Occasionally, other physicists ask me what the goal of amplitudes research is. What’s it all about?
I want to give my usual answer: we’re calculating scattering amplitudes! We’re trying to compute them more efficiently, taking advantage of simplifications and using a big toolbox of different approaches, and…
Usually by this point in the conversation, it’s clear that this isn’t what they were asking.
When physicists ask me about the goal of amplitudes research, they’ve got a longer view in mind. Maybe they’ve seen a talk by Nima Arkani-Hamed, declaring that spacetime is doomed. Maybe they’ve seen papers arguing that everything we know about quantum field theory can be derived from a few simple rules. Maybe they’ve heard slogans, like “on-shell good, off-shell bad”. Maybe they’ve heard about the conjecture that N=8 supergravity is finite, or maybe they’ve just heard someone praise the field as “demoting the sacred cows like fields, Lagrangians, and gauge symmetry”.
Often, they’ve heard a little bit of all of these. Sometimes they’re excited, sometimes they’re skeptical, but either way, they’re usually more than a little confused. They’re asking how all of these statements fit into a larger story.
The glib answer is that they don’t. Amplitudes has always been a grab-bag of methods: different people with different backgrounds, united by their interest in a particular kind of calculation.
With that said, I think there is a shared philosophy, even if each of us approaches it a little differently. There is an overall principle that unites the amplituhedron and color-kinematics duality, the CHY string and bootstrap methods, BCFW and generalized unitarity.
If I had to describe that principle in one word, I’d call it minimality. Quantum field theory involves hugely complicated mathematical machinery: Lagrangians and path integrals, Feynman diagrams and gauge fixing. At the end of the day, if you want to answer a concrete question, you’re computing a few specific kinds of things: mostly, scattering amplitudes and correlation functions. Amplitudes tries to start from the other end, and ask what outputs of this process are allowed. The idea is to search for something minimal: a few principles that, when applied to a final answer in a particular form, specify it uniquely. The form in question varies: it can be a geometric picture like the amplituhedron, or a string-like worldsheet, or a constructive approach built up from three-particle amplitudes. The goal, in each case, is the same: to skip the usual machinery, and understand the allowed form for the answer.
From this principle, where do the slogans come from? How could minimality replace spacetime, or solve quantum gravity?
It can’t…if we stick to only matching quantum field theory. As long as each calculation matches one someone else could do with known theories, even if we’re more efficient, these minimal descriptions won’t really solve these kinds of big-picture mysteries.
The hope (and for the most part, it’s a long-term hope) is that we can go beyond that. By exploring minimal descriptions, the hope is that we will find not only known theories, but unknown ones as well, theories that weren’t expected in the old understanding of quantum field theory. The amplituhedron doesn’t need space-time, it might lead the way to a theory that doesn’t have space-time. If N=8 supergravity is finite, it could suggest new theories that are finite. The story repeats, with variations, whenever amplitudeologists explore the outlook of our field. If we know the minimal requirements for an amplitude, we could find amplitudes that nobody expected.
I’m not claiming we’re the only field like this: I feel like the conformal bootstrap could tell a similar story. And I’m not saying everyone thinks about our field this way: there’s a lot of deep mathematics in just calculating amplitudes, and it fascinated people long before the field caught on with the Princeton set.
But if you’re asking what the story is for amplitudes, the weird buzz you catch bits and pieces of and can’t quite put together…well, if there’s any unifying story, I think it’s this one.
Would it be fair to paraphrase that the long term hope is that there is a deep way of thinking about amplitudes that leads to much simpler math and new insights as well?
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That’s a true statement, it’s just a bit broader than I was going for here.
I would expect any community that studied scattering amplitudes to have that kind of long-term hope, of finding simpler math and new insights. What I’m trying to address in this post is a more specific shared philosophy. When physicists in other subfields ask me about amplitudes, they usually seem to think that we’re “up to something”, that there’s a message we’re pushing beyond the obvious things an amplitudes community would care about. And if there is such a message, I think it’s something like this “minimality” concept: find some minimal building blocks for a quantum field theory (with X property…) and classify what can be built from them, with the hope of seeing something new.
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You can amplify this somewhat: the usual approach through a path integral introduces a truly huge redundancy since you have to pick field coordinates to define the integrals (even perturbatively). None of this should matter to the output of the theory such as for scattering amplitudes, form factors, correlation functions, i.e. also for standard model predictions. If you don’t need it, why introduce it?
Illustrative example: start with a free field theory and do an arbitrarily complicated field transformation. The lagrangian is now a horrible mess. If you compute its scattering amplitudes, you magically find zero all the way through. Diagrammar contains some analysis if you are curious (look for “equivalence theorem”).