Loops and Strings and Stuff

If you tie your laces, loops and strings might seem like parts of the same equation, but when it comes to quantum gravity they don’t have much in common. String Theory and Loop Quantum Gravity, both attempts to consistently combine Einstein’s General Relativity with quantum theory, rest on entirely different premises.

String theory posits that everything, including the quanta of the gravitational field, is made up of vibrating strings which are characterized by nothing but their tension. Loop Quantum Gravity is a way to quantize gravity while staying as closely as possible to the quantization methods which have been successful with the other interactions.

The mathematical realization of the two theories is completely different too. The former builds on dynamically interacting strings which give rise to higher dimensional membranes, and leads to a remarkably complex theoretical construct that might or might not actually describe the quantum properties of space and time in our universe. The latter divides up space-time into spacelike slices, and then further chops up the slices into discrete chunks to which quantum properties are assigned. This might or might not describe the quantum properties of space and time in our universe...

String theory and Loop Quantum Gravity also differ in their ambition. While String Theory is meant to be a theory for gravity and all the other interactions – a “Theory of Everything” – Loop Quantum Gravity merely aims at finding a quantum version of gravity, leaving aside the quantum properties of matter.

Needless to say, each side claims their approach is the better one. String theorists argue that taking into account all we know about the other interactions provides additional guidance. Researchers in Loop Quantum Gravity emphasize their modest and minimalist approach that carries on the formerly used quantization methods in the most conservative way possible.

They’ve been arguing for 3 decades now, but maybe there’s an end in sight.

In a little noted paper out last year, Jorge Pullin and Rudolfo Gambini argue that taking into account the interaction of matter on a loop-quantized space-time forces one to use a type of interaction that is very similar to that also found in effective models of string interactions.

The reason is Lorentz-invariance, the symmetry of special relativity. The problem with the quantization in Loop Quantum Gravity comes from the difficulty of making anything discrete Lorentz-invariant and thus compatible with special relativity and, ultimately, general relativity. The splitting of space-time into slices is not a priori a problem as long as you don’t introduce any particular length scale on the resulting slices. Once you do that, you’re stuck with a particular slicing, thus ruining Lorentz-invariance. And if you fix the size of a loop, or the length of a link in a network, that’s exactly what happens.

There has been twenty years of debate whether or not the fate of Lorentz-invariance in Loop Quantum Gravity is really problematic, because it isn’t so clear just exactly how it would make itself noticeable in observations as long as you are dealing with the gravitational sector only. But once you start putting matter on the now quantized space, you have something to calculate.

Pullin and Gambini – both from the field of LQG it must be mentioned! – argue that the Lorentz-invariance violation inevitably creeps into the matter sector if one uses local quantum field theory on the loop quantized space. But that violation of Lorentz-invariance in the matter sector would be in conflict with experiment, so that can’t be correct. Instead they suggest that this problem can be circumvented by using an interaction that is non-local in a particular way, which serves to suppress unwanted contributions that spoil Lorentz-invariance. This non-locality is similar to the non-locality that one finds in low-energy string scattering, where the non-locality is a consequence of the extension of the strings. They write:

“It should be noted that this is the first instance in which loop quantum gravity imposes restrictions on the matter content of the theory. Up to now loop quantum gravity, in contrast to supergravity or string theory, did not appear to impose any restrictions on matter. Here we are seeing that in order to be consistent with Lorentz invariance at small energies, limitations on the types of interactions that can be considered arise.”

In a nutshell it means that they’re acknowledging they have a problem and that the only way to solve it is to inch closer to string theory.

But let me extrapolate their paper, if you allow. It doesn’t stop at the matter sector of course, because if one doesn’t assume a fixed background like they do in the paper one should also have gravitons and these need to have an interaction too. This interaction will suffer from the same problem, unless you cure it by the same means. Consequently, you will in the end have to modify the quantization procedure for gravity itself. And while I’m at it anyway, I think a good way to remedy the problem would be to not force the loops to have a fixed length, but to make them dynamical and give them a tension...

I’ll stop here because I know just enough of both string theory and loop quantum gravity to realize that technically this doesn’t make a lot of sense (among many other things because you don’t quantize loops, they are the quantization), and I have no idea how to make this formally correct. All I want to say is that after thirty years maybe something is finally starting to happen.

Should this come as a surprise?

It shouldn’t if you’ve read my review on Minimal Length Scale Scenarios for Quantum Gravity. As I argued in this review, there aren't many ways you can consistently introduce a minimal length scale into quantum field theory as a low-energy effective approximation. And pretty much the only way you can consistently do it is using particular types of non-local Lagrangians (infinite series, no truncation!) that introduce exponential suppression factors. If you have a theory in which a minimal length appears in any other way, for example by means of deformations of the Poincaré algebra (once argued to arise in Loop Quantum Gravity, now ailing on life-support), you get yourself into deep shit (been there, done that, still got the smell in my nose).

Does that mean that the string is the thing? No, because this doesn’t actually tell you anything specific about the UV completion, except that it must have a well-behaved type of non-local interaction that Loop Quantum Gravity doesn’t seem to bring, or at least it isn’t presently understood how it would. Either way, I find this an interesting development.

The great benefit of writing a blog is that I’m not required to contact “researchers not involved in the study” and ask for an “outside opinion.” It’s also entirely superfluous because I can just tell you myself that the String Theorist said “well, it’s about time” and the Loop Quantum Gravity person said “that’s very controversial and actually there is also this paper and that approach which says something different.” Good thing you have me to be plainly unapologetically annoying ;) My pleasure.