Professor Steve Simon discuses Knots, World-Lines, and Quantum Computation at TEDxOxford.
Listen to the MP3 Audio here: Knots, World-Lines, and Quantum Computation by Steve Simon at TEDxOxford
Theoretical physics. What does that make you think of? Maybe you had physics in school, or maybe you think of one of the greats like Albert Einstein. Maybe you think of fundamental particles: the elementary building blocks of our universe.
I am a theoretical physicist, and I think of these things, but I spend an awful lot of time thinking about knots. What I usually want to know about knots is whether one knot is the same or different from another knot. What I mean by this is: can the knot on the right be twisted and turned around and turned into the knot on the left without cutting without using scissors? If you can do this, we say they are equivalent knots, and otherwise we say they’re inequivalent.
Surprisingly enough, this question of equivalence of knots is very important for certain types of fundamental particles. Furthermore, it’s important for the future of technology. This is what I am going to tell you in the next 15 minutes.
To get started we need some of the results from relativity. Now relativity is a pretty complicated subject, I am not going to explain much of it. But one of the themes that we learn from it is that space and time are mostly the same thing. So, I’ve a little story to explain this, it’s a story of Einstein’s world and his day.
So, we have his home, his work, his cinema on the screen, and there’s a clock in the upper right hand corner, so keep your eye on the clock during the day. So Einstein starts his day, and he goes to work, then after a while, he comes home for lunch, the clock keeps ticking, he goes back to work, the clock keeps ticking, in the afternoon he decides to go to cinema, he goes to the cinema, the clock keeps ticking, and then eventually he goes home. Well, the physicists would look at this and would want to treat time more similarly as space, and the way we do this is we plot space on an axis, and we plot time on another axis. Einstein’s so called “World Line” is this dark red line which tells you where in space is he at any given time. It’s called his World Line because it tells you where in the world is he at any given time.
Now we can go through the day, and keep your eye on the dark red ball. The ball goes up one step every hour as we go through the day. It goes back and forth in space, tracing Einstein’s position. So the world line is just a convenient way of keeping track of where Einstein is at any point in the day.
We can do the same thing with a more complicated world. So here we imagine looking down on Einstein’s neighborhood from a helicopter above. So Einstein starts his day at home, he goes to work, he goes to the cinema, he goes back home. A student on the same day starts at home, goes to the library, goes to the pub and goes home.
Now if we follow them both on the same day, Einstein goes to work, the student goes to the library, Einstein goes to the cinema, the student goes to the pub, Einstein goes home, the student goes home, it starts to look pretty complicated. But we can simplify it by looking at the space-time diagram of what happened. We do that by turning the neighborhood sideways, plotting time vertically and notice I’ve drawn a blue vertical line at the position of every object in the neighborhood that doesn’t move, such as the library or the pub; they stay fixed in space and they move through time.
Einstein and student’s world lines move around in the neighborhood as they go through time. Now you can kind of see where I am going with this. Einstein and the student’s world lines have wrapped around with each other. If you pull those tight, you’ll discover you have it knotted.
We need one more thing from the theory of relativity. We need E = mc². Again, this is a thing that I am not going to explain to you in much depth. But roughly what it means is that energy and mass are the same thing. So if we have a particle in our world like an electron — that’s a particle of matter. Now each particle of matter has an opposite particle of antimatter. In the case of electron, the antimatter particle is called the positron. Both the electron and the positron have mass. If you bring them together, however, they can annihilate each other, giving off their mass as energy, usually as light energy. The process works in reverse just as well. You can put in the energy and get out the mass of the particles.
Now, we are going to do the same thing we did with Einstein’s neighborhood. We are looking down on the neighborhood, we put in energy to create a particle and antiparticle. We put in energy to create a particle and antiparticle. Then maybe we move one particle around another, and we bring them back together, we re-annihilate them, we annihilate them; releasing the energy again.
Now if we’d look at that in the space-time diagram, it looks a little bit like this; time running vertically, we put in the energy, we put in the energy, we wrap one particle around the other, and we annihilate them, and we annihilate them again. And you can see quite clearly here that the world lines have knotted around with each other.
Do the same thing with more particles, by putting in more energy, move them around in some very complicated way and bring them back together. The space-time diagram would look a little bit like this, making a very complicated knot.
Now here’s the amazing fact upon which the rest of my talk relies. Certain particles, called Anyons exist in 2+1 dimensions. Now I should probably say what I mean by 2+1 dimensions. Two dimensions mean we’re talking about a flat surface, so these particles live on flat surfaces. We say +1 dimensions, we mean also time. So we are just saying that the particles on the flat surfaces move around in time. So, these particles called Anyons exist with the properties at the end depend on the space-time knot that their world lines have formed. So you can kind of see now why I am so concerned with whether two knots are the same or whether they are different. We can conduct an experiment by which we create some particles, move them around to form a knot and then they have some property at the end of the knot. Let me do the experiment again, create the particles, make another knot, and I want to know whether the properties of the particles at the end of the knot are the same as the properties of the particles at the end of a different knot. This is why I am concerned with whether the knots are the same or whether they are different.