Why String Theory is Wrong (Script) (Patreon)

There’s this idea that beauty is a powerful guide to truth in the mathematics of physical theory. String theory is certainly beautiful in the eyes of many physicists. Beautiful enough to pursue even if it’s wrong?

Hermann Weyl once said, “If I have to choose between beauty and truth, I choose beauty.” It was in reaction to a rebuke by Einstein. Weyl had tried to explain electromagnetism by imposing on Einstein’s general theory of relativity very first gauge symmetry, Weyl invariance. Einstein pointed out it the proposal led to some absurd results, and so the idea went down in flames. It just couldn’t be true, despite the elegance of the math. But sometimes it can be hard to let go of the sense that a beautiful theory must be right. Could this also be the case with string theory?

As it happens, Weyl’s old idea DID work when translated to the very particular case of the quantum string, which is part of what got string theory going in the first place. We talked about this in detail in our episode on “why string theory is right” which itself was a sequel to our primer on the basics of string theory. In those episodes we saw some of the remarkable ways that string theory promised to converge on a theory of everything. It seemed so beautiful. The effortlessness of its inclusion of quantum gravity, its promise to unify all particles under one umbrella. There’s also the convergence of many versions of string theory to single picture with a very specific number of extra dimensions. I’ll talk more about that today. So why, with all this promise of being so right, do more and more physicists think that string theory is, after all that, either woefully incomplete, or just plain wrong?

Modern string theory is the convergence of many beautiful ideas in physics, each of which feel right in their own way. To see where string theory ultimately fails we need to rewind to look at some of these a bit closer. To start with, to a precursor to string theory and the origin of all this extra dimension stuff. In 1919, not long after Einstein published his great theory of gravity, Theodor Kaluza discovered something strange. He was playing around with the new-fangled general relativity in 5 dimensions – 4 space and one time. Because why not? He found that in the right sort of 5-D spacetime, you can separate the resulting Einstein equations into a 4-D component that looks exactly like the familiar general relativity of our universe. Plus a bit of extra math from the extra spatial dimension. Crazily, that math also looked familiar. It looked like Maxwell’s equations for electromagnetism. It appeared that gravity acting in this 5th dimension looks like electromagnetism to beings trapped in our 4-D spacetime.  Einstein himself was supposedly jubilant at the idea – a rather better reaction that was received for the electromagnetism of poor old Hermann Weyl.

The mild inconvenience of there very clearly being no extra spatial dimension was solved by Oscar Klein in the late 20’s. Klein realized you can get a sensible quantum theory if you compactify that extra dimension – shrink it down to around 10^-30th of a meter so it’s only visible to things equally miniscule. In the resulting Kaluza-Klein theory, the 5th dimension is looped into a tiny circle. At every point in space, there’s another direction to move – up, down, left, right, forward, back, and … around? Momentum in that looped dimension has the exact behavior of electric charge, with the direction of rotation determining the sign of the charge.

It was an incredible discovery, and a beautiful one. It even made a prediction – the ratio between the mass and electric charge of the electron. Assuming the experimentally measured value for the electric charge, the corresponding electron mass should be … around 5 kilograms. Uh, probably wrong. And it’s not the only problem with the first version of Kaluza-Klein theory. It also predicted an unknown field – the dilaton field, and a corresponding particle that had never been seen. It also didn’t give anything beyond electromagnetism, but to be fair the other fundamental forces hadn’t been discovered at that stage. 

These seem like fatal flaws, but we can thank this wrongness for the later development of string theory. People tried various things to fix the issues – for example by adding more compact dimensions of various shapes, and, of course, strings. There are many Kaluza-Klein-inspired theories out there. String theory is just the most famous.

So, start with Kaluza-Klein, add vibrating strings and exactly the right extra spatial dimensions and you have string theory. The last critical ingredient is supersymmetry. This is a theoretical symmetry between bosons and fermions which very elegantly explains certain anomalies like the vast differences between the forces. It also introduces fermions to the boson-only version of string theory to give superstring theory. The introduction of “SUSY” along with the discovery of the right symmetries for the extra dimensions, sparked the first superstring revolution in the mid-80s. Roughly coinciding with the theatrical release of Weird Science. Just saying.

Superstrings started out with incredible promise, and so there was a proliferation of different versions of superstring theory. It turns out there are 5 different ways to tie a superstring. Type I, type IIA & B, and heterotic SO(32) or E8xE8. Five different approaches to getting all of the desired particles out of the basic premise of strings wiggling in 10 dimensions. All require six compactified extra dimensions of space. What differs is the detailed geometries and symmetries of those spaces and the way strings vibrate within them. In fact, these versions appeared fundamentally different from each other – divergent rather than convergent. Contradictory even. Hardly elegant. One might even say ugly. Or wrong.

But the key to their convergence and the return to beauty had already been glimpsed. The various superstring theories exhibited what we call dualities. A duality in physics is when two apparently different mathematical theories prove to represent the same physical process. These dualities revealed that certain classes of string theory were actually different ways of expressing exactly the same theory. Perhaps there was a glimmer of hope for these divergent versions of string theory after all. 

To give you a sense of what a duality looks like, let’s go back to the good old simplicity of Kaluza-Klein theory. Or at least to the simplicity of just one extra circular spatial dimension. In fact, let’s simplify even further. Imagine only one extended and one compactified spatial dimension. If the latter is circular we get a tube. Our tiny quantum strings can roam the small dimension. They can even wind around it, perhaps multiple times, and in either direction, before forming a closed loop. The number of times a string winds around this compactified dimension is called its winding number.

The energy of such a string depends on the winding number times the radius of the compact dimension – that makes sense – it basically gives the length of the string.  These strings are vibrating with standing waves like guitar strings, and the energy also depends on the frequency of that vibration. That frequency depends on the density of wave cycles on the string, That’s just the number of wave cycles around each coil – the mode number - DIVIDED by the radius.

So there are two ways to get a high-energy string: have a large winding number along with a large radius – that gives you a long string; OR have a large mode number with a small radius – that gives you a high frequency vibration. It turns out that, mathematically, these two are completely equivalent. At least they give exactly the same physics. Either winding number TIMES radius or mode number DIVIDED by radius can be used to define the momentum of a particle produced by this string.  So you can construct a theory in which momentum increases with the size of the compact dimension, or where momentum decreases with that size, and both give the same results. Sounds weird, but this may just save string theory.

I just described a type of duality – in this case a T-duality – short for target-space duality. In a duality, two apparently contradictory ways of describing the mechanics of the universe can lead to exactly the same results. The appearance of dualities tells us that we probably can’t take our geometric interpretations of the math as seriously as we’d like to. T-dualities proved that some of these different versions of string theory were actually different expressions of the same theory.

The other main type of duality in string theory is S-duality – strong-weak duality. In this case it’s a duality between strongly versus weakly interacting strings. This seems even more contradictory, but it’s incredibly powerful. We’ll glimpse the mechanics and the implications of S-duality as we look deeper into M-theory and holography in the future.

S-duality provided the final lynchpin that demonstrated that the 5 different types of string theory were all manifestations of the same theory. The guy who ultimately brought it together was Ed Witten. At a string conference in ‘95, Witten showed that the disparate string theories were all just different perspectives – different limits or special cases of a single overarching theory. This was M-theory, where the M stands for … really whatever you want it to. Membrane, magic, mother-theory … According to Witten, it’s to be decided when the full nature of the theory is understood.

We’ll come back to M-theory in real detail, but the important thing here is that it adds a single extra dimension connects all of the 5 string theory types via S-duality. That duality did require the addition of an extra dimension – 11 in M-theory compared to the 10 in the disparate superstring theories.

Well that sounds a bit arbitrary. Your theory not working? Just add another dimension! Actually, the realization that superstring theory could be 11-dimensional was a revolution. It sparked the second superstring revolution. See, in parallel to the development of superstring theory, other physicists had been working on supergravity. For independent reasons that we don’t have time to get into, 11 is also the magic number for supergravity dimensions. Supergravity should be the low-energy limit, large-scale limit to superstring theory. So it was incredibly exciting that string theory appeared to have an 11-dimensional version – M-theory.  – to correspond to everyone’s favourite 11-D supergravity. This convergence of the superstring theories with each other and with supergravity restored the sense of beauty to string theory. It appeared to be on the track to rightness again.

So where did things go wrong? Well, in a sense they were never really right. This M-theory thing? It is still not well defined. It’s not solvable using perturbation theory, which doesn’t leave much room to explore its implications. In all superstring theories, the extra spatial dimensions are wrapped not in simple loops but in complicated geometries called Calabi-Yau manifolds. The behavior of strings on these hyper-dimensional surfaces is only understood in idealized cases – for example sections of these manifolds that can be approximated as simple tubes, like in Kaluza-Klein theory. But more worrying, there are countless possible geometries – countless Calabi-Yau manifolds to choose from. The standard number given is 10^500 different topologies. The actual number is a lot higher.

This is the string landscape. Each geometry for the compactified dimensions implies a different set of properties for vibrating strings, and so a different family of particles and different laws of physics to go with them. It seems an impossible task to find which one corresponds to OUR universe, if any do. This is the impasse.  In principle, the standard model lives somewhere in the string landscape, but without knowing the geometry of that it can’t be verified, nor can it make testable predictions beyond the standard model. Well, there is supersymmetry. Essentially all string theory requires supersymmetry to work. Physicists at the Large Hadron Collider had expected to find the supersymmetric particles by now. They haven’t. There’s a hint from cosmic rays – we talked about it earlier – but string theorists are still rightly concerned. 

Their elegant theory, which was converging so beautifully, has stalled. Will they follow the example of Hermann Weyl, and choose beauty over truth? One last word on Weyl: his idea of explaining electromagnetism by adding his gauge symmetry to general relativity was wrong, but it inspired the entire field of gauge theory upon which much our understanding of the quantum world depends. It also gave us the sought-after quantum electromagnetism in the end, just with a slightly different symmetry. So perhaps string theorists should also stick to their guns. As wrong or incomplete as current string theory may be, it may also be the inevitable early step as we seek an even more beautiful, and ultimately more right understanding of space time.