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.
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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 Tommy 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 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 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.”
https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/
