# A Non-Amplitudish Solution to an Amplitudish Problem

There was an interesting paper last week, claiming to solve a long-standing problem in my subfield.

I calculate what are called scattering amplitudes, formulas that tell us the chance that two particles scatter off each other. Formulas like these exist for theories like the strong nuclear force, called Yang-Mills theories, they also exist for the hypothetical graviton particles of gravity. One of the biggest insights in scattering amplitude research in the last few decades is that these two types of formulas are tied together: as we like to say, gravity is Yang-Mills squared.

A huge chunk of my subfield grew out of that insight. For one, it’s why some of us think we have something useful to say about colliding black holes. But while it’s been used in a dozen different ways, an important element was missing: the principle was never actually proven (at least, not in the way it’s been used).

Now, a group in the UK and the Czech Republic claims to have proven it.

I say “claims” not because I’m skeptical, but because without a fair bit more reading I don’t think I can judge this one. That’s because the group, and the approach they use, isn’t “amplitudish”. They aren’t doing what amplitudes researchers would do.

In the amplitudes subfield, we like to write things as much as possible in terms of measurable, “on-shell” particles. This is in contrast to the older approach that writes things instead in terms of more general quantum fields, with formulas called Lagrangians to describe theories. In part, we avoid the older Lagrangian framing to avoid redundancy: there are many different ways to write a Lagrangian for the exact same physics. We have another reason though, which might seem contradictory: we avoid Lagrangians to stay flexible. There are many ways to rewrite scattering amplitudes that make different properties manifest, and some of the strangest ones don’t seem to correspond to any Lagrangian at all.

If you’d asked me before last week, I’d say that “gravity is Yang-Mills squared” was in that category: something you couldn’t make manifest fully with just a Lagrangian, that you’d need some stranger magic to prove. If this paper is right, then that’s wrong: if you’re careful enough you can prove “gravity is Yang-Mills squared” in the old-school, Lagrangian way.

I’m curious how this is going to develop: what amplitudes people will think about it, what will happen as the experts chime in. For now, as mentioned, I’m reserving judgement, except to say “interesting if true”.

## 4 thoughts on “A Non-Amplitudish Solution to an Amplitudish Problem”

1. itaibn

I had a double-take when I looked at the abstract of this paper and saw mention of N=0 supergravity. Isn’t the N variable a measure of how much supersymmetry the theory has? Doesn’t N=0 supergravity mean just ordinary non-supersymmetric gravity?

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1. 4gravitons Post author

Heh. That’s a fun one. It looks like a ridiculous flourish, but the distinction actually means something. Supergravity isn’t just gravity with supersymmetric partners: it has two extra fields, the Kalb-Ramond field and the dilaton, that come in alongside gravity. When they say “N=0 supergravity”, they mean that those fields are still there in addition to the graviton, it would actually be inaccurate for them to call it “pure gravity”.

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2. Physicsnewbie

Hello,

How can you formulate the non-perturbative phenomena using only on-shell degrees of freedom? For instance, much of current research on theoretical physics involve instantons, magnetic monopoles, etc. Do these appear in the on-shell S-matrix bootstrap literature? Can you study the phases of gauge theory e. G. Quark confinement using your methods?

Thank you

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1. 4gravitons Post author

For the most part, you can’t. Most of the things people do with on-shell methods are explicitly perturbative, and nonperturbative phenomena are more or less out of reach.

I say “more or less” because there are a few ways out of this. One involves adding in solitons or the like on an ad hoc basis, as if they were just more on-shell states in your theory. There’s a bit of research along those lines, though not a ton.

The other thing to keep in mind is that, for those mainly focused on planar N=4 super Yang-Mills, these concerns don’t come up. The planar limit suppresses instantons, UV finiteness means you avoid renormalons, so in that particular case the perturbation series is expected to have a finite radius of convergence, and to capture the full nonperturbative behavior of the theory.

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