Why String Theory is Right - Script (Patreon)

Some see string theory as the one great hope for a theory of everything – that it will unite quantum mechanics and gravity and so unify all of physics into one glorious theory. 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?

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 you get different particles – analogous to how different vibrational modes on guitar strings gives different notes. And, by the way, strings exist in 6 compact spatial dimensions on top of the familiar 3.

In this episode I’m going to tell you why string theory is right – or 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. It’s 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 or ruled out. It’s unfalsifiable.  But string theorists might disagree. They might say, maybe half jokingly, that string theory does makes 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 fit gravity into string theory – in fact it would 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 smallest scales! We did talk about this and other problems with developing a quantum theory of gravity in a recent episode. But before we get into 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 point particles of the standard model. When a point particle moving through space and time in traces a line. On a spacetime diagram – time versus one dimension of space –  this is called its worldline. 

In quantum theories of gravity, the gravitational force is communicated by the graviton particle. When the graviton acts on another particle, it exerts is effect at an intersection in their worldlines, 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 1-D surface, but as a 2-D sheet called a worldsheet. 

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, and so you avoid the danger of black-hole-creating infinities.

OK, put a pin in these worldsheets. We’re going to need them again 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 worldsheets will help us see why string theory predicts gravity in the first place. And this is a second point in its favor. 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 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”, but this is a tricky process, and 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 how Paul Dirac developed a wave equation for the electron that took into account Einstein’s special relativity. It was a mathematical mess until Dirac added some nonsense terms to the electron wavefunction 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 can bring us closer to the truth.

Quantizing the motion of strings also starts out ugly, and 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 a particular coordinate or quantity, we say 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 here it’s worth the review. So we expect the phase of the quantum wavefunction 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 you CAN add a special corrective term to Schrodinger that fixes these phase differences, preserving local phase invariance. That term looks just 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, 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, it didn’t work. Turns out in 4-D spacetime it DOES care if you change the scale of space - the separation of its tracks - differently in different places. But it turns out that there’s a very particular geometric situation that does have Weyl invariance. That’s on the 2-dimensional worldsheet of a quantum string. Remember that?

Mysteriously, the 2-D sheet traced out in spacetime by a vibrating 1-D string has this symmetry that lets us redefine the scale on its surface however we like. That means we can we easily smooth out that surface and write a nice simple quantum wave equation from the equations of motion. But only for 1-D strings making a 2-D worldsheet! Not for any other dimensional objects. This is part of what makes strings so compelling. They’re quantizable in a way other structures aren’t. 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 2-D gravity on the worldsheet. It’s a projection of the 3-D gravitational field.

So with our quantized equations of motion in hand, you can predict the quantum oscillations for our string. These are particles, and the first mode looks just like the graviton – a quantum particle in the afore-mentioned 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. 9 to be precise. In fact if string theory makes any prediction, it’s the existence of these extra dimensions. And this is where string theory starts to look less attractive. Our universe has 3 spatial dimensions. String theorists hypothesize that the extra dimensions are coiled on themselves so can’t be seen. But that seems like a hell of an extra thing to add in order to make the theory work. There’s also no experimental evidence of their existence. And that’s just the first of many problems of string theory – like I said, we’re going to need a whole episode for that.

Physicists were led 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 string theories by M-theory and in the discovery of AdS/CFT correspondence – again, for future episodes. Can such an elegant and rich mathematical structure really have nothing to do with reality? There’s plenty of historical precedent for it, but there’s no fundamental principle that says that pretty math must lead to truth. 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 space time.