Theoretical Physics

A Jewel at the Heart of Quantum Physics

Artist’s rendering of the amplituhedron, a newly discovered mathematical object resembling a multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated — the probabilities of outcomes of particle interactions.

Illustration by Andy Gilmore

Artist’s rendering of the amplituhedron, a newly discovered mathematical object resembling a multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated — the probabilities of outcomes of particle interactions.

Physicists have discovered a jewel-like geometric object that dramatically simplifies calculations of particle interactions and challenges the notion that space and time are fundamental components of reality.

“This is completely new and very much simpler than anything that has been done before,” said Andrew Hodges, a mathematical physicist at Oxford University who has been following the work.

The revelation that particle interactions, the most basic events in nature, may be consequences of geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-term expression.

“The degree of efficiency is mind-boggling,” said Jacob Bourjaily, a theoretical physicist at Harvard University and one of the researchers who developed the new idea. “You can easily do, on paper, computations that were infeasible even with a computer before.”

The new geometric version of quantum field theory could also facilitate the search for a theory of quantum gravity that would seamlessly connect the large- and small-scale pictures of the universe. Attempts thus far to incorporate gravity into the laws of physics at the quantum scale have run up against nonsensical infinities and deep paradoxes. The amplituhedron, or a similar geometric object, could help by removing two deeply rooted principles of physics: locality and unitarity.

“Both are hard-wired in the usual way we think about things,” said Nima Arkani-Hamed, a professor of physics at the Institute for Advanced Study in Princeton, N.J., and the lead author of the new work, which he is presenting in talks and in a forthcoming paper. “Both are suspect.”

Locality is the notion that particles can interact only from adjoining positions in space and time. And unitarity holds that the probabilities of all possible outcomes of a quantum mechanical interaction must add up to one. The concepts are the central pillars of quantum field theory in its original form, but in certain situations involving gravity, both break down, suggesting neither is a fundamental aspect of nature.

In keeping with this idea, the new geometric approach to particle interactions removes locality and unitarity from its starting assumptions. The amplituhedron is not built out of space-time and probabilities; these properties merely arise as consequences of the jewel’s geometry. The usual picture of space and time, and particles moving around in them, is a construct.

“It’s a better formulation that makes you think about everything in a completely different way,” said David Skinner, a theoretical physicist at Cambridge University.

The amplituhedron itself does not describe gravity. But Arkani-Hamed and his collaborators think there might be a related geometric object that does. Its properties would make it clear why particles appear to exist, and why they appear to move in three dimensions of space and to change over time.

Because “we know that ultimately, we need to find a theory that doesn’t have” unitarity and locality, Bourjaily said, “it’s a starting point to ultimately describing a quantum theory of gravity.”

Clunky Machinery

The amplituhedron looks like an intricate, multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated, “scattering amplitudes,” which represent the likelihood that a certain set of particles will turn into certain other particles upon colliding. These numbers are what particle physicists calculate and test to high precision at particle accelerators like the Large Hadron Collider in Switzerland.

United States Postal Service

The iconic 20th century physicist Richard Feynman invented a method for calculating probabilities of particle interactions using depictions of all the different ways an interaction could occur. Examples of “Feynman diagrams” were included on a 2005 postage stamp honoring Feynman.

The 60-year-old method for calculating scattering amplitudes — a major innovation at the time — was pioneered by the Nobel Prize-winning physicist Richard Feynman. He sketched line drawings of all the ways a scattering process could occur and then summed the likelihoods of the different drawings. The simplest Feynman diagrams look like trees: The particles involved in a collision come together like roots, and the particles that result shoot out like branches. More complicated diagrams have loops, where colliding particles turn into unobservable “virtual particles” that interact with each other before branching out as real final products. There are diagrams with one loop, two loops, three loops and so on — increasingly baroque iterations of the scattering process that contribute progressively less to its total amplitude. Virtual particles are never observed in nature, but they were considered mathematically necessary for unitarity — the requirement that probabilities sum to one.

“The number of Feynman diagrams is so explosively large that even computations of really simple processes weren’t done until the age of computers,” Bourjaily said. A seemingly simple event, such as two subatomic particles called gluons colliding to produce four less energetic gluons (which happens billions of times a second during collisions at the Large Hadron Collider), involves 220 diagrams, which collectively contribute thousands of terms to the calculation of the scattering amplitude.

In 1986, it became apparent that Feynman’s apparatus was a Rube Goldberg machine.

To prepare for the construction of the Superconducting Super Collider in Texas (a project that was later canceled), theorists wanted to calculate the scattering amplitudes of known particle interactions to establish a background against which interesting or exotic signals would stand out. But even 2-gluon to 4-gluon processes were so complex, a group of physicists had written two years earlier, “that they may not be evaluated in the foreseeable future.”

Stephen Parke and Tomasz Taylor, theorists at Fermi National Accelerator Laboratory in Illinois, took that statement as a challenge. Using a few mathematical tricks, they managed to simplify the 2-gluon to 4-gluon amplitude calculation from several billion terms to a 9-page-long formula, which a 1980s supercomputer could handle. Then, based on a pattern they observed in the scattering amplitudes of other gluon interactions, Parke and Taylor guessed a simple one-term expression for the amplitude. It was, the computer verified, equivalent to the 9-page formula. In other words, the traditional machinery of quantum field theory, involving hundreds of Feynman diagrams worth thousands of mathematical terms, was obfuscating something much simpler. As Bourjaily put it: “Why are you summing up millions of things when the answer is just one function?”

“We knew at the time that we had an important result,” Parke said. “We knew it instantly. But what to do with it?”

The Amplituhedron

The message of Parke and Taylor’s single-term result took decades to interpret. “That one-term, beautiful little function was like a beacon for the next 30 years,” Bourjaily said. It “really started this revolution.”

Arkani-Hamed et al.

Twistor diagrams depicting an interaction between six gluons, in the cases where two (left) and four (right) of the particles have negative helicity, a property similar to spin. The diagrams can be used to derive a simple formula for the 6-gluon scattering amplitude.

In the mid-2000s, more patterns emerged in the scattering amplitudes of particle interactions, repeatedly hinting at an underlying, coherent mathematical structure behind quantum field theory. Most important was a set of formulas called the BCFW recursion relations, named for Ruth Britto, Freddy Cachazo, Bo Feng and Edward Witten. Instead of describing scattering processes in terms of familiar variables like position and time and depicting them in thousands of Feynman diagrams, the BCFW relations are best couched in terms of strange variables called “twistors,” and particle interactions can be captured in a handful of associated twistor diagrams. The relations gained rapid adoption as tools for computing scattering amplitudes relevant to experiments, such as collisions at the Large Hadron Collider. But their simplicity was mysterious.

“The terms in these BCFW relations were coming from a different world, and we wanted to understand what that world was,” Arkani-Hamed said. “That’s what drew me into the subject five years ago.”

With the help of leading mathematicians such as Pierre Deligne, Arkani-Hamed and his collaborators discovered that the recursion relations and associated twistor diagrams corresponded to a well-known geometric object. In fact, as detailed in a paper posted to in December by Arkani-Hamed, Bourjaily, Cachazo, Alexander Goncharov, Alexander Postnikov and Jaroslav Trnka, the twistor diagrams gave instructions for calculating the volume of pieces of this object, called the positive Grassmannian.

Named for Hermann Grassmann, a 19th-century German linguist and mathematician who studied its properties, “the positive Grassmannian is the slightly more grown-up cousin of the inside of a triangle,” Arkani-Hamed explained. Just as the inside of a triangle is a region in a two-dimensional space bounded by intersecting lines, the simplest case of the positive Grassmannian is a region in an N-dimensional space bounded by intersecting planes. (N is the number of particles involved in a scattering process.)

It was a geometric representation of real particle data, such as the likelihood that two colliding gluons will turn into four gluons. But something was still missing.

The physicists hoped that the amplitude of a scattering process would emerge purely and inevitably from geometry, but locality and unitarity were dictating which pieces of the positive Grassmannian to add together to get it. They wondered whether the amplitude was “the answer to some particular mathematical question,” said Trnka, a post-doctoral researcher at the California Institute of Technology. “And it is,” he said.

Nima Arkani-Hamed

A sketch of the amplituhedron representing an 8-gluon particle interaction. Using Feynman diagrams, the same calculation would take roughly 500 pages of algebra.

Arkani-Hamed and Trnka discovered that the scattering amplitude equals the volume of a brand-new mathematical object — the amplituhedron. The details of a particular scattering process dictate the dimensionality and facets of the corresponding amplituhedron. The pieces of the positive Grassmannian that were being calculated with twistor diagrams and then added together by hand were building blocks that fit together inside this jewel, just as triangles fit together to form a polygon.

Like the twistor diagrams, the Feynman diagrams are another way of computing the volume of the amplituhedron piece by piece, but they are much less efficient. “They are local and unitary in space-time, but they are not necessarily very convenient or well-adapted to the shape of this jewel itself,” Skinner said. “Using Feynman diagrams is like taking a Ming vase and smashing it on the floor.”

Arkani-Hamed and Trnka have been able to calculate the volume of the amplituhedron directly in some cases, without using twistor diagrams to compute the volumes of its pieces. They have also found a “master amplituhedron” with an infinite number of facets, analogous to a circle in 2-D, which has an infinite number of sides. Its volume represents, in theory, the total amplitude of all physical processes. Lower-dimensional amplituhedra, which correspond to interactions between finite numbers of particles, live on the faces of this master structure.

“They are very powerful calculational techniques, but they are also incredibly suggestive,” Skinner said. “They suggest that thinking in terms of space-time was not the right way of going about this.”

Quest for Quantum Gravity

The seemingly irreconcilable conflict between gravity and quantum field theory enters crisis mode in black holes. Black holes pack a huge amount of mass into an extremely small space, making gravity a major player at the quantum scale, where it can usually be ignored. Inevitably, either locality or unitarity is the source of the conflict.

Puzzling Thoughts

Locality and unitarity are the central pillars of quantum field theory, but as the following thought experiments show, both break down in certain situations involving gravity. This suggests physics should be formulated without either principle.

Locality says that particles interact at points in space-time. But suppose you want to inspect space-time very closely. Probing smaller and smaller distance scales requires ever higher energies, but at a certain scale, called the Planck length, the picture gets blurry: So much energy must be concentrated into such a small region that the energy collapses the region into a black hole, making it impossible to inspect. “There’s no way of measuring space and time separations once they are smaller than the Planck length,” said Arkani-Hamed. “So we imagine space-time is a continuous thing, but because it’s impossible to talk sharply about that thing, then that suggests it must not be fundamental — it must be emergent.”

Unitarity says the quantum mechanical probabilities of all possible outcomes of a particle interaction must sum to one. To prove it, one would have to observe the same interaction over and over and count the frequencies of the different outcomes. Doing this to perfect accuracy would require an infinite number of observations using an infinitely large measuring apparatus, but the latter would again cause gravitational collapse into a black hole. In finite regions of the universe, unitarity can therefore only be approximately known.

“We have indications that both ideas have got to go,” Arkani-Hamed said. “They can’t be fundamental features of the next description,” such as a theory of quantum gravity.

String theory, a framework that treats particles as invisibly small, vibrating strings, is one candidate for a theory of quantum gravity that seems to hold up in black hole situations, but its relationship to reality is unproven — or at least confusing. Recently, a strange duality has been found between string theory and quantum field theory, indicating that the former (which includes gravity) is mathematically equivalent to the latter (which does not) when the two theories describe the same event as if it is taking place in different numbers of dimensions. No one knows quite what to make of this discovery. But the new amplituhedron research suggests space-time, and therefore dimensions, may be illusory anyway.

“We can’t rely on the usual familiar quantum mechanical space-time pictures of describing physics,” Arkani-Hamed said. “We have to learn new ways of talking about it. This work is a baby step in that direction.”

Even without unitarity and locality, the amplituhedron formulation of quantum field theory does not yet incorporate gravity. But researchers are working on it. They say scattering processes that include gravity particles may be possible to describe with the amplituhedron, or with a similar geometric object. “It might be closely related but slightly different and harder to find,” Skinner said.

Courtesy of Jaroslav Trnka

Nima Arkani-Hamed, a professor at the Institute for Advanced Study, and his former student and co-author Jaroslav Trnka, who finished his Ph.D. at Princeton University in July and is now a post-doctoral researcher at the California Institute of Technology.

Physicists must also prove that the new geometric formulation applies to the exact particles that are known to exist in the universe, rather than to the idealized quantum field theory they used to develop it, called maximally supersymmetric Yang-Mills theory. This model, which includes a “superpartner” particle for every known particle and treats space-time as flat, “just happens to be the simplest test case for these new tools,” Bourjaily said. “The way to generalize these new tools to [other] theories is understood.”

Beyond making calculations easier or possibly leading the way to quantum gravity, the discovery of the amplituhedron could cause an even more profound shift, Arkani-Hamed said. That is, giving up space and time as fundamental constituents of nature and figuring out how the Big Bang and cosmological evolution of the universe arose out of pure geometry.

“In a sense, we would see that change arises from the structure of the object,” he said. “But it’s not from the object changing. The object is basically timeless.”

While more work is needed, many theoretical physicists are paying close attention to the new ideas.

The work is “very unexpected from several points of view,” said Witten, a theoretical physicist at the Institute for Advanced Study. “The field is still developing very fast, and it is difficult to guess what will happen or what the lessons will turn out to be.”

Note: This article was updated on December 10, 2013, to include a link to the first in a series of papers on the amplituhedron.

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  • trying to comprehend this hypothesis this is like a blind man trying to grab smoke with a mall in each hand…for us laymen that is..

  • Buky Fuller’s Synergetic’s might prove useful after 6 decades of playing second fiddle to mainstream science.

  • Good job Natalie Wolchover! Your article is a lucid explanation of an arcane topic. It is clear and interesting to read, and it gives relevant perspective from the history of modern physics. Thank you!

  • Using geometry to predict the behavior of particles is nothing new. The standard model is geometric. Dr. Garrett Lisi has been positing the use of the E8 Lie Group geometry to predict particle behavior since 2007.

  • It is a far cry to say that a theory doesn’t explicitly rely on unitarity or space-time than saying that we need to give up on these concepts. I don’t read anything in this article that captures the essence of that statement.

    Both of those issues seem to be stumbling blocks in quantum gravity but I don’t see how it helps to say that “unitarity is not fundamental” without giving an alternative interpretation of the sum over histories approach to QM.

    I would frankly prefer that the authors do not make such statements until they do have a consistent framework that gives up on unitarity in particular.

    There was a lot of similar talk at the fuzz or fire conference in Santa Barbara and it was equally unsatisfying.

  • These are exciting times. Great article; it managed to strike a good balance for those of us familiar with physics but not deeply well-educated about it.

  • Normally in write-ups of new physics results, there’s always some physicist quoted who points out some issues with the approach or that it’s too early to make any firm conclusions. This article doesn’t really have that.

  • I suppose they mean conceptually geometric rather than physically geometric.

    This reminds me of a proposition that reality is a holographic projection on the surface of a hypersphere. The article even mentions, “a master amplituhedron with an infinite number of facets, analogous to a circle in 2-D, which has an infinite number of sides.” Sounds like a hypersphere to me.

    Also, regarding gravity, if they’re willing to throw out unitarity and locality, why not gravity? It could equally be a “property that merely arises as consequences of the jewel’s geometry.

  • Jose, that was my thought too. Fuller’s geometric approach was quite elegant, and this article kept reminding me of it.

  • “But the new amplituhedron research suggests space-time, and therefore dimensions, may be illusory anyway.” Yeah I read Michael Talbot’s The Holographic Universe. So not surprised.

  • Thank you very much for this deeply satisfying article. It is rare to find an example of scientific journalism that is both comprehensible to one not in the field and not simplified to the point of uselessness.

    Greatly simplifying the calculation of particle interactions makes the simulation of our universe more feasible, right? To my mind, any discoveries in this direction increase the likelihood that that the universe we inhabit is itself a simulation.

  • This is exciting indeed. I’ve recently proposed that quantum mechanics is telling us about a domain that is not contained in spacetime — a domain from which space and time are emergent and approximate notions. This work lends further support to the idea that we must let go of spacetime as a fundamental substance. Rather, it is a construct based on our everyday experience.

  • Clifford Pickover ~~ “Mathematics is the Loom upon which God weaves the Fabric of the Universe.”

  • Great article Natalie. Very interesting stuff. I can’t wait to see how Sheldon and Leonard deal with this new revelation. 🙂

  • If I’m understanding this correctly, they’re saying they may have found an object that contains and dictates our entire universe. So if this theory proves reliable, and if their search for a gravityhedron works out, we would be able literally to see the shape of our universe.

    This is so cool.

  • Does this complement string theory, or negate it? There’s a paragraph that mentions ST, but is confusing to my layman’s brain.

  • I was struck by the analog that to calculate the area under a curve, you can add up a large sum of rectangles (Feyman), but in the limit you can get a closed form solution (jewel thing).

  • Suggests that a multidimensional fractal formulation may also be profitable, where
    the geometric object representing the scattering amplitudes is the surface calculated in n-space.
    It always seemed like feynman diagrams had a recursive fractal aspect…..

  • While this may end up being smoke and mirrors, a hand-waiving exercise, or my favorite ‘I think you should be more explicit here in Step 2.’ Then again, it may not be.

    Just as Algebra is an abstraction of arithmetic. And calculus is an extension of Algebra using infinities. Geometry is a method of representing Algebra, arithmetic and calculus graphically – I view this as just a new method of representing something else. It’s a tool that can be used to discern the tautology of possible outcomes. Rather than just throwing numbers at formula via brute force and trying to sift the wheat from the chaff you can narrow the range down to those that have a good chance of working (lie within the outcome set) and those that do not.

    If it helps to develop GUT – great. If it helps to reduce the amount of computing power needed at the LHC, among other facilities, by identifying calculating short cuts. Even better. If it doesn’t seem to be able to do any of these – it’s still a pretty cool picture. One that easily could be framed and hung on the wall of the Guggenheim or used in some display at MSI in Chicago. That alone makes it a work of art. If it does what the authors claim – all the better. Especially if it can, as this wonderful article did, help bring an immensely complex subject such as this to the unwashed masses. Such as myself.


  • So maybe I misunderstand. But it seems they are taking the structure of a Mathematical Geometric Proof or the rules for creating one and applying those to construct simpler rules for estimating Scattering Amplitudes.. based on the evidence, or supported by the outcomes.
    The “help” the new model seems to provide are the “Intuitive” leaps followed by the confirmation of those intuitive leaps in making calculations.
    Building up a framework of “Intuitive” axioms or rules “like those used in Geometry..” extended to a space not limited by previous assumptions about how geometry corresponds to familiar space and time so far is proving more efficient at making more correct rules which can be proved correct through experimentation.
    Intuiting that a useful theorem for Quantum Gravity is very interesting.. but either its too difficult to explain to the lay person in this article.. or so far has actually proved elusive.. kind of reminds me of Fermat’s Last Theorem. We could could intuit its True and that someday we would have a Proof.. but for over 300 yrs it was still an elusive goal.
    I very much liked the article. And the history behind the underpinnings driven by the SuperCollider.. the perspective was great.

  • It’s stunning to contemplate that we – us right now reading this – live in this split second of human history when mankind is capable of even attempting to calculate and discover the very fabric of our universe and reality.

    Our ancestors for thousands, millions, and even billions of years back did little more than muck around looking for enough food and shelter to stay alive one more day.

  • This is reminiscent of Lisi’s proposed unified field theory based on the E8 Lie group. However, although both approaches are inventive and suggestive, neither make (falsifiable) predictions regarding new physics. It seems an intriguing, fortuitous and very useful discovery, but so far nothing more. Here’s hoping that will change.

  • I have often wonder if space time is an emergent property, quantum physics only seems odd if you try to imagine particles all getting there spacial properties by being defined relative to a single unifying space time. But quantum physics seems saying that spatial properties are relative, for an entwined pair the spin of one particle is defined relative to the other, but it isn’t defined to anything out side the system, until it interacts with something outside the system, this not a problem if you forget about space time, and just treat them as individual relationships between particles. This also solves the cat in the box paradox, if spacial properties can be relative, then whether the cat is dead or alive is determined by the relationship between mater in the box, and when we look in the box we only resolve relationships between the inside of the box and the outside, just as the spin between the two particles is already defined so is the relationship between matter in the box and therefore the state of the cat. You can also be in the box with the cat and then the universe out side the box becomes the cat in the box experiment, so the whole specialness of the observer is removed.

  • Space time as you all put it is just an illusion. A construct of our world. Space time is the exception in the universe. In the rest of the universe and higher densities space time has no real meaning. Everything exists simultaneously. It still does here but we have a very limited perception of reality.

  • A wonderfully written piece. I hope it wins an award for science journalism.

    My math is 30 years out of date, but I’ve been wondering about a potential relationship between the necessary existence of dark matter and dark energy based upon the apparent total mass of the Universe and the possibility of parallel realities and a multiverse. Now, if we add the amplituhedron as a simplifying structure, I wonder if it can turn out that taking several (I believe it may be six or eight) symmetric universes as a whole in this geometry where each of the symmetries could be a universe, could we account for the mass and remove the need for dark matter? I wish I had the capabilities to answer this. I would love to hear from real physicists if my concept makes any sense. If it does, it has enormous ramifications.

  • Gilmore’s illustration is something one can enjoy without understanding those things that laymen like myself will never understand. Beautiful.

  • A very well written article. After reading this I was left wondering if the properties of M theory’s Branes could potentially be analogous to this amplituhedron if completed in its final form. Perhaps with the amplituhedron we are complimenting M theory by describing the geometry of our universes P-Brane. Having said that, this leaves me with the feeling of the holographic principle (us being 3d representations of 2d information on the walls of our hypersphere, or more disconcerting, that we are starting to describe the programming of the simulation.

  • “You can easily do, on paper, computations that were infeasible even with a computer before.”
    Ha! Yeah right, Jake. YOU can easily do it. I’m going to go get a sandwich and try to forget that my experience of reality is illusory.

  • I dropped out of a class in grad school because of a homework assignment that required me to compute some 50 Feynman diagrams. This article makes me feel vindicated yet nostalgic for the days when I could actually study things like this.

  • To me, this is beauty. The fact that the behavior of the world can be described with some fair degree of accuracy with a simple, elegant solution is truly awe inspiring.

    Every major step forward has been simple and beautiful in their time. Newton’s original equation for gravity and his laws of motion. Einstein’s relativity. Even Feynman’s diagrams that were the center of his QED had a simplicity that won him the Nobel. As we learned more, these theories grew more complex until there came another return to simplicity and beauty. Quantum Field theory is pretty complex, and String theory even more so. Maybe this is the next leap forward, the next simple, beautiful idea.

  • Fascinating stuff!

    Also, I love that the comments on articles such as this, are so rife with positivity and humor! I wonder if there’s a connection between education and curiosity, and happy comments? ;P

  • Linearized physics of discrete space geometry is not the same as discrete physics of discrete space geometry.

    Linearization error makes all the difference in the world. Linearized physics breaks down at discrete scales. Discrete physics doesn’t.

    The greatest advantage of quantum gravity over other theories is that it tries to build the model of the world from ground up. That is the only proper way to build a model of a discrete world.

  • What I got from the article was that they have a new method of computing scattering amplitudes that is insanely efficient. Good news for grad students who forget to add a one diagram in a calculation of 120 diagrams and then curse the day they got into theoretical physics. What makes their method special is that the formulation seems to indicate that there is merit in thinking of spacetime as an emergent phenomenon rather than as a background. This is very good as talking about gravity as an emergent phenomenon by itself is tricky but if spacetime is emergent along with gravity it solves some of the issues involved in formulating quantum gravity along this track.

    To make an analogy, Feynman figured out a calculation method with diagrams but Wilson gave us a conceptual shift with the idea of renormalization group and the resulting EFT’s. These guys might be on track to give us a calculation method AND a conceptual shift in one package. Thats big.

    Also, I have a name suggestion for the ‘master amplituhedron’: the one ring to rule them all.

  • I once heard a lecture / talk by Feynman . It still is the most impressive and interesting talk I ever heard. He could grasp technical subjects and explain them in a powerful way.

  • Tony & Jose – thanks for bringing Buckminster Fuller into this – his search was for Nature’s geometry which he called the geometry of thinking, a geometry which works equally well in describing the initial structures and movement in Universe, and everything else. He closed the gap between his large and small-scale pictures 50 years ago and predicted how long it would take more conventional science to catch up – bravo.

  • I think the missing link that this amplituhedron needs is Erik Verlinde’s entropic gravity, not Garrett Lisi’s work. Alex Wissner-Gross has also shown that entropy is how (artificial) intelligence works, consolidating (perhaps) thousands of lines of code from several different fields of AI into a single fuction based on thermodynamics just like how amplituhedron consolidates hundreds of Feynman diagrams (and thousands of mathematical terms) into a single function as well.

  • Extremely minor, but the video from the conference in 16:9 (instead of 4:3): (16:9)

  • I am very surprised that no one mentioned it.
    This is a result for Steven Wolframs ‘A New Kind Of Science’
    Premise: Simple laws produce profoundly complex results.
    Simple law: Simplifying the calculations of particle interactions, produces the results of amplitude calculations.

    A unit circle is a geometric solution to many trigonometry problems. Same principal, much more complex solution to a very very complex problem.

    Mark these words: There will be rooms full of 5th graders able to use this, in the next generation.

  • First, this is a phenomenally well-written science article, kudos to the author!

    I also saw a couple comments here that I, as a moderately-informed layperson, may be able to clarify, mostly stemming from confusion about what this object actually means; contrasting it with string theory may help.

    String theory proposes that the universe is made of strings, and calculates what that would look like. There are other “theories of everything” (which is a terrible name) that make different assumptions, and then make predictions. This object is not that; as far as I understand, it is a mathematical tool designed for a specific purpose, not a predictive theory.

    Wizard Garber mentioned, “Using geometry to predict the behavior of particles is nothing new… Dr. Garrett Lisi has been positing the use of the E8 Lie Group geometry to predict particle behavior since 2007.” This is true, but the E8 geometry is used to predict individual particle properties (some of which are interaction particles), while this object is used to predict scattering probabilities. Both are incomplete, both have problems with gravity (amongst other things), and both are just tools to help learn more, rather than fundamental descriptions of reality.

    Ignacio Mosqueira says, “It is a far cry to say that a theory doesn’t explicitly rely on unitarity or space-time than saying that we need to give up on these concepts.” This is not what the authors are saying. They have dropped the assumptions of unitarity and locality in their math, and produced an abstract object. This object just so happens to give the same answers as the equivalent Feynmann sum. The insight is simply that, if getting the right answer doesn’t depend on those two properties, maybe reality also doesn’t?

    Jake says, “If I’m understanding this correctly, they’re saying they may have found an object that contains and dictates our entire universe,” which is unfortunately not correct. They have found a neat mathematical object, no more descriptive of our actual physical universe than, say, a slice of angel food cake – it just has slightly more predictive power.

    And finally, if I’m wrong about any of this, for the love of the FSM tell me! Being wrong when you think you’re right is just the worst…

  • Very well-written, clear article on some difficult topics. A couple of points seem to me to indicate that the unitarity and locality issues may not be completely intractable. Unitarity shouldn’t necessarily require an infinite number of observations of a particular particle interaction because there are only a finite number of ways such interactions can go, determined by the selection rules appropriate for given quantum numbers for the particles involved, considering CPT conservation etc. (hence the “master amplituhedron” may not have exactly an infinite number of facets due to similar conservation rules for baryons and so on). Locality seems completely different but if the Heisenberg Uncertainty principle doesn’t eventually fall, the complementarity of momentum and position should mean that two of mass, time, and space are quantizable, but all three need not be. Since mass empirically doesn’t seem to be quantized, that may indicate that time and space are, so that the infinities in measuring space and time separations below the Planck limits become “merely” Very Large. That in turn might resolve the dimensional difference disconnect between string theory and QFT.

  • An amazing piece of science. What does this mean to humans? Of course it has so many applications – the mind boggles.

    As the article states the amplituhedron is “not built out of space-time and probabilities; these properties merely arise as consequences of the jewel’s geometry. The usual picture of space and time, and particles moving around in them, is a construct.”

    Grammatically this is akin to using a word to define itself. Geometrically speaking , space and time are not to be considered essential for geometry but merely something that dimensionionally emergesas the shape takes form.

    Breathtaking in its beauty, scattering in its complexity, and counter logical in its form.

  • Harrison, I don’t think that would remove dark anything, but rather confirm it as extremely exotic. As opposed to something, at least a little more mundane, like finding ten time more black holes than expected.

  • luke lea, I gather that the stuff is grassmannian+. you put it in a big enough euclidean space and space time MIGHT show up as some sort of a solution(s) of einstein equation (at lower energy limit)?

    I am wondering what the vacuum looks like. no particles. so no legs. at least we should be able to describe vacuum to be able to say quantum gravity or whatever may have been found?

  • Awesome Article. Totally excited about the advances that may come from this line of research. Must read more of her articles.

  • Thank you, Natalie, for this well-written and (for the most part) general-audience-friendly article on what is a rather arcane topic, and a mind-bending way of perceiving reality.
    Your work here (and as I read further, throughout Quanta) reminds me of the stellar book by Isaac Asimov, “Atom: Journey Across the Subatomic Cosmos”.
    Please, if you are so inclined, the world desperately needs a similar tome on the latest advances in subatomic physics. I would love to see such a book by you on my shelf.

  • Time does not exist. It is not an object that can interact with other objects. It is just a human construct. I’m glad to see Big Physics coming around to this idea. Additionally, there is only one dimension, that of volume. Length (1D) is not independent of width (2D), nor width to depth (3D), therefore we do not live in three dimensions, only one. We use the Euclidean concept of dimensions, and they work well enough at the crude macroscopic level. However, the concept of one “place,” i.e. a fixed location based on coordinates, at the quantum level has repeatedly shown that multiple dimensions and therefore “place” is just another human construct that does not stand up to scrutiny. It appears that after decasdes of hitting one’s head against the wall Physics has decided that they are getting no where and it is time to look and the very basic foundations of our understanding of Physical “Laws.”

  • This is an amazing development. Having had to learn Feynman diagrams and the accursed renormalization in grad school, I just knew there had to be something better! I really hope this is it.

    But I think it’s philosophically incorrect to call spacetime ‘illusory’, no matter what comes of this. That’s indisputably how the universe presents itself from a particular point of view; the amplituhedron (among other things) shows us that other points of view are possible. But I don’t see any reason to label one point of view as more ‘real’ than another. The amplituhedron is a model, like other models; a very beautiful one, perhaps a very useful one. But it isn’t and can’t be the universe, full stop.

    As C. S. Lewis said (don’t have the exact quote handy, but this is close), “The hill that looks green when you are standing on it looks blue from ten miles away. Both are equally good facts.” What’s the ‘real’ color of the hill? To answer that question, I first ask, “Where are you standing?”

  • Oh, and Greg: While you are correct to say that time is not an object that can interact with other objects, I would argue that is not at all the same as the statement, ‘Time does not exist.’

    In much the same way, I think Eddington was dead wrong to move from the statement, “The atom is mostly empty space,” (though one can quibble whether electric fields are really ’empty space’) to “Solidity is an illusion.” Of course it isn’t. At the human scale it is very much a reality.

    We can to some degree explain the experience of solidity from what we know of atoms, true. But it’s only by using solid instruments with our solid bodies that we ever came to know of atoms in the first place. ‘Emergent’ does not mean ‘illusion’ – the emergence is an ‘equally good fact’ as the level it emerges from.

  • What a fascinating discussion. I do firmly beleive that Time is a human construct, like numbers. We NEED numbers and time to make sense of our surroundings but they do not exist separate from our own minds. Time is not a force. Time does not have energy. Time does not have volume. The universe is an infinite series of waves at infinite frequencies that coalese into energy, matter, and objects and is constantly changing to different frequencies, matter and objects. The universe is an uncaring thing and it certainly doesn’t care what we tink of it.

  • I wonder if the finding is relate to this history from Predrag Cvitanovic:

  • Ah-oh, theoretical physicists are discovering that everything is exponentially simpler than theoretical physicists thought it was…

  • Thanks for once again providing yet another resource for my introductory physics course. “Time,” after all, really is only a four-letter word representing an “idea” which is also a four-letter word which is … aw shucks, four-letter words fail me. To quote Robin Williams “Reality–Man what a concept!”

  • Some of the best science writing I’ve read in a while. Thank you.

    My interpretation of unitarity: asserting that probability always sums to one is similar to stating an outcome always occurs.

    Self-evident and superfluous, perhaps?

  • What does this mean exactly?

    “In a sense, we would see that change arises from the structure of the object,” he said. “But it’s not from the object changing. The object is basically timeless.”

    The object just exists and as a result of its structure time arises from its shape?

  • This is so well written and so interesting, that I actually stop texting a pretty girl who likes me, just to read it hahaha

  • This may be the most beautiful object I’ve ever seen.
    This immaculate rainbow of fractal facets may be the first visually comprehensible representation of god. Well, at least to me. This is looking behind the curtain. A freshly discovered geometric form in which is encoded the basic features of reality. My mind is blown.

    Congratulations to the scientists, as well as the author of the article.

  • all of this looks incredibly interesting but i don’t understand the relationship between a polygon and the understanding of our universe. Seriously I’d like to have this enthusiasm but I’m so lost ! Anyway I’m happy to see that humanity have a new tool to work with !

  • I think my brain needs a Cigarette, a Shower and a cold bucket of ice after reading that. Gorgeously written, and the information was quite easy to understand, and I am not a huge math person. Finding that it may actually be all geometrically based may give credence to all those experiments into symbolism in the 15th century. Were they on the right track back then?

  • Very well written article. Admittedly, the theories are beyond my level. As a layman, and one in the business world, the words that jumped out at me were ones like “efficient”, “accurate” and “simple”. (not necessarily exact quotes). If we can get 20% more efficiency out of all of our theoretical physics grad students, post docs and full researchers, we will likely be able to advance more discoveries faster. Whether or not time is “real”, that efficiency will be of great benefit. 😉

  • 1) The sketch says it is for 8 gluons, but only has 7 points, and the title says “7 point”, though the numbers below go from 1 to 8.

    2) Any chance you could actually show us the “one-term, beautiful little function” rather than just refer to it? We are not all mathematically challenged.

  • Here in Beijing I recently showed my students a 1995 (?) video on the Mandelbrot Set, no, not for mathematical purposes, but to try to help illustrate the idea of “figure,” and how it relates to understanding virtue in Plato’s “Meno.” Socrates seems intent on finding a simple, fundamental virtue from which other virtues are “emergent,” to use the language used in Arkani-Hamed’s lecture video. I will leave it at this. I am no mathematician. Is this amplituhedron, like the Mandelbrot set, a fractal of some sort? Both seem to be trying mathematically to convey the idea that something of infinite complexity, such as particle interactions, can be understood very simply.

    I would like to join my voice with others in complementing Ms. Wolchover on the clarity and comprehensibility — to a certain extent — of this article.

  • Quote: “And unitarity holds that the probabilities of all possible outcomes of a quantum mechanical interaction must add up to one. The concepts are the central pillars of quantum field theory in its original form, but in certain situations involving gravity, both break down, suggesting neither is a fundamental aspect of nature”

    I am not sure if stating probabilities as property of nature is a correct statement. Probabilities quantify state of knowledge about nature and not nature itself.

  • David said “They have found a neat mathematical object, no more descriptive of our actual physical universe than, say, a slice of angel food cake – it just has slightly more predictive power.” The same could be said for quantum mechanics, that was the point of Schrödinger’s cat, and a fundamental aspect of Einstein’s criticisms of quantum mechanics, that quantum mechanics is not descriptive, merely predictive. For many physicists, even for most, the predictive value of a theory is good enough regardless of it’s ability to actually describe reality. Some people, many of them not physicists, interpret a theory as descriptive of reality merely because it is predictive thus leading to belief in absurdities and logical contradictions such as the possibility of a Schrödinger’s cat being real, or the idea that unitarity could be violated.

  • This may be a naive question, but if “the new amplituhedron research suggests space-time, and therefore dimensions, may be illusory anyway,” then shouldn’t gravity also be “illusory?” If space and time are illusions, then shouldn’t adhering to the surface of an illusion (as gravity keeps me pinned to the Earth) must therefore also be an illusion? If true, maybe there IS no other shape that describes it.

  • Sorry to throw water on the party, here (I’m as excited as anyone here, by the way), but we do need to keep in mind that this is, after all, only a mathematical construct, until it is proven otherwise.

  • The use of a physical construct to re formulate exceedingly complex equations and simplify them to the point where infinities are eliminated and solution appears possibly is amazingly similar to the relationship between M Theory and String Theory.
    Could one observe that it is possible that the relationship suggests a relationship to a larger unitary theory with a geometric solution that may apply to both theories .

  • Coincident with this article’s release is my reading of Mario Livio’s “Is God a Mathematician?” In this book Livio lucidly develops the amazing correspondence between mathematical abstractions and perceived reality. As I read this article I was particularly struck by the Greek’s fascination with geometry. In asserting that “all is number” might the Pythagoreans have intuited something very close to the truth? Here’s hoping this exciting new model is as fruitful as anticipated!

  • What a beautiful jewel with 2 missing pieces. I predict that when connect gravity to the equation then the jewel will be complete

  • “…time and space are only sensible forms of our intuition, but not determinations given for themselves or conditions of objects as things in themselves.”

    – Immanuel Kant

  • Brilliant article on all counts. The concept reminds me of a historical observation that we only understood the mechanics of the planets when Copernicus changed the center of our calculations from the earth to the sun. This concept appears breaks our coordinate system from our space-time to a broader “whatever”.

  • All,
    To me, an electrical engineer, this looks like the physics world has been stuck in discrete mathematics, digital mathematics since Feyman.  If you digitized a sine wave (what happens when you create a CD or MP3 file from an analog LP record) , you would (1) have an approximation of the sine wave, (2) a Taylor Series of infinite products, and (3) be left with an inability to use calculus for problem solving and instead would have to use arithmetic, matrix mathematics and other discrete methods.  You would have to do all analysis using computers (as in digital processing) and would not be able to use ‘analog’ mathematics: calculus & geometry.  You would be pursuing greater and greater computing speed, parallelism and memory because you are stuck in the digital, discrete world.

    It looks like physics love affair with discrete mathematics may be coming to a close with a focus on geometry leading back to analog formulas, leading to calculus and the ability to perform analysis on paper.

    Unless with think the universe is a big discrete processing machine.  Nothing in the universe seems discrete.  It seems to crave exponential and trigonometric terms.


  • This is some of the best science writing I’ve ever seen. As a layman (elementary spcecial education teacher) I still enjoy keeping up on the newest ideas and research. Natalie’s level of skill in helping a novice like me to begin to grasp the basic concepts of the article is laudatory. Thank you so much!

  • As a non-physicist, I found this article interesting and exciting. I love profound learning. Looking forward to the implications in future practice and research`

  • Cool! Definitely reinforces my desire to major in math and/or physics. Can’t wait to actually fully understand this.

  • I’m a lay person when it comes to physics and mathematics so please forgive me if these three questions are very stupid. I’m just trying to wrap my head around things. Perhaps someone has some insight:

    1. Why is N-dimensions described where N= the number of particles involved in the scattering process? Does this mean that each particle is a separate dimension? How is the number of particles involved in scattering equal to the number of dimensions?

    2. What are the implications for the different theories about how the universe started? Does this mean that if you say two branes collided to form the universe you’re really describing a mathematical formulation of equivalency wherein the actual, and equivalent, means of creation was a facet of this geometrical jewel form?

    3. So reality is described by a geometrical shape. Can that tie into the theory that reality as we understand and see it is simply a projection of other underlying mathematical processes? Are we a projection of the mathematical expression of simplified amplituhedrons on the master amplituhedron?

    Again, sorry if these are ridiculous questions.

  • Thank you for the historic perspective – how the thirty-year old discovery graduately lead to this beautiful result. It will be exiting to see the predictive value of the discovery. The LHC must be perfect for that purpose.

  • Beautifully written… and something you save to read again… taking the time to muse on the progress of the thoughts here. It certainly brought the work of Buckminster Fuller to my mind… and a particular comment he made (best I can remember it here on the spot): “When I develop an idea – if it is not simple, or beautiful… then I know that it is wrong.” Michio Kaku is quite public with his quest for a “theory of everything”… perhaps this is part and particle of that. This is simple and beautiful in it’s presentation and understanding – I do like the thought of the Mandelbrot potential in this exploration of the “Amplituhedron”. A device with a strange banner…. Excelsior!

  • Second the emotion that Stephen Wolfram’s Mathematica
    may be the language used to describe the jewel in the lotus;
    The amplituhedron is even more beautiful than the seashell
    on the cover of ‘A New Kind of Science’.

  • I certainly hope none of my tax dollars was spent on wasting time on this nonsense. When are scientists going to concentrate on real problems like feeding everybody and quit worrying about what causes gravity and other ridiculous studies? Let these “mathematicians” get real jobs that actually benefit real people.

  • This is a fascinating development, but I am posting mainly to commend the author of this article explaining it…

    Usually science reporting leaves me disappointed or disgusted in how incomplete or just plain wrong it is- the author of this article did an astoundingly good job explaining this extremely complicated subject- without dumbing it down. Actually, “astoundingly good” is an understatement- I feel comfortable in saying this is quite likely the BEST example of science reporting I have ever seen – and as a scientist I have read (and usually winced while reading) many examples of science journalism…

    I was impressed- Natalie Wolchover is the epitome of what science journalism SHOULD be. I plan to look up whatever else she has written today…

    Thank you for the work you put into this Natalie- I hope to read a LOT more from you!

    BRILLIANT work!

  • Interesting that this construct appears to be derived from the tetrahedron. R. Buckminster Fuller proposed this in “Synergetics,” that the entire structure of the universe is explained by the fact that the tetrahedron is the simplest possible structure, and that there is no other way to arrange four points in a structure, other than as a tetrahedron. (A square is unstable unless it is triangulated, in which case it becomes a flat tetrahedron.)

  • The conclusions do not bare out to the hypothesis. Imagine you did not know about, circles, sin, cos, tan, radius, … All you knew was the a function that was an infinite summation. Then suddenly someone discovers the circle, and that your summation was nothing but a taylor expansion of the sin function. Suddenly your math becomes much easier with this new understanding. That does not mean everything you previously understood was wrong. It just means you have a much better tool to think about problems. It is reasonable to expect and summation formula, including feynman diagrams represent some sort of geometry. Finding that geometry gives a new and powerful tool. But it does not mean what you know about feynman diagrams is wrong. In fact just like a small angle approximation, the first order and usually second order diagrams can still be increadably useful for understanding and solving a certain class of problems.

  • They say that the better one understands a subject, the better they can explain it in simple terms. Thank you Natalie for having such a keen grasp of your field!

  • Very interesting article. From what I understand, a little Aristotelian philosophy could have debunked the concepts of “locality” and “unitarity” (the “central pillars of quantum field theory”) from the very beginning. Both concepts violate the Law of Identity (that an existent is what it is). “Locality” suffers from the same fallacy as Zeno’s Paradox (e.g. that space can be infinitely divided thus we can never travel from point A to point B). Namely, an existent has identity, i.e., is something with defined limits (however construed). Thus, “infinity” as such does not have (metaphysical) existence, so there is no problem of traversing from A to B (the distance is what it is, whether one inch or a Planck length). “Unitarity” also violates the Law of Identity because it seems to assume (?) an infinite number of possible outcomes. However, A is A, so a particle only can act in ways limited by its nature (the Law of Causality). . . .Gee, it’s too bad these poor quantum physicists had to go through all that tortuous mathematics and mind experimentation before realizing that the foundations of their field may be made of sand. But maybe I should study physics before casting any quantum stones.

  • The basic idea is that we still aren’t sure what the fundamentals of our universe are. We understand the over arching workings at certain levels but lack the basic building blocks.

    I think of it like seeing the building instead of the structure of the building. The more complete our knowledge of the fundaments, the simpler the outer workings appear to be when looked at this way.

    I think of this as a wonderful leap forward and that we may soon integrate the large and small scales… we seem to know the fields of mathematics and physics required to do so, and that lends to focused study that will draw brilliant minds. We have a path!

  • I’m amused by how many people are jumping on this to claim that it somehow vindicates their particular intellectual hero (Garrett Lisi, Buckminster Fuller, Stephen Wolfram, Benoit Mandelbrot, Immanuel Kant*…). Nope: whether or not it turns out to be as significant as the article suggests, this is new stuff, and the credit for it goes to Nima Arkani-Hamed and his colleagues.

    *Roger Penrose is named with rather more justification, as the originator of twistor theory, but the amplihedron work appears to go well beyond that.

  • Ryan S, I had the same exact thought. I’ve pondered, almost ceaselessly, over the nature of gravity; especially considering that it appears to become loosened or to disappear, altogether, at the subatomic level.
    One hypothesis I’ve formulated on the subject is that gravity does not necessarily exist independent of matter, but may be a reaction thereof. As particles collide, the energy they produce may be tantamount to the fusion of weak and strong forces, giving rise to electromagnetism. The energy bonds that then form may be what constitutes gravity, the result of nuclear energy producing electromagnetic energy. This would explain why, generally speaking, the more mass a body contains, the stronger its gravitational force; along with the fact that most matter doesn’t simply scatter away once, strong enough, bonds have been formed.
    This may actually also explain the nature of black holes. Upon exceeding the Schwarzschild Radius, it is possible that the amount of energy produced at the core of said structure produces, hypothetically speaking, a planck temperature that results in infinite energy and, as a result, infinite gravity.

    Of course, I’m open to discussing and or being refuted on the subject. I’m here to learn, after all.

  • A wonderful article I just wish “they” had called it “amplihedron” and not “amplitehedron” the latter being almost impossible for me to get my tongue around.

  • @Casey SLedge
    “I certainly hope none of my tax dollars was spent on wasting time on this nonsense. When are scientists going to concentrate on real problems like feeding everybody and quit worrying about what causes gravity and other ridiculous studies?”

    Are you speaking of scientists in general? …because it would be horribly ignorant to think that they don’t play an important role in helping the real world.

    As for *theoretical physicists* in particular, then I can understand where you’re coming from. It’s important to realize though that theoretical science comprises only a fraction of the scientific community as a whole, in research or in practice. All in all, the ‘useless crap’ that theoretical scientists do today is no different in essence than what the innovative thinkers of the past have done. Who knew that a bunch of freaks studying ‘little animals’ under a microscope would give us modern medicine and disease prevention? Who knew that Einstein’s Theory of Relativity is the reason why our GPS systems work and modern transportation is even feasible? Theoretical science mostly serves its purpose in the ‘long long-term’, but it does have its quick benefits.

    Not all scientists are theorists, far from it. Indeed, many devote themselves to solving real-world problems. Even pure physics plays a part if you try to understand it. Each scientist does his/her thing and serves a purpose.

  • This article is beautifully written. It takes obscure concepts and put them in terms that the rest of us can *almost* comprehend. Well done Natalie!

  • I agree with those above commenting on the lack of attribution of the “twistor” concept to Roger Penrose. I await with interest his comments on the amplituhedron and its implications.

  • Natalie’s writing is spell binding, judging by the laudatory comments here;
    too good, in fact, as it gives us the illusion of grasping the arcane problems
    tackled by the experts working in this field. Clearly, Natalie’s writing alone
    has sufficient power of attraction, and I wish her article was devoid of pictures
    and images of any kind; the artist’s colorful depiction of the amplituhedron,
    and even the slides from Arkani-Hamed’s lecture, tickle our minds into believing
    that we can attain some easy, intuitive understanding of the nasty math underneath.
    As a physicist, I can honestly say that this is so far down the rabbit hole,
    that it should come with a warning, when popularized in blogs and magazines.

  • OK – very well written article on a very complex and very cool theory. Also, very interesting comments from just about everyone! The use of geometry to describe interaction models s not surprising. But the lack of space/time or unity principles as basic tenets – in effect moving them to outcomes of the model rather than basic principals – is awesome. The ability to finally include gravity in a coherent model along with other observed implications of the model is truly amazing. This is certainly the start of the perception change necessary to move our understanding forward. But I suspect it isn’t the last. Still – very cool!

  • I rarely comment on articles, but would like to add my own thanks to the many others for Natalie Wolchover’s incredibly well written article. Thank you.

  • [quote]
    Virtual particles are never observed in nature, but they were considered mathematically necessary for unitarity — the requirement that probabilities sum to one.

    How is this meant? Because scientists have made virtual photons real by giving them extra energy with magnetic fields.

    And also the Casimir effect is explained with virtual particles.

  • On my third read-through of this interesting article. Thanks to the author!

    A question / comment regarding this minor point in the article: “… Virtual particles are never observed in nature, but they were considered mathematically necessary for unitarity…”

    My understanding is that the existence of virtual particles is extremely fleeting, but the reality of their existence has been confirmed:
    1. by their effect on the fine structure of lines found in the optical spectra of the hydrogen atom
    2. as reported here:
    “Chalmers scientist, Christopher Wilson and his co-workers have succeeded in getting photons to leave their virtual state and become real photons, i.e. measurable light.”

  • How would one validate the accuracy of this model, if current model requires calculations “that even a computer cant do”?

  • One more “beautifully written” to add to the prior ones. You made an extremely complex story as simple as it could be made yet not simpler than it needed to be. A work of art.

  • It is not clear that this geometric object is anything other than the expression of a class of complementarities. It happens to facilitate mathematical calculation but should not be considered any more “real” than the visualization of backwards time-travel in Feynman diagrams. To call it “jewel-like” may therefore be misleading, as it invites the reader to consider physicality. The headline, juxtaposing “jewel” with “heart”, is an example of how this misleading may be subtle, suggesting, however unintentionally and metaphorically, some sort of tangibility. The poetic bow to the Eightfold Way may be lost on many readers.

  • Does the thought experiment on locality imply that space-time doesn’t exist in environments smaller than the Planck length?

  • Great Article. I have been following the work of Richard Feynman, Leonard Susskind, and Edward Witten for many years. It is interesting to see how string theory has developed into the Holographic projection theory of matter being projeted on to a two dimensional hypersphere. it is also interesting to note that the data is based on how a Black hole works based not on volume, but on the information being preserved on the the surface structure. What comes to my mind is that geometry is the key player both in the micro and macro realms of the universe. And that our thinking has to change with regards to space-time and gravity being fundamental. They all may well be emergent properties of the amplituhedron.

  • Much of the foundation for this, starting with the triangle, is spelled out in Buckminster Fuller’s magnum opus, Synergetic Geometry, which you can find free, in the public domain, the way Bucky wanted it, at It’s breathtaking!

  • Come on guys, enough with the “Simulation” talk. It just pushes everything back one square – our universe is Simulated by some process in some OTHER Uber-Universe, which why can’t we say that universe is simulated by… etc. etc. yadayadayada. It is no better than saying “well god created it, because it could not have occurred otherwise” and then wait, what created god?
    As for the holographic representation in “higher” dimensions, the novel thing about the Holographic theory is that it does not involve higher dimensions, it shows that a three-dimensional portion of the universe can always be represented in two dimensions by someone outside that three-dimensional volume, basically for the same reason that the information in a black hole depends on the area of its event horizon, not on its “volume”.

  • Thank you Pterosaur for raining on our parade, we need that sometimes. Yes, the headline sucks. But as the comment before yours says, this article is a work of art and something that tells laymen like me as much as we can absorb without totally confusing us. I have “read” Penrose’s “Road to Reality” four times, getting more out of it each time but still not fully understanding half of it (my graduate “work” was in math, not physics, and it was really just study, not research), and I am gratified to see that his decades-long work on Twistors is maybe going to be useful.

  • Mark, a two-dimensional “hypersphere” is what we in the field call… a sphere. Let’s not complicate things, they are confusing enough for us guys with three-dimensional brains.

  • More than one Comment has mentioned the possibility that our Universe is just a Simulation.
    Well, let’s consider that. Is it possible that there is some Uber-Universe where someone, for god only knows what reason, has decided to program our kind of “reality” into his massive computer? That Universe might not even HAVE Quantum Physics, or even Gravity; it could work according to entirely different Laws, and this maniac has programmed such things into OUR Simulation, and somehow we experience it.
    Okay then, we probably have to assume that Time is not the same there either, since there are a LOT of calculations to be made for each nanosecond of our Universe. Fine, maybe this guy’s computer takes a trillion years to simulate one nanosecond of our universe, or may a trillion trillion trillion years, it doesn’t matter because all WE experience is that nanosecond. Okay but then let’s imagine that this guy’s computer is some kind of simple Babbage Difference Engine, some very simple device that can run very complicated programs. Heck, it could be made of WOOD and run very slowly and require lots of maintenance, yet given enough Time it would still make all the calculations necessary.
    I hope you can see how absurd is the direction this is going.

  • And if we are just some Simulation or Emulation in some other guy’s computer, why would my consciousness experience any of what his computer calculates? I mean, what if TWO guys simulate our Universe, which one are we supposed to experience? With my luck, I would choose the wrong one and lose contact with all you other guys. Oh wait, it looks like that already happened. Darn.

  • I found the article(s) on the “new”? concept of what I assume to be a “lesser”God particle fascinating with regard to possibilities outcomes…at least this theory has high probability factors of being feasible. It is good to see new theories challenging doctrine! However, the amplituhedron does little to explain itself with regard to energy and gravity. This “origami” particle seems both riddle and paradox, and has no propensity to “show its true face” to the accepted protocols of quantum/mass reality. (perhaps they are both shy). The real question at hand is a simple one. In what manner does the amplituhedron serve as a “causation” for the “actuality” of particle energy? Or perhaps it leans more toward atom formation? In any event, I very much enjoyed reading the article. Please keep “tilting at windmills”.

  • The universe (galaxies) are all traveling at the speed of light (gravity)which is inherent in each particle’s existence in the form of resistance. We witness the galaxtic lens at 90 degrees to the speed of gravity and a current resistance of 75 Kps (Velocity of galaxies on average). In other words, when Newton saw the proverbial apple fall he was witnessing the resistance to gravity, in more explanation, all of this is at the speed of light (gravity) minus a resistance of ~0 to ~c.

  • I wonder when these scientists are going to call Nassim Haramein, or if they already have. lol

    I mean, he was theorizing this decades ago, but was laughed at by the know-it-alls. It’s sad when scientists make science as dogmatic as religion.

    Core fundamental of science: Question everything. What you “know” now is ALWAYS open for change, whether you like it or not.

  • I notice in this thread a tendency for readers to think too conventionally about the nature of space, time, and gravity. So here are some things to keep in mind when considering the nature of time and gravity, courtesy of Einstein’s Theories of Relativity…

    First, your local matter evolves more slowly relative to other matter the faster you go, and especially more slowly relative to other matter as you approach the speed of light. The clock on your spaceship will show 5 minutes of time while the clock on Earth might show several hours. Time in this sense is only inferred from change and comparison. For example, “Light moves from here to there in the same amount of time it takes for the ‘second hand’ on my atomic clock to change.” Change occurs. Rates of change correspond, or differ. This comparison in the amount of change is what we call “relative time.”

    Second, gravity in relativity is not a force acting across space-time, it is an actual curvature of space-time, proportional to the masses of the objects in the continuum. Photons moving at the speed of light in the vacuum travel in straight lines, but are diverted by traveling in curved space. In this sense, gravity is not a “force” but rather just a fundamental relationship between matter and space-time.

    As we formulate our predictive models we might ask: Is matter and energy localized in a “real” spatial continuum, or is the space-time continuum just a reflection of certain relationships among wavicles, as seen from the point of view of recording machines like ourselves, which are made of those interacting wavicles? If that which we call “space” is just a multivalent relationship, the continuum —and therefore distance itself— may be seen as abstract.

  • I have been reading this kind of stuff for many years, and yes, the idea of such a geometric representation did come out of Roger Penrose’s Twister Space Theory, when he tried to unify gravity with quantum mechanics. Its very interesting to see the emergence of the amplituhedron, as I wasn’t aware twistor space theory was going anywhere. Deeply embedding complex numbers as twistor space theory does to describe physical reality seems to place it closer to quantum theory than to general theory of relativity (and its description of gravity). So I am not really surprised that gap has yet to be closed. But I believe there are some really cool experiments being designed that will put quantum mechanics and gravity into the same experiment that will stir things up a lot.
    The issue of how Time is perceived in physics has always generated a lot of argument. The Relativity believers (“Einsteinians “) think time is purely an artifact, The QM people seem to think otherwise. My own hypothesis is that time results from the dynamics of the human brain as it interacts with the physical universe, Most likely it results from a bifurcation phenomena in the neural dynamics of the brain. This results in the perception of one-dimensional time and three-dimensional of space. I am actively working to figure out the details.

  • It was Richard Feynman who likened particle collider research to smashing Swiss watches together & then trying to figure out how the mechanical escapement worked from the parts that fell out. How ironic then that Skinner should pull out the busted Ming vase analogy for Feynman diagrams.

  • Several comments questioned the material the amplihedron is made of. Well, it is a mathematical object, so it’s made of the same stuff as a circle – information, pure and simple!
    I want a master origamist to make a model of an amplihedron. We may be approaching Einstein’s prediction that the Theory of Everything would be so simple that a kindergarten child would understand it.
    I wonder what Lee Smolin and Lisa Randall think about this. Hey breaker breaker, got your ears on?
    Lee believes that a TOE would explain spacetime as well as mass and energy. He thinks superstrings are not fundamental because they assume spacetime and operate in it.

  • The ‘Amplituhedron’ is essentially a hyper-symmetric view of measured relationships among quantum particles. As implied in the root article (as i understand it), like other symmetry pictures there are sub-structures that contain known and as yet unknown physical relationships in our mundane and higher subset of ‘space-time-?. One supposes that, for instance, Maxwell’s Equations are the descriptors of one such subset; if so, then one further imagines that ‘c’ the velocity of light need not be a constant except in the limited-boundaries of our as yet ‘space-time’. ‘Above’ 4 space, there may be Amplituhedron super volumes (canonical spaces) where ‘c’ is not a constant nor is a limiting ‘velocity’. Further, one seems moved/motivated to posit that Relativity itself is a phenomenon of our subset symmetry of the Grand Grassian? But, not to entirely eliminate relativity, one imagines if there is a correlative concept to super symmetry like relativity, and so on…maybe the hyper form of Amplituhedral physics will reveal a ‘Relativity’ principle having to do with the boundaries of time, the extensions of multi-verses, etc.

  • Seems to me that it would mean space/time is crystalline in nature =/
    …. I had a thought of Strings interacting locally with each other, could this cause a gravity like effect when expanded into the macro scale?

    I’m not a scientist, just read a lot…

  • This paper in PRL by a postdoc at MIT shows via AdS/CFT that the creation of nonlocal quantum entanglement yields wormholes:

    Also, this paper published in Nature Physics is just one of multiple different papers which show that nonlocal quantum entanglement must originate outside of or beyond 4D spacetime:

    It might be interesting to think about the above findings in terms of the new amplituhedron found by Arkani-Hamed and Trnka.

  • Also, this paper published in “Science” shows that the sudden death of nonlocal quantum entanglement (ESD) is common in Nature but neither clearly deterministic nor stochastic:

  • I am not a scientist so please forgive me if my question is stupid.

    Could this multidimensional shape be a possibility for a calabi yau manifold?

  • Probability is not reality; it’s just the result of using measuring tools and procedures that are too gross and inefficient to bring into full resolution what they’re trying to measure. Don’t mistake the ruler for the measurement. Physics hasn’t yet caught up with reality, but that doesn’t mean there IS no reality.

  • The real story here is Hermann Grassman’s. Published in 1844, his masterwork on linear extension theory is profoundly revolutionary. It was largely ignored during his lifetime, only to re-emerge time and time again in different guises. One person did grasp completely its full implications, and that was William Clifford. He incorporated the core of Grassmann’s theory into what he called geometric algebra. That is just what it sounds like, a compete algebraic system for manipulating objects in n-dimentional space. As you work through the algebraic manipulations, you can be simultaneously rotating, translating, mirroring, projecting, and mapping the objects you modeled with the algebra, depending on the problem and the version of Clifford’s algebra that you use to solve it. Unwrap the algebraic package when you’re done, and your objects are transformed accordingly, inhabiting their new positions.

    It’s taken over 150 years but Grassmann’s vision, as David Hestenes calls it, has emerged from the shadows to shine a light on what we perceive as physical reality.

  • Nice article. Explains it about as well as can be done without getting too technical.
    The geometry in question here is not “real” space (spacetime), but is rooted in energy-momentum space, the complement of spacetime after a Fourier transform.
    The amplituhedron doesn’t require giving up unitarity or locality. But it doesn’t require them, which is nice. It’s also independent of perturbation theory. Thus the method might be useful for including gravity. It means as well that “effective” field theories can be studied with the same method.
    What remains to be seen is if real dynamical content is there, or if the amplituhedron ends up like the S-matrix and “bootstrap” theories of the 1960s — all kinematics and global symmetries, no dynamics.

  • Off the technical aspect, this discovery brings back a book read decades ago by Hermann Hesse – Der Glasperlenspiel – aka, the Glass Bead Game – aka Magister Ludi. How every other subject is based on mathematics…..

  • Obviously,multidimensional arrays provide mathematical and inteligible(sometimes) basis for explaining the nature of what otherwise(in our dull 3-dimension euclidean space)would remain paradoxical .Epistemologically,this trend is no news.Just consider Einstein´s Space-Time conception or de Broglie ´s Wave nature of matter .The thing is a NEW WAY of thinking about consistent representation of “what is given” to us .The promising suspect of finding a sustainable explanation for GRAVITY at quantum-physics level gives this “ARTISTIC” Amplituhedron a very beautiful bonus anyway.

  • Incredibly written.
    Seems to me that it is heading in a direction to explaining how spin states can result in geometric shapes. Structure/function of quantum (maybe the forbidden ‘aether’) causes motion to exist only in spin states which may be why spherical geometrical shapes are the most efficient. Charge, magnetism, with angular motion creating torus shapes of integer and half-integer spins (inverting/reverting of the laterally spinning skein of the flowing torus reminiscent of Klein aspects), with an overlapping magnetic field spins that create areas of synergy through that when resonating properly create something similar to standing waves.

    It would be similar to the research done here, Emergent Collective Phenomena in a Mixture of Hard Shapes through Active Rotation: concerning interacting crystal spins that behave as if ‘living’. Which I could see manifesting from chaos to fractal and organization, as seen here in the article, Probing the Edge of Chaos:

  • Basically this is Descartes, saying, there is no empty space, meaning, there is no non-geometrically structured space.

    Of course, the Amplituhedron is extended, it’s volume can be calculated, but it is not extended in a pre-existing empty space.

    Ok, next step: when there is no preexisting empty space, there is no locality. Why not? Well, things are not situated in a pre-existing empty space, so locality cannot arise, in anyway, as a problem.

  • I would say that there are types of locality, but they are temporary. As an example, double layers would define ‘bubbles’ of localization. The Earth’s magnetosphere being one. The heliosphere another one, etc. Temporary closed systems (locality) within other temporary closed systems, etc etc, all existing within an open system (non-locality).
    Oops, but that would have a possible conclusion being that the Big Bang never happened.

  • Yes!! geometry seems to be the key player in sub quantum and astronomical realms of the universe.

  • Exceptionally well written article; much better than 98% of the science journalism I read. I hope the author stays in the field and writes some books.

  • “They say scattering processes that include gravity particles may be possible to describe with the amplituhedron, or with a similar geometric object.”

    Pardon my question, but does this statement mean that they are using the Amplituhedron to attempt to prove the existence of gravitons?

  • One thing that concerned me was the requirement of the “Positive Grassmanian,” for the Amplituhedron. However, if you consider that our perception of Space-Time is a positive , with gravity possibly being the main reason for the concavity of the surface (and the boundaries for the , then the requirement actually makes sense, in those terms. Given those assumptions, then gravity is a fundamental aspect of the Amplituhedron.

    It does make you wonder about whether Negative Grassmanians could exist for Amplituhedrons, and what they signify…

  • As many of you I’m not “deeply” educated on physics. That said, I would suggest you to not be afraid if you’re not getting all of understanding on the matter. My thought is, if you(as do I) like physics, keep reading and you’ll day-after-day better understanding. Open your mind, be receptive for this weird and amazing world.
    Thanks for the article!!!

  • Isn’t all of this based on the premise the math describes the physical reality as opposed to just a tool to predict it? In other words Occam’s razor, that the simplest equation is the true physics? Perhaps this isn’t the case and there is no one to one correspondence. That might be easier to swallow than giving up space-time.

  • This is an interesting approach. I've long held that geometry holds a special significance in theory's attempting to describe the universe's design and evolution. These efforts are best understood as approximations of how the universe works. Although several are finely detailed, none are 100% accurate. Some are fortunate enough to have experimental or observational evidential support , all have unresolved issues. This one kind of reminds me of how dna expresses proteins by exposing specific segments of code. (I digress).

  • "Time" is not a "thing". Time is a human thought concept that we use to measure a sequence of events. When a star goes nova there is nothing that can make it "un-happen" in any dimension. There was not a "time substance" in the original matter that exploded at the big bang. Trying to find a physical force that comprises time is a waste of time. Just because humans base everything on "time" doesn't make it real, only a perception. Energy and matter is the only reality in our universe.

  • The pre-history of the "amplituhedron" is long and curious. I'm surprised, reading this article again after two years, that there's no mention of the long years of toil put in by Bern, Kosower, and Dixon on calculating amplitudes in gauge theory. The whole idea is there in ovo. For example, see:

    My bet is that it's related to fluxes and a non-coordinate-way way of quantizing fields. We make a mistake when we start with things already embedded in spacetime with a specified coordinate system. We're looking at the shards of the Ming vase in that case. That's always been the problem with many (but not all) approaches to quantum gravity, definitely including string theory, a point made very well by Lee Smolin.

  • This object is very similar in some respects to the 'Platonia' framework proposed by Julian Barbour in his book The End of Time.

  • To me it's all reminiscent of ancient Pythagorean philosophy and Kepler's neo-pythagorean mysterium cosmographicum in which he made the attempt to define planetary distances using the 5 platonic solids. Ancient metaphysical Renaissance? Or a shift in our concept of complexity? Interesting either way

  • This theory is a perturbative one, correct me if I'm wrong please. And perturbative theory of strong couplings will not work?
    In addition, because the Amplituhedron is built on Supersymmetric Yang Mills, it will only encompass Supersymmetric Yang Mills theories, non-abelian, while the world has a big chunk from abelian theory such as QED?

    By that, these amplitudes, might not have a physical equivalent?

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