MATT O'DOWD: Thanks
to the Great Courses Plus for supporting
PBS Digital Studios. Some see string theory
as the one great hope for a theory of
everything that will unify quantum mechanics and
gravity and so unify all of physics into one great,
glorious theory of everything. Others see string theory
as a catastrophic dead end, one that has consumed a
generation of geniuses with nothing to show for it. So why are some of the
most brilliant physicists of the past 30-plus
years so sure that string theory is right? [MUSIC PLAYING] Why has string theory been
the obsession of a generation of theoretical physicists? What exactly is so compelling
about tiny vibrating strings? In our last
string-theory episode, I talked about what
these things really are and covered some history. In short, the strings
of string theory are literal strands
and loops that vibrate with
standing waves simply by changing the vibrational mode
and you get different particles analogous to how different
vibrational modes on guitar strings give different notes. And, by the way,
these strings exist in six compact
spatial dimensions on top of the familiar three. In this episode,
I'm going tell you why string theory is
right, at least why so many of those
geniuses think it is. Maybe I can summarize. It's pretty, or at least
it started out that way. Its mathematics seem to
come together so neatly towards a unified description
of all forces and particles, and most importantly that
unification includes gravity. I want to try to
give you a glimpse into this mathematical elegance. I also want to give you a
teaser on why string theory is actually wrong. Don't worry, that topic will
get its own whole episode. The greatest criticism
of string theory is that it's never made
a testable prediction. The space of possible
versions of string theory is so vast that nothing can
be calculated with certainty, so string theory can neither
be verified nor ruled out. It's unfalsifiable. But string theorists
might disagree. They might say,
maybe half jokingly, that string theory does
make one great prediction. It predicts the existence
of gravity, which is stupid, of course. Everyone knows that
Isaac Newton discovered gravity when he fell
out of an apple tree, or something like that. There was definitely
an apple tree involved. But the fact is when
you start to work out the math of string theory,
gravity appears like magic. You don't need to try to fight
gravity into string theory. In fact, it will be
difficult to remove it, and the quantum gravity
of string theory is immune to the main difficulty
in uniting general relativity with quantum mechanics. It doesn't give you
tiny black holes when you try to describe
gravity on the smaller scales. We did talk about this
and other problems with developing a quantum theory
of gravity in a recent episode, but before we get to the
nuts and bolts of how string theory
predicts gravity, it's worth taking a moment to see
how stringy gravity avoids the problem of black holes. Let's actually start with the
regular old point particles of the standard model. When a point particle is
moving through space and time it traces a line. On a spacetime diagram, time
versus one dimension of space, this is called its world line. In quantum theories of gravity,
the gravitational force is communicated by
the graviton particle. When the graviton acts
on another particle, it exerts its effect at an
intersection in their world lines over some distance. But in very strong gravitational
interactions, that intersection itself becomes more
and more point like. The energy density at that
point becomes infinite. More technically, you start to
get runaway self-interactions, infinite feedback effects
between the graviton and its own field. If you even try to
describe very strong gravitational interactions,
you get nonsense black holes in the math. OK, let's switch
to string theory where particles are not points. They're loops or
open-ended strands. The graviton in
particular is a loop. When strings move on
a spacetime diagram, they trace out
sheets or columns. In fact, you can think of a
string not as a 1D surface but as a 2D sheet
called a world sheet. Now let's look at the
interaction of two strings. The vertex is no
longer point like. It can't be point like. Even the most
energetic interactions are smeared out over
the string, so you avoid the danger of black
hole creating infinities. OK, put a pin in
these world sheets. We're going to need them later. They illustrate why quantum
gravity isn't hopelessly broken in string theory, and that's a
huge point in favor of string theory, but these
world sheets will also help us see why
string theory predicts gravity in the first place. And this is the second point
in string theory's favor. You see, it turns out that
tiny vibrating quantum strings automatically reproduce the
theory of general relativity and, in the same mechanism,
seem to promise to reproduce all of quantum theory too. This is part of the
elegance I spoke of earlier. This stuff appears a little too
naturally in the math of string theory to be a coincidence,
or so a string theorist might tell you. For some reason,
vibrating strings are bizarrely well
suited to quantization. By quantization I mean taking
a classical large-scale description of something
like a ball flying through the air or a
vibrating rubber band and turning it into a
quantum description. To do this, you basically
take the classical equations of motion and follow a
standard recipe to turn them into wave equations with
various quantum weirdness added in like the
uncertainty relation between certain variables. I say basically. This is a tricky
process, and it only works if your equations of
motion are especially friendly. Schrodinger's equation is the
first and easiest example. It quantizes the
equations of motion of slow-moving,
point-like particles. A while ago, we talked about
Paul Dirac developed a wave equation for the electron that
took into account Einstein's special theory of relativity. It was a mathematical mess until
Dirac added some nonsense terms to the electron-wave function
that caused a lot of the mess to cancel out. Those nonsense terms turned out
to correspond to antimatter. The resulting Dirac equation
is incredibly elegant, and in the pursuit
of that elegance Dirac predicted the
existence of antimatter. This is a powerful
example of how following mathematical
prettiness could bring us closer to the truth. Quantizing the motion of
strings also starts out ugly, but there are also some
math tricks to make it work. A big part of it is
making use of symmetries. If the physics of
a system doesn't care about how you define
particular coordinates or quantities, we say
that that parameter is a symmetry of the
system or that the system is invariant to transformations
in that parameter. Finding symmetries
can massively reduce the complexity of the math. A really important type of
symmetry in quantum mechanics is gauge symmetry. It's when you can redefine
some variable in different ways everywhere in space and
still get the same physics. I want to remind you of one
particularly crazy result of gauge symmetries. It's a reminder
because we covered it, but it's so relevant that
it's worth the review. So, we expect the phase
of the quantum wave function to be a gauge
symmetry of any quantum theory. That means you should be able to
shift the location of the peaks and valleys in different ways
at different points in space without screwing up the physics. And guess what? In the raw Schrodinger
equation, you can't. It breaks various
laws of physics. But it turns out that
you can add a very special corrective term to
the Schrodinger equation that fixes these phase differences
preserving local phase invariance. That term looks
like what you would get if you added the
electromagnetic field to the Schrodinger equation. So in a way,
electromagnetism was discovered in its quantum form
by studying the symmetries of quantum mechanics. It turns out that exploring
a very different symmetry of string theory both makes it
possible to quantize the theory and gives us a very different
field, the gravitational field. So, like I was
saying, when we try to quantize string theory,
of course it's a huge mess. Applying the usual
old symmetries got physicists some of the
way, but to succeed, they needed an extra
weird type of symmetry. That symmetry is Weyl
symmetry or Weyl invariance. This is a weird one. It says that changing
the scale of space itself shouldn't affect
the physics of strings. Hermann Weyl actually came
up with this symmetry right after Einstein proposed his
general theory of relativity. He tried to use it to
unify general relativity with electromagnetism. Fun story-- Weyl invented
the name gauge symmetry to describe this scale
invariance inspired by the gauge of railroad
tracks which measures the separation of the tracks. Anyway, Weyl symmetry
doesn't work. Turns out that in 4D
spacetime it does matter whether you change
the scale of space and the separation
of its tracks. But it turns out that
there's a very particular geometric situation that
does have Weyl invariance. That's on the 2D
dimensional world sheet of a quantum string. Remember that? Mysteriously, the 2D sheet
traced out in spacetime by a vibrating 1D
string has this symmetry that lets us redefine the scale
on its surface however we like. That means we can smooth out
that surface mathematically and write a nice,
simple quantum wave equation from the
equations of motion, but only for 1D strings
making a 2D world sheet, not for any other
dimensional object. This is part of what makes
strings so compelling. They are quantizable in a way
that other structures aren't. But there's a cost to
using this symmetry. Just as local phase
invariance required us to add the electromagnetic
field to the Schrodinger equation, adding
Weyl invariance means we need to add a new field. That field looks like a 2D
gravity on the world sheet. It's a projection of the
3D gravitational field. So, with our quantized
equations of motion in hand, you can predict the quantum
oscillations of our string. These are particles,
and the first mode looks like the
graviton, a quantum particle in the aforementioned
gravitational field. If you use string
theory to write down the gravitational field in what
we call the low-energy limit, which just means not in
places like the center of a black hole,
then it looks just like the gravitational
field in Einstein's theory. OK, a caveat-- you
can only get the right particles, including the
graviton and the photon, out of string theory for
a very specific number of spatial dimensions,
nine to be precise. In fact, if string theory
makes any predictions, it's the existence of
exactly this number of extra dimensions. And this is where string theory
starts to look less attractive. Our universe has three
spatial dimensions. String theorists hypothesize
that the extra dimensions are coiled on themselves
so they can't be seen, but that seems like a
hell of an extra thing to add in order to
make your theory work. There's also no
experimental evidence of the existence of
these dimensions. And that's just the first of
many problems of string theory. But like I said, we're going to
need a whole episode for that. Physicists were lead
to string theory by the elegance of
the math and the fact that it appeared, at
least in the beginning, to converge on
the right answers. That convergence is also seen
in the union of different string theories by M theory
and in the discovery of AdS/CFT correspondence-- again, for future episodes. But can such an elegant and rich
mathematical structure really have nothing to do with reality? There's plenty of
historical precedent for mathematical beauty
leading to truth, but there's no fundamental
principle that says it has to. Perhaps we're now
overly distracted by the elegance
of string theory. Philosophical points to
consider as we continue to follow the mathematical
beauty hopefully towards an increasingly true
representation of spacetime. Thanks to the Great Courses
Plus for supporting PBS Digital Studios. The Great Courses Plus is
a digital learning service that allows you to learn about
a range of topics from educators and professors from
around the world. You can go to thegreatcoursesp
lus.com/spacetime to get access to a library of different video
lectures about science, math, history, literature, or even
how to cook, play chess, or become a photographer. New subjects, lectures,
and professors are added every month. For more information,
visit thegreatcoursesp lus.com/spacetime. Last week we talked about one of
the most misunderstood concepts in quantum mechanics, the
idea of virtual particles and their tenuous
connection to reality. You guys asked pretty much
every question that I avoided. Uri Nation asks
about the photons that mediate the magnetic
field or the contact force between two bodies. Aren't they virtual? Well, they are, but
they don't exist. These fundamental
forces are mediated by fluctuations in the quantum
fields of the relevant forces. Those fluctuations
can be approximated as the sum of many
virtual particles, but the particles
themselves are just convenient mathematical
building blocks to describe a messy
disturbance in the field. Eddie Mitch asked whether the
virtual particles are required to explain the Casimir force. So the Casimir
effect is sometimes explained as resulting from the
exclusion of virtual particles between two very closely
separated conducting plates which results in the plates
being drawn together. So, if the Casimir effect
really is due to a change in the zero-point energy-- and there are those
who say it isn't-- but if it is, then it's still
misleading to attribute it to virtual particles. More accurately, the
conducting plates create a horizon in what would
otherwise be a perfect infinite vacuum. In fact, you create two
horizons between the plates and one horizon on the outside. Those horizons
perturb the vacuum which can lead to the creation
of very real particles, as in Hawking radiation. But in the Casimir
effect, the double horizon between the plates restricts
what real particles can be produced
there whereas there's less restriction on the
outside of the plates with their single horizon. That leads to a net pressure
pushing the plates together. David Ratliff asks if a
quantum tree falls in a vacuum and nobody is around to measure
it, does it still have energy? Well, believe it or not,
it's a serious question as to whether the universe has
counter-factual definiteness, whether or not we can make
a meaningful statement about the state of the
universe without conducting an experiment. To address this
seriously, I want you to imagine this
gedankenexperiment. You have a box containing
a vial of poison connected to a radioactive isotope
that could either decay or not, releasing the poison. You put a mime in the box. Quantum mechanics can't tell
us whether anyone cares.
