Nature Is a Lazy Mathematician, Part 3

We are here to decode the sentence: "A spinor is the most basic mathematical object that can be Lorentz-transformed", because it tells us something about nature's character. In Part 1, we familiarized ourselves with 4-dimensional spacetime, the Minkowski metric and Lorentz transformations. In Part 2, we talked about the connections to group theory. In Part 3, we need to first take a complete detour before we bring this all together.

The light switch

Forget everything about group theory and focus for a moment on the following setup: Imagine a light switch. We want a light switch that just has two settings: up and down. We could model this light switch as a vector in a two-dimensional space:

If you’re into computers, you can think of this as a two-bit system. If the light switch is in the up position, then the top bit is 1. If the light switch is in the down position, then the bottom bit is 1.

Now we are going to make this a much stranger system. Imagine that your light switch can rotate around in space. But curiously, it loses its memory each time you turn it and look at it. Look at the image below:

If you first look at the light switch when it is oriented along the x-axis and you put the switch in the “up” direction, it will always remain in the “up” direction, no matter how many times you look at it. But if you then turn it and lay it down in the y-direction, and then you look at it, the switch has a 50/50 chance of either being in the “left” or “right” setting. But, if you don’t move it, it will perfectly preserve its memory. Same in the other direction: if it’s lying down in the y-direction, and you put the switch in “left”, and then you swivel back and measure in the x-direction, you have a 50/50 chance of getting “up” or “down”. It never loses its memory if you don't move it, but it always loses its memory when you do.

How would we model such a system in terms of probabilities of measurements? Let’s first do another example: consider modeling a die with 6 faces, 1 through 6. Each face has the same probability of coming up when I throw the die. So we can write this down as a 6-dimensional vector, where each dimension carries the probability of that face coming up. We have to multiply the whole thing by 1/6 because we want all the probabilities to sum to 1, and 1 = 6 * (1/6).

How would we model a loaded die? Like this:

So this would be a die where the number 1 has a 50% chance of coming up when I throw it, and the numbers 3 and 5 have a 1/4 chance each of coming up, and the numbers 2, 4 and 6 can never come up. Again, the probabilities sum to 1. So again: the way this model works is that before we throw the die, we know the probabilities of what might happen. But after we throw the die, only one of those components of that 6-dimensional vector will “light up”. The die vector will go from having all those numbers in it (before throwing it) to having just one 1 in one of the components, and the rest 0s.

Now back to our light switch system with the peculiar memory-loss behavior. Let’s see if we can model this just with our up and down vectors from above. The first thing we want to model is that a light switch cannot both be up or down once we look at it - it needs to be just one of those. Obviously, that’s how light switches work. For vectors, that means they have to be “orthogonal”, or if you want to picture them, they have to be perpendicular to each other. The vectors up = (1,0) and down = (0,1) are just that. See how if we just draw them into a simple 2D coordinate system. “Up” and “down” seems like a weird naming scheme, but remember that to model a die, we would have had to draw a 6-dimensional vector, so these dimensions here are really more about the “information” in the light switch (is it up, or down?), and not a real direction. Let's check on the modeling of probabilities: if the light switch is in the up position, we have the vector up = (1,0), so the probability of it being in the up position is 100%. Checks out.

Ok, now let’s move on to our swiveling light switch. If we lay down the switch on the side, we put the switch to “right”, and then we turn it up, we know that we have a 50/50 chance of seeing it in the “up” or “down” position. How would we model that? With a vector “right” that looks like this:

Does this work? Yes: remember how that upper number in the vector models the “up” switch position, and the lower number in the vector models the “down” switch position. Wherever the vector “lights up” (is equal to 1), that’s where we expect the light switch to be. So here, we have an equal chance of seeing it up or down. (Why the square roots? Well, for no particular reason, we’re just going to agree from here on out that we need to first square these numbers before they become probabilities.) Also, do these probabilities sum to 1? They do: square root of 2 to the power of 2 plus the same thing again = 1.

What about “left”? Can we just use the same vector as for “right”, because that one worked so nicely? No! Remember that we have the other constraint that when we look at it, the switch has to either be left or right, so the “left” and “right” vectors have to be orthogonal to each other. Try this one:

And indeed this vector is orthogonal to the “right” vector, when we look at it in our 2D coordinate system.

With a little bit of math, how do you calculate “orthogonality”, that two vectors are perpendicular? You take the so-called dot product of two vectors, and that has to be 0. First, this is just a shorter way to write the left and right vectors, where we just package together their components into one vector, no other change:

Now here is their dot product, where you multiply each pair of components in the same dimension, and then add up all those products:

And that shows mathematically that indeed, the vectors “left” and “right” are orthogonal to each other, because their dot product is equal to 0. You can try that with up = (1,0) and down = (0,1) as well: up * down = 1 * 0 + 0 * 1 = 0.

Now we’re going to make this really complicated. We give the light switch another direction that we can swivel it into: in the z-axis in space. We’re going to keep all the other rules the same. So when we lay down the light switch and set it to “right”, and then we turn it into the x-direction and measure, we get a 50/50 chance of seeing it in the “up” or “down” position. Or, if instead we turn the light switch to face in the z-direction, we get a 50/50 chance of seeing it in the “back” or “front” position. And vice versa: switch it to “front”, and then swivel it upwards, and you get 50/50 up or down when you look.

It should be easy to model this with our “up, down, left, right” vectors, correct? Here are the steps: introduce the “back” and “front” vectors, make sure they’re orthogonal to each other (so back * front = 0), and make sure they have a 50/50 chance of coming up when we previously see either up or down, or left or right.

But it turns out that something really interesting happens, when you try to do that. You will find no real solution for it. Well, no “real” solution - but an “imaginary” one. Here are the front and back vectors:

First of all, these vectors look very similar to the ones we’ve been playing with. Except for the peculiar “i”. That is the imaginary constant. It is defined as the square root of -1. But a good way to imagine it is as just another dimension. When you see an i in an equation, you follow a simple rule: you can combine all equation components with i with all other components with i, and all components without i with all other components without i. (And when you square i, it becomes -1.) So, they kind of really behave like just another vector dimension - they “co-exist” with the non-i numbers in this equation, and you can’t merge them together.

That also means we can’t really draw them into our 2D coordinate system anymore. They add not just one dimension, but two dimensions: because both our upper vector coordinate and our lower vector coordinate now have an i. That means we’d have to draw an axis popping out of the screen for both of our previous x and y directions, and both of those would have to be orthogonal to each other. That’s 4-dimensional, and we can't draw that on a piece of paper. Except, here is a very clever way to visualize it: spinors live in two dimensions, but have two additional degrees of freedom (parameters) you can control (the two imaginary parts of the complex numbers in their two components). That's enough parameters to describe a 3-dimensional sphere, with the spinor itself being a little flagpole, and the angle of the flag being described by the 4-th dimensional parameter.

Now we’re very close to the end. If you think about it, what happened here, and why did our little vector model of our mysterious light switch fall apart, and the strange i showed up? Because we started in just two dimensions: “up” and “down”, or (1,0) and (0,1). Those are two bits that are either on or off. That was enough to model two swivel directions, in the x- and y-direction, of our light switch, and the memory-wiping conditions that we put on it. But then we introduced a third dimension of our light switch, when we allowed for it to also swivel in the z-direction! So effectively, we tried to model a 3-dimensional system, but only by using 2 coordinates. Of course that’s going to break. And the math simply forced us to make use of that other “dimensionality trick” - starting to use complex numbers. And because each complex number is made up of a real and an imaginary number, that’s 2 more dimensions, so now we have 2 * 2 = 4 dimensions, and that’s enough to model our 3-dimensional system with weird memory-wiping constraints.

Bringing it all together

Now: we spent a lot of time first talking about groups. Then we talked about this light switch model. Here is how we bring them together.

Go back all the way to the place where we talked about the Lorentz group. That is the group of all transformations in 4-dimensional spacetime which are “legal” under the maxim that the speed of light is the limit for everything: all rotations, and all shifts; an observer jumping on a train while watching an experiment, or swiveling the experiment around in her hand. Spacetime is 4-dimensional, so all those transformations were 4-dimensional matrices.

But then we invoked group theory, which lets us discover the most fundamental encodings of these kinds of operations. In group theory, we turned the Lorentz group into its Lie algebra, which enshrines everything that is legal about 4-dimensional spacetime transformations, simply by setting the speed of light to finite and declaring that the world has to look the same from all vantage points, no matter how fast they’re moving.

And then it turns out that once you have a Lie algebra, you can use it in other dimensions. Which we will now finally do. What is the smallest possible dimensionality you can use the Lorentz group’s Lie algebra in? One dimension, but that’s just a point and is boring. Let’s try two dimensions. The rotation matrix for a rotation in two-dimensional space which adheres to the principles of a finite speed of light and the world looking the same from all vantage points is:

There it is again, the imaginary number i. This is not at all the same rotation matrix that we would get if we just rotated some two-dimensional vector in a two-dimensional space, this is something much weirder. And in fact, this turns out to exactly model that weird light switch behavior we showed above. There is another giveaway in these numbers: look at how the rotation angle is divided by 2, directly in the rotation formula. What happens if you take our “up” light switch, up = (1,0), and we rotate it by 360 degrees? In our real world, that should just make it what it was before. In this bizarro world, it rotates the up = (1,0) vector into the weirdo-up = (-1,0) vector. What happens when you measure the probability of looking at the light switch in this state? Nothing: it’s up. Because remember how we’re squaring the numbers, and (-1)*(-1) = 1. So looking at a light switch in the (-1,0) state is just a light switch in the up position. But imagine you have two light switches meeting, and one is in up = (1,0), the other is in weirdo-up = (-1,0): they... disappear. They really do. Only if you rotate an “up” light switch by 720 degrees, two full rotations, does it get back to its original state, (1,0). (In that world, the old joke "he did a complete 360" actually has meaning.)

Spin

But surely such a light switch doesn’t exist in the real world? Yes it does: it is very hard to believe, but this is exactly the behavior of the electron. Electrons have a property called “spin”, which makes them a little like tiny spinning tops, and thus they have a little magnetic moment. You would expect that when electrons tumble out of an electron source, where their magnetic moment points would be all over the place:

No: when you measure it, all electrons "point" either strictly up or down. The famous Stern-Gerlach experiment showed this for the first time in 1923, and it was a huge surprise to everyone. They shot silver atoms (in which their orbiting electrons have spin) from a source (1 in the image below) through a magnetic field, and expected to see a random deflection by electrons (line 4 in the image below). Instead, they saw two distinct point (5 in the image below): electron spin was clearly quantized, and either pointed straight up or down.

That is "two-dimensional behavior", and now you know where it is coming from. Electrons are particles in our three-dimensional space, but they have an aspect that lives decidedly in a two-dimensional space. The old science fiction trope where aliens from another dimension visit us: that is actually the case in the form of electron spin, they are visitors from the second dimension.

A spinor is the most basic mathematical object that can be Lorentz-transformed

Now we have collected all the pieces and can put them together.

1. The world has to look the same from every vantage point.
2. The speed of light is finite.
3. Space and time contort to keep the distance between two events in spacetime constant.
4. The Lorentz transformations are all rotations and shifts that are allowed while spacetime distance constant.
5. The Lorentz group encodes these transformations in its Lie algebra.
6. A group can be represented in different dimensions: we defined the Lorentz group as transformations in 4-dimensional spacetime, but we can ask how its representation looks in other dimensions.
7. The lowest dimension for which the Lorentz group has a representation is 2.
8. When we derive the Lorentz transformations in 2 dimensions, it turns out that they describe spinors, with the strange behavior we demonstrated above.
9. Electron spin is described by a spinor.

So: an unbelievably simple starting point - a) the world looks the same from everywhere and b) the speed of light is finite - leads to symmetry principles which lead to the postulation that pure mathematics provides for the theoretical possibility that oddly behaving two-dimensional beings exist. And lo and behold, we find them, and they're all around us. Nature is given an opportunity in an entirely mathematical fact, and it jumps on it, in the simplest possible way.

A historical curiosity is that to most people, Einstein is probably famous for his 1905 theory of special relativity. But he won his Nobel Prize for his paper on the photoelectric effect, which would turn into quantum mechanics. The electron spin is a famous quantum mechanical observation. As far as it seems to me, you don't even need any quantum mechanical discussion - it follows from a finite speed of light and symmetry constraints.

Books

Great books to read on this: Physics from Symmetry (Jakob Schwichtenberg), on special relativity and group theory. An Introduction to Spinors (Andrew M. Steane), includes the flagpole picture. Quantum Mechanics - The Theoretical Minimum (Leonard Susskind), on modeling spin behavior.