*[This is a transcript of the video embedded below.]*

The space as we experience it has three dimensions. Left-right, forwards backwards and up-down. But why three? Why not 7? Or 26? The answer is: nobody knows. But if no one knows why space has three dimensions, could it be that it actually has more? Just that for some reason we didn’t notice? That’s what we’re going to talk about today.

The idea that space has more than three dimensions may sound completely crazy, but it’s a question that physicists have been seriously investigating for more than a century. And since there is a lot to be said, this video will have two parts. In this part we are going to talk about the origins of the idea of additional dimensions, the Kaluza-Klein theory, and all that. And in the next part we will talk about recent work, string theory and black holes on the Large Hadron Collider, etc.

Let’s start by remembering how to describe space and objects in it. In two dimensions we can put a grid on a plane, and then each point is a pair of numbers that indicate how far you have to go from zero in the horizontal and vertical directions to get to that point. The arrow pointing to this point is called the “vector”.

This construction is not two-dimensional. You can add a third direction and do the exact same thing. And why stop there? You can no longer draw a grid for four dimensions of space, but you can definitely write down the vectors. They are just a series of four numbers. Indeed, you can construct vector spaces in any number of dimensions, even in an infinite number of dimensions.

And once you have vectors in those higher dimensions, you can use them to create geometry, e.g. B. construct higher-dimensional planes or cubes and calculate volumes or curve shapes etc. And while we cannot draw these higher dimensional objects directly, we can draw their projections in lower dimensions. This is, for example, the projection of a four-dimensional cube in two dimensions.

It might seem perfectly obvious today that you can create geometry in any number of dimensions, but it is actually a relatively recent development. It was not until he was forty-three that the British mathematician Arthur Cayley wrote about “Analytical Geometry of (n) Dimensions”, in which n could be a positive integer. Higher-dimensional geometry sounds innocent, but it was a big step towards abstract mathematical thinking. It was the beginning of what is now called “pure mathematics,” that is, mathematics that is done for its own sake, not necessarily because it has an application.

However, abstract math concepts often prove useful in physics. And these higher dimensional geometries have proven to be very useful to physicists, since in physics we normally deal not only with things that sit in certain places, but also with things that move in certain directions as well. For example, if you have a particle, to describe what it does you need both position and momentum, where momentum indicates the direction the particle is moving. In fact, every particle is described by a vector in a six-dimensional space with three entries for the position and three entries for the momentum. This six-dimensional space is called phase space.

By dealing with phase spaces, the physicists got used to dealing with higher-dimensional geometries. And, of course, they wondered if the actual space we live in couldn’t have more dimensions. This idea was first pursued by the Finnish physicist Gunnar Nordström, who tried in 1914 to use a 4th dimension of space to describe gravity. But it didn’t work. The person who found out how gravity worked was Albert Einstein.

Yeah, that guy again. Einstein taught us that gravity does not need an additional dimension of space. Three dimensions of space are sufficient. All you have to do is add a dimension of time and allow all of those dimensions to curve.

However, if you don’t need extra dimensions for gravity, you might be able to use those for something else.

Theodor Kaluza certainly thought so. In 1921 Kaluza wrote a paper in which he tried to use a fourth dimension of space to describe electromagnetic force in a very similar way to how Einstein described gravity. But Kaluza used an infinitely large additional dimension and didn’t really explain why we don’t usually get lost in it.

This problem was solved a few years later by Oskar Klein, who assumed that the 4th dimension of space had to be rolled up to a small radius so that you could not get lost in it. You just wouldn’t notice if you stepped in, it’s too small. This idea that electromagnetism is caused by a coiled 4th dimension of space is now called the Kaluza-Klein theory.

I’ve always found it amazing that this works. You take up an extra dimension of space, roll it up and gravity comes out along with electromagnetism. You can explain both forces completely geometrically. This is probably why Einstein was convinced in his later years that geometry was the key to a unified theory for the fundamentals of physics. But at least so far this idea has not worked.

Does the Kaluza-Klein theory make predictions? Yes, it does. All electromagnetic fields going into this 4th dimension must be periodic in order to fit on the coiled dimension. In the simplest case, the fields just don’t change when you go into the additional dimension. And that reproduces normal electromagnetism. But you can also have fields that vibrate once, then twice, and so on. These are called higher harmonics, as you did in music. So the Kaluza-Klein theory predicts that all of these higher harmonics should also exist.

Why didn’t we see them? Because you need energy to shake that extra dimension. And the more it wobbles, that is, the higher the harmonics, the more energy you need. How much energy Well, that depends on the radius of the additional dimension. The smaller the radius, the smaller the wavelength and the higher the frequency. So a smaller radius means you need more energy to figure out if the extra dimension is there. The theory does not say how small the radius is, so we do not know what energy is required to study it. The short summary, however, is that we’ve never seen any of these higher harmonics, so the radius must be very small.

Incidentally, Oskar Klein himself was very modest with regard to his theory. He wrote in 1926:

“Whether behind these hints of possible something can be realized must of course leave the future.”

(“Whether these possibilities are based on reality must of course be decided by the future.”)

But we don’t use Kaluza-Klein theory instead of electromagnetism, and why is that? That’s because the Kaluza-Klein theory has some serious problems.

The first problem is that the geometry of the additional dimension gives correct electric and magnetic fields, but no charged particles such as electrons. You still have to enter this. The second problem is that the radius of the additional dimension is not stable. If you bother with it, it can start to gain weight, and that can have observable consequences that we haven’t seen. The third problem is that the theory is not being quantized and no one has figured out how to quantize geometry without encountering problems. However, you can quantize simple ancient electromagnetism with no problem.

Today we of course also know that the electromagnetic force actually combines with the weak nuclear force to form the so-called electroweak force. Interestingly, it turns out that this is not a problem for the Kaluza-Klein theory. Indeed, it was shown by Ryszard Kerner in the 1960s that the Kaluza-Klein theory can be applied not only to electromagnetism, but to any similar force, including strong and weak nuclear forces. All you need to do is add a few more dimensions.

How many? You need two more for weak nuclear power and four more for strong nuclear power. In total we now have one dimension of time, 3 for gravity, one for electromagnetism, 2 for weak nuclear force and 4 for strong nuclear force, making a total of 11.

In 1981 Edward Witten noticed that 11 happened to have the same number of dimensions, which is the maximum for supergravity. What happened after that, we’ll talk about next week.