I want to tell you about an adventure and that is something that is here, you all have heard about the Higgs Particle.
What does it do?
It is supposed to give mass to other particles.
And you wonder what the hell is that important for.
Everyone gives mass to everybody, so what the hell?
The Higgs by itself predicts nothing except its own existence.
It does not even predict its own mass.
And I will try to tell you what the importance, the real importance of the Higgs particle is.
In the end, let’s get something clear, too, when you do theoretical physics,
in the end you want to be able to do calculations, the outcome of which agrees with experiment.
That’s the only and one criterium that you apply to physics theory.
Now, revolutionary ideas have been in this world since a long time.
For example, in the old times there were people who invented glasses, some monks near Venice.
And you can imagine our life also when there were no glasses.
But even more stunning is the invention of music.
The way you do music, which happened by some monk, someone in the 1000s or something.
And very few things have been so important as this one here is.
It’s due to this that we could have people like Mozart and Beethoven, who would write music in their own house.
And if you don’t know how to write music, you have a problem.
And therefore that happened in Europe.
And outside Europe nothing much in the way of development of music happened.
Well, quantum field theory was something that was created around 1930.
But it remained stuck till 1948.
So 18 years long, nothing much happened.
And then the very simple idea, of which you probably have heard very little, happened,
and that idea was due to a person named Kramers, so the idea of renormalisation.
And I will explain to you how renormalisation has been on the basis of any progress that was made since then.
Now, in 1904, talking about the energy of the electron, the self-energy of the electron,
there was a theory about that of Abraham and Lorentz.
And the idea that they had, these people, the question is what do you think of an electron?
An electron has the charge concentrated in a point, how must I see this?
Ok, you pile all the charge together in one point.
One thing to do it is to assume that the charge is in the little sphere.
So the idea of Abraham and Lorentz is the electron is a little sphere.
And there it is, that’s the electron.
And then you see that the energy, the little sphere, you think of the charge coming from infinity and then being pressed together.
The question how much energy does it cost you to press all the charge together in a little sphere?
And the answer is, the energy that you need for that is inversely proportional to the magnitude, to the radius of the sphere.
So as you went smaller and smaller the energy you need to apply is bigger and bigger.
So you cannot really go to a point because that means that the energy becomes infinite.
So this is the problem, it starts surfacing in those old days.
Now, the smearing out, I use this model as old as it is, in order to introduce some terminology.
The smearing out of the charge of the electron over a little sphere, that’s called regulator mechanism.
You regulate something that otherwise would be infinite.
So it’s regulated infinity.
And so here you have something of a little, with radius r.
And the next thing, the radius r is to regulate the parameter.
It allows you, to tell you precisely how you regulate, how you keep things finite.
And if that regulator parameter goes to zero, meaning the sphere shrinks, then the energy going with it will go to infinity.
And you speak of it goes proportional to that.
So as the radius becomes twice as small the energy becomes twice as big, and you describe that as a linear divergence.
So the classical electro magnetism has a linear divergence.
And this is, I would say, a rather naïve description of the theory that would remain part of the theory for 35 years.
And then in the meantime many things happened.
Quantum mechanics was born.
And so people started to worry about the self-energy of the electron within the context of quantum mechanics,
and that changed things.
Here you see Abraham and Lorenz doing their first steps in the unknown of the electron.
And then we got quantum field theory.
Quantum field theory was born at the end of the 1920s, made by Heisenberg and Pauli.
And they gave us the quantum version of the theory of the electron and the electric force and everything.
And the calculation of the electron self-energy was done by a Victor Weisskopf.
Victor Weisskopf was the director of CERN when I got there.
So that’s the link that I feel with that time period.
And Weisskopf discovered that, at least it was in the quantum theory, the divergence of the electron,
you know that thing inversely proportionate to the radius, that disappeared and the only divergence was logarithmic.
I’ll expand on this a little bit.
You have something called a Feynman diagram.
And in the Feynman diagram you picture the electron as a particle that can emit a photon and reabsorb it.
And when you try to calculate it, out comes infinity.
So that’s what you have.
And the corresponding expression which you can write down in field theory, you don’t have to understand it very well.
That corresponding expression is like that, and you see it’s odd in the momentum p,
meaning that a linear divergence by symmetric integration disappears.
And that you can see in the last line.
So there’s not much of progress, but it’s some progress,
the divergence of the self-mass of the electron in that scheme was logarithmic.
And now you see the following, if you want to deal, which you have to,
if you want to deal with infinities you have to invent a way to make that infinity finite.
You have to make your charge in the sphere or what have you.
And you have to introduce a regulator parameter in such a way that if that regulator parameter goes this way or that way,
you approach the situation that you think happens in nature.
And then a new complication arises.
The regulator mechanism may violate important properties and you have some real difficulties,
for example, in the theory of Abraham and Lorentz.
The sphere, that an electron is, doesn’t remain a sphere if the electron starts moving according to the theory of relativity.
It becomes an elliptic thing.
So then you would say the regulator method violates Lorentz invariance.
And so you get this problem, which is really a problem and you have other things, too.
An important property in quantum electrodynamics is gauge invariance.
You wanted gauge invariance because it guarantees you that the theory will produce something that will conserve its charge,
and we all believe very much in the conservation of charge, experimentally it is something that works very well.
So you have to have gauge invariance.
Now, Pauli and Villars is a sort of an abstract thing.
So people thinking about the problem didn’t realise the complexities of it.
Pauli and Villars understood it.
And they developed a regulator mechanism that respected gauge invariance and that respected Lorentz invariance.
So they did something in that aspect.
And then finally after that there was the great revolution of 1948.
In 1948 there was a conference in Shelter Island in America and at that conference two people presented results
which were absolutely surprising and new.
And these two things were the Lamb shift, that’s a small shift in the energy spectrum of the hydrogen atom,
the Lamb shift discovered by Mr. Lamb, using equipment that was developed in World War II.
And then there was the magnetic moment of the electron that also was measured with, for that time, unprecedented precision.
And they found a little bit of difference from the magnetic moment that the Dirac theory predicts.
That’s called the anomalous magnetic moment.
And so these two things were there and people knew, well, where do they come from?
Probably it’s a quantum effect, but how?
Quantum theory was not developed and not only that, whenever one tried to do a calculation, you got infinity.
So this was a real difficulty.
And then Mr. Kramers from Leiden, he was the chairman at some point or some session,
and he got up and he said what you really should do, and he gave us the idea of renormalisation.
And that I am going to explain to you.
This is the very simple idea that I want you to understand as something that has been crucial in the development of the theory.
The idea of renormalisation is as simple as it can be, and you wonder why they didn’t think of it before.
That’s always with big ideas.
Once you know them you say, yeah, well, of course!
But before you know them you don’t seem to be able to think of it.
So Kramers says, let’s look to the mass of the electron.
Here is an electron and it has a mass.
What we see is the net result of a lot of things or at least some things.
Surely the mass of the electron has as an ingredient, the mass that the electron has before you give it an electric charge.
So that’s what you call the bare mass.
And then the electron gets its mass.
Then you turn on the electric field and you get the self-energy of the electron sitting in its own field,
that little sphere, remember.
A bit more complicated with Pauli and Villars but that’s it.
So the mass that you see experimentally is the sum of the two.
of what goes on in the very inner part of the electron.
But for sure the mass that we see is the sum of two masses.
And we don’t really know what the mass of the electron is before we started with the theory.
So the bare mass of the electron is a fictitious parameter.
It is not something that you can measure.
But then added to this bare mass of the electron comes the mass due to the self-energy of the field, that field mass.”
So you get the addition.
The only thing that you can observe experimentally is the sum of the two.
just make that self-mass minus infinity, so that together they come out at the desired value.”
He said, “I know the result”, he says, “why are you arguing?
I don’t have to know the interior of the electron.
The only thing I have to know is the sum total, that mass.” So this was the idea of renormalisation.
And it electrified the people, the participants on the conference, and calculations started to go on.
And people started to produce calculations of the Lamp shift and the anomalous magnetic moment.
And quantum field theory, as we know it today, was essentially born.
But no one has gotten this idea before Kramers.
It’s sort of an amazing thing.
So Kramers says, “Ok, a few corrections to the self-mass of the electron is evident,
well, just make the mass initially minus infinity.
Let’s not worry where it comes from.” That’s the idea.
Now, the idea of Kramers initiated calculation of many people.
Feynman, Bethe, Schwinger, Tomonaga.
But Feynman, I would say, did something very special.
And again it’s something special, you have to appreciate it.
We appreciated it only much later.
But at that time Feynman came up with the idea of Feynman diagrams.
And Feynman diagrams are this:
You make a little drawing and then corresponding to the drawing you write down an equation, a formula.
That’s the Feynman diagram.
And that picture that you make can be a very easy way of reflecting on what that value should be.
So what Feynman’s method does, is make it very easy for you to write down what actually is going on.
The actual value of that thing, whatever, the theory.
In other words, for Feynman’s diagrams you then have the Feynman rules.
And Feynman rules does show how you should make those diagrams.
So knowing the Feynman rules, you have the theory and it is easy, it’s wonderfully easy.
And Feynman himself, when he came from Stockholm, when he got the Nobel Prize,
he came to CERN and at the end he stood at CERN, he stood and said, “But what did I do?
All I did is inventing a method of doing bookkeeping.” The point is, that’s true.
But sometimes that’s more important than anything else because it makes what you do transparent.
So that’s what Feynman did.
And in a complicated theory such as field theory, having a good method and having a simplification of rules
and a nice way of putting it, is essential.
That’s what Feynman did.
So from here on I will sort of work with Feynman diagrams, and just remember when I write down a Feynman diagram,
they’re intuitive things.
And what they put down there gives you an idea of what’s going on in terms of a process.
But the essential idea is that you write down that thing and it’s easy to make it up
and then you can write an equation that goes with it and that works fine.
Diagrams are made up of lines and vertices.
And you see in this diagram here an example of a Feynman diagram with two vertices.
There’s an experiment I do, and some guys say, my god, you see how wonderful, who invented this is a genius.
There we have the two vertices, here is vertex number 1 and that’s vertex number 2, and these are the lines.
So that’s a very simple diagram and gives you an idea.
This is an electron and a positron scattering.
And the arrow tells you which is the electron and which is the positron.
Given now this diagram and given the Feynman rules, specifying the momenta of the ingoing particles,
you can actually compute an equation that gives you precisely, by squaring it and something,
that gives you the probability of this happening.
So it’s a very nice way of doing a calculation.
And here you see the Feynman rules, and I won’t go any further in it.
Just believe me when I say that once you know the rules, you can do the calculation.
And there you saw something, the self-energy of the electron.
And you can in your mind present it as something, a photon goes away from the electron, travels a little bit and comes back.
And it does it so fast that it escapes your detection.
You cannot measure it, but it is there, it happens.
Quantum mechanics allows that.
And a diagram like this is called a one-loop diagram.
And then you do the calculation and, for god sake, out comes infinity.
So there is the infinity of quantum field theory.
And then you get the remark of Mr. Kramers, of renormalisation that infinity,
this one here which is created by the electron sitting in its own field.
And then you have here the bare mass, the mass that the electron would have if there was no electric field.
And all you want is the sum to be finite.
And that precisely is the idea of Kramers.
And in quantum electrodynamics all infinities of the theory, and that is the wonderful thing, can be taken care of this way.
And once you have done that, absorbed the infinities in the free parameter of the theory,
you get the finite theory, you can compute everything.
This is really very wonderful.
And so then we go on to the next.
Here is Mr. Feynman.
He was very proud of his invention because we see here he painted his car with it.
And the Feynman rules are quite simple.
But then we go on to another type of interaction which is weak interactions.
And weak interaction has had a lifetime in some sense coinciding with my time that I was in the field.
When I entered the field we knew next to nothing of it.
The only thing you knew, and I’m now speaking about 1960 roughly, were certain processes, the decay of the neutron,
the decay of the K-meson, you saw them decaying.
You could measure the energy and the momentum of the particles coming out, but you had no idea what was going on.
And so you looked at these things and you didn’t know what the hell is this?
And the only thing we knew at that time was that these processes were not going very strongly.
They were what we called weak interactions.
Later on we have understood they are far from weak, but in those days, for the things that we knew, they were weak.
Then there was the thing which was very peculiar about the interactions.
We discovered, the physicists of that time discovered, that the interactions looked like there was an intermediary particle.
So you see where you look for the neutron decay, here, then you think to yourself make a particle that is in between there.
And this imposes a certain structure on the combination of electron and neutrino, so the kinematics, it has spin 1.
And that was seen.
So we saw special effects in the spectrum of those decays indicating that there was a spin 1 particle.
And I remember very well that was the situation.
And you looked at it and you thought, what the hell is going on here?
What is this spin 1 particle?
How heavy is it?
You didn’t know.
The same thing happened here with the K-meson and the W, as we called it by that time.
That was going that way.
So this was the only thing known to those interactions.
And so at this point what do you do, you do theory.
You try to go further.
But where the hell do you go?
What do you do?
Well, I tell you what you do, what you do is you start assuming things about these intermediary particles.
So you have something that comes from experiment.
And then you start playing with it.
And what kind of game are we playing?
First of all we make this thing interacting with itself, so we introduce another.
We have the W plus, and the W minus that you could see here from the neutron decay had the W minus,
and K there was a W plus, so you knew these two were there.
We didn’t know anymore.
But we invented another one.
Not me, people invented another one, the W zero, and introduced an interaction between the three, completely hypothetical.
So what the hell do we do with this interaction?
Well, start doing calculations.
For a theoretical physicist doing calculation is, you know, like inventing the paradise.
So we started doing calculations like that diagram over there.
That you could have a vector boson, the scattering of two of these Ws, they exchange the W,
they go on and they reabsorb and out they come.
So this is some process, you can compute it.
What do we find?
Infinity.
So that’s the first thing, you did this calculation, you found infinity.
But then you discovered, or you realised there could be other possibilities.
For example, you could have changed the 2 vertices below and you would get another diagram.
And that also is a possibility.
So you get many more diagrams.
So you keep on doing the calculation, adding all these diagrams.
And you hope, you pray every evening, you go to bed and you pray that the result is finite.
But it ain’t.
So what happened then is there are infinities, and you couldn’t absorb them in the known constant,
in the strengths, for example, of the coupling of the Ws to themselves.
And so you had bad infinities that you could not absorb in the free parameters of the theory.
And if you had such bad infinities, we called the theory not renormalisable.
And so we got a new goal in life.
Can we make that theory renormalisable?
Change it a little bit so that the infinities cancel and can be absorbed in the free parameters.
And so that starts the hunt for what to do here where you want to make the theory finite.
This is actually how it happened.
I did this and I didn’t know what to do.
And so the next step you introduce a new interaction.
And that new interaction is 2 Ws interacting in a point.
That can happen in Feynman diagrams and then you get new diagrams.
So what happens is you introduce a new interaction
and try to make it in such a way that it makes the theory more decent, has less infinities.
And then you choose the strength of that interaction.
You choose very carefully in such a way there is cancellation of infinities.
That was a new thing that was not happening in quantum electrodynamics.
So that’s the first idea that you put in.
Maybe I can, by having different kinds of interactions, make them work together and make something that’s finite.
So that’s what you can try and, by god, that works.
And it turns out that if you have this thing here you can get rid of almost all infinities.
And the theory that you get then, that’s 4 point interaction, 2 point interaction, was known.
Not to me at the time, I didn’t know what the hell I was doing.
I mean most of the time you don’t know what you are doing.
But then the theory had this 4 point interaction, a 3 point interaction.
And that was already studied before in something called the Yang-Mills theory,
who had done it as a sort of an exercise in elegance.
It did something that had nothing to do with infinities.
It was symmetry.
No one knew that it had anything to do with infinities.
And it was done long ago, long before we started this work here.
But in any case it was what we call Yang-Mills theory.
That was pointed out to me by somebody and I thought, well, we are lucky we’re teaming up with some known stuff.
So, yeah, when you get the Yang-Mills theory,
and you get that purely out of the requirement that things sum up so that you don’t get too many infinities,
that you get a renormalisable theory.
Did we get the renormalisable theory?
Answer: almost.
What happened is there was just one little infinity somewhere left.
And you couldn’t get rid of that.
You tried that, that, that, didn’t work.
You were wrestling with it for a month this way, that way and so on.
Couldn’t do it.
And then it turns out another idea was needed.
And the other idea that was needed was that you introduce,
you had to introduce another particle which I have indicated with red lines there.
And that particle couples to the W and adds by the fact that it is here, it adds to the complications of the situation.
And then it turns out that that particle was known also.
This particle and the couplings that you could deduce was the Higgs particle, introduced long ago, long before that.
That is to say about 5 or 6 years.
So out of the requirement – now, this is Higgs particle – you made it in such a way that the theory,
the infinities would cancel as much as possible.
And that in fact the resulting theory would be what we call a renormalisable theory, that idea of Mr. Kramers.
And so you see not only has there new interactions been introduced between the vector bosons.
In addition, first we started by introducing in the W zero.
That was not seen.
Then you started making interaction with 4 Ws.
You got almost there.
Then the idea came to introduce the Higgs and the Higgs resolved the whole problem.
And there was only one parameter, only one new particle which was the part of the Higgs particle.
That one you couldn’t figure out from the situation.
Renormalisability requires there to be a Higgs, it doesn’t tell you how heavy.
So that was the situation.
And so you can see this long train of speculation which could have broken down at any point,
but it is just continuing the idea of Kramers, observing to the end, as far as you can.
If you do that, you get something and whether that something is meaningful or not has to become clear.
And the only way that it can become clear here is finding the Higgs.
And, by god, that happened at CERN.
They found Higgs, not Mr. Higgs.
That’s Mr. Higgs there.
But they found it in the experiment done at CERN.
And you see here Mr. Higgs looking proudly to us, more or less at the age when he discovered or when he postulated his particle.
And that was done by Mr. Kramers.
So you stand there as if I have to stop.
Ok, I’ll stop.