My post on the “physics of decimals” a couple of weeks back caught physics blogger Luboš Motl’s attention, with predictable results. Mostly, this led to a rather unproductive debate about semantics, but he did bring up one thing that I think deserves some further clarification.
In my post, I asked you to imagine asking a genie for the full consequences of quantum field theory. Short of genie-based magic, is this the sort of thing I think it’s at all possible to know?
A Candle of Invocation? Sure, why not.
In a word, no.
The world is messy, not the sort of thing that tends to be described by neat exact solutions. That’s why we use approximations, and it’s why physicists can’t just step in and solve biology or psychology by deriving everything from first principles.
That said, the nice thing about approximations is that there’s often room for improvement. Sometimes this is quantitative, literally pushing to the next order of decimals, while sometimes it’s qualitative, viewing problems from a new perspective and attacking them from a new approach.
I’d like to give you some idea of the sorts of improvements I think are possible. I’ll focus on scattering amplitudes, since they’re my field. In order to be precise, I’ll be using technical terms here without much explanation; if you’re curious about something specific go ahead and ask in the comments. Finally, there are no implied time-scales here: I’ll be rating things based on whether I think they’re likely to eventually be understood, not on how long it will take us to get there.
Let’s begin with the most likely category:
Probably going to happen:
Mathematicians characterize the set of n-point cluster polylogarithms whose collinear limits are well-defined (n-1)-point cluster polylogarithms.
The seven-loop N=8 supergravity integrand is found, and the coefficient of its potential divergence is evaluated.
The dual Amplituhedron is found.
A general procedure is described for re-summing the L-loop coefficient of the Pentagon OPE for any L into a polylogarithmic form, at least at six points.
We figure out what the heck is up with the MHV-NMHV relation we found here.
Likely to happen, but there may be unforeseen complications:
N=8 supergravity is found to be finite at seven loops.
A symbol bootstrap becomes workable for QCD amplitudes at two or three loops, perhaps involving Landau singularities.
Something like a symbol bootstrap becomes workable for elliptic integrals, though it may only pass a “physicist” level of rigor.
Analogues to all of the work up to the actual Amplituhedron itself are performed for non-planar N=4 super Yang-Mills.
Quite possible, but I’m likely overoptimistic:
The space of n-point cluster polylogarithms whose collinear limits are well-defined (n-1)-point cluster polylogarithms that also obey the first entry condition and some number of final entry conditions turns out to be well-constrained enough that some all-loop all-point statements can be made, at least for MHV.
The enhanced cancellations observed in supergravity theories are understood, and used to provide a strong argument that N=8 supergravity is perturbatively finite.
All-multiplicity analytic QCD results at two loops, for at least the simpler helicity configurations.
The volume of the dual Amplituhedron is characterized by mathematicians and the connection to cluster polylogarithms is fully explored.
A non-planar Amplituhedron is found.
Less likely, but if all of the above happens I would not be all that surprised:
A way is found to double-copy the non-planar Amplituhedron to get an N=8 supergravity Amplituhedron.
The enhanced cancellations in N=8 supergravity turn out to be something “deep”: perhaps they are derivable from string theory, or provide a novel constraint on quantum gravity theories.
Various all-loop statements about the polylogarithms present in N=4 are used to make more restricted all-loop statements about QCD.
The Pentagon OPE is re-summed for finite coupling, if not into known functions than into a form that admits good numerics and various analytic manipulations. Alternatively, the sorts of functions that the Pentagon OPE can sum to are characterized and a bootstrap procedure becomes viable for them.
Irresponsible speculations, suited to public talks or grant applications:
The N=8 Amplituhedron leads to some sort of reformulation of space-time in a way that solves various quantum gravity paradoxes.
The sorts of mathematical objects found in the finite-coupling resummation of the Pentagon OPE lead to a revival of the original analytic S-matrix program, now with an actual chance to succeed.
Extremely unlikely:
Analytic all-loop QCD results.
Magical genie land:
Analytic finite coupling QCD results.
4 gravitons 
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The generalization of super-Yang-Mills amplitudology to QCD is certainly an interesting topic. But can you imagine a connection to nonperturbative QCD?
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As I mentioned, finite coupling QCD (and thus nonperturbative) seems to be in magical genie land. Maybe if the resurgence people manage a few unexpected miracles there’ll be a connection there.
One thing I could see happening on the amplitudes side is that we come to understand the functions that appear in finite coupling N=4 amplitudes well enough to say something about nonperturbative QCD amplitudes…but amplitudes probably just aren’t the right observables to be thinking about in QCD when the coupling is strong, at least not by themselves.
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Hi 4gravitons!
In light of the progress in Amplitudes over the past 9 years, has your classification of the difficulty of fulfiling any of these prophecies shifted? If so, could you tell us which ones? Which developments led to those shiftings?
Are there any trails of papers which have brought a wrecking ball to the Amplitudes assembly line?
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Heh, nice to check in on this, yeah!
Out of the “probably going to happen” section:
Mathematicians have not characterized the space of cluster polylogarithms with collinear limits that are also cluster polylogarithms. In part this is a more difficult task than I had appreciated, but I also think there hasn’t really been a serious attempt, it turns out to be a fair ways outside of how mathematicians are thinking about these things.
For N=8 supergravity, there’s no seven-loop integrand yet. Progress has slowed, but not stopped. I had an extended post about the state of the efforts back in 2018, and a brief update from the Amplitudes 2022 conference.
There’s still no dual amplituhedron, though Nima’s recent work is plausibly progress towards that kind of thing.
Resumming the OPE definitively stalled past one loop, with the infinite sums out of reach of the kinds of tricks I’ve managed to explore.
And the MHV-NMHV relation didn’t continue to the next loop order so it was just an extremely persistent coincidence. 😉
You can then follow most of those threads up the prediction tree, but there are a couple of exceptions where people found different paths.
There is now progress towards bootstrapping QCD amplitudes using Landau singularities. There were a lot of deep mathematical subtleties that I hadn’t anticipated, but it’s looking like people are overcoming them. They’re still not super close to having something that directly useful but it’s now actually looking like there’s a realistic path there.
An elliptic symbol bootstrap now also exists, though as I predicted I would argue it’s only “up to a physicist level of rigor”.
And all-multiplicity QCD results don’t exactly exist yet, but it seems like there might be a viable path towards them through the prescriptive unitarity approach of Jake Bourjaily’s group, combined with a better understanding of elliptic integrals.
And there is a revival of the old analytic S-matrix program, though it doesn’t have much to do with the Pentagon OPE and its goals are much more modest. 😉
As to the amplitudes assembly-line, it’s mostly churning along the same way it was ten years ago, just with shifts of focus (the biggest one being the huge surge of interest in gravitational waves). There are some attempts at disruptive ideas, with the ones that look coolest from my perspective being a couple of numerical techniques that can sometimes sidestep the need for integration, an aspiring rival to integration-by-parts called intersection theory, and the still vague possibility that AI does something ridiculous.
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Thank you for the update! Awesome!
Are you talking about Borinsky et al’s work?
I profusely apologise for the barrage of questions:
1. From what I’ve gathered, the S-matrix bootstrap program is about reverse-engineering the constraints on amplitudes. How does one usually classify these constraints?
2. Where is the elliptic symbology bootstrap program today, relative to the lifecycle of the polylog symbology program? Roughly 2010-era polylog symbology, perhaps?
3. Has Landau analysis been applied to elliptics?
4. I’ve heard amplitudes folk say that cluster algebras cannot describe three-particle amplitudes. Will three-particle amplitudes ever be amenable to an alphabet-like treatment?
5. In which projects would one reach for symbology? Where might one reach for surfaceology/curve integrals instead? Are they complementary? my question is really what are curve integrals good for
6. As a follow-up to mitchellporter’s question, have any results in the Resurgence program caught your attention recently?
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For the numerical techniques, Borinsky et al’s work is one line I had in mind, yeah. There’s another method that I’ve found impressive but don’t remember the name of that builds series expansions off of differential equations.
There’s some reverse-engineering, but mostly it’s a matter of using plausible constraints and trying to see what they buy you. Typically one can only impose some of the implications of the constraints, so advances in the field involve finding ways to impose them more completely. I don’t think there’s a nice classification of everything people apply. Causality and unitarity are the big ones, and there are more specific principles that partially encode those like positivity. There are also more “empirical” principles that people occasionally try to prove from more fundamental ones, like crossing symmetry. And you sometimes get the more “amplitudeology” principles like the double-copy showing up.
Elliptic symbology is in a different situation that makes it weird to compare. Early polylog symbology came to physics with much more cooperation from mathematicians: Goncharov was on the first paper. Elliptics started with some conversation, but not direct collaboration. So I think we’re plausibly at a 2010’s-level understanding, but with the caveat that we could be getting some much more fundamental things wrong, but that on the other hand we’re doing a lot more with it (more akin to what people were doing in 2011…ok, that’s not that far along!)
I think there is still some confusion about how to do this right. I tried to hash it out with Cristian Vergu a few years back. I think he understands it better now but I don’t remember them publishing an example yet. There are weird half-empirical principles that seem to be doing something related, called Schubert analysis.
So, three-particle amplitudes per se don’t make sense: you can’t have all three particles be physical on-shell states without going to complex kinematics. Usually when people talk about symbols for three-particle amplitudes they’re talking about something called a form factor, which is a slightly different beast. If you can’t use cluster algebras on them I’m not sure anything especially deep is going on, but regardless three-particle form-factors do have symbol alphabets you can bootstrap, you just don’t have a cluster algebra story for where they come from.
I think the selling point of surfaceology is supposed to be that you can get the whole amplitude for a realistic process in a form suitable for numerical evaluation (say with Borinsky and co’s methods). With symbology at the moment, you’re either bootstrapping something that’s not realistic, or simplifying something that is. So as a candidate practical technique, it’s supposed to be more applicable. I don’t think it has really been applied practically yet though, and like all Nima-stuff it’s mostly motivated by vague long-term stuff with quantum gravity and so on.
I keep up with resurgence a lot less than with amplitudes. I haven’t heard about anything huge, just incremental progress. But I can’t guarantee I would have heard of anything if there was.
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Are Calabi-Yau manifolds specific to the structure of conformal field theories? Do they also show up in QCD Feynman integral calculations?
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They show up pretty much everywhere, they’re actually harder to find in conformal theories because those theories have simpler amplitudes. There were some recently found in gravitational wave calculations for example, and there are QCD-EW amplitudes that will need them at two loops.
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Ah, and the numerical method I forgot the name of: AMFlow
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