When string theorists are out of luck, will Loop Quantum Gravity come to rescue?

Tl;dr: I don’t think they want rescuing.

String theorists and researchers working on loop quantum gravity (LQG) like to each point out how their own attempt to quantize gravity is better than the others’. In the end though, they’re both trying to achieve the same thing – consistently combining quantum field theory with gravity – and it is hard to pin down just exactly what makes strings and loops incompatible. Other than egos that is.

The obvious difference used to be that LQG works only in 4 dimensions, whereas string theory works only in 10 dimensions, and LQG doesn’t allow for supersymmetry, which is a consequence of quantizing strings. However, several years ago the LQG framework has been extended to higher dimensions, and they can now also include supergravity, so that objection is gone.

Then there’s the issue with Lorentz-invariance, which is respected in string theory, but its fate in LQG has been subject of much debate. As of recently though, some researchers working on LQG have argued that Lorentz-invariance, used as a constraint, leads to requirements on the particle interactions, which then have to become similar to some limits found in string theory. This should come as no surprise to string theorists who have been claiming for decades that there is one and only one way to combine all the known particle interactions...

Two doesn’t make a trend, but I have a third, which is a recent paper that appeared on the arxiv:

A note on quantum supergravity and AdS/CFT
Norbert Bodendorfer
arXiv:1509.02036

Bodendorfer argues in his paper that loop quantization might be useful for calculations in supergravity and thus relevant for the AdS/CFT duality.

This duality relates certain types of gauge theories – similar to those used in the standard model – with string theories. In the last decade, the duality has become exceedingly popular because it provides an alternative to calculations which are difficult or impossible in the gauge theory. The duality is normally used only in the limit where one has classical (super)gravity (λ to ∞) and an infinite number of color charges (N_c to ∞). This limit is reasonably well understood. Most string theorists however believe in the full conjecture, which is that the duality remains valid for all values of these parameters. The problem is though, if one does not work in this limit, it is darned hard to calculate anything.

A string theorist, they joke, is someone who thinks three is infinitely large. Being able to deal with a finite number of color charges is relevant for applications because the strong nuclear force has 3 colors only. If one keeps the size of the space-time fixed relative to the string length (which corresponds to fixed λ), a finite N_c however means taking into account string effects, and since the string coupling g_s ~ λ/N_c goes to infinity with λ when N_c remains finite, this is a badly understood limit.

In his paper, Bodendorfer looks at the limit of finite N_c and λ to infinity. It’s a clever limit in that it gets rid of the string excitations, and instead moves the problem of small color charges into the realm of super-quantum gravity. Loop quantum gravity is by design a non-perturbative quantization, so it seems ideally suited to investigate this parameter range where string theorists don’t know what to do. But it’s also a strange limit in that I don’t see how to get back the perturbative limit and classical gravity once one has pushed g_s to infinity. (If you have more insight than me, please leave a comment.)

In any case, the connection Bodendorfer makes in his paper is that the limit of N_c to ∞ can also be obtained in LQG by a suitable scaling of the spin network. In LQG one works with a graph that has a representation label, l. The graph describes space-time and this label enters the spectrum of the area operator, so that the average quantum of area increases with this label. When one keeps the network fixed, the limit of large l then blows up the area quanta and thus the whole space, which corresponds to the limit of N_c to infinity.

So far, so good. If LQG could now be used to calculate certain observables on the gravity side, then one could further employ the duality to obtain the corresponding observables in the gauge theory. The key question is though whether the loop-quantization actually reproduces the same limit that one would obtain in string theory. I am highly skeptical that this is indeed the case. Suppose it was. This would mean that LQG, like string theory, must have a dual description as a gauge theory still outside the classical limit in which they both agree (they better do). The supersymmetric version of LQG used here has the matter content of supergravity. But it is missing all the framework that in string theory eventually give rise to branes (stacks thereof) and compactifications, which seem so essential to obtain the duality to begin with.

And then there is the problem that in LQG it isn’t well understood how to get back classical gravity in the continuum limit, which Bodendorfer kind of assumes to be the case. If that doesn’t work, then we don’t even know whether in the classical limit the two descriptions actually agree.

Despite my skepticism, I think this is an important contribution. In lack of experimental guidance, the only way we can find out which theory of quantum gravity is the correct description of nature is to demonstrate that there is only one way to quantize gravity that reproduces the General Relativity and the Standard Model in the suitable limits while being UV-finite. Studying how the known approaches do or don’t relate to each other is a step to understanding whether one has any option in the quantization, or whether we do indeed already have enough data to uniquely identify the sought-after theory.

Summary: It’s good someone is thinking about this. Even better this someone isn’t me. For a theory that has only one parameter, it seems to have a lot of parameters.