A friend of mine from St. John's College, who was recently accepted to a physics doctoral program at Penn State, asked me what my opinion of Loop Quantum Gravity is. I replied be email, and then I decided, why not tell the world!
Now, Loop Quantum Gravity is the main rival to String Theory as an attempt to quantize gravity, although it only commands about a tenth of the resources that String Theory does. The people who work on it tend to have more of a General Relativity background than a Particle Physics background, and this tends to influence what types of problems they are trying to solve.
Warning: Unlike my other physics posts, I have made no attempt to make my commentary here accessible to non-physics people. (Yes, that means every other time I wrote a physics post and nobody understood it, I was the one to blame for not making it accessible enough!)
Einstein's theory of general relativity is background free, meaning that it does not start with any absolute background space or time, but instead allows the spacetime geometry to be dynamically constructed from the evolution of the metric. A theory of quantum gravity ought to be similar---it ought to be expressed in a way which doesn't depend on the prior specification of any spacetime metric. I think this is really important, but no one really knows how to do this. There are many ideas, but they all have various difficulties.
In principle, I think the idea of LQG---to build spacetime out of a discrete, quantum structure---is a very elegant and moving idea. (I first got interested in quantum gravity by reading the online writings of John Baez, who used to work on LQG.) Also, the LQG people have a very beautiful quantization of space at one time, in terms of spin networks. Essentially, by doing a step-by-step quantization of GR at one time (minus the dynamics), making only a few arbitrary choices, they were able to obtain spin networks. I'm sure you [i.e. the friend I was writing to---AW] know what these are, but let me assure you that they are beautiful and have some deep connections to geometrical ideas.
The next step in the construction of LQG is to decide what the dynamics are. Technically, this is done either (A) by choosing a "Hamiltonian constraint" in parallel with the Hamiltonian formulation of GR, or (B) in the spin-foam formalism, by postulating some sort of sum over histories assigning an action to each spin foam. It is here which we encounter the major problem: There is no agreement over how to implement the dynamics! There are many ideas, but no consensus on what to do. Implementing dynamics seems to involve some arbitrary choices. Some of the proposed solutions seem to me obviously wrong (e.g. see Smolin's criticism of Thiemann's Hamiltonian constraint: arXiv:gr-qc/9609034). There is also a serious danger that by choosing the wrong dynamics, one breaks the diffeomorphism invariance of the theory. In the Hamiltonian approach this manifests itself in so-called "anomalies in the constraint algebra", while in the spin foam approach it is unclear whether the inner product obtained from the sum over histories really has the necessary gauge invariance. I summarized these problems in passing, with citations, in the
Introduction to this article of mine: arxiv:1201.2489.
Thus---even leaving aside the critical hard problem of whether and how a continuum spacetime can emerge from a discrete description (a problem aggravated by the fact that it is difficult to see how any discrete model of spacetime besides causal sets could possibly preserve Lorentz invariance, see arXiv:gr-qc/0605006)---I would say that LQG really doesn't exist yet as a well-defined theory. Unless you consider dynamics to be an unimportant part of a theory. And finding sensible dynamics is a really hard problem, perhaps impossible.
Yet, despite the lack of dynamics, there's no end of papers where people do specific applications, like count black hole entropy, or even attempt to do quantum cosmology (basically by truncating the theory to a finite number of degrees of freedom, and then quantizing those degrees of freedom in a way which is "loopy" in spirit). But all of these things are totally provisional until one can embed them in an actual theory with dynamics. People used to be really interested in solving these hard problems, but I feel like a lot of them have now given up and are seeking more limited goals. This is a shame, since I think progress can only come by facing the hard issues head on. And maybe by showing some flexibility in how the theory is formulated.
Once one has the dynamics, again one can say nothing about the real world until one has identified the correct vacuum state. An arbitrarily constructed "weave" state that happens to look like some Riemannian geometry doesn't cut it. You have to figure out how to identify the right vacuum state---the one with lowest energy (once you figure out how to define that!). Many deep questions here! I think most people in LQG are asking all the wrong questions.
One can put too much emphasis on quantizing gravity---really that's backwards, we need the classical theory to emerge from the quantum theory, not vice versa. When people calculate discrete area and volume spectra for spin network edges and vertices, they've got things backwards. These are just some operators at the Planck scale. The really interesting question is not, how much "area" is associated with each spin, but how many of each type of spin crosses a given area of the vacuum state (if such a thing even exists).
I despise the ignorant bigotry which most string theorists show towards LQG, even though LQG barely exists as a theory. Their contempt is undeserved. The LQG people are trying to do something genuinely harder---to reconstruct spacetime from first principles. We don't know how to formulate string theory except by means of strings propagating in some background spacetime, or via dualities like AdS/CFT. Since the theory has gravitons, with a diffeomorphism gauge symmetry, it's clear to me there has to be some background free formulation of string theory, but no one has any idea what this would look like. And most string theorists don't even understand why it is important.
Personally (and unexpectedly for me) I've found that as someone who studies black hole theormodynamics, I can interface better with string theorists than with LQG people---the ones who are really interested in fundamental concepts, like Don Marolf and others at UCSB, for example---even though I don't really consider myself a string theorist. This may be a bit of a conceit at this point, since I've now written multiple papers on AdS/CFT. My heart is more strongly devoted to the types of ideas LQG people explore, but my mind
recognizes that they really haven't made all that much progress.