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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?
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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.
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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.