SuperStrings vs. Quantum Loop Gravity

Two competing Theories of Everything based on the Planck Scale
Superstrings describing matter and energy, QLG describing space and time.

Seems like the two theories oughta dovetail into each other; wouldn’t that be nice?

Alas, the theories are incompatible.

IIRC, Superstrings MUST have extra curled-up dimensions, whereas QLG just dosen’t work in higher dimensions.

What other barriers to unification are there between these theories?

First of all, loop quantum gravity isn’t a theory of everything, it’s an attempt at finding a quantum theory of gravity—that is, a quantum theory that will reproduce the phenomenology of general relativity at the larger scale (or more properly, in the limit of large action). There have been some attempts to incorporate particles into LQG, but none have gained much traction to the best of my knowledge. So the first difference is one in scope: string theory aims to quantize gravity while unifying it with the rest of the forces and particles (the content of the standard model of elementary particles), while LQG has the more modest aim of finding a theory of quantum gravity.

Furthermore, they’re different from the outset: in string theory, you start basically with a theory of a (relativistic, that is, obeying special relativity) string moving through a background spacetime, then apply a process of quantization to that, and find that you get a theory that seems able to mimic theories of certain particles (bosons, i.e. particles with whole-integer spin), albeit (due to some technical constraints—essentially, the quantization ‘breaks’ a symmetry that the classical theory possesses, which however leads to an inconsistency of the theory in most cases) only in 26 dimensional spacetime.

Then there’s this thing called supersymmetry that you can add to the theory. Essentially, supersymmetry is a symmetry between bosonic (force-carrying) and fermionic (matter) particles, such that any boson gets a fermionic superpartner (and vice versa). That means you suddenly don’t just have a theory of bosons anymore, but also one capable of incorporating matter. Also, again for technical reasons, this becomes possible for 10 dimensional spacetime.

LQG takes a wholly different tack. First of all, you don’t start out with a theory of things in space (e.g. strings or particles), but rather, with a theory of the dynamics of space itself (Einstein’s general relativity). Essentially, what you do then is to reformulate that theory in a different language—basically, you change the dynamical variables of the theory from the thing that gives you distances between objects (the metric) from something that can be represented in terms of loops, whence the theory takes its name.

So a crucial difference here is that LQG is background independent: we don’t start out with some fixed spacetime, but the spacetime itself is the theory’s dynamical content. This is in line with Einstein’s general relativity, but in contrast to string theory, where you start with a fixed background; however, proponents of the latter would be quick to remind you that this is probably an artifact of the current formulation of the theory—ultimately, what we have is a so-called ‘perturbation series’, i.e. a series of approximations that are hoped to reproduce the full theory in the appropriate limit—however, nobody yet has a complete formulation of the latter.

To be fair, it should also be noted that the String Model was also not originally developed as a Theory of Everything. It was originally aiming to be a mere Grand Unified Theory, unifying the electroweak interaction with the strong interaction (i.e., what’s currently considered part of particle physics). But the early researchers noticed that any version of the model which reproduced everything that we know about particle physics also predicted the existence of a massless spin 2 fundamental particle, which matches what a graviton (the hypothetical fundamental particle of gravity) should look like.

The idea of a Grand Unified Theory is quite reasonable. The electric and weak interactions have already been fully unified, to an extent that you can’t really understand either one alone, but only both together. And there is solid experimental evidence that strongly suggests that the strong force (which we don’t understand nearly as well) could also be unified, or at least tied closer together with, them, if we could just figure out how. But outside of the String Model, there’s no real reason beyond wishful thinking to expect that gravity should be unifiable with the other three interactions. It must be possible to quantize gravity somehow, but there’s nothing that says that that quantization must connect in any way to the quantization of the other forces.