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.
Illustration by Andy Gilmore

That’s right ladies and gentlemen, quantum mechanics just got easier to understand. A team of physicists have released a paper showing their discovery of a jewel-like geometric structure that takes equations, which can be thousands of terms long, and simplifies them to a single term. This discovery is poised to dramatically simplify the equations particle physicists use when calculating particle interactions. It also proposes the uncomfortable idea that space and time are not fundamental aspects of our reality, and it brings us much closer to unifying gravity and quantum theory under one comprehensive model.

The discovery comes on the heels of decades of research in particle interactions. Particle interactions are some of the most basic and common events found in nature. Traditionally, these interactions have been very difficult or even impossible to calculate. Scientists required the use of the world’s most powerful computers to calculate even the simplest interactions. This new geometric structure, called the amplituhedron, is so simple that a particle physicist could calculate these interactions, by hand, on a single sheet of paper.

That, in case you were wondering, is insanely impressive. Harvard University theoretical physicist, Jacob Boujaily, and founder of this idea, said, “The degree of efficiency is mind-boggling. You can easily do, on paper, computations that were unfeasible even with a computer before.”

The Basic Idea

This theory is revolutionary on a number of fronts. At the moment, it’s being catapulted into the forefront of grand unified theory research. Some physicists suspect that a geometric object similar to the amplituhedron could eventually lead to a bridge that connects the physics of the very large with the physics of the very small. To date, all of the unified theories that are proposed are riddled with serious and deep-rooted problems, such as paradoxes and infinities. To unify macro and micro physics, the amplituhedron is paving the way to eliminate two of physics deeply rooted points and some of quantum theory’s central pillars: locality and unitarity.

Image Credit: Georg Johann
Image Credit: Georg Johann

Simply put, unitarity is the idea that the sum of all probabilities describing every potential outcome of any quantum event is always equal to one (yes, that was the simple was of saying it). This places an inherent restriction on the amount of evolution that is allowed in any quantum system. Following the same “simple” trend, locality is basically the idea that particles can only interact with, and be influenced by, particles occupying space immediately surrounding them. It’s important to note that locality exists in quantum mechanics largely because special relativity insists upon it. Experimentally, we have shown through quantum entanglement that there seems to be a way to get around locality in the quantum world. In contrast, unitarity is a mathematical construction that helps to make nice round equations. In quantum field theory, both locality and unitarity are central concepts, but there is a catch. When attempting to add gravity to quantum theory, under certain situations, these two pillars (locality and unitarity) break down and stop working. This presents some amount of evidence that neither principle is a fundamental aspect of nature.

This is where the amplituhedron comes in. This geometric shape isn’t constructed by using the probabilities innate to spacetime, but instead suggests that the nature of spacetime is an attribute of the geometry of the amplituhedron. Our idea about the fabric of reality is just that– fabricated, an imaginary construct we have laying over the deeper and more fundamental construction of spacetime. According to David Skinner, a theoretical physicist who calls the Cambridge University home, “It’s a better formulation that makes you think about everything in a completely different way.”

The Complicated World of Particle Interactions

The amplituhedron is a very menacing, beautiful, complicated, multifaceted object that exists in higher dimensions. In principle, you can use the volume of this object to calculate all of the most basic features of reality, known in quantum mechanics as “scattering amplitudes.” This computation describes the probabilities of particles changing into other particles when colliding. These types of calculations are routinely made and tested at particle colliders such as the Large Hadron Collider (LHC). To understand the importance of the amplituhedron, we must first look at where it all began, 60-years ago with the development of Feynman diagrams.

Image <a href="">source</a>
Image source

Named after the Nobel winning Richard Feynman, these diagrams describe all of the ways a particle could scatter, and then the likelihood of any given outcome actually occurring. Feynman diagrams range from the trivially simple to the impossibly difficult. The simplest Feynman diagrams resemble trees, while the more complicated ones have one or more loops that explain particles turning into a virtual particle. A virtual particle is interesting because they aren’t observed in nature, but many physicists have regarded them as a mathematical necessity because they were required to achieve unitarity.

Though Feynman’s diagrams were a stroke of genius , they were simply the wrong tool to use to calculate nuclear particle interactions. In fact, the fact that we are able to compute anything at all is the prime discovery of the computer age; the number of diagrams required to describe something as simple as the 2-gluon to 4-gluon interaction gets so explosively large that scientists didn’t start those computations until the age of computers.  You see, to describe the collision of two gluons that result in four gluons in a lower energy state, particle physicists at the LHC require the use of 220 Feynman diagrams. Together, these diagrams represent thousands of terms involved in the computation that are necessary to determine the scattering amplitude. In short, scientists have realized that Feynman diagrams, though beautiful, are effective ways to calculate a single mathematical object–they are laborious, require many different pieces, and are so numerous that it makes it difficult to do computations even with computers. Physicists are trying to move from that “incalculable” process to a single calculation that (thought difficult) is possible for humans to do (and certainly much easier for computers).

This started with theoretical work preparing for the completion of the Superconducting Super Collider (SSC) that was to be built in Texas (but eventually canceled). Physicists wanted to create a background framework describing scattering amplitudes with which to test the SSC and look for exotic or interesting signals. Physicists quickly determined that creating such a framework for even simple 2-gluon to 4-gluon interactions was so complicated that “they may not be evaluated in the foreseeable future.” Then, in the 1980s, this gluon interaction was simplified from an equation containing several billion terms to a single formula 9-pages long. This was an expression computers of the time could handle, and quantum field theory got a little more manageable. This type of simplifying laid the groundwork for the amplituhedron.

Enter: The Amplituhedron

Though the gluon simplification was achieved in the mid-1980s, it took a couple of decades for particle physicists to really start putting that revolution to use. This started in the mid-2000s when physicists started to find patterns in the scattering amplitudes – and you know how much physicists like patterns. This started the general trend of thought that an underlying mathematical structure might be supporting quantum field theory.

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.  Credit: Arkani-Hamed et al.
Twistor diagrams
Credit: Arkani-Hamed et al.

Eventually, twistor variables and their corresponding diagrams were developed, which attempted to simplify Feynman diagrams even further. These diagrams moved away from describing particle interactions in familiar variables, such as time and position, and used twistor variables instead. These diagrams worked, and gained rapid acceptance among particle physicists, but scientists didn’t understand how they worked, why they worked, or what made them so simple. Arkani-Hamed provides a colorful description by saying, “The terms in these relations were coming from a different world, and we wanted to understand what that world was.”

The amplituhedron didn’t start coming to light until December of 2012 with the discovery of the positive Grassmannian. This geometric object is the result from studying the relationship between recursion relations and their corresponding twistor diagrams. According to the paper, these diagrams act as an instruction manual for calculating the volume of portions of the positive Grassmannian. This object consists of a region of N-dimensional space bounded intersecting planes (where N is the number of interacting particles).

This geometric structure was exciting, but incomplete. The positive Grassmannian’s construction was being restricted by locality and unitarity. Instead of falling together as eloquent things tend to fall together, something was missing. The prevailing idea was that determining the scattering amplitude had to be the answer to some other mathematical question. It turns out, that idea was right.

Credit: Nima Arkani-Hamed
Credit: Nima Arkani-Hamed

The scattering amplitude was determined to be the volume of the amplituhedron. Natalie Wolchover from the Simon Foundation best describes this mathematical structure,”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.”

To reiterate the awesomeness of this achievement, the diagram pictured here is a sketch of an amplituhedron depicting an 8-gluon particle interaction. If you were to attempt to use Feynman diagrams to represent this, you’d be dealing with about 500 pages of algebra.

If the discovery of the amplituhedron wasn’t cool enough, physicists have also discovered a “master amplituhedron.” This object has an infinite number of sides (similar to how a circle has an infinite number of sides in two dimensions) and it can, in theory, describe every possible physical process. All of the amplituhedra that exist in lower dimensions should exist on one of the master’s facets. Skinner describes this structure as having powerful calculational ability and talks of it’s incredible suggestiveness since “they suggest that thinking in terms of spacetime was not the right way of going about this.”

Quantum Gravity: The Future of Physics

This idea has very profound implications. Thus far, all of our theories attempting to unify gravity with quantum mechanics have failed. Because of this, scientists have an impossible time describing the internal workings of black holes, the singularity that started the big bang, and other important objects and events. Ideas like string theory are at the forefront of this research, but they tend to be confusing or unproven/unprovable (or both). According to Arkani-Hamed, ” We can’t rely on the usual familiar quantum mechanical space-time pictures of describing physics. We have to learn new ways of talking about it. This work is a baby step in that direction.”


Image Credit: <a href="">Charles Imes</a>
Image Credit: Charles Imes

It’s very important to note that the amplituhedron, even though it doesn’t include unitarity and locality, also doesn’t include gravity. Physicists are in the middle of working on that very problem. It’s possible the amplituhedron contains the answer to quantum gravity, finally unifying the four fundamental forces of physics, or it’s possible the final geometric shape we seek is a little different.

This work is fantastic, very exciting, and moving along very quickly. As physicists seek to understand the meaning of the amplituhedron the rest of the world gets to wait with bated breath to learn of their findings. It’s possible we could have another Einsteinian-type revolution of our understanding of the nature of reality within our lifetimes. Wouldn’t that be exciting?