Molecules are made of atoms. Atomic nuclei are made of neutrons and protons. And the

neutrons and protons are made of quarks and gluons. Many physicists think that this is

not the end of the story, but that quarks and gluons are made of even smaller things,

for example the tiny vibrating strings that string theory is all about. But then what?

Are strings made of smaller things again? Or is there a smallest scale beyond which

nature just does not have any further structure? Does nature have a minimal length?

This is what we will talk about today.

When physicists talk about a minimal length, they usually mean the Planck length, which

is about ten to the minus thirty-five meters. The Planck length is named after Max Planck,

who introduced it in 1899. Then to the minus thirty-five meters sounds tiny and indeed

it is damned tiny. To give you an idea, think of the tunnel of the Large Hadron Collider.

It’s a ring with a diameter of about 10 kilometers. The Planck length compares to

the diameter of a proton as the radius of a proton to the diameter of the Large Hadron

Collider. Currently, the smallest structures that we can study are about ten to the minus

nineteen meters. That’s what we can do with the energies produced at the Large Hadron

Collider and that is still sixteen orders of magnitude larger than the Planck length.

What’s so special about the Planck length? The Planck length seems to be setting a limit

to how small a structure can be so that we can still measure it. That’s because to

measure small structures, we need to compress more energy into small volumes of space. That’s

basically what we do with particle accelerators. Higher energy allows us to find out what happens

on shorter distances. But if you stuff too much energy into a small volume, you will

make a black hole.

More concretely, if you have an energy E, that will in the best case allow you to resolve

a distance of about hbar c over E. I will call that distance Delta x. Here, c is the

speed of light and hbar is a constant of nature, called Planck’s constant. Yes, that’s

the same Planck. This relation comes from the uncertainty principle of quantum mechanics.

So, higher energies let you resolve smaller structures.

Now you can ask, if I turn up the energy and the size I can resolve gets smaller, when

do I get a black hole? Well that happens, if the Schwarzschild radius associated with

the energy is similar to the distance you are trying to measure. That’s not difficult

to calculate. So let’s do it. The Schwarzschild radius is approximately M times G over c squared

where G is Newton’s constant and M is the mass. We are asking, when is that radius similar

to the distance Delta x. As you almost certainly know, the mass associated with the energy

is E equals M c squared. And as we previously saw, that energy is just hbar c over Delta

x. You can then solve this equation for Delta x. And this is what we call the Planck length.

It is associated with an energy called the Planck energy. If you go to higher energies

than that, you will just make larger black holes. So the Planck length is the shortest

distance you can measure.

Now, this is a neat estimate and it’s not entirely wrong, but it’s not a rigorous

derivation. If you start thinking about it, it’s a little handwavy, so let me assure

you there are much more rigorous ways to do this calculation, and the conclusion remains

basically the same. If you combine quantum mechanics with gravity, then the Planck length

seems to set a limit to the resolution of structures. That’s why physicists think

nature may have a fundamentally minimal length.

Max Planck by the way did not come up with the Planck length because he thought it was

a minimal length. He came up with that simply because it’s the only unit of dimension

length you can create from the fundamental constants, c the speed of light, G, newton’s

constant, and h-bar. He thought that was interesting because, as he wrote in his 1899 paper, these

would be natural units that also aliens would use. Check the information below the video

for references.

The idea that the Planck length is a minimal length only came up after the development

of general relativity when physicists started thinking about how to quantize gravity. Today,

this idea is supported by attempts to develop a theory of quantum gravity, which I told

you about in an earlier video. In string theory, for example, if you squeeze too much energy

into a string it will start spreading out. In Loop Quantum Gravity, the loops themselves

have a finite size, given by the Planck length. In Asymptotically Safe Gravity, the gravitational

force becomes weaker at high energies, so beyond a certain point you can no longer improve

your resolution.

When I speak about a minimal length, a lot of people seem to have a particular image

in mind, which is that the minimal length works like a kind of discretization, a pixilation

of an photo or something like that. But that is most definitely the wrong image. The minimal

length that we are talking about here is more like an unavoidable blur on an image, some

kind of fundamental fuzziness that nature has. It may, but does not necessarily come

with a discretization.

What does this all mean? Well, it means that we might be close to finding a final theory,

one that describes nature at its most fundamental level and there is nothing more beyond that.

That is possible, but. Remember that the arguments for the existence of a minimal length rest

on extrapolating 16 orders magnitude below the distances what we have tested so far.

That’s a lot. That extrapolation might just be wrong. Even though we do not currently

have any reason to think that there should be something new on distances even shorter

than the Planck length, that situation might change in the future.

Still, I find it intriguing that for all we currently know, it is not necessary to think

about distances shorter than the Planck length. Thanks for watching. See you next week.