# How to Get a “Minimum Scale” Without Pixels

Zoom in, and the world gets stranger. Down past atoms, past protons and neutrons, far past the smallest scales we can probe at the Large Hadron Collider, we get to the scale at which quantum gravity matters: the Planck scale.

Weird things happen at the Planck scale. Space and time stop making sense. Read certain pop science articles, and they’ll tell you the Planck scale is the smallest scale, the scale where space and time are quantized, the “pixels of the universe”.

That last sentence, by the way, is not actually how the Planck scale works. In fact, there’s pretty good evidence that the universe doesn’t have “pixels”, that space and time are not quantized in that way. Even very tiny pixels would change the speed of light, making it different for different colors. Tiny effects like that add up, and astronomers would almost certainly have noticed an effect from even Planck-scale pixels. Unless your idea of “pixels” is fairly unusual, it’s already been ruled out.

If the Planck scale isn’t the scale of the “pixels of the universe”, why do people keep saying it is?

Part of the problem is that the real story is vaguer. We don’t know what happens at the Planck scale. It’s not just that we don’t know which theory of quantum gravity is right: we don’t even know what different quantum gravity proposals predict. People are trying to figure it out, and there are some more or less viable ideas, but ultimately all we know is that at the Planck scale our description of space-time should break down.

“Our description breaks down” is unfortunately not very catchy. Certainly, it’s less catchy than “pixels of the universe”. Part of the problem is that most people don’t know what “our description breaks down” actually means.

So if that’s the part that’s puzzling you, maybe an example would help. This won’t be the full answer, though it could be part of the story. What it will be is an example of what “our description breaks down” can actually mean, how there can be a scale beyond which space-time stops making sense without there being “pixels”.

The example comes from string theory, from a concept called “T duality”. In string theory, “extra” dimensions beyond our usual three space and one time are curled up small, so that traveling along them just gets you back where you started. Instead of particles, there are strings, with length close to the Planck length.

Picture a loop of string in a small extra dimension. What can it do?

One thing it can do is move along the extra dimension. Since it has to end up back where it started, it can’t just move at any speed it wants. It turns out that the smaller the extra dimension, the more energy the string has when it spins around it.

The other thing it can do is wrap around the extra dimension. If it wraps around, the string has more energy if the dimension is larger, like a rubber band stretched around a pipe.

The string can do either or both of these multiple times. It can wrap many times around the extra dimension, or move in a quicker circle around it, or both at once. And if you calculate the energy of these combinations, you notice something: a string wound around a big circle has the same energy as a string moving around a small circle. In particular, you get the same energy on a circle of radius $R$, and a circle of radius $l^2/R$, where $l$ is the length of the string.

It turns out it’s not just the energy that’s the same: for everything that happens on a circle of radius $R$, there’s a matching description with a circle of radius $l^2/R$, with wrapping and moving swapped. We say that the two descriptions are dual: two seemingly different pictures that turn out to be completely physically indistinguishable.

Since the two pictures are indistinguishable, it doesn’t actually make sense to talk about dimensions smaller than the length of the string. It’s not that they can’t exist, or that they’re smaller than the “pixels of the universe”: it’s just that any description you write down of such a small dimension could just as easily have been of a larger, dual dimension. It’s that your picture, of one obvious size of the curled up dimension, broke down and stopped making sense.

As I mentioned, this isn’t the whole picture of what happens at the Planck scale, even in string theory. It is an example of a broader idea that string theorists are investigating, that in order to understand space-time at the smallest scales you need to understand many different dual descriptions. And hopefully, it’s something you can hold in your mind, a specific example of what “our description breaks down” can actually mean in practice, without pixels.

## 5 thoughts on “How to Get a “Minimum Scale” Without Pixels”

1. ohwilleke

This is the best description I’ve seen of the T-duality in string theory and about why talking about the size of an extra dimension may be indeterminate or a category error, and for that I sincerely thank you for this post.

But, honestly, it doesn’t leave me feeling that I have advanced my understanding of “how there can be a scale beyond which space-time stops making sense without there being “pixels”.”, by reading it. I think that this is because an extra dimension seems like a very different concept than a tiny distance in a large, effectively infinite dimension that can’t be traversed or wrapped around by a Planck scale string.

If anything, the notion of small extra dimensions tends to reinforce the pixel concept, because one common way that small extra dimensions are illustrated visually is to show a multitude of little bubbles projecting themselves from points in space that are adjacent to each other.

If there is a connection between alternative descriptions of small extra dimensions which descriptions are related by the T-duality in string theory, and how to explain how there can be a minimum length scale without “pixels”, I’m missing a leap of logic somewhere and failing to connect the dots.

Also, it seems a bit odd to discuss the breakdown of space-time at minimum distance scales in string theory, which I normally think of (perhaps incorrectly) as providing a potential quantum gravity theory in which there is not minimum length and distances in large ordinary dimensions don’t lose meaning at tiny scales, as opposed to quantum gravity theories in the loop quantum gravity and casual dynamical triangles family that seek to quantize the space-time background, rather than being based on a graviton as a carrier boson for the gravitational force that quantizes the exclusively wave-like gravitational waves of general relativity.

I certainly don’t disagree that it is possible that our description can break down on a scale beyond which space-time stops making sense without there being “pixels”. But, the only example I know of which seems to illustrates this notion, or at least something closely analogous to it, in a way that does seem to explain how a minimum scale could arise that doesn’t involve “pixels”, is Heisenberg’s uncertainty principle.

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So, I’m not saying that T duality is “what happens at the Planck scale” so much as it’s a clean example of a space-time description breaking down. The picture of what happens in noncompact space is going to be more complicated, less clean, and most relevantly is currently unknown.

I don’t think it’s a good idea to think of small extra dimensions as “pixel”-like. I can understand why certain depictions would give you that idea, but think about the simplest example, a rolled up piece of paper. It’s not that there’s a separate circle at each point along the line, there’s one continuous circle. People sometimes draw pictures of a Calabi-Yau bubble at a bunch of adjacent points, but that’s just because you can’t really draw it continuously like the curled up paper example, at least not with that many dimensions. It’s kind of a bad way to draw it but it’s hard to find a better alternative.

String theory doesn’t have a minimum length in the same way as LQG and CDT, things go smoothly even to smaller scales. However, the geometric picture of space is still expected to break down at the Planck scale, just in a smooth way. You’ll occasionally hear people who work with entanglement entropy and the like talking about a maximum density of information, or ideas like that. In general, the expectation is that once you get down to a certain scale it stops making sense to just think in terms of a straightforward space-time, at those scales various sorts of duality (including T and S duality, but perhaps more relevantly holography, maybe something amplituhedron-like, etc) mean that other descriptions work better than pure space-time ones.

How exactly that story works isn’t known yet. But I think you can get an intuition for how that sort of story could work, how another description can take primacy over the straightforward space-time one, by looking at more well-studied cases where it just doesn’t make sense to keep scaling spacetime forever. And that’s the role T duality is playing in this post.

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