Do the same thing you would with any other theory, and you get infinity. You get repeated infinities, an infinity of infinities. And while you could fix one or two infinities, fixing an infinite number requires giving up an infinity of possible predictions, so in the end your theory predicts nothing.
String theory fixes this with its own infinity, the infinite number of ways a string can vibrate. Because this infinity is organized and structured and well-understood, you’re left with a theory that is still at least capable of making predictions.
(Note that this is an independent question from whether string theory can make predictions for experiments in the real world. This is a much more “in-principle” statement: if we knew everything we might want to about physics, all the fields and particles and shapes of the extra dimensions, we could use string theory to make predictions. Even if we knew all of that, we still couldn’t make predictions from naive quantum gravity.)
Are there ways to fix the problem that don’t involve an infinity of vibrations? Or at least, to fix part of the problem?
That’s what Zvi Bern, John Joseph Carrasco, Henrik Johansson, and a growing cast of collaborators have been trying to find out.
They’re investigating N=8 supergravity, a theory that takes gravity and adds on a host of related particles. It’s one of the easiest theories to get from string theory, by curling up extra dimensions in a particularly simple way and ignoring higher-energy vibrations.
Bern, along with Lance Dixon and David Kosower, invented the generalized unitarity technique I talked about last week. Along with Carrasco and Johansson, he figured out another important trick: the idea that you can do calculations in gravity by squaring the appropriate part of calculations in Yang-Mills theory. For N=8 supergravity, the theory you need to square is my favorite theory, N=4 super Yang-Mills.
Using this, they started pushing forward, calculating approximations to greater and greater precision (more and more loops).
What they found, at each step, was that N=8 supergravity behaved better than expected. In fact, it behaved like N=4 super Yang-Mills.
N=4 super Yang-Mills is special, because in four dimensions (three space and one time, the dimensions we’re used to in daily life) there are no infinities to fix. In a world with more dimensions, though, you start getting infinities, and with more and more loops you need fewer and fewer dimensions to see them.
N=8 supergravity, unexpectedly, was giving infinities in the same dimensions that N=4 super Yang-Mills did (and no earlier). If it kept doing that, you might guess that it also had no infinities in four dimensions. You might wonder if, at least loop by loop, N=8 supergravity could be a way to fix quantum gravity without string theory.
Of course, you’d only really know if you could check in four dimensions.
If you want to check in four dimensions, though, you run into a problem. The fewer dimensions you’re looking at, the more loops you need before N=8 supergravity could possibly give infinity. In four dimensions, you need a forbidding seven loops of precision.
Still, Bern, Carrasco, and Johansson were up to the challenge. Along with Lance Dixon, David Kosower, and Radu Roiban, they looked at three loops, calculating an interaction of four gravitons, and the pattern continued. Four loops, and it was still going strong.
At around this time, I had just started grad school. My first project was a cumbersome numerical calculation. To keep me motivated, my advisor mentioned that the work I was doing would be good preparation for a much grander project: the calculation of whether the four-graviton interaction in N=8 supergravity diverges at seven loops. All I’d have to do was wait for Bern and collaborators to get there.
I named this blog “4 gravitons and a grad student”, and hoped I would get a chance to contribute.
And then something unexpected happened. They got stuck at five loops.
The method they were using, generalized unitarity, is an ansatz-based method. You start with a guess, then refine it. As such, the method is ultimately only as good as your guess.
Their guesses, in general, were pretty good. The trick they were using, squaring N=4 to get N=8, requires a certain type of guess: one in which the pieces they square have similar relationships to the different types of charge in Yang-Mills theory. There’s still an infinite number of guesses that can obey this, so they applied more restrictions, expectations based on other calculations, to get something more manageable. This worked at three loops, and worked (with a little extra thought) at four loops.
But at five loops they were stuck. They couldn’t find anything, with their restrictions, that gave the correct answer when “cut up” by generalized unitarity. And while they could drop some restrictions, if they dropped too many they’d end up with far too general a guess, something that could take months of computer time to solve.
So they stopped.
They did quite a bit of interesting work in the meantime. They found more theories they could square to get gravity theories, of more and more unusual types. They calculated infinities in other theories, and found surprises there too, other cases where infinities didn’t show up when they were “supposed” to. But for some time, the N=8 supergravity calculation was stalled.
And in the meantime, I went off in another direction, which long-time readers of this blog already know about.
Recently, though, they’ve broken the stall.
What they realized is that the condition on their guess, that the parts they square be related like Yang-Mills charges, wasn’t entirely necessary. Instead, they could start with a “bad” guess, and modify it, using the failure of those relations to fill in the missing pieces.
It looks like this is going to work.
We’re all at an amplitudes program right now in Santa Barbara. Walking through the halls of the KITP, I overhear conversations about five loops. They’re paring things down, honing their code, getting rid of the last few bugs, and checking their results.
They’re almost there, and it’s exciting. It looks like finally things are moving again, like the train to seven loops has once again left the station.
Increasingly, they’re beginning to understand the absent infinities, to see that they really are due to something unexpected and new.
N=8 supergravity isn’t going to be the next theory of everything. (For one, you can’t get chiral fermions out of it.) But if it really has no infinities at any loop, that tells us something about what a theory of quantum gravity is allowed to be, about the minimum necessary to at least make sense on a loop-by-loop level.
And that, I think, is worth being excited about.