Nima AKANI-HAMED & Jaroslav TRNKA
Amplituhedron : A description of a way to
solve maximally supersymmetric (i.e. N=4) Yang-Mills theory in
An amplituhedron is a geometric structure that enables simplified
calculation of particle interactions in some quantum field
theories. In planar N = 4 supersymmetric Yang–Mills theory, an
amplituhedron is defined as a mathematical space known as the
Amplituhedron theory challenges the notion that space-time
locality and unitarity are necessary components of a model of
particle interactions. Instead, they are treated as properties
that emerge from an underlying phenomenon.
The connection between the amplituhedron and scattering amplitudes
is at present a conjecture that has passed many non-trivial
checks, including an understanding of how locality and unitarity
arise as consequences of positivity.
Research has been led by Nima Arkani-Hamed. Edward Witten
described the work as “very unexpected" and said that "it is
difficult to guess what will happen or what the lessons will turn
out to be."
In the approach, the on-shell scattering process "tree" is
described by a positive Grassmannian, a structure in algebraic
geometry analogous to a convex polytope, that generalizes the idea
of a simplex in projective space. A polytope is a kind of
higher dimensional polyhedron, and the values being calculated are
scattering amplitudes, and so the object is called an
Using Twistor theory, BCFW recursion relations involved in the
scattering process may be represented as a small number of Twistor
diagrams. These diagrams effectively provide the recipe for
constructing the positive Grassmannian, i.e. the amplituhedron,
which may be captured in a single equation. The scattering
amplitude can thus be thought of as the volume of a certain
polytope, the positive Grassmannian, in momentum twistor space.
When the volume of the amplituhedron is calculated in the planar
limit of N = 4 D = 4 supersymmetric Yang–Mills theory, it
describes the scattering amplitudes of subatomic particles. The
amplituhedron thus provides a more intuitive geometric model for
calculations whose underlying principles were until then highly
The twistor-based representation provides a recipe for
constructing specific cells in the Grassmannian which assemble to
form a positive Grassmannian, i.e. the representation describes a
specific cell decomposition of the positive Grassmannian.
The recursion relations can be resolved in many different ways,
each giving rise to a different representation, with the final
amplitude expressed as a sum of on-shell processes in different
ways as well. Therefore any given on-shell representation of
scattering amplitudes is not unique, but all such representations
of a given interaction yield the same amplituhedron.
The twistor approach simplifies calculations of particle
interactions. In a perturbative approach to quantum field theory,
such interactions may require the calculation of hundreds of
Feynman diagrams. In contrast, twistor theory provides an approach
in which scattering amplitudes can be computed in a way that
yields much simpler expressions.
The twistor approach was relatively abstract. The amplituhedron
provides an underlying model. Its geometric nature suggests the
possibility that the nature of the universe, both classical
relativistic spacetime and quantum mechanics, can be described
with geometry. Calculations can be done without assuming the
quantum mechanical properties of locality and unitarity. In
amplituhedron theory, locality and unitarity arise as a direct
consequence of positivity. They are encoded in the positive
geometry of the amplituhedron, via the singularity structure of
the integrand for scattering amplitudes.
Since the planar limit of the N = 4 supersymmetric Yang–Mills
theory is a toy theory that does not describe the real world, the
relevance of this technique for more realistic quantum field
theories is currently unknown, but it provides promising
directions for research into theories about the real world.
New Discovery Simplifies Quantum Physics: Introducing the
a b c d e f Arkani-Hamed & Trnka 2013.
a b c Nima Arkani-Hamed; Bourjaily, Jacob L.; Freddy Cachazo;
Goncharov, Alexander B.; Alexander Postnikov; Jaroslav Trnka
(2012). "Scattering Amplitudes and the Positive Grassmannian".
Ryan O'Hanlon (September 19, 2013). "How to Feel About Space and
Time Maybe Not Existing". Pacific Standard.
Natalie Wolchover (September 17, 2013). "A Jewel at the Heart of
Quantum Physics". Quanta Magazine.
a b Trnka, Jaroslav. "The Amplituhedron". Retrieved 19 September
4 gravitons and a grad student; The Amplituhedron and Other
Excellently Silly Words
Kevin Drum (September 18, 2013). "Maybe Space-Time Is Just an
Illusion". Mother Jones.
Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov and Trnka,
Scattering Amplitudes and the Positive Grassmannian, Arxiv paper
1212.5605 (Dec 2012)
Arkani-Hamed, Nima; Trnka, Jaroslav (2013). The Amplituhedron.
Nima Arkani-Hamed (2013-08-30). "The Amplituhedron" (video). SUSY
2013 Conference Video Archive.
Scattering Without Space-Time Subrahmanyan Chandrasekhar Lecture,
25 September 2012 on YouTube
N = 4 D = 4 super Yang–Mills theory from nLab
Arxiv paper on Total positivity, Grassmannians, and networks (Sept
4 gravitons and a grad student; The Amplituhedron and Other
Excellently Silly Words
Nima Arkani-Hamed, Jaroslav Trnka
[ PDF ]
Perturbative scattering amplitudes in gauge theories have
remarkable simplicity and hidden infinite dimensional symmetries
that are completely obscured in the conventional formulation of
field theory using Feynman diagrams. This suggests the existence
of a new understanding for scattering amplitudes where locality
and unitarity do not play a central role but are derived
consequences from a different starting point. In this note we
provide such an understanding for N=4 SYM scattering amplitudes in
the planar limit, which we identify as ``the volume" of a new
mathematical object--the Amplituhedron--generalizing the positive
Grassmannian. Locality and unitarity emerge hand-in-hand from
September 17, 2013
A Jewel at the Heart of Quantum
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
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
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
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.
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
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
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 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
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
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
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 arXiv.org 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
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.
A sketch of the amplituhedron representing an 8-gluon particle
interaction. Using Feynman diagrams, the same calculation would
take roughly 500 pages of algebra.
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
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
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
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,”
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
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
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.”
Arkani Hamed's Lecture on Amplituhedron
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