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In the 1960s, the charismatic physicist
Geoffrey Chew espoused a radical vision of the universe, and with it, a new way
of doing physics. Theorists of the era were struggling to find order in an
unruly zoo of newfound particles. They wanted to know which ones were the
fundamental building blocks of nature and which were composites. But Chew, a
professor at the University of California, Berkeley, argued against such a
distinction. “Nature is as it is because this is the only possible nature
consistent with itself,” he wrote at the time. He believed he could deduce
nature’s laws solely from the demand that they be self-consistent.

Scientists since Democritus had taken a reductionist
approach to understanding the universe, viewing everything in it as being built
from some kind of fundamental stuff that cannot be further explained. But
Chew’s vision of a self-determining universe required that all particles be
equally composite and fundamental. He conjectured that each particle is
composed of other particles, and those others are held together by exchanging
the first particle in a process that conveys a force. Thus, particles’
properties are generated by self-consistent feedback loops. Particles, Chew
said, “pull themselves up by their own bootstraps.”

Chew’s approach, known as the bootstrap philosophy,
the bootstrap method, or simply “the bootstrap,” came without an operating
manual. The point was to apply whatever general principles and consistency
conditions were at hand to infer what the properties of particles (and
therefore all of nature) simply had to be. An early triumph in which Chew’s
students used the bootstrap to predict the mass of the rho meson — a particle
made of pions that are held together by exchanging rho mesons — won many
converts.

But the rho meson turned out to be something of a
special case, and the bootstrap method soon lost momentum. A competing theory
cast particles such as protons and neutrons as composites of fundamental
particles called quarks. This theory of quark interactions, called quantum
chromodynamics, better matched experimental data and soon became one of the
three pillars of the reigning Standard Model of particle physics.

But the properties of individual quarks seemed
arbitrary, and in another universe they might have been different. Physicists
were forced to recognize that the set of particles that happen to populate the
universe do not reflect the only possible consistent theory of nature. Rather,
an endless variety of possible particles can be imagined interacting in any
number of spatial dimensions, each situation described by its own “quantum
field theory.”

Geoffrey Chew giving a seminar in Berkeley,
California, in 1961.

The bootstrap languished for decades at the bottom
of the physics toolkit. But recently the field has been re-energized as
physicists have discovered novel bootstrap techniques that appear to solve many
problems. While consistency conditions still aren’t much help for sorting out
complicated nuclear particle dynamics, the bootstrap is proving to be a
powerful tool for understanding more symmetric, perfect theories that,
according to experts, serve as “signposts” or “building blocks” in the space of
all possible quantum field theories.

As the new generation of bootstrappers explores this
abstract theory space, they seem to be verifying the vision that Chew, now 92
and long retired, laid out half a century ago — but they’re doing it in an
unexpected way. Their findings indicate that the set of all quantum field
theories forms a unique mathematical structure, one that does indeed pull
itself up by its own bootstraps, which means it can be understood on its own
terms.

As physicists use the bootstrap to explore the
geometry of this theory space, they are pinpointing the roots of “universality,” a
remarkable phenomenon in which identical behaviors emerge in materials as
different as magnets and water. They are also discovering general features of
quantum gravity theories, with apparent implications for the quantum origin of
gravity in our own universe and the origin of space-time itself. As leading
practitioners David
Poland of Yale University and David
Simmons-Duffin of the Institute for Advanced Study in Princeton, New
Jersey, wrote in a recent
article, “It is an exciting time to be bootstrapping.”

##
__Bespoke
Bootstrap__

__Bespoke Bootstrap__
The bootstrap is technically a method for computing “correlation
functions” — formulas that encode the relationships between the particles
described by a quantum field theory. Consider a chunk of iron. The correlation
functions of this system express the likelihood that iron atoms will be
magnetically oriented in the same direction, as a function of the distances
between them. The two-point correlation function gives you the likelihood that
any two atoms will be aligned, the three-point correlation function encodes
correlations between any three atoms, and so on. These functions tell you
essentially everything about the iron chunk. But they involve infinitely many
terms riddled with unknown exponents and coefficients. They are, in general,
onerous to compute. The bootstrap approach is to try to constrain what the
terms of the functions can possibly be in hopes of solving for the unknown
variables. Most of the time, this doesn’t get you far. But in special cases, as
the theoretical physicist Alexander
Polyakov began to figure out in 1970, the bootstrap takes you all the
way.

Polyakov, then at the Landau Institute for
Theoretical Physics in Russia, was drawn to these special cases by the mystery
of universality. As condensed matter physicists were just discovering, when
materials that are completely different at the microscopic level are tuned to
the critical points at which they undergo phase transitions, they suddenly
exhibit the same behaviors and can be described by the exact same handful of
numbers. Heat iron to the critical temperature where it ceases to be
magnetized, for instance, and the correlations between its atoms are defined by
the same “critical exponents” that characterize water at the critical point
where its liquid and vapor phases meet. These critical exponents are clearly
independent of either material’s microscopic details, arising instead from
something that both systems, and others in their “universality class,” have in
common. Polyakov and other researchers wanted to find the universal laws
connecting these systems. “And the goal, the holy grail of all that, was these
numbers,” he said: Researchers wished to be able to calculate the critical
exponents from scratch.

What materials at critical points have in common,
Polyakov realized,
is their symmetries: the set of geometric transformations that leave these
systems unchanged. He conjectured that critical materials respect a group of
symmetries called “conformal symmetries,” including, most importantly, scale
symmetry. Zoom in or out on, say, iron at its critical point, and you always
see the same pattern: Patches of atoms oriented with north pointing up are
surrounded by patches of atoms pointing downward; these in turn are inside
larger patches of up-facing atoms, and so on at all scales of magnification.
Scale symmetry means there are no absolute notions of “near” and “far” in
conformal systems; if you flip one of the iron atoms, the effect is felt
everywhere. “The whole thing organizes as some very strongly correlated
medium,” Polyakov explained.

The world at large is obviously not conformal. The
existence of quarks and other elementary particles “breaks” scale symmetry by
introducing fundamental mass and distance scales into nature, against which
other masses and lengths can be measured. Consequently, planets, composed of
hordes of particles, are much heavier and bigger than we are, and we are much
larger than atoms, which are giants next to quarks. Symmetry-breaking
makes nature hierarchical and injects arbitrary variables into its correlation
functions — the qualities that sapped Chew’s bootstrap method of its power.

Alexander Polyakov receiving the Physics Frontiers
Prize in Geneva, Switzerland, in 2013.

But conformal systems, described by “conformal field
theories” (CFTs), are uniform all the way up and down, and this, Polyakov discovered, makes
them highly amenable to a bootstrap approach. In a magnet at its critical
point, for instance, scale symmetry constrains the two-point correlation
function by requiring that it must stay the same when you rescale the distance
between the two points. Another conformal symmetry says the three-point
function must not change when you invert the three distances involved. In a landmark 1983 paper known simply
as “BPZ,” Alexander Belavin, Polyakov and Alexander Zamolodchikov showed that
there are an infinite number of conformal symmetries in two spatial dimensions
that could be used to constrain the correlation functions of two-dimensional
conformal field theories. The authors exploited these symmetries to solve for
the critical exponents of a famous CFT called the 2-D Ising model — essentially
the theory of a flat magnet. The “conformal bootstrap,” BPZ’s bespoke procedure
for exploiting conformal symmetries, shot to fame.

Far fewer conformal symmetries exist in three
dimensions or higher, however. Polyakov could write down a “bootstrap equation”
for 3-D CFTs — essentially, an equation saying that one way of writing the
four-correlation function of, say, a real magnet must equal another — but the
equation was too difficult to solve.

“I basically started doing other things,” said Polyakov, who went on to make seminal contributions to string theory and is now a professor at Princeton University. The conformal bootstrap, like the original bootstrap more than a decade earlier, fell into disuse. The lull lasted until 2008, when a group of researchers discovered a powerful trick for approximating solutions to Polyakov’s bootstrap equation for CFTs with three or more dimensions. “Frankly, I didn’t expect this, and I thought originally that there is some mistake there,” Polyakov said. “It seemed to me that the information put into the equations is too little to get such results.”

##
__Surprise
Kinks__

__Surprise Kinks__
In 2008, the Large Hadron Collider was about to
begin searching for the Higgs boson, an elementary particle whose associated
field imbues other particles with mass. Theorists Riccardo Rattazzi in Switzerland, Vyacheslav Rychkov in
Italy and their collaborators wanted to see whether there might be a conformal
field theory that is responsible for the mass-giving instead of the Higgs. They
wrote down a bootstrap equation that such a theory would have to satisfy.
Because this was a four-dimensional conformal field theory, describing a
hypothetical quantum field in a universe with four space-time dimensions, the
bootstrap equation was too complex to solve. But the researchers found a way to put bounds on
the possible properties of that theory. In the end, they concluded that no such
CFT existed (and indeed, the LHC found the Higgs boson in 2012). But their new
bootstrap trick opened up a gold mine.

Their trick was to translate the constraints on the
bootstrap equation into a geometry problem. Imagine the four points of the
four-point correlation function (which encodes virtually everything about a
CFT) as corners of a rectangle; the bootstrap equation says that if you perturb
a conformal system at corners one and two and measure the effects at corners
three and four, or you tickle the system at one and three and measure at two
and four, the same correlation function holds in both cases. Both ways of
writing the function involve infinite series of terms; their equivalence means
that the first infinite series minus the second equals zero. To find out which
terms satisfy this constraint, Rattazzi, Rychkov and company called upon
another consistency condition called “unitarity,” which demands that all the
terms in the equation must have positive coefficients. This enabled them to
treat the terms as vectors, or little arrows that extend in an infinite number
of directions from a central point. And if a plane could be found such that, in
a finite subset of dimensions, all the vectors point to one side of the plane,
then there’s an imbalance; this particular set of terms cannot sum to zero, and
does not represent a solution to the bootstrap equation.

Physicists developed algorithms that allowed them to
search for such planes and bound the space of viable CFTs to extremely high
accuracy. The simplest version of the procedure generates “exclusion plots”
where two curves meet at a point known as a “kink.” The plots rule out CFTs
with critical exponents that lie outside the area bounded by the curves.

*Lucy Reading-Ikkanda/Quanta Magazine. Graph adapted from Nature Physics 537 (2016)*

Surprising features of these plots have emerged. In
2012, researchers used Rattazzi and Rychkov’s trick to home in on the valuesof the critical
exponents of the 3-D Ising model, a notoriously complex CFT that is in the same
universality class as real magnets, water, liquid mixtures and many other
materials at their critical points. By 2016, Poland and Simmons-Duffin had calculated the two main critical
exponents of the theory out to six decimal places. But even more
striking than this level of precision is where the 3-D Ising model lands in the
space of all possible 3-D CFTs. Its critical exponents could have landed
anywhere in the allowed region on the 3-D CFT exclusion plot, but unexpectedly,
the values land exactly at the kink in the plot. Critical exponents
corresponding to other well-known universality classes lie at kinks in other
exclusion plots. Somehow, generic calculations were pinpointing important
theories that show up in the real world.

The discovery was so unexpected that Polyakov
initially didn’t believe it. His suspicion, shared by others, was that “maybe
this happens because there is some hidden symmetry that we didn’t find yet.”

“Everyone is excited because these kinks are unexpected and interesting, and they tell you where interesting theories live,” said Nima Arkani-Hamed, a professor of physics at the Institute for Advanced Study. “It could be reflecting a polyhedral structure of the space of allowed conformal field theories, with interesting theories living not in the interior or some random place, but living at the corners.” Other researchers agreed that this is what the plots suggest. Arkani-Hamed speculates that the polyhedron is related to, or might even encompass, the “amplituhedron,” a geometric object that he and a collaborator discovered in 2013 that encodes the probabilities of different particle collision outcomes — specific examples of correlation functions.

Researchers are pushing in all directions. Some are
applying the bootstrap to get a handle on an especially symmetric
“superconformal” field theory known as the (2,0)
theory, which plays a role in string theory and is conjectured to exist in
six dimensions. But Simmons-Duffin explained that the effort to explore CFTs
will take physicists beyond these special theories. More general quantum field
theories like quantum chromodynamics can be derived by starting with a CFT and
“flowing” its properties using a mathematical procedure called the
renormalization group. “CFTs are kind of like signposts in the landscape of
quantum field theories, and renormalization-group flows are like the roads,”
Simmons-Duffin said. “So you’ve got to first understand the signposts, and then
you can try to describe the roads between them, and in that way you can kind of
make a map of the space of theories.”

Tom
Hartman, a bootstrapper at Cornell University, said mapping out the space
of quantum field theories is the “grand goal of the bootstrap program.” The CFT
plots, he said, “are some very fuzzy version of that ultimate map.”

Uncovering the polyhedral structure representing all
possible quantum field theories would, in a sense, unify quark interactions,
magnets and all observed and imagined phenomena in a single, inevitable
structure — a sort of 21st-century version of Geoffrey Chew’s “only possible
nature consistent with itself.” But as Hartman, Simmons-Duffin and scores of
other researchers around the world pursue this abstraction, they are also using
the bootstrap to exploit a direct connection between CFTs and the theories many
physicists care about most. “Exploring possible conformal field theories is
also exploring possible theories of quantum gravity,” Hartman said.

##
__Bootstrapping
Quantum Gravity__

__Bootstrapping Quantum Gravity__
The conformal bootstrap is turning out to be a power
tool for quantum
gravity research. In a 1997 paper that is now one of the most highly
cited in physics history, the Argentinian-American theorist Juan Maldacenademonstrated a mathematical
equivalence between a CFT and a gravitational space-time environment
with at least one extra spatial dimension. Maldacena’s duality, called the
“AdS/CFT correspondence,” tied the CFT to a corresponding “anti-de Sitter
space,” which, with its extra dimension, pops out of the conformal system like
a hologram. AdS space has a fish-eye geometry different from the geometry of
space-time in our own universe, and yet gravity there works in much the same
way as it does here. Both geometries, for instance, give rise to black holes —
paradoxical objects that are so dense that nothing inside them can escape their
gravity.

Existing theories do not apply inside black holes;
if you try to combine quantum theory there with Albert Einstein’s theory of
gravity (which casts gravity as curves in the space-time fabric), paradoxes
arise. One major question is how black holes manage to preserve quantum
information, even as Einstein’s theory says they evaporate. Solving this
paradox requires physicists to find a quantum theory of gravity — a more
fundamental conceptualization from which the space-time picture emerges at low
energies, such as outside black holes. “The amazing thing about AdS/CFT is, it
gives a working example of quantum gravity where everything is well-defined and
all we have to do is study it and find answers to these paradoxes,”
Simmons-Duffin said.

If the AdS/CFT correspondence provides theoretical
physicists with a microscope onto quantum gravity theories, the conformal
bootstrap has allowed them to switch on the microscope light. In 2009,
theorists used the bootstrap to
find evidence that every CFT meeting certain conditions has an
approximate dual gravitational theory in AdS space. They’ve since been working
out a precise dictionary to translate between critical exponents and other
properties of CFTs and equivalent features of the AdS-space hologram.

Over the past year, bootstrappers like Hartman and Jared Kaplan of
Johns Hopkins University have made quick progress in understanding how black
holes work in these fish-eye universes, and in particular, how information gets
preserved during black hole evaporation. This could significantly impact the
understanding of the quantum nature of gravity and space-time in our own
universe. “If I have some small black hole, it doesn’t care whether it’s in AdS
space; it’s small compared to the size of the curvature,” Kaplan explained. “So
if you can resolve these conceptual issues in AdS space, then it seems very
plausible that the same resolution applies in cosmology.”

It’s far from clear whether our own universe
holographically emerges from a conformal field theory in the way that AdS
universes do, or if this is even the right way to think about it. The hope is
that, by bootstrapping their way around the unifying geometric structure of
possible physical realities, physicists will get a better sense of where our
universe fits in the grand scheme of things — and what that grand scheme is.
Polyakov is buoyed by the recent discoveries about the geometry of the theory
space. “There are a lot of miracles happening,” he said. “And probably, we will
know why.”

This post was written by Usman Abrar. To contact the
writer write to iamusamn93@gmail.com.
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