their using triangle building blocks corresponds to the series of line segments in Feynman's original path integral.
Keeping track of all the building blocks does lead to a lot of bookkeeping, but eventually you let their size go to zero. It was just a temporary way to finitize the problem.
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Well that sort of responds your #1 question. That is why someone who likes the CDT approach might like it. I could also tell you why I like Loop Quantum Cosmology. And someone else who has a favorite QG might contribute another part of the response.
I like LQC (loop quantum cosmology) because it gets rid of singularities and runs the model back before the big bang. That is another thing that quantizing spacetime geometry (i.e. quantizing gravity, or Gen Rel) can be expected to do. So it is another approach that seems to be working out. If you want more detail on that, let me know.
). Is a weighted sum sort of an average? You said the pick arbitrary point in the 'universe' and run the diffusion experiment but it seems to me (probably because I don't understand it right) that there would be infinite paths? 
