Physics is the simplest and most central of all branches of human knowledge.
I know that.
You know that.
Everybody knows that.
And it’s true!
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 around this single equation
you see here.
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.
So that might sound a little dodgy and, I admit, when I was a much younger student,
I sort of rebelled at the idea.
But 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 kilometers or about 3,884 miles.
But suppose I needed to know that number quite precisely- say to an accuracy of a kilometer
or about half a mile.
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
or about 1,000 miles per hour for my American audience, 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, 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’ll just let that
one slide.
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 center of the Earth to the poles is 6,357 kilometers,
while the distance from the center to the equator is 6,378 kilometers.
If you talk about diameter and not the radius, the Earth is 42 kilometers 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.
This picture here exaggerates what is going on, but essentially the Earth is a bit pear-shaped.
Well, that’s okay, the planet is getting older and it happens to the best of us.
Notice that I said meters here when we were talking about kilometers before.
The changes in the Earth’s geometry that gives 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.
So this gives you the 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.
If you’re truly allergic to math, I’m, well- I’m sorry.
But this is just kind of cool, so hang in there.
It’s really not that hard.
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.
You’re going to have to trust me that I did it right.
But, you do trust me, don’t you?
Of course you do!
We’re old friends.
So- this equation looks positively yucky.
I mean, 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.
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.
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?
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.
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.
Okay- so now that you’re an expert on what perturbation theory is all about, we can talk
in general terms about how it applies to particle physics calculations.
Basically, all we do is to take the complicated equation we started out this video with and
replace it by a simpler and approximate one.
And voila!
Instant success.
Now this topic has a lot of facets to it, so I’ve broken the conversation up into
several different videos.
To understand the whole thing, you’ll have to watch all of them.
So let me suggest that you also watch my video on Feynman diagrams.
Then you can combine the knowledge that you learned here with the Feynman diagram video
and you’ll be ready to tackle particle physics theory.
And the video that ties this all together is the one on theoretical quantum electrodynamics.
So, to recap, perturbation theory is a key trick that lets you get a handle on particle
physics calculations.
And if you combine this video with the Feynman diagram and the QED one, then you will be
a particle expert.