Saturday, August 8, 2015

Amplituhedron (For the Rest of Us)

There has been a lot of buzz over the last two years about a geometric shape called the amplituhedron. It has its own Facebook page (with the bold statement "I am the shape of the universe") and more than one jewelry vendor sells trinkets designed after an artist's rendering of its shape shown in several articles. In more scientific circles there are "amplituhedron" discussion groups, where the nature of space-time itself is energetically debated.

The amplituhedron is a new mathematical object coined by its discoverers Nima Arkani-Hamad and Jaroslav Trnka in 2013. The excitement about this object is not only that it might be a first tentative glimpse into the secret nature of space-time. It also questions our concept of space-time itself, the brainchild of Albert Einstein and others and one firmly entrenched in how we think the universe works. When we look into the amplituhedron, we might be peering into something deeper and more fundamental than space-time.

As exciting as this breakthrough is, it is very challenging. There is a wide chasm between the extremely technical and difficult to understand original article and popular accounts of it in magazines like Discover and Scientific American. It's not easy to know the science behind the amplituhedron, or how it fits into modern theoretical physics and there are misconceptions online about exactly what this object can and cannot tell us about the nature of reality at its most fundamental level.


I usually start my investigations at Wikipedia. In this case Wiki offers perhaps the most concise treatment you're going to find, to a fault really. It also tells us nothing about why it's caught our scientific attention like it has, why it's so special. I think it's much better to start with Natalie Wolchover's very good article in Quanta magazine, called A Jewel at the Heart of Quantum Physics. Here she tells the compelling scientific story of the amplituhedron. I hope to build on that story here and try to work out why and how you get to the structure itself.

The amplituhedron is a mathematical object that exists in a branch of theoretical physics called quantum field theory (QFT). Quantum field theory is a conceptual and mathematical framework for studying the behaviours of subatomic particles. It is in some ways an extension of quantum mechanics (QM), the theory that governs how individual particles behave. In QFT, you can describe not just the individual behaviour of particles but how they behave together in space and time, how they act in a field that has spatial dimensions in other words. Think of electrons moving in an electric field for example. We can use QM to describe the workings of a single electron. We move on to describe its motion and behaviour with other electrons and with photons if we use the simplest and best understood branch of QFT called quantum electrodynamics (QED). QFT is a very powerful and very quickly expanding field of study. Like a block party that's getting out of hand, the branches of the theory extend outward to talk to other theories and there are many cases in which QFT's do not agree with each other and some do not even speak the same language. Usually, the very idea of what the particles "live in" varies with the QFT you choose.

As newcomers to theoretical physics we can be happy just to understand the basics of Einstein's 4-dimensional space-time. Yet even here, once we dig a bit deeper, we find that space-time comes in different forms depending on what kind of action we want to describe. If we want to talk about a problem in special relativity, the space-time we use is a simplified type of 4-dimensional smooth Lorentzian manifold called Minkowski space. If we want to describe how space-time stretches and bends under gravity, we use tensor fields in the Lorentzian manifold described by general relativity. We can also assign various symmetries to the mathematical structure of space-time. Usually this is done to simplify the solution to a particular problem (the link above is an excellent lecture all about symmetry in physics). Many space-time theories are physical theories - they are abstractions or models of the physical (real) entity of space and time, but they are not the entity itself, a point that I wish to stress in this article.

Space-time is just one of many mathematical models that attempt to describe the dimensions of space. In theoretical physics, space as a real entity no longer exists. It becomes one of a large number of possible mathematical models, which can house specific particle fields, symmetries and dimensions. They can also have unique dynamics. General relativity space is stretchy but special relativity space isn't. Many of these models are inconceivable in our everyday world.

The amplituhedron exists in a very specific mathematical space described by N=4 supersymmetric Yang-Mills theory. This space is a mathematical model and the amplituhedron is mathematical object within it, and it is defined by its own mathematical space called the positive Grassmannian. It is not important to memorize these terms but instead to grasp that the space we are talking about is not real physical space. You cannot go out and find Yang-Mills space somewhere. It is a simplified system or approach to tackling a complex problem in a new theory. For example, a physicist can put a new theory into N=4 supersymmetric Yang-Mills space and see how it evolves in it. Is it solvable for example? We can think of N=4 supersymmetric Yang-Mills theory is a simplified unreal universe that is perfectly symmetrical and has no gravity. We don't know if the supersymmetry (this article link is an especially good introduction) this theory describes exists in reality or not. No supersymmetric particles have been found yet - particle physicists are looking for them. In this case supersymmetry does the all-important job of simplifying matters - when solving for this theory you can swap out bosons, fermions and scalar fields and the predictions of the theory don't change. N=4 super Yang-Mills, derived from a simple 10-dimensional theory, is closely related to the most popular 11-dimensional string theory called M-theory or matrix theory. The amplituhedron is the result of working string theory into twistor space or twister geometry (this is how we get to the geometrical object), which happens to have the same dimensionality as 3+1 Minkowski space-time.

When this was done, researchers noticed something very unexpected and that is what I wish to focus on here. The real jewel is not so much the pretty colourful object sometimes dubbed the "shape of the universe" but instead the hint that it is making to us - that a geometric approach to understanding space-time might reveal new insights into how it works.

Two Pillars of Theoretical Physics Re-Examined

Like the space-time in which it is formulated, the amplituhedron is not a real object. It is a tool, a spookily efficient tool that greatly simplifies how physicists calculate something called scattering amplitudes. You can try out an introductory lecture on scattering amplitudes by the lead author himself, Nima Arkani-Hamad, at Cornell University. There, you can get a sense of why he is so passionate about this research in general as well. We'll get to this most important part but first, when physicists look at the calculations involved in creating the object it appears that two things we take for granted in physics, two conceptual pillars if you will - locality and unitarity - no longer look fundamental but instead seem to be emergent properties of reality. Much of the internet buzz focuses on this.


Locality has been on shaky ground for several decades now. Non-locality (Einstein's spooky action at a distance) is probably one of the first stinging slaps we feel when we begin to explore quantum physics. Quantum mechanics is to blame. The principle of locality states that an object is only directly influenced by its immediate surroundings. For an action at one point to influence an action at a distant point something like a wave, a particle or an energy field must carry that influence. However, a measurement on one of a pair of entangled particles causes simultaneous collapse of the wave function of the other particle no matter how far away it is, even it it's across the universe. This phenomenon, called quantum entanglement, has been experimentally verified. Therefore, QM might not be a local theory (if you get into this, you will find controversy around this. A number of experts do not accept non-locality despite the evidence, and some have found ways around non-locality, hence my use of "might").

But special relativity (SRT) is a local theory. It requires that no influence can travel faster than the speed of light. If we get back to our opening QED example (electrons and photons in space) of a QFT, we get our first taste of just how utterly complex QFT really is. Under most interpretations of quantum entanglement, understanding QM demands that we discard locality at the subatomic particle level. Understanding SRT demands that we must accept locality, however, because we must have it in order for the theory to make sense. It prohibits any influence from traveling faster than the speed of light through space. And yet QED, which gives a complete mathematical description of how light and matter interact, is built on QM and seamlessly incorporates SRT. To see this contradiction between QED and SRT in action so to speak, I recommend that you examine the famous double-slit experiment.  In two past articles I wrestle with the implications of this experiment: What Is An Electron REALLY? - scroll down to the subheading Young's Double Slit Experiment Peers Into a Secret Quantum World and The Universe Is Real and Not Real - scroll down to subheading What Young's Double Slit Experiment Says. As you mull over this experiment yourself, I don't think it is too much of a stretch to accept the possibility that locality could be emergent rather than fundamental.

Before I go on, I should mention that not all researchers agree that locality and unitarity are in fact emergent phenomena in the amplituhedron theory. Some experts prefer the term "derived" instead because, they argue, when we look at N=4 supersymmetric Yang-Mills theory by itself, both locality and unitarity are preserved exact and they can both be derived from the mathematical description of the amplituhedron developed in this theory. However, as the paper's authors argue, locality and unitarity are not required in the mathematical description of the amplituhedron itself so they can be thought of therefore as emergent phenomena in that they don't necessarily exist at the quantum scale but they show up at the macro scale.

Emergence is any property, law or phenomenon that occurs at macroscopic scales but not at microscopic scales, even though a macroscopic system can be thought of as a conglomerate of microscopic elements. In this case, both locality and unitarity might arise in the system as we move from the microscopic quantum or particle scale up to the macroscopic or everyday scale we ourselves experience and can test. There is a subtle but important distinction between emergent and mathematically derived. The paper's authors go on to make a case for a connection between these "emergent" phenomena and the indeterminism in QM. This is an important little point because much of the excitement in the physics community stems from these two being emergent. I invite you to be the judge. As food for thought, try this exchange on, where the question of emergence versus derived is tackled by a physicist from England.


This phenomenon - the second of two widely accepted beliefs in physics - is just as deeply ingrained in our logical minds. Like locality it makes sense and to consider otherwise jolts us. To describe unitarity we must delve into probabilities. The amplituhedron, and quantum mechanics in general, are all about probabilities, thanks to the uncertainty principle that underscores the theory. You have probably heard of the electron cloud inside an atom. It's a region around the nucleus where an electron might be because we cannot know exactly where it is at any given moment. Put more precisely we can know either the electron's momentum (and velocity) or its position but we cannot know both simultaneously no matter how carefully we measure them. Uncertainty is built into the quantum system. We can however assign probabilities to these values, and we can sum up or integrate the probability densities at all points inside the electron cloud (a tedious process!) and they will add up to one - meaning that there is absolute certainty the electron is in there somewhere.

Particle physicists, those people who smash particles together to see what happens, rely heavily on probabilities, or probability amplitudes or scattering amplitudes as they are most often called, in order to anticipate which particles will come out in any given particle collision. For example, smashing two gluons together in a collider does not give you a single standard result, a consistently identical set of particles every time. The particles, and their momenta, can vary each time even under identical conditions. This is the experimental verification of uncertainty. You might be most likely to get a particular outcome (there is a high probability density for example that two gluons will come out) but there are always less likely possible outcomes as well to consider (a gluon and two photons might come out instead, adding up to the same total energy) and all of these, which approach an infinite number, must be added to get that perfect value of one. You can call it the conservation of probability. We are going to delve deeper into this kind of scenario because atom smashing probability calculations are exactly how the amplituhedron came to be, and exactly why it is such a remarkable breakthrough.

The amplituhedron theory suggests that at a very fundamental level of reality unitarity doesn't have to be conserved. Like locality, it is emergent. Here is a possible explanation of why that might be the case, and it comes from how Arkani-Hamad himself talks about it in his paper: in any collision event, all the probabilities of what will come out and with what energy is actually infinite (meaning unitarity is conserved), but the system of particles in the collider must be finite in reality and this is where unitarity falls down and is not conserved.

You could also argue that both locality and unitarity fail because the measurement itself fails, in the case of black holes in particular. If you could measure a phenomenon and go smaller in scale, down to Planck scale (which is about 1.6 x 10-35 m), the energy you would need to make your measurement would approach infinity. To measure or "see" something you need to detect it - hit it with a particle and let the particle bounce back. A (big wavelength) visible photon definitely won't cut it. A smaller wavelength electron won't do; even a tiny wavelength ultrahigh-energy gamma photon won't work. Even if you could measure it, infinite energy focused on an infinitely small spot would form a black hole, making any measurement attempt impossible, and therefore you can't address the validity of locality or unitarity at Planck scale and lower - the scale in which subatomic particles live. For those black hole enthusiasts, one can also argue that black hole information loss also destroys unitarity.

Keep in mind that mathematically speaking the amplituhedron does not require either locality or unitarity yet in the Yang-Mills theory in which it is formulated both are preserved. Does the possibility that locality and unitarity are emergent phenomena hint that the fundamental reality of the universe is different than what we know of it through quantum mechanics and general relativity? One of the most mystifying things in theoretical physics is that these two vastly successful theories do not talk to each other (they don't commute) and yet the universe seems to be a single seamless reality. Many theorists hope that the amplituhedron or a related theory might be a first (baby) step toward a new framework that can describe both the quantum world and gravity. What the geometric structure of the amplituhedron can do in terms of particle physics calculations offers perhaps the most compelling hint that this theory or one like it might someday reveal a more fundamental simpler unified reality. By discarding Feynman diagrams and using the geometry of the amplituhedron instead, one can drop even the familiar notions of position and time to calculate how particles behave. What does this mean for our notion of space-time?

Is the Amplituhedron a Look at Reality Through A New Kind of Magnifying Glass?

When particle physicists try to predict the outcomes of various particle collisions, they must preserve the unitarity of the system. In a way it is like balancing a chemical equation by adjusting the number of atoms on each side of the equation except that in this case probability is balanced or conserved rather than the number of atoms. Unlike chemical equations, the process of calculating all the probability amplitudes is almost impossibly difficult. Even a very simple 2-particle gluon-gluon collision into 4 less energetic gluons requires a computer to do the work. As the number of particles involved in a collision increases, so does the complexity of the calculations.

1) The Amlituhedron as a Powerful New Method

The amplituhedron is all about particle physics, the branch of physics that studies the nature of particles that make up matter and energy. We often think of subatomic particles as the smallest possible physical objects, like specks of dust. However, subatomic particles are currently understood as excitations of quantum fields, and that is why quantum field theory is used to investigate these particles. The ultimate quantum field theory is the Standard Model of particles. The supersymetric and string theories mentioned, including N=4 supersymmetric Yang-Mills theory, are extensions of the Standard Model.

Particle physicists study the behaviours of subatomic particles and look for new theoretical particles by smashing atoms and particles together in high-energy colliders. Those theoretical particles that belong to supersymmetric theories, for example are suspected to be very massive and very unstable. We don't observe them in everyday matter because they should live only in a very high-energy environment, the kind of environment a high-energy collider can replicate. This is how physicists hunt for these exotic particles and finding them will help answer fundamental questions about why matter and energy is the way it is in our universe.

2) Feynman Diagrams

When two particles are smashed into one another at almost light speed, they are annihilated and a multitude of new particles scatter from the impact, created from the relativistic mass of the original particles (recall that mass is equivalent to energy). In order to predict and identify specific particles, many of which do not exist outside of atoms such as gluons, physicists use something called scattering amplitudes. These are complex numbers that, when they are squared, give probabilities for incoming particles to scatter into particular outgoing particles. In theory you can calculate scattering amplitudes by drawing Feynman diagrams of the collision, using a set of schemes called perturbation theory. It's a way of describing a complex system in terms of a much simpler one. You take a simple system that has a known mathematical solution, add "disturbances" to it, and then measure what happens to the system. Your results are called corrections.

The calculation of scattering amplitudes requires large complex integrals over many variables. They are complicated but they do have a regular structure and they can be represented by Feynman diagrams. The diagram offers a simple visual representation of a huge arcane and abstract formula.  A generic Feynman diagram is shown below.

This one happens to represent a particle-particle interaction. Particles (of momentum p) correspond to the solid lines and the dotted line corresponds to a virtual particle of momentum k. This diagram could represent two electrons interacting through the exchange of a virtual photon, or two neutrons interacting through a virtual pion or two quarks interacting through a virtual gluon. The inner dotted line represents a virtual particle, which is never directly observed and never stays behind as a product. What it does is mediate the interaction. Most often you will see it drawn as a blue squiggly line.

We would be done here if this were the whole story to particle interactions but according to Feynman's work on path integrals, there is actually an infinite number of paths a particle can take from the left side to the ride side. Theoretically it could take off, go around the universe, interact with an enormous number of other particles and come back. Not surprisingly, this introduces complexity to the Feynman diagrams. The diagram above describes the case where the momentum (or energy) of the virtual particle is fixed. It's completely determined by the momenta of the external particles. It is also the sum of their momenta, and most of the contribution to the amplitude comes from this single interaction (called a tree diagram). However, there are many possible ways to get from a to b. There are cases (in fact an infinite number of cases) where the momentum of the virtual mediating particle is not fixed. These are higher order corrections to the perturbation theory. They can be drawn as loop diagrams (the loop or loops is/are drawn where the dotted line is located). These diagrams - one loop (2nd order), two loops (3rd order) and so on - are notorious for being difficult to solve. As you go up in order your corrections contribute a smaller and smaller fraction of the total amplitude. At the same time, the number of equations goes up - in alarming fashion. Loop diagrams indicate collisions where virtual particles interact with each other before branching out as products and these virtual particles can have very high momenta and very complicated interactions.

3) Beyond Feynman Diagrams

In 1948, the Feynman diagram method was a huge breakthrough in particle physics. Finally you had a way to visualize what was going on at the particle level, but what you get is a frustratingly complex picture for all but the simplest of particle interactions. N=4 supersymmetric Yang-Mills theory beautifully simplifies the way-too-heavy Feynman diagram method. How it does this is a question of math. Why it does this is a source of wonder. This theory has three symmetries in it and we can eliminate a large number of scattering amplitudes (corrections) by requiring that each amplitude conform to the three symmetries. Ordinary Yang-Mills theory describes electrodynamics and quantum chromodynamics. It works very well to describe the behaviours of electrons and quarks in other words, and it can be thought of as a very good physical theory. The supersymmetric extension of it, though very useful, has not been found in nature because no supersymmetric particles have been discovered (yet). It is a drastically simplified version of reality and therefore it is considered to be a toy theory rather than a physical theory. I say this as a reminder that neither Feynman's infinite integrals nor the geometric structure of the amplituhedron are meant to be understood as physical processes in nature. If you could look at what's going on with an impossibly powerful magnifying glass (and not make a black hole) you would not see infinite summation arrows or Feynman diagrams or amplituhedrons. I hope I am not insulting you but I continually find myself wanting to do this.

To do the amplituhedron simplification, two mathematical tools are used - twistors and Grassmanians.

First, Andrew Hodges discovered in 2005 that scattering amplitudes could be modeled using a construction called twistor diagrams. You can see examples of them in Wolchover's article. By using a set of formulas called recursion relations, discovered by Ruth Britto, Freddy Cachazo and Bo Feng, hundreds of Feynman tree diagrams could be translated into just a few twistor diagrams. These diagrams encode scattering information as something called multidimensional contour integrals. They basically provide the instructions to construct the positive Grassmanian, which is a high-dimensional structure called the amplituhedron.

The most interesting part here is that the physicists working on this found that the twistors encoded the information in a different way than Feynman diagrams do. Locality and unitarity are not required in the twistor formulation, while both are essential to calculating the Feynman diagrams. The mathematics behind twistor diagrams is difficult because it deals with many-dimensional spaces. Twistor-string theory lives in 9 + 1 space-time dimensions. Twistor space has 6 dimensions and the space-time in which the Feynman diagrams live is 3 + 1. The startling point is that all these theories map onto each other under specific supersymmetric conditions (a surprise in itself). You can reverse the process to get the scattering amplitude by mapping twistor space back onto 3 + 1 space-time. In other words, the amplituhedron is an unreal structure living outside of ordinary space-time, but the particle collisions described by it are real and do live in space-time. For a technical account of this process try Freddy Cachazo's The Geometry of Trees.

The numerous Feynman tree diagrams are encoded in the volume of a positive Grassmanian, a structure in algebraic geometry that is called a convex polytop. An example of a typical one, in three dimensions, is shown below.

The amplituhedron is the particular geometric structure you get when you input scattering amplitudes. To see an artist's rendering of it, see Natalie Wolchover's article A Jewel at the Heart of Physics.


The Feynman diagrams, the twistor diagrams and the positive Grassmanian (the volume of the amplituhedron itself) all encode the scattering amplitude information of particles when they collide with each other, but they live only inside drastically simplified quantum field theory. The relatively simple and elegant structure of the amplituhedron, living in the unique and unreal multidimensional space of N=4 supersymmetric Yang-Mills theory, encodes the same information as the over-encumbered Feynman diagrams that live in our familiar Einsteinian 3 + 1 space-time.

It is hard to ignore the fact that the amplituhedron is a pared down theoretical space where locality, unitarity and even our notions of time and space do not necessarily exist and yet this geometry does the most efficient job of describing particle behaviour. If we could magically look directly at the particles interacting would we see something purely geometric in nature? This is the over-step I try to warn about even though it is very alluring. What the amplituhedron really means is hard to know but it does kind of tap familiar imperfect space-time on the shoulder and ask, "What gives?"