• Yahya Ashraf

What is perturbation theory?

Physics is the simplest and most central of all branches of human knowledge. Of all of the branches of physics, the truly deepest and most fundamental of all is particle physics. Our current theory, which is called the Standard Model, is based on this single equation you see here.

the standard model equation

In principle, you could simply take it, throw some math skills at it and figure out the answer to all of particle physics’ questions. But there’s actually a problem- and that problem is that nobody knows how to solve the equation exactly. The equation is simply too difficult. So that might make you wonder how it is that scientists can claim that they know what they’re doing when they can’t even solve the equation that is central to the theory.

Something sounds kind of shady. And you’d be right to wonder about that. However theoretical physicists are a clever bunch and they employ a mathematical trick called perturbation theory. The basic idea is to replace the unsolvable equation with one that is approximately correct. While the calculation won’t be perfect, it’ll be pretty close; and, if you need a more accurate calculation, you employ a more accurate approximation.

Examples of Perturbation theory

Let me give you some examples that might make you feel better about this approach. So- let’s start with a familiar one. Suppose I needed to know with some precision the shape of the Earth. Depending on just how precisely you need to know it, you’ll get a different answer. So the simplest answer to the shape of the Earth is that it’s a sphere. We’ve seen pictures of it from NASA and there’s no doubt that this is a very good approximation.

The Earth is basically a sphere with a radius of 6,367 kilometres But suppose I needed to know that number quite precisely- say to an accuracy of a kilometre Well, under that fairly precise requirement, one has to do a bit better than the whole “Earth is a sphere” approximation. Because the Earth is spinning with a speed at the equator of about 1,670 km per hour the Earth actually isn’t a perfect sphere. We can see how this works by this demo here.

When the object is stationary, it has a circular shape. However, once the object is spinning, we see that the shape deforms. It gets wider across the equator and shorter across the polls. The spinning creates centrifugal forces. Yes, I know that centrifugal forces are fictitious, and if my main point was rotation, I’d be a bit more careful here. But the idea of centrifugal forces is useful in this situation, so we’d just go with it.

Well, the Earth is also spinning and this actually causes the globe to distort. The diameter of the Earth across the equator is actually larger than the diameter connecting the North and South poles. And the difference is pretty big. It turns out that the distance from the centre of the Earth to the poles is 6,357 kilometres, while the distance from the centre to the equator is 6,378 kilometres. If you talk about diameter and not the radius, the Earth is 42 kilometres fatter than it is tall. That’s a difference of about 25 miles. So that sounds big, but is it? Well, if you think of it in terms of a percentage, this difference is only about 0.3% or about one part in 300. So it’s actually just a tiny correction. Unless you need to know the shape of the Earth to an accuracy of under a percent, calling it a sphere is just fine. And this is a crucial point. Calling the Earth a sphere is a very good approximation and it works for most purposes.

And the story doesn’t end there. If you need to know the shape of the Earth even more accurately, it turns out that the Antarctic ice sheet squishes the bottom of the Earth. This and a few other effects causes the North Pole to bulge upwards about 17 meters and makes a bulge of 7 meters in the mid-southern latitudes.

Essentially the Earth is a bit pear-shaped. Notice that I said meters here when we were talking about kilometres before. The changes in the Earth’s geometry that give it a tiny pear-like shape are about 1/1,000 of the equatorial bulge distortion, which was already a small one-in-three-hundred distortion on the basic spherical shape of the Earth. And, of course, if you want to get even more precise, there are smaller distortions still, not to mention issues of mountains, hills, trees, buildings, et cetera. But none of these ever-smaller corrections changes the fact that the idea that the Earth is a sphere is a really good approximation for most situations. Each successive approximation is just a small perturbation on the overall and dominant shape. For most purposes, the approximation of a sphere is good enough.

A mathematical example of perturbation theory

So this gives you a basic idea of how scientists perform particle physics calculations. But I want to show you how it’s done in a slightly more mathematical way. Suppose you wanted to do a calculation that involved a sine wave. You’ve seen pictures like this, where a mathematical function squiggles up and down- that’s a sine wave. Well, it turns out that it can be hard to do calculations with such a complicated function- but there are ways to approximate it. Now, I’m going to show you an equation that is another way to write a sine wave.

The sine wave equation

On the left-hand side, we have a nice compact sine function and on the other side, we have lots of terms- an x term, an x-cubed term, an x to the fifth term and so on to infinity. But here is the beauty of perturbation theory. Maybe you don’t need all of those terms. Maybe just the first one will do or maybe the first and second. And, if you can use just one or two, the whole calculation can be a lot easier. So let’s take a look and see how many terms we might need. Well, let’s plot the sine wave and then zoom into near the place where x = 0. Then let’s compare the sine curve to just the first term in the series- the x term.

Graph 1 - y = x
Graph 1 - y = x

The first thing we see is that the two actually agree pretty well. So that’s already looking good. However, when we zoom out a bit, we see that the first term doesn’t do so well.

The x term and the sine wave diverge quite a bit. So what happens if we add the second term- the x-cubed term?

graph 2 - y = x- x^3/3!
graph 2 - y = x- x^3/3!

Well, we see that it improves things a lot. But it’s not perfect. If we add the third term, the x to the fifth power term, we see that the approximation looks even more like a sine wave.

graph 2 - y = x- x^3/3! + x^5/5!
graph 2 - y = x- x^3/3! + x^5/5!

In fact, if we added more and more terms, it gets better and better. So that’s the beauty of perturbation theory. Sometimes you don’t have to solve the hard problem. Instead, you can take an approximate equation and solve that. The result will be a good enough answer for many cases. And, if you need more precision, you just make a more accurate approximation.

Now let’s talk about how it applies to particle physics. Basically, all we do is to take the complicated equation we started out with earlier in the blog and replace it with a simpler and approximate one. Most of the things in that equation do not contribute significantly and if we take approximate values we could get a pretty good approximate of the answer which will be more or less near to the real value so we don’t have to really do the impossible math. We could just use this clever technique called perturbation theory and still get a pretty good answer, if we want a more accurate answer then we could just use more accurate data. Physicists use this theory to get answers in almost all the fields of physics, ranging from QCD to classical physics, and we get pretty accurate results just by using this intelligent technique. If you want to know more about it then you can watch MIT’s open course lecture on the same.

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