Scott Aaronson wrote a satirical, condescending parody of the news stories about the amplituhedron called
The Unitarihedron: The Jewel at the Heart of Quantum Computing
I had to laugh because I found it somewhat funny – and because the "revolutionary tone" Nima chooses when he talks about his and Jaroslav Trnka's wonderful new insight is detectably stronger than the tone I would choose. At the end of this essay, I will explain why.
But that's where my agreement with Scott Aaronson ends.
He wrote about the unitarihedron, diaperhedron, and other polyhedrons that are equally or more unifying, beautiful, revolutionary etc. as the amplituhedron that they contain. Aaronson's concepts are meant to overthrow all the paradigms in computer science but they are claimed to contain physics and the amplituhedron, too. You may imagine the stuff he probably wrote.
He seriously asked the readers what they thought about the reasons why he had posted that silly comical post. Your humble correspondent only offered some satirical explanations
I think that the goal of this funny text is to collect some money from a stupid enough billionaire who will think that you have achieved almost the same, if not more, than the players of the twistor minirevolution – so that you may buy more diapers.
Incidentally, there’s another solution to your diaper problem that I know and that was invented by a family I once met. They didn’t have enough money to buy the diapers for their baby, either. So instead, they bought a villa in Hollywood and wiped the baby’s buttocks by rubbing it against the grass on their garden.
but Wolfgang wrote the first meaningful answer, "physics envy", and I think he is right.
As the discussion continued, it became increasingly clear that Scott Aaronson feels jealous because his own field can't quite produce the same groundbreaking revolutions and universal, beautiful, elegant mathematical structures that are relevant "everywhere". So he invented his own fictitious polyhedrons to calm down his jealousy.
The Wolfgang-Scott exchange proceeded like this:
Wolfgang: Scott, I don’t think your re-branding effort will work. While the physicists find the jewels at the heart of our existence, all you ever do is figure out how “hard” everything is (but every traveling salesman knows that already). But all you have to offer are conjectures, while the physicists find beauty and simplicity (although it can be difficult for the laymen to see that).
Scott: ROTFL! Have you looked recently at beyond-Standard-Model theoretical physics? It’s a teetering tower of conjectures (which is not to say, of course, that that’s inherently bad, or that I can do better). However, one obvious difference is that the physicists don’t call them conjectures, as mathematicians or computer scientists would. Instead they call them amazing discoveries, striking dualities, remarkable coincidences, tantalizing hints … once again, lots of good PR lessons for us! As for beauty and simplicity, I dare say that those are the common currency of all mathematical fields. We all look for those, and are all thrilled when we occasionally find them.
I agree with Wolfgang, of course.
Complexity theorists are working on a totally legitimate piece of applied mathematics. Many of the results are rigorously established, many of them are useful, many of them may be applied in a wide variety of situations, I like some of the results, I know some of the results, I have been trained in some of the related skills, and I am not hiding at all that e.g. Scott Aaronson knows those things 50 times better than I do.
But nothing in the complexity theory and similar branches of maths can be at least remotely compared to the beauty, uniqueness, rigidity, and universal importance of the mathematical structures that were unmasked underneath the theories of modern physics. Modern physics is learning how Nature works and what is beating inside Her chest. Physicists are learning about the laws of Nature and absorb the "natural" beauty of mathematical structures. In contrast with that, complexity theory largely studies "man-made" problems and "man-made" solutions to these problems.
Man-made problems are much less unique, much more messy, and much less universally important. When it comes to the detailed formulation of problems, there's always a lot of room for deformations and modifications. As you apply these deformations and modifications, the character of the problem is continuously changing and there's no "universal invariant heart" inside this continuous family that would remain intact.
When it comes to the beauty and rigidity, it is nonsensical to sell the Standard Model as the "last bastion of elegance" – something that Scott seems to be doing – because the Standard Model isn't really among the most elegant theories. Many features of this effective theory look contrived and arbitrary. We surely know much more beautiful theories in physics, especially in the Beyond-the-Standard-Model physics.
I am also amazed that Aaronson misunderstands the difference between mathematics and natural sciences. In physics and other natural sciences, one is simply not supposed to prove things rigorously. Even if one proved some results rigorously, he wouldn't know with certainty that the structures and theories apply to the phenomena around us simply because there's no rigorous way to establish that the axioms or assumptions of the theorems are obeyed by Nature. That's why physicists aren't trying to be perfectly rigorous. It's not their business and it wouldn't be too useful, anyway. They have their own standards of rigor that are optimized to obtain deep results and minimize the potential for errors along the way.
But back to the beauty. Scott must completely misunderstand the character of the beauty in theoretical physics if he says that this beauty is "everywhere in mathematics". It's surely not. Some mathematical objects, concepts, relationships, and proofs are elegant, others are not. It's not quite possible to accurately quantify the elegance but all genuine experts recognize it when they see it. Elegant mathematical structures smoothly connect their parts, they can't be deformed, there are no useless parts in them that could be changed, and they are often indispensable in a wide range of situations. Moreover, the indispensability is never just a formality or a bookkeeping device.
A field whose bulk is all about the proof of inequalities (I mean inequalities saying that the number of operations needed to solve XY is greater than a function CD of the number of nodes etc.) – inequalities that can typically be improved and the progress is never final – just doesn't have and can't have the kind of mathematical beauty we know from physics. It's a collection of partial results that always depend on all the details we insert. The details are arbitrary and the "core structure" that would be independent of them is nearly non-existent.
One could offer tons of examples but take the newest one, the so-far officially unpublished amplituhedron by Arkani-Hamed and Trnka. It's not just some PR. It's an actual shape in a space whose volume, when calculated through a very simple volume form, precisely generates all the scattering amplitudes of the maximally supersymmetric gauge theory in four dimensions.
Those amplitudes aren't just some random messy assorted numbers. They have lots of amazing mathematical patterns, hidden symmetries, appearance of special functions with some special parameters, transcendentality or the lack of it. They obey lots of conditions whose origin we understand – linked to locality, unitarity, and the Yangian symmetry. They admit various seemingly inequivalent expansions using the Feynman diagrams and many other similar but different forms of expansions. All of them are completely rigid but all of them agree.
The volume of a totally well-defined polytope in a multi- or infinite-dimensional space reproduces all these amplitudes with everything we knew it. So the beauty of the previous mathematical relationships is guaranteed to sit somewhere inside. And what is essential is that the object whose beauty we are worshipping is a completely particular shape – at least for any fixed external data. It is not a bureaucratic document that achieves something that could also be achieved differently. It is a unique object "out there" in the real world.
What I am saying is that the amplituhedron exists like a particular object (some generalization of a dodecahedron, imagine something like that, or a skyscraper in higher dimensions whose construction depends on the properties of the external particles) while the diaperhedron or any other computer-science-based objects Aaronsons talks about don't exist as objects. They're nonsense. Even if he gave some meaning to these words, they wouldn't exist as objects, and if they existed as objects with fixed properties, these objects wouldn't be indispensable in any important, universal enough procedure.
So it's not just P.R. Nima Arkani-Hamed and Jaroslav Trnka have found something important that has "beef" in it while Scott Aaronson has found nothing of the sort.
I promised you to say why I think that Nima's way of talking about these insights carries more "revolutionary excitement" than what your humble correspondent would choose. We hear that the picture proves the death of the spacetime, the death of the unitarity, we're entering a completely new era. I am not quite getting the justification for these big claims.
What I see is that the methods to calculate the scattering amplitudes in the maximally supersymmetric gauge theory in four dimensions have developed to a new level of simplicity that is superior relatively to the others, that explains why particular terms (non-Feynman diagrams) were summed in the previous iterations of the program, and that gives us a totally new geometric visualization of the conditions that the amplitudes obey due to locality and unitarity. The picture makes the Yangian symmetry manifest which is great, too.
But otherwise, what I have seen seems consistent with the opinion that what they have found is a formula that results from a "change of variables". If you trust that Nima and Jaroslav are not lying to us, their claimed checks of some amplitudes up to six loops shows that their new simpler formula is right.
One should still ask Why is it right? In my opinion, it should be explained. It is a new duality of a sort. I can imagine that a relatively mundane derivation of this Arkani-Hamed-Trnka formula will be found – perhaps someone among us here will find it. It seems plausible to me that the amplituhedron is just a polytope living in a space of some degrees of freedom that may be defined on the Feynman diagrams or world sheets relevant for this most famous example of the AdS/CFT. The world sheet with the Feynman diagram drawn on it may perhaps be triangulated differently, new monodromies around the loops or other degrees of freedom may be found, and the whole path integral may be reorganized using these objects.
Where I really differ from Nima's interpretation is the status of the claims that the spacetime was shown not to be important. At the end, we want to interpret these amplitudes in a spacetime. That's why these particular polytopes are more important than many other polytopes: their volumes are relevant for a theory we knew thanks to the spacetime, a theory that was derived in a spacetime.
Moreover, we should ask: What is the space where the amplituhedron lives? Clearly, it's not the usual spacetime. It's not even the momentum space or the twistor space. It is some other auxiliary space because its dimension depends on the number of helicity flips (how far we are from the maximally helicity violating amplitudes). And as long as it is legitimate to call it an "auxiliary space" – because we haven't seen the space in any other physics application than the construction of the amplituhedron – we may also say that it is not fundamental.
It's interesting that the amplituhedron is a polytope of its own type. It is defined by inequalities. But they may result from some inequalities guaranteeing that some stringy vertices in a string diagram don't conflict with each other or something like that. The inequalities defining the amplituhedron are unusually complicated and structured but they're not the first inequalities that have appeared in the definition of intervals or regions of integration in important enough physics calculations.
Once you call the amplituhedron an "auxiliary space", you may also realize that physics has seen many other auxiliary spaces. Any integral is a "volume with some volume form". Any path integral, for example, is a "volume" of a structure in a particular auxiliary space, the configuration space for all the degrees of freedom in the spacetime. The volume form includes the exponential of the action and it's very different than one found in the amplituhedron. But in some sense, it is a technical difference.
The path integrals may often be rewritten as integrals over finite-dimensional auxiliary spaces. For example, multiloop diagrams in perturbative string theory are rewritten as the integrals over the finite-dimensional moduli spaces of Riemann surfaces. The integrands involve correlators of the vertex operators and the details look different. But I don't see any evidence that the integral computing the volume of the amplituhedron is something "qualitatively different" from the integral over the moduli space of Riemann surfaces in string theory, for example. These moduli spaces are also auxiliary spaces – we are not integrating over the spacetime or the momentum space, either. For some reason, people didn't hype this fact as the last nail in the spacetime's coffin. The fact that the space is auxiliary and hasn't been encountered before kind of makes the formula with it less groundbreaking, not more groundbreaking.
It's plausible that Nima disagrees with me about that point; this is suggested by one of his remarks during his talk at SUSY 13. He said we have learned all the dualities and relationships between things we knew but we finally need to learn something completely new we didn't learn at all. Well, this sounds like a call to spark a revolution. Quantum mechanics did something along these lines.
On the other hand, they are still calculating scattering amplitudes, just using a new formula. So what they have are insights about the same things we already knew; they are not really freshly discovered brand new objects analogous to quantum mechanics.
In the amlituhedron organization of the calculations, the unitarity and locality are derived (geometric) properties of the formulae for the amplitudes. Something else is "fundamental". But what is it? Well, the fundamental things are a simple volume form, a rule for the dimension of the amplituhedron's ambient space, and especially numerous inequalities (saying that the minors [determinants of submatrices] of many matrices etc. are positive).
Let's simplify the discussion and talk about the inequalities only. These inequalities play the role of the new physics principles that replace unitarity of locality. That's fine but I just don't understand in what sense they are "brand new principles to be universally important in physics" rather than "just some technical constraints needed to obtain a particular class of quantities through integrals". It just seems strange to me to promote some inequalities in a technical calculation to something equally physical and fundamental as locality or unitarity.
The inequalities underlying the amplituhedron have to be what they are for the locality and unitarity to hold – or, if our constraints are stronger, they have to be what they are to match the results of a particular known theory that was built from the assumptions of locality and unitarity. So in this sense, the unitarity and locality are still fundamental assumptions of the story. They're just being obscured a little bit, transformed into a less familiar condition.
In physics, we have some important universal principles like the principle of least action in classical physics; Feynman's principle that all histories should be summed over in quantum mechanics; locality; symmetries; unitarity, and so on. All these things are very physical because they ultimately talk about things we can observe. If they use auxiliary spaces and unobservable quantities, then those play a secondary role. So I have a trouble to classify inequalities defining a region in an auxiliary space – the amplituhedron – to be a similarly fundamental, physical principle. If someone told us a reason why these positive conditions must hold – and it wouldn't be the reason that we need to reproduce the theory we have picked partly because it was local and unitary – the situation would change. But as long as we are only given the system of inequalities as a result that was derived and we should believe, I just can't treat this system as a physical principle. For example, I don't know why such minors should generalize to any other theory than the maximally supersymmetric gauge theory that was studied. To be honest, I would bet that the inequalities won't keep their form. They will have to be updated to get other theories' amplitudes if that's possible at all. We don't know what the deformation will be. For that reason, no new fixed principle of physics has been found so far.
With these comments, I must still emphasize that I think that the amplituhedron development is extremely exciting and "supersedes" and "explains" a large fraction of the technical results in the twistor minirevolution literature. The new formulae seem extremely elegant and unified – much more elegant, robust, unique, universally applicable, and unified than Scott Aaronson's proof that the travelling salesman serving \(M\) families with \(N\) babies, using iWatches with a quantum computer he may choose to distrust in \(M/N\) percent of the time, and thinking about the diaperhedrons whenever he runs out of gasoline will need to buy less than \(M^{1.917}N^{1.63}(MN)!\) diapers for the babies and himself (that's how I understood Aaronson's revolutionary theory of the unitarihedron).
But if something is far more elegant and indispensable than Scott Aaronson's proofs, it still doesn't have to be a revolution comparable to the quantum revolution. I may be wrong but so far I don't see a revolution of a comparable magnitude although the results are extremely powerful and bring the 10-year-old research in this are to a brand new level.