
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.
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.
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
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
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.”
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?