A few weeks back, Quanta Magazine had an article about a new discovery in my field, called antipodal duality.
Some background: I’m a theoretical physicist, and I work on finding better ways to make predictions in particle physics. Folks in my field make these predictions with formulas called “scattering amplitudes” that encode the probability that particles bounce, or scatter, in particular ways. One trick we’ve found is that these formulas can often be written as “words” in a kind of “alphabet”. If we know the alphabet, we can make our formulas much simpler, or even guess formulas we could never have calculated any other way.
Quanta’s article describes how a few friends of mine (Lance Dixon, Ömer Gürdoğan, Andrew McLeod, and Matthias Wilhelm) noticed a weird pattern in two of these formulas, from two different calculations. If you flip the “words” around, back to front (an operation called the antipode), you go from a formula describing one collision of particles to a formula for totally different particles. Somehow, the two calculations are “dual”: two different-seeming descriptions that secretly mean the same thing.
Quanta quoted me for their article, and I was (pleasantly) baffled. See, the antipode was supposed to be useless. The mathematicians told us it was something the math allows us to do, like you’re allowed to order pineapple on pizza. But just like pineapple on pizza, we couldn’t imagine a situation where we actually wanted to do it.
What Quanta didn’t say was why we thought the antipode was useless. That’s a hard story to tell, one that wouldn’t fit in a piece like that.
It fits here, though. So in the rest of this post, I’d like to explain why flipping around words is such a strange, seemingly useless thing to do. It’s strange because it swaps two things that in physics we thought should be independent: branch cuts and derivatives, or particles and symmetries.
Let’s start with the first things in each pair: branch cuts, and particles.
The first few letters of our “word” tell us something mathematical, and they tell us something physical. Mathematically, they tell us ways that our formula can change suddenly, and discontinuously.
Take a logarithm, the inverse of 
Mostly, this complex logarithm still seems to be doing what it’s supposed to, changing in a nice slow way. But there is a weird “cut” in the graph for negative numbers: a sudden jump, from 
As physicists, we usually don’t like our formulas to make sudden changes. A change like this is an infinitely fast jump, and we don’t like infinities much either. But we do have one good use for a formula like this, because sometimes our formulas do change suddenly: when we have enough energy to make a new particle.
Imagine colliding two protons together, like at the LHC. Colliding particles doesn’t just break the protons into pieces: due to Einstein’s famous 
So the beginning of our “words” represent branch cuts, and particles. The end represents derivatives and symmetries.
Derivatives come from the land of calculus, a place spooky to those with traumatic math class memories. Derivatives shouldn’t be so spooky though. They’re just ways we measure change. If we have a formula that is smoothly changing as we change some input, we can describe that change with a derivative.
The ending of our “words” tell us what happens when we take a derivative. They tell us which ways our formulas can smoothly change, and what happens when they do.
In doing so, they tell us about something some physicists make sound spooky, called symmetries. Symmetries are changes we can make that don’t really change what’s important. For example, you could imagine lifting up the entire Large Hadron Collider and (carefully!) carrying it across the ocean, from France to the US. We’d expect that, once all the scared scientists return and turn it back on, it would start getting exactly the same results. Physics has “translation symmetry”: you can move, or “translate” an experiment, and the important stuff stays the same.
These symmetries are closely connected to derivatives. If changing something doesn’t change anything important, that should be reflected in our formulas: they shouldn’t change either, so their derivatives should be zero. If instead the symmetry isn’t quite true, if it’s what we call “broken”, then by knowing how it was “broken” we know what the derivative should be.
So branch cuts tell us about particles, derivatives tell us about symmetries. The weird thing about the antipode, the un-physical bizarre thing, is that it swaps them. It makes the particles of one calculation determine the symmetries of another.
(And lest you’ve heard about particles with symmetries, like gluons and SU(3)…this is a different kind of thing. I don’t have enough room to explain why here, but it’s completely unrelated.)
Why the heck does this duality exist?
A commenter on the last post asked me to speculate. I said there that I have no clue, and that’s most of the answer.
If I had to speculate, though, my answer might be disappointing.
Most of the things in physics we call “dualities” have fairly deep physical meanings, linked to twisting spacetime in complicated ways. AdS/CFT isn’t fully explained, but it seems to be related to something called the holographic principle, the idea that gravity ties together the inside of space with the boundary around it. T duality, an older concept in string theory, is explained: a consequence of how strings “see” the world in terms of things to wrap around and things to spin around. In my field, one of our favorite dualities links back to this as well, amplitude-Wilson loop duality linked to fermionic T-duality.
The antipode doesn’t twist spacetime, it twists the mathematics. And it may be it matters only because the mathematics is so constrained that it’s forced to happen.
The trick that Lance Dixon and co. used to discover antipodal duality is the same trick I used with Lance to calculate complicated scattering amplitudes. It relies on taking a general guess of words in the right “alphabet”, and constraining it: using mathematical and physical principles it must obey and throwing out every illegal answer until there’s only one answer left.
Currently, there are some hints that the principles used for the different calculations linked by antipodal duality are “antipodal mirrors” of each other: that different principles have the same implication when the duality “flips” them around. If so, then it could be this duality is in some sense just a coincidence: not a coincidence limited to a few calculations, but a coincidence limited to a few principles. Thought of in this way, it might not tell us a lot about other situations, it might not really be “deep”.
Of course, I could be wrong about this. It could be much more general, could mean much more. But in that context, I really have no clue what to speculate. The antipode is weird: it links things that really should not be physically linked. We’ll have to see what that actually means.
4 gravitons 
This is the kind of content I come here for. New discoveries, new mysteries, explained in simple enough terms that I can get a sense of what’s happening without worrying about not knowing any QM whatsoever.
The duality between two-to-four-gluons and two-gluons-to-gluon-and-Higgs sounds like some symmetry of the gluon (or strong force?) forces the Higgs to exist.
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This is something I should have covered but yeah, it’s not. The “two gluons goes to gluon and Higgs” doesn’t really have a Higgs in there. It’s a lot closer to “two gluons goes to gluon and two virtual gluons”. It’s just that in practice the main thing people use that kind of thing for at the LHC is by “plugging in” the virtual gluons to another diagram that produces a Higgs boson. So it gets summarized as “two gluons goes to gluon plus Higgs” because that’s what the calculation would be used for (were it in the Standard Model and not N=4 super Yang-Mills), but the actual calculation being compared in the duality doesn’t have a Higgs in it.
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What’s the source of the branch cut in this case then? You only have massless particles yet both parts of the antipode have some kind of discontinuity.
I will accept “these are vague analogies and you can’t expect to use them as building blocks for anything” as an answer 😉
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Amplitudes with massless particles have branch cuts when (combinations of momenta)^2 vanish (hence equal to the zero mass). So the branch cut starts at zero, and the relevant question is in which variables it does so in, which is determined by what diagrams you can draw.
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Doesn’t that mean life exists to resist gravity using the least amount of force possible? Meaning life exists to cool or heat the universe as needed.
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There’s no gravity involved in any of this, let alone life, so I think you’re confused.
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Apologies, I see I have not shown my work.
Let’s ignore life. Say reality exists against gravity because it must. NASA is currently measuring dark energy from black holes based on the same theories you are describing here. I want to understand how the universal spiral, gravity, our fingertips, rings on a tree are related. They are related because reality must spiral to resist gravity. If you assume DNA is also effected by the gravitational forces there could be something there.
Theory: Black holes exist to hold reality together – relative to the gravitational pull of the original big bang. Assuming the inverse is effected on a subatomic level than so could DNA itself. I know science do not like to involve dreams but there is a connection there, specifically the handoff between autonomy and nonautonomy. Relating to brain waves and the golden ratio. https://www.researchgate.net/publication/222143648_The_golden_mean_as_clock_cycle_of_brain_waves
Experiment: I will try to give a very simple example. I have a bernese mountain dog who’s tail is curly. All vets have said this is because my dog’s DNA, simply put he was always destined to have a curly tail. If let’s say I try a new form of “dog yoga” and his diaphragm starts to redistribute to new areas of the body. Let’s say he starts to feel good after a few weeks. Let’s say his once curly tail is not so curly. Let’s say after a few years of PT – the dogs tail goes straight and the dog feels better than he has in the past – no longer limping – is jumping around like puppy. Assuming the end result in the dog’s tail is straight – would that not mean I have altered the dog’s DNA?
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This still has nothing to do with the topic of this post. Putting unrelated comments on a page to promote your own work is a form of spam, and I don’t permit spam on this blog.
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Thanks for the post and the clarifications.
One thing that worries me about dualities is what you mentioned in the last paragraphs, i.e. how far one can go with ( some of ) these dualities, or the domain of their validity.
The holographic duality that works well for Ads/ CFT , does not seem to make much sense for an asymptotically deSitter universe like ours, or for other “bubble universe” cosmological models, or ” bag of gold” kind of geometries, or the expanding volume of non extremal black holes that could be ” confined” inside a horizon that has the same area , approximately, for a long time, etc…
So , are these dualities part of more fundamental, or deeper principles, or are they appear only in certain specific situations ( that have certain symmetries e.g.)?
Most people that are actively working on these are seem to be convinced that dualities (like the Holographic hypothesis ) have the status of a fundamental principle, but others ( that are in the minority), are not thinking so..
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I think the results of Princeton’s Global Consciousness Project, which found a found a trillion-to-one odds ratio that collective thought can affect the output of a global network of quantum random number generators, provides a compelling hypothesis for why the antipodal duality exists: https://avshrikumar.medium.com/the-global-princeton-experiment-that-found-trillion-to-1-odds-that-a-collective-consciousness-e94c911f79f8
Quoting from the article:
“””There is a very interesting implication in the fact that an XOR mask is applied in the GCP project, because the Global Consciousness Project proved the existence of correlations between the final RNG output, not with the raw RNG output. There is only one way to explain how this happened: the quantum tunneling processes that provided the raw bitstream were conditioned on the XOR mask that would be applied to it. In other words, a constraint was applied on the physical processes behind the quantum tunneling such that RNGs at different locations would produce similar output (after XORing with a bitmask). Note that this is the same type of constraint that applies in quantum entanglement: a constraint on the similarity between two observations. The “hidden variables” of quantum mechanics don’t specify individual measurements; they specify constraints on the outcomes.
In other words, these results reconcile free will with determinism: “fate” or deterministic outcomes are authored by our free will/“consciousness”, which retro-causally influences events so that particular coincidences occur. By extension, could we argue that the very laws of physics are a retro-causal explanation for the universe designed by the “free will” of the collective consciousness? I wouldn’t be surprised if that is the reason for the antipodal duality: the laws of physics must be set up to allow coincidences to occur, and this manifests as a duality between seemingly unrelated aspects of physics.”””
Put that in your pipe and smoke it 😉
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Ok, so I should preface this by saying that I don’t always let comments like this through, because some people use them as a form of spam: they find every post they can that can be loosely related to some pet topic, and then put comments on them every time they find them. I don’t think you’re doing that, I think you genuinely thought that this duality could be an example of the coincidences that you’ve heard about in other contexts.
It isn’t, though, because the kinds of coincidences that the proposals you’re describing, in order to have relevance to quantum mechanics, have to be coincidences of a very particular type. Those types of coincidences are already built into the rules of quantum field theory, and we know what they look like (for a related example, see the results described in this other post), Mathematically speaking, they’re actually pretty simple. The antipode, by contrast, is quite mathematically complicated. It’s not the sort of mathematical operation that appears in the coincidences of quantum mechanics, that’s part of why it’s so hard to understand what kind of explanation it could possibly have.
(More generally, coincidences you hear about on this blog will generally not be that type of coincidence, precisely because that type of coincidences is built in to the rules of quantum field theory in a fairly straightforward way already. So I’d advise you not to do more “this seems related to consciousness!” comments unless you have a better reason to think it really is related, otherwise I might actually think you’re trying to spam.)
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Hey Matt,
I couldn’t help but notice that the logarithm plot looks awfully similar to the region around the critical point of a temperature-pressure phase diagram. Would there be an insightful analogy in either direction?
Might the antipodal duality tell us something interesting about the transition between a fluid’s liquid and gaseous phase?
Could a stream of insights flow from the even more recent developments around the antipode self-duality?
Perhaps phase transitions would reveal something interesting about particles? I suspect, though this is a topic that has already been thoroughly explored.
Kind regards from a curious spectator
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There is indeed a connection between the mathematical thing happening in that plot (called a branch cut) and the critical points and phase transitions of phase diagrams. If you plot a function across that phase diagram, it will often have a branch point precisely on the critical point. Here’s a stack exchange thread with more discussion about this, and to what extent the analogy works.
I don’t think antipodal duality/self-duality would say much about this though, because these aren’t really that type of system…there’s not really thermodynamics going on in the same way, these are interactions of single particles. But conformal field theories come up in both contexts, so maybe there’s a connection I’m not thinking of.
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