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It’s one of
the most brilliant, controversial and unproven ideas in all of physics: string
theory. At the heart of string theory is the thread of an idea that’s run
through physics for centuries that at some fundamental level, all the different
forces, particles, interactions and manifestations of reality are tied together
as part of the same framework.

Instead of
four independent fundamental forces — strong, electromagnetic, weak and
gravitational — there’s one unified theory that encompasses all of them. In
many regards, string theory is the best contender for a quantum theory of
gravitation, which just happens to unify at the highest-energy scales. Even
though there’s no experimental evidence for it, there are compelling
theoretical reasons to think it might be true.

Back in
2015, the top living string theorist, Ed Witten, wrote a piece on what every
physicist should know about string theory. Here’s what that means, even if
you’re not a physicist. When it comes to the laws of nature, its remarkable how
many similarities there are between seemingly unrelated phenomena. The
mathematical structure underlying them is often analogous, and occasionally
even identical.

The
difference between standard quantum field theory interactions (L), for
point-like particles, and string theory interactions (R), for closed strings.
(Wikimedia Commons user Kurochka)

The way that
two massive bodies gravitate, according to Newton’s laws, is almost identical
to the way that electrically charged particles attract-or-repel. The way a
pendulum oscillates is completely analogous to the way a mass on a spring moves
back-and-forth, or the way a planet orbits a star. Gravitational waves, water
waves, and light waves all share remarkably similar features, despite arising
from fundamentally different physical origins.

And in the
same vein, although most don’t realize it, the quantum theory of a single
particle and how you’d approach a quantum theory of gravity are similarly
analogous. The way quantum field theory
works is that you take a particle and you perform a mathematical “sum over
histories.” You can’t just calculate where the particle was and where it is and
how it got to be there, since there’s an inherent, fundamental quantum
uncertainty to nature.

A Feynman
diagram representing electron-electron scattering, which requires summing over
all the possible histories of the particle-particle interactions. (Dmitri
Fedorov)

Instead, you
add up all the possible ways it could have arrived at its present state (the
“past history” part), appropriately weighted probabilistically, and then you
can calculate the quantum state of a single particle. If you want to work with
gravitation instead of quantum particles, you have to change the story a little
bit. Because Einstein’s General Relativity isn’t concerned with particles, but
rather the curvature of space-time, you don’t average over all possible
histories of a particle.

In lieu of
that, you average instead over all possible space-time geometries. Working in
three spatial dimensions is very difficult, and when a physics problem is
challenging, we often try and solve a simpler version first. If we go down to
one dimension, things become very simple.

Gravity,
governed by Einstein, and everything else (strong, weak and electromagnetic
interactions), governed by quantum physics, are the two independent rules known
to govern everything in our Universe. (SLAC National Accelerator Laboratory)

The only
possible one-dimensional surfaces are an open string, where there are two
separate, unattached ends, or a closed string, where the two ends are attached
to form a loop. In addition, the spatial curvature — so complicated in three
dimensions — becomes trivial. So what we’re left with, if we want to add in
matter, is a set of scalar fields (just like certain types of particles) and
the cosmological constant (which acts just like a mass term): a beautiful
analogy.

The extra
degrees of freedom a particle gains from being in multiple dimensions don’t
play much of a role; so long as you can define a momentum vector, that’s the
main dimension that matters. In one dimension, therefore, quantum gravity looks
just like a free quantum particle in any arbitrary number of dimensions.

A graph with
trivalent vertices is a key component of constructing the path integral
relevant for 1-D quantum gravity. (Phys. Today 68, 11, 38 (2015))

The next
step is to incorporate interactions, and to go from a free particle with no
scattering amplitudes or cross-sections to one that can play a physical role,
coupled to the Universe. Graphs, like the one above, allow us to describe the
physical concept of action in quantum gravity. If we write down all the
possible combinations of such graphs and sum over them — applying the same laws
like conservation of momentum that we always enforce — we can complete the
analogy.

Quantum
gravity in one dimension is very much like a single particle interacting in any
number of dimensions. The next step would be to move from one spatial dimension
to 3+1 dimensions: where the Universe has three spatial dimensions and one time
dimension. But this theoretical “upgrade” for gravity may be very challenging.

The
probability of finding a quantum particle at any particular location is never
100%; the probability is spread out over both space and time. (Wikimedia
Commons user Maschen)

Instead,
there might be a better approach, if we chose to work in the opposite
direction. Instead of calculating how a single particle (a zero-dimensional
entity) behaves in any number of dimensions, maybe we could calculate how a
string, whether open or closed (a one-dimensional entity) behaves. And then,
from that, we can look for analogies to a more complete theory of quantum
gravity in a more realistic number of dimensions.

Instead of
points and interactions, we’d immediately start working with surfaces,
membranes, etc. Once you have a true, multi-dimensional surface, that surface
can be curved in non-trivial ways. You start getting very interesting behavior
out; behavior that just might be at the root of the spacetime curvature we
experience in our Universe as General Relativity.

Feynman
diagrams (top) are based off of point particles and their interactions.
Converting them into their string theory analogues (bottom) gives rise to
surfaces which can have non-trivial curvature. (Phys. Today 68, 11, 38 (2015))

While 1D
quantum gravity gave us quantum field theory for particles in a possibly curved
spacetime, it didn’t describe gravitation itself. The subtle piece of the
puzzle that was missing? There was no correspondence between operators, or the
functions that represent quantum mechanical forces and properties, and states,
or how the particles and their properties evolve over time. This
“operator-state” correspondence was a necessary, but missing, ingredient.

But if we
move from point-like particles to string-like entities, that correspondence
shows up. As soon as you upgrade from particles to strings, there’s a real
operator-state correspondence. A fluctuation in the spacetime metric (i.e., an
operator) automatically represents a state in the quantum mechanical
description of a string’s properties. So you can get a quantum theory of gravity
in spacetime from string theory.

Deforming
the spacetime metric can be represented by the fluctuation (labelled ‘p’), and
if you apply it to the string analogues, it describes a spacetime fluctuation
and corresponds to a quantum state of the string. (Phys. Today 68, 11, 38
(2015))

But that’s
not all you get: you also get quantum gravity unified with the other particles
and forces in spacetime, the ones that correspond to the other operators in the
field theory of the string. There’s also
the operator that describes the spacetime geometry’s fluctuations, and the
other quantum states of the string. The biggest news about string theory is
that it can give you a working quantum theory of gravity.

That doesn’t
mean it’s a foregone conclusion, however, that string theory is The path to
quantum gravity. The great hope of string theory is that these analogies will
hold up at all scales, and that there will be an unambiguous, one-to-one
mapping of the string picture onto the Universe we observe around us.

Brian Greene
presenting on String Theory. (NASA/Goddard/Wade Sisler)

Right now,
there are only a few sets of dimensions that the string/superstring picture is
self-consistent in, and the most promising one doesn’t give us the
four-dimensional gravity of Einstein that describes our Universe. Instead, we
find a 10-dimensional Brans-Dicke theory of gravity. In order to recover the
gravity of our Universe, you must “get rid of” six dimensions and take the
Brans-Dicke coupling parameter, ω, to infinity.

If you’ve
heard of the term compactification in the context of string theory, that’s the
hand-waving word to acknowledge that we must solve these puzzles. Right now,
many people assume that there exists a complete, compelling solution to the
need for compactification. But how you get Einstein’s gravity and 3+1
dimensions from the 10-dimensional Brans-Dicke theory remains an open challenge
for string theory.

A 2-D
projection of a Calabi-Yau manifold, one popular method of compactifying the
extra, unwanted dimensions of String Theory. (Wikimedia Commons user Lunch)

String
theory offers a path to quantum gravity, which few alternatives can truly
match. If we make the judicious choices of “the math works out this way,” we
can get both General Relativity and the Standard Model out of it. It’s the only
idea, to date, that gives us this, and that’s why it’s so hotly pursued. No
matter whether you tout string theory’s successes or failure, or how you feel
about its lack of verifiable predictions, it will no doubt remain one of the
most active areas of theoretical physics research.

At its core,
string theory stands out as the leading idea of a great many physicists’ dreams
of an ultimate theory.

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