Stanford University. Before we start, we're going to talk about quantum chromodynamics today. Quantum chromodynamics, and we're not going to talk about the deep mathematics of it and go into it.

We have basically one lecture to do quantum chromodynamics, which is a huge subject. It's a subject which easily takes a quarter to, but I'll stick with highlights and tell you what the basic intuitive picture is. Quantum chromodynamics is similar to quantum electrodynamics.

Quantum electrodynamics is the theory of electrons and photons. It's also the theory of the Coulomb force, the quantum mechanical theory of the Coulomb force. If applied to the problem of atoms in which you do not worry about the motion or the nature of the nucleus,

the structure of the nucleus, you simply think of the nucleus as a fixed point, creating an electric field, then quantum electrodynamics is also the theory of atoms. In the same sense, quantum chromodynamics is the theory of quarks and

gluons and the things that quarks and gluons make. The things that quarks and gluons make are called hadrons. H-A-D-R-O-N-S, hadrons or hadrons.

And they consist of things like protons and neutrons, which are called baryons, which have a net quark number or a net baryon number. That means an imbalance of quarks and anti-quarks. Three quarks for the proton, three quarks for the neutron. And mesons, which are quark, anti-quark pairs.

And all of them are full of gluons. Gluons are the electrically neutral stuff, which is the glue and the binding stuff that holds the quarks and gluons together. Much the same way that the electromagnetic field or

Coulomb field is the binding agent that holds atoms together. Atoms, other electrostatically bound objects. All right, but before we go into it, I want to remind you a little bit about the mathematics of spin.

Because first of all, that particular mathematics will come up in another guise, but also a simply related mathematics to it. Will also come up in the guise of a quantity called color, or a concept called color.

We're not going to do group theory in this class, all right? We'll try to finesse it. But the basic mathematics of what we're talking about is group theory. As I said, I'm not going to use group theory, or at least not call it group theory, as we use it.

For spin, the theory of spin is the theory of a group. A group, if you don't know what a group is, just ignore it, all right? But the group is the group of rotations of space. The theory of spin is also the theory of the symmetry of physics

with respect to rotations of space. But we finessed that by not talking about the theory of rotations, but by talking about the theory of angular momentum. I'll just remind you very quickly that we derived everything about angular

momentum from some basic commutation relations, which were of the form Lx with Ly equals I times Lz. I think there's an h-bar in there, but yeah, there is an h-bar in there.

Where those are the components of angular momentum. We worked that out last quarter, and we found some rather surprising and interesting things. Oh, of course, there are three other relations like this, which are gotten by permuting x to y, y to z, and z back to x.

I won't write them down. But when we worked out the consequences of this, first of all, we broke the symmetry. Now, when I say we broke the symmetry, I'm not talking about some symmetry breaking in nature.

Just we broke the symmetry in our mathematical description. The symmetry being the symmetry of rotation or the symmetry which interchanges x, y, and z and so forth. We broke it in our heads, maybe not in the real physics, but just by focusing on the z component of spin, or

the z component of angular momentum. We basically focused on it. We cannot measure all three components simultaneously. They don't commute with each other. Here they are. They don't commute with each other. At most, one of them commutes with itself. Well, of course, it commutes with itself. And no two of them commute. So at most, you can talk about one of them, or

measuring one at a time, and we arbitrarily chose Lz. When we did that and used these commutation relations, what we found was, first of all, that L is quantized. And the difference between neighboring values of L was

always one unit in units of h-bar, number one. Number two, we focused on spin. And we described different spin possibilities. The first set of spin possibilities were the half-spin objects, or

the things which we call fermions. Those were the objects whose spin was centered about zero in such a way that the first level, the first z component of spin was a half. And the first one in the opposite direction was minus a half. Three halves, minus three halves, and so forth.

Those were fermions. But right now, the main point was that the mathematics of spin, or angular momentum, gave rise to some multiplets. There was first the spin-half multiplet. That was the multiplet with a spin minus a half and

a spin plus a half, like the spin of the electron. That was associated with half-spin particles and fermions. Bosons, which for our purposes now, just think integer spin. The first one was just spin zero.

There are particles with spin zero, they have no spin, that's all. They just have no spin at all. Then there are particles with spin one. The photons have spin one particle. There are other spin one particles. There are nuclei and atoms of spin one. And they have three states, plus one,

minus one in units of h-bar, and zero. And then the next interesting family has spin three halves. Spin three halves again goes on the half-spin column here, and that has one half, minus one half, three halves, minus three halves.

There are particles with spin three halves. There are certainly nuclei with spin three halves. And atoms with spin three halves. So these things also exist. Likewise, what comes after spin one? Spin two. Spin two, spin zero, spin one, spin minus one.

Or the z-component of spin one, minus one, two, and minus two. These labels here are the z-components of the spin. Now, when you speak about an object of spin l, l is the notation that denotes the highest value of z-component of spin.

So this would be, or m, you can use m. Ordinarily, it's used m, m or l, it doesn't matter. Let's use l, let's use l, l is a little bit different than m.

When you speak of spin l, you're speaking about an object whose maximum spin along the z-axis is l. So this would be spin a half, l equals a half. This one would be l equals one, and so forth.

How many states are there? How many independent states are there for a spin, a particle with spin l? Two l plus one. Two l plus one. For example, for spin a half, twice l is one, plus one is two.

Two states. If l is one, twice one is two, add one more, that's three. Three states. And so two l plus one is the number of states, is the number

of independent spin states for a given value of, for a given value of l. Okay, so that's something I want you to remember.

Now, another thing, if you have more than one particle, such as an electron, then you have two spins. Spins can be combined. Let's consider what you can make in the way of spin by combining together the spins of two electrons.

Let's forget the fact that electrons can orbit around each other. Let's ignore what's usually called orbital angular momentum, and just concentrate on the spin angular momenta. What kind of angular momentum can you make for two half-spin particles?

Well, the maximum value of the z component of spin is what? One, in units of h-bar. Let's forget h-bar now. When I say unit, the maximum spin, I mean in units of h-bar. The maximum spin you can have along the z-axis is one.

That's when both spins are pointing along the same, when both spins are pointing upward along the z-axis. That axis is the z-axis. So you can have angular momentum one. You can have angular momentum minus one, and you can also have angular

momentum, or z component of angular momentum, zero. But in fact, you can have two states where the z component of angular momentum is zero, this one and that one.

This one and that one. So there are two states, two independent states, whose z component of angular momentum is zero. How many states are there for a spin one object? Three. Two l plus one.

Okay. So, there must be a spin one combination, because we can make total spin up equal to one. That's the maximum. So there must be somewhere there some combination, must correspond to spin one. What does the spin one have?

It has spin one along the axis, spin minus one along the axis, and spin zero. But there are two states with spin zero, with z component of spin equal to zero. They can't both be part of the spin one multiplet. What could the extra one be?

It's got to be either spin zero, spin one, spin two, spin three, spin four, spin a half, spin minus, spin a half, spin three halves. So here's what we found. Let's write them down.

There's both spins up, or both z components of spin. I'm going to stop saying z components. Both spins up, that's got z component equal to one. Both spins down, that's got z component equal to minus one. And then, there seem to be two with z component equal to zero.

But if I look at spin one, all it has is plus, minus, and zero. There are only three of them. So this cannot be a multiplet.

These four states cannot all correspond to spin one. What else could be there besides spin one? Spin zero. Can't be spin two, because spin two has to have five states.

The only other thing is spin zero. So there must be quantum states here, three quantum states corresponding to spin one, and when you put the spins together, and you ask what kind of angular momentum can you make, you can make spin one and spin zero. The only combination, the only question is what combination,

what quantum mechanical combination of these two correspond to spin zero, and what combination correspond to spin one, to the missing spin one combination. And I will tell you right now, the answer is that the symmetric combination,

with a plus sign, incidentally, this state over here is symmetric between interchange. You can think of it, if you like, as one spin over here, one spin over here. They're labeled, there's the one over here and the one over here. The spin states, if they're both up, is symmetric. That means the spin state is the same if you interchange the two spins.

Obviously, they're both up. This one is also symmetric. Up down by itself is not symmetric. If you interchange spin, if you interchange the two of them, this one goes to this one, and this one goes to this one. They swap.

But there's one linear combination, one quantum mechanical combination, which is symmetric under the interchange of the two. Plus, minus, plus, minus, plus. This is the symmetric state. What should you do to it to make it have total probability equal to one?

Divide it by the square root of two. These are the three states which together form zero, one, and minus one. If you rotate the axes, they transform into each other.

This is a spin one up. This is a spin one down. Where's the ones with spin one in the other directions? Made up out of this. Okay, so this is the spin one multiplet. And what about the other one? We're missing a state. There were four altogether.

Here we have three linear combinations. What's the other linear combination? The other orthogonal linear combination with a minus one. So there's another one here with up down minus down up over square root of two.

This one, these three correspond to L equals one. This one corresponds to L equals zero. So these are the ways that you can put two spin halves together to make the two possible combinations.

This combination, L equals zero. What is it like with respect to its angular momentum? The answer is it doesn't have any angular momentum. It's a thing without angular momentum. As far as angular momentum goes, it's like a nothing. I mean, it may have some energy. It may have other things, charge, whatever.

But as far as angular momentum goes, it's like empty space. It's got none. Okay? You might ask, can a up spin and a down spin come together and annihilate and disappear? Well, there's all kinds of reasons why electrons can't disappear,

but just for pure angular momentum reasons. If we're just worried about conservation of angular momentum and nothing else, then which of these combinations can just disappear if, as I said, if we're worried about nothing but angular momentum conservation? Can this disappear?

No, because it has two units of angular momentum up. How about down? How about this one? Yes or no? Let's take a vote. How many think yes? How many think no? Okay, the no's win because this one is not without angular momentum.

It looks like it has no Z component of angular momentum, and it doesn't. But the angular momentum about the X and Y axis of this is not equal to zero. It does not have L equals zero. It has L equals one. This is the missing thing that allows you to rotate the angular momentum into other directions.

What about this one? Yeah, that one has angular momentum zero, and if the only consideration was angular momentum conservation, it could decay. It could just disappear. Now, one way of thinking about it is you could say, well, yeah, that's good enough.

That's close enough. This one here can disappear. Okay. So if you measure the angular momentum of the symmetric one with respect to Z, you'd get one or minus one.

Yes. Wait. In which case? In the symmetric case. No. Along the Z axis, you would get zero in either case. Yeah. Along the Z axis. But if you measure the angular momentum along some other axis, you would get one or minus one.

All right. So that's the theory of angular momentum. Now, why am I bringing that up now, or the theory of spin? In particular, spin a half. Spin a half and how you build out of it, other spins. Oh, incidentally. Let's try to build spin three halves.

Let's ask what happens if you put together three spins. Well, if they're all pointing upward in the same direction, and that would be the maximum, what would you get? Three halves.

How many states are there with spin three halves? Oh, incidentally, how many states are there altogether of three spins, each with spin a half? Eight. Two times two times two.

Okay. So there are eight states altogether. How many states are there for a spin three halves object? Four. So how many are left over? Four. What can those other four be?

Could they be another spin three halves object? No. Why not? It'll be zero on the other axis. Okay. So let's go through it again. If you have a spin three halves multiplet,

there will be one state which has z component of spin three halves and one with minus three halves. We've already used those up. When we said there were four states which formed the spin three halves multiplet, we used them up. Now all that's left now is spins whose maximum value is

plus or minus a half. So what can be there? What in addition can be there? Only spin one half. Anything else? No, there can't be spin zero because making three spin

halves can never make spin zero. All it can make is spin three halves and spin one half. But we seem to have four states left over. The implication is there are two distinct ways to make spin one half.

So there are combinations which correspond to the spin three halves. The easiest ones are all three spins up or three spins down. And then there are more complicated ones. And then there are two distinct ways to make spin one half, which altogether adds up to four states. All right, so when you take three spins, three half

spins, you get spin three halves and you get spin one half twice. Two distinct ways of doing it. Let's keep that in mind. Okay, now why am I talking about spin now? What I really want to talk about is a concept called isospin, isotopic spin.

Isotopic spin was the first internal symmetry group or the first internal quantum number, the first analog of a spin, which occurred in particle physics that did not have to do with the rotation of space.

But it had to do with the rotation of an imaginary space or an internal space. Or if you like, it just gave rise to another set of quantum mechanical variables which were basically isomorphic, completely similar to spin, to spin.

But it didn't have to do with spin. It didn't have to do with the rotation of space. You could imagine, if you like, that it had to do with the rotation of some internal directions. Internal directions, imaginary directions of space, mathematical directions. So let me tell you where it comes from. We've already actually described it, although we haven't said it.

We talked about quarks. And we talked about there's a whole variety of different kinds of quarks, but most of the quarks were rather heavy. They were rather heavy and had an appreciable mass in

units of GEVs or hundreds of MEVs. The natural mass scale for hadron physics is somewhere, is hundreds of MEVs. The mass of a typical meson is a couple of hundred MEVs. The binding energy of particles is MEVs, sorry,

hundreds of MEVs, not MEVs, hundreds of MEVs. What object in nature has binding energies of order of a few MEVs? Nuclei. Things that hold protons and neutrons together.

But what holds a neutron together as three quarks has a binding energy of a few hundred MEV. All right, so the natural energy scale is a few hundred MEV, and there were only two quarks which were very light or whose masses were very light by comparison with

that, and they were the up quark and the down quark. The other quarks are heavy, and to make particles out of them, there's a cost in energy, and so for the lightest objects, the most stable, typically heavier objects will decay to lighter ones, the light objects of nuclear physics, the lightest objects and the ones which are

stable are the ones made of up and down quarks. Now, up and down quarks have almost the same mass. Of course, the down quark is about twice as heavy as the up quark, but they both have very small mass by comparison with the hundreds of MEVs of nuclear physics.

So in some first approximation, you can say that the up quark and the down quark have no mass. More important, their masses are close to being equal. That's asymmetry.

What does that mean? That means if you took all up quarks and replaced them by all down quarks, everything would be pretty similar. Now, of course, the up quark is slightly more massive than the down quark. So typically, things made of down quarks will be a little more heavy than the things made of the corresponding

up quarks, but the difference is small. For example, two up quarks and a down quark make a proton. Two down quarks and an up quark make a neutron. The mass difference between a neutron and a proton is very small. So that's an example of the almost symmetry between

up quarks and down quarks. So in first approximation, you just forget the difference. Up quarks and down quarks are symmetric with respect to each other. In the same way or mathematically the same way that

a spin up and a spin down are symmetric. In the case of spin, it really has to do with the symmetry of space, rotating the axes. In the case of up quarks and down quarks, it's just a mathematical manufactured space that you can imagine

where you take an up quark to a down quark by flipping some direction in some imaginary direction in your head, but all you're doing is interchanging up quarks and down quarks. Thinking of up quarks and down quarks as mathematically the same or mathematically isomorphic to up spins and

down spins, you come to the concept of isotopic spin. Isotopic spin is the analog of spin, but not up and down in the sense of the Z axis and the flipping spin, but up and down in the sense of up quarks and down quarks,

because there's nothing really up or down about it. It's just an interchange of two labels, up quarks and down quarks. And so it invents the concept of isotopic spin, and for all mathematical purposes, the replacement of up quarks and

down quarks is very analogous to the replacement of an up spin by a down spin. Okay, so for spin, we have two states, up and down, and that defines spin.

For quarks, we have up and down, and that defines isospin. Incidentally, what does the iso come from? Isotope. Isotope. We're going to see in a moment that isotopic spin interchanges, when you interchange up quarks and down

quarks, you interchange proton and neutron. Of course, if you interchange proton and neutron in a nucleus, you'll wind up making an isotope of something. So it comes from the word isotope, isotopic spin. But what it is is in nuclear physics in the old

days, it was just a replacement of the interchange of a proton by a neutron. And thinking of the proton and neutron as a spin multiplet in a mathematical spin space. It was traced eventually just to the fact that there were two quarks, up quarks and down quarks. That was the origin of it. Yeah.

Did isospin predate quarks? Oh yeah. Let's do the Heisenberg. Yeah. Well, something has to provide the extra charge or. Yeah. OK, so isotopic spin is not a precise symmetry of nature.

It's not a precise symmetry of nature in that first of all, the up quark and down quark don't have exactly the same mass. But even if they did, there would still be a distinction, a physical distinction which would make them different, and that is their electric charge. All right. Electric forces within the nucleus, except for a big

nucleus, when a nucleus gets big, electric forces become strong, but for small nuclei and especially for protons and neutrons and hadrons, electromagnetic forces are negligible by comparison with the other forces holding

protons and neutrons together. So, an approximation. Ignore the mass difference between protons and neutrons or ignore the mass difference between up quarks and down quarks and ignore the fact that they have electric charge altogether and concentrate on the other forces of nature, in particular the strong interaction forces,

which we'll come to. And then there is a precise symmetry relating up quarks to down quarks, and that symmetry is very much like ordinary spin symmetry. It is analogous to spin symmetry. Okay, let's talk about what you can make.

Supposing, well, one quark by itself, we will come to understand, is not a physical object that we can examine in the laboratory. The simplest object that we can examine in the laboratory is three quarks, and that's a proton or a neutron.

That's like having three spins. What can you make out of three spins? What you can make out of three spins is either spin three-halves or spin one-half, and you can make the spin one-half in two distinct ways. But let's forget the two ways.

Let's just say we can make spin three-halves and spin one-half. In the same way, taking three quarks, we can make an object of isotopic spin one-half or an object of isotopic spin three-halves. Let's first concentrate on the object of isotopic spin one-half.

An object of isotopic spin one-half, how many states should it have? Just by pure mathematical analogy with ordinary spin. Two. A spin-a-half object has two states. An isospin isospin is the simplified word. An isospin state of one-half also would have two states.

What are those two states? Proton and neutron. So let's write down what the proton and neutron are in

quark language.

Now, that's a proton. OK. Let's imagine that we've labeled the quarks. By label the quarks, let's imagine they're really located at three distinct spots inside the proton. This, of course, is not really true.

But I mean, they move around. But let's just simplify the story and say there are three distinct quarks, and we're going to label them with three distinct labels. Here's a down quark, and here's two up quarks. All right. The down quark might be the quark at position one.

All right. So let's call it at position one. The up quark might be at position two, and the other up quark might be at position three. Is there anything wrong with this state?

Quarks are fermions, right? Quarks are fermions. Because quarks are fermions, their states should be anti-symmetric. Their states should be anti-symmetric. Now, that suggests that maybe the right combination for

the U quarks is to anti-symmetrize it, and that would be correct. That's not an important issue for us right now, but I'm just being very precise. That down quark and two up quarks in an anti-symmetric state is a proton.

An anti-symmetric state of two spins makes spin zero. So this is an isospin zero object times an isospin a half object that has isospin a half, but never mind. This is a bit of mathematics. It's a little too fancy.

We don't need to worry about it. This is the proton. It has charge plus one, two thirds plus two thirds, it's four thirds, and minus a third makes charge one. So that's the proton, and it is the combination of

three quarks which makes an isospin a half state. Same as three spins making a spin a half state. So the proton is also a member of an isospin a half object, and the other one is gotten by simply interchanging ups and downs, and that's up, down, down.

Let me not belabor this point about the symmetry here. And that's the neutron. The proton and the neutron in this language are symmetric with respect to each other, and they form an

isotopic spin a half doublet. In the same sense that you can put three spins together to make a spin a half, and there are only two such states. So isotopic spin invades the theory of protons and

neutrons, and in this language just gives us another, just like the quark has isotopic spin a half, so does the neutron and proton. Okay. There's another object which is very similar to a proton

and neutron in many ways. It's also made of three quarks, but it's the combination, oh incidentally, what is the spin of the proton and neutron, actual spin? One half, okay? So you're taking three quarks with the same spin,

sorry, three quarks with spin and three quarks with isospin, and you've made a spin a half and an isospin a half, okay? Now, there's another object which has three quarks in

it which has spin three halves and isospin three halves. If you like, the three ordinary spins are lined up in it in the same direction, and the three isospins are

lined up. So what kind of thing are they? Well, if the three isospins are lined up, that must mean that there are objects with three up quarks. Three up quarks, three down quarks.

Also, the three spins are aligned, just the ordinary spins. The three spins are also aligned to form spin three halves, or down, or whatever. Three quarks up, three quarks, three up quarks, or

three down quarks. This is not a neutron. This is not a neutron or a proton. This is a new object which is a little bit more massive than a proton or neutron. It has a name. It's called the delta. It has isospin three halves, and it has spin three halves.

So it's sometimes called the delta three halves. It has both isospin and spin three halves. Let's forget its spin. Its spins are just aligned. If we line them up all along the z-axis, then they're all

aligned. Let's forget that. What about this? Can this be all there is to an isospin three halves state? How many states does a spin three halves object have? Four, right? So there must be two more, and there are two more.

There are two more delta three halves states, two more of them, and the two more are UUD and UDD. Four states, four objects, all of which are related by symmetries.

They all have very close to the same mass. The only thing which distinguishes their mass is the little small difference between masses of up and down quarks, and they're all very similar to each other. I'm telling you this for a reason. This played an important historical role that I'm going to tell you about in a moment. These four states are called the delta three halves.

And what's their charges? Let's go through their charges. What's the charge of this one? Excuse me. What's the difference between UUD and DUU? All right, let's work out the charges. Pardon? Let's work. What? The third one down. What's the difference between that and the first one?

Ah, the spinny. Spin. Spin. That's spin three halves, not spin one half. Oh, OK. The isospin is? The z component, no, it's part of the multiplet which is an isospin three halves object, right? Iso-spin three halves.

So it's both spin and isospin three halves. But by looking at these two, you couldn't tell that. You have to know a bit more about the nature of the way they're combined. All right. But let's just- The bottom two are isospin and one half, are they not? No, they have a component of isospin, just as the analog

of the z component of spin would be one half, but the full isotopic spin would be three halves. Yeah. Remember, every object which has a spin three halves comes in four states. Four states.

Here they are. One half, one half, minus three halves, minus three halves. Same is true for isospin, OK? But let's concentrate on spin for a moment.

These two states, that's these two. And the ones of maximum and minimum spin, that's these two. So there's got to be four of them. We know that a thing with spin three halves has four

states. Z component of isospin can be one half and minus one half, but by analogy with ordinary spin, there will have to be four of them.

OK, but let's come to the important points. Let's first of all label their charges. This one has three charge two thirds object. It has charge two.

This one has charge minus one. And UUD has charge plus one. And charge, what is this one? Zero. So the charges are different. Well, these two do have the same charges.

Sorry, did I get this right? UUD, no, yeah, yeah, yeah. These two happen to have the same charge as proton and neutron, but these are different. All four of these objects have to within a small discrepancy the same mass.

Just as the proton and neutron have the same mass, these have the same mass, but that mass is not the same as the proton and neutron. Proton and neutron, the mass is about 940 in units of MEVs,

millions of electron volts. This is the mass of the proton and neutron, with the neutron being slightly heavier, not by much. The mass of the delta is about 1200 MEV.

The delta can decay. It can decay into a proton and a neutron and a pion. Let's just see how that would work. Let's take the easy ones. The easy ones have all three quarks the same, up, up, up.

So here we have three up quarks moving along. And how can that decay? What is it going to decay into? It's going to decay into a proton or neutron and a meson. A meson is a quark-antiquark pair. Let me just draw for you how this happens.

This would be up, up, up. The two up quarks go off. This up quark also goes off, but quarks can't separate like that. Quark and an antiquark appear in between, like that.

The quark and the antiquark in between could either be a down quark or an up quark. So one possibility is that this is another up quark. And then this would be an anti-up quark and this would be an up quark. An up quark and an anti-up quark have zero charge.

This would be the decay of the delta three halves. What do we have here? This one, no, this one can't happen. This one actually can't decay. Not possible. We have to put a down quark here. There's just not enough energy for it to happen.

This would be an anti-down quark and an up quark. So what would this thing be? Up, up, and down. That would be a proton. The proton is lighter than the delta. So there's enough leftover energy. How much energy is left over?

Let's see, 260 MeV about, right? Something like that, 300 MeV roughly, 300 MeV. What's the mass of a pion? About 140, about 140 MeV.

So there's enough energy for this to happen and still leave over some kinetic energy for the pion to fly away. What kind of pion is this with an up and a down bar? What's its charge? Two thirds and one third. So this is a pi plus and a proton.

Total charge two. This is the object with charge two. So these delta objects are not stable. They all can decay into a proton or neutron and a pion.

And they're very short lived. Their lifetime, oh, we could work out the numeric. I'll tell you what their lifetime is, order of magnitude. Order of magnitude, it's the time that light would take to cross the proton. So you can figure out that the proton is about 10 to the minus 13 centimeters.

What's the speed of light, 10 to the 10th centimeters per second? So 10 to the minus 23rd seconds or something. They don't last long, but they do last long enough to be identifiable as distinct objects when they're produced, these deltas.

These deltas and collisions, particle collisions, deltas are produced. And they're real objects, but they're very, very short lived. Excuse me. As a matter of notation, does it make any difference which order you put these and these? No, not for purposes here.

Okay, that's exactly what we have to come to now. There's something wrong here.

Something's rotten in the state of Denmark. What is it? Well, we have three up quarks whose spins are all in the same direction. We could put those spins, we could if we like, choose those spins to be all up and

make a delta three halves with its spin, ordinary spin up with all three in the same direction, and they'd all be three up quarks. You're not allowed to put two fermions in the same state. If you could fiddle around with a spin, maybe you could do something with a spin,

but all the three spins are the same. Why are they the same? Because it has spin three halves. Three quarks with spin three halves have the same spin. They also have the same iso-spin, namely they're all up quarks.

So we seem to have found an object which consists of three identical up quarks. That's a violation of the principle that you cannot put two fermions into the same state. This was the clue that led eventually to what is called quantum chromodynamics.

It was realized that quarks have to have another property. They must have another property so that the three quarks in the delta three halves can be different from each other.

The fermion statistics, the fermionic property of them requires that they be different. So there must be a label, another label that was hidden from view for some reason which is there but not apparent in experiments.

That label is called color. Color is, again, a highly arbitrary term. It has nothing whatever to do with ordinary color. And it was just a label to a name, a name.

In different places, the three colors of quarks are different. In some places, they're red, white, and blue, mostly in the southern parts of the United States. Other places are red, green, and blue.

Red, green, and blue being the primary colors of light that you see with your eyes. I will use red, green, and blue. I haven't heard red, white, and blue being used for a long time, not for the last nine years or so.

But they're just labels. They're just labels and in no sense are they physically different one from another in any sort of interesting measurable way. The fact that there are three distinct ones and that they are not the same is important. They have exactly the same mass. They have exactly the same properties.

So let's now write down what we know about the labeling of quarks, all the various quarks we know about. There are up quarks, down quarks, that's the lightest one, and there are charmed quarks,

strange quarks, and after charm comes top quarks and bottom quarks. They're not listed in order of mass. The up is the lightest, the down is next. The charm is much heavier than the strange by a factor of about eight or something like

that, and the top is vastly heavier than the bottom, but I've just put them in. I've been perverse and put up on top of down. I don't know why. These are the six distinct types of quarks, but now for each type of quark, there is

another label. Quarks are labeled by two labels, and the two labels are red, green, and blue. As I said, I don't want anybody to think they're really red, green, and blue, but that's

the label. And where did that come from? How did we know it was there? Well, it's one of these things where physicists simply followed their noses. They simply said, look, there's something wrong with this quark model. You can't have three quarks all in the same state.

They must differ by something else. Now, they could differ by position, but it was known that for one reason or another that it didn't have to do with position and momentum. That was known to be irrelevant for dynamical reasons, for energetic reasons, so that was

not the issue. There were three quarks all in the same state. They must have, in fact, the situation was actually quite similar to what happened in atomic physics. Spin was discovered by Uhlenbeck and Goudsmit because of atomic spectroscopy.

Helium had two electrons in the same state. No good. Violates the Pauli principle. Well, actually, the timing here, I think, may be a little bit confused, but it could have worked this way. They could have said, well, Pauli tells us that you can't put two things in that. Maybe it did work.

I can't remember now. I wasn't around. I don't remember. One of the reasons that Goudsmit and Uhlenbeck might have, since I don't know the history in detail, might have invented spin was because in the helium atom, there were two electrons apparently in the same orbital state.

So you've got to attach to the electron another quantum number, another label. That worked. That worked just fine. It worked a second time. History repeated itself, and various people, Nambu, Nambu was the one who realized this,

Yochiro Nambu, Nambu, the Japanese American physicist, Nambu, realized that this called for another quantum number, and he said every quark has to have another degree of freedom. I don't think he called it color. I suspect it was Gelman, but I'm not sure, but a good deal later.

And then what he said is, look, all that's going on here is that you put three quarks together, one red, one green, and one blue, and now you're in business. No more violation of the Heisenberg uncertainty principle. The same trick incidentally works for protons and neutrons, that you can understand the

proton and neutron also as a red, what do I call this again? Red, green, and blue. Those are the primary colors, right? A question. So is there a physical property that can be measured or? No, no. In every possible way, they are the same, except we know there are three distinct ones.

So if you had one, you could tell the other. Why? I mean, how can you say there are? How can you say that maybe the, you know, the crystal doesn't apply? Huh? Well. If you can't detect the difference, right? You can, if you have one, you can tell the other one is different from you.

How? What are you measuring? Oh, that's, that's, that's a little bit complicated, and in collisions, you can tell that if you have one, if you have two of them, you can tell that they can't be the same. And that you can tell. Couldn't you, and in, are they conserved, and the color conserved, and the action

Well, yeah, the color is conserved, but it's also always zero. It's not only conserved, but it's always zero. So it's a little bit funny conservation law. But look, you could say the same thing about The function doesn't quite tell you that you need all three of the colors for a UUD.

Say it again. No. But you need, do need something to distinguish the two U's. Right. But it gives you more than you need in that case. Yeah, it gives you more than you need. That's why, that's why historically the delta three halves played an important role.

It was looking at this guy here, which was unambiguous. Three quarks, parallel spin, parallel isospin, something's wrong. It would have been possible to analyze the protons and neutrons, but it would have been less convincing. When the delta three halves was interpreted in terms of quarks, that really hit you over the head.

So the ultimate resolution is, and we now know really experimentally extremely well that the three quarks, the three different kinds of colors are really there.

You know, even with ordinary spin, what's the difference between spin up this way and spin that way? There's no difference. They have exactly the same property. All you have to do is turn your head over or spin this way. Unless you have another one to set a direction or something to set a direction, then you can't tell the difference between them.

On the other hand, if you have some object which picks out a direction and you bring a spin up to it, you can tell whether the spin is along a particular axis or not.

In the same way, if you have a proton, a proton being different than a neutron, and the proton and the neutron are identifiable objects in the laboratory. If you have a proton, it picks out a direction in this isospin space.

It's up in the isospin direction, not down. And if you have another quark, you can discover whether that quark has its isospin parallel to or anti-parallel to the proton. So yes, you can tell, not it's isospin, you can tell whether it's, I said something wrong, but close enough.

If you have one object, it can provide a kind of frame of reference for the other ones and test whether they're the same or not. But the mathematics of quantum chromodynamics simply requires that you have these three different things.

You could say maybe it's a violation of the Pauli principle. You could. There's no known mathematical framework for discussing that. And there's no quantum field theory

that has anything but fermions and bosons. So you'd have to invent something new. Yeah, I thought the isospin is a kind of observer. You can call it observable, you know, like momentary position. And you can make a case that isospin

is a kind of observer. Absolutely. What would be the appropriate observable for Pauli? Much, much more subtle. Much more subtle. Much more subtle. Let's come back to it. I'll tell you when we get there,

after we've talked a little bit about. I have a question. It was known that quarks and fermions is composed of the skin of the proton which is in the area of the quarks. Is that out of your mind? Mm-hmm, mm-hmm. Okay. Yeah. I guess I have the same question in a different form. If you have a down, up, up, a proton,

one of them, one of the three quarks is green. And since the down, there's only one of, I mean the question is how can you really associate with one color with one of the quarks? No, you have to think of quantum superpositions of states.

So you might write, let's take the case of the proton. Up, down, down. This could be green, red, blue. But we have to write all possibilities,

all ways of combining them to make a real proton. So we have to use some quantum mechanics to symmetrize the wave functions and so forth. Symmetrize and anti-symmetrize them appropriately. But the big advantage of this delta three halves is

you didn't have to do anything fancy. Just the three all in the same direction, no. Okay, sure, you could say, well, maybe there was something wrong with our ideas about quantum field theory. But by now, the theory of quantum chromodynamics,

which is exactly, chromo has to do with color. Color is the important quantity in quantum chromodynamics. Just like charge is the important quantity in electrodynamics. This theory is a highly accurate description of experimental data associated with the collision of hadrons.

And its accuracy is way beyond what can be questioned now. So the ultimate answer is that it works. Okay, now let's talk about gluons. The missing ingredient now, we have the analog of electrons, the quarks.

They're fermions, they have some attributes, they have some electric charge, like electrons. They stick together, not quite the way electrons stick to a nucleus, perhaps a little more

the way electrons stick to positrons. But they stick together somehow. What's missing is what sticks them together. What sticks together atoms is the electrostatic field. The electrostatic field is associated with photons.

We can either think of it in field language, that every electron creates a field around it. Or we can think of it in particle language, that electrons emit and absorb photons. The exchange of photons back and forth between charged particles creates the forces between them.

What about quarks? What sticks them together? Particles very, very similar to the photon. At first it was a speculation, maybe such objects exist. Then a theory was built,

a mathematical theory was built with quarks and gluons. Gluons being the analog of photons. They're very similar to photons. They have spin one, just like the photon, which means they have the same kind of polarization states. They're massless, like photons, very similar, but with one big difference that we'll come to.

And they jump back and forth, on the one hand, a quark is a source of the field that's associated with gluons, the gluon field, in the same sense that the electron is the source of the electromagnetic field.

On the other hand, not the other hand, but a similar hand, the quarks can emit and absorb gluons. What are gluons like? What quantities do gluons have? Photons are pretty, in a certain sense, uninteresting, except for the fact that they have a polarization.

They have a spin, they have a polarization. They have a momentum, and they have a polarization, and that's about all. They don't have any charge. By itself, a photon, if it collides with another photon, there are no forces.

Now, that's not exactly true. There are forces between photons, but they're secondary effects. They're not electrostatic forces because the photon is charged. They're secondary effects which come from quantum electrodynamics and loops of complicated Feynman diagrams involving electrons, but the primordial interaction

between photons is there is none. They move freely past each other, and that's why a beam of light moving in one direction will pass through a beam of light moving in the other direction with no interaction, unless you're in some material. Photons are not very interesting.

And in the sense that they don't interact with each other, they are interesting from the point of view of their interaction with electrons. And basically, all of quantum electrodynamics is summarized by one diagram. And that diagram, which we've drawn several times, is the emission of a photon from an electron.

Electrons are drawn as having a directionality, the direction along which the charge is moving. You can flip lines around, and every time you see an arrow going downward, that indicates a positron.

But it's all one basic vertex. That's it. And out of that, you can build forces. You can build collisions, everything else. Just for the purposes of bookkeeping, think of a photon as having the same charge

as an electron and a positron. In fact, a photon, if it's given a hit and given a little extra energy, can decay into an electron and a positron. It's not that in any sense a photon is made of an electron and a positron. That's not the point. But it happens to have the same properties

as an electron and a positron. In particular, it's electric charge. It also has a angular momentum. It has a spin, a spin of one. And with an electron and positron, if you line up their spins, you can also make a spin of one. So in many ways, a photon is similar to an electron

and a positron. That's sometimes indicated by thinking of the photon as a fictitious composite of an electron and a positron. And then this diagram, this diagram of the emission of a photon can be drawn just by saying

the electron comes along, becomes an electron over here. The electron over here was really a positron moving backward, which turned around. Now, there's no content in this other than to say,

without losing any electric charge, you can emit a photon. And you can see it directly. You certainly don't need this to see that an electron can emit a photon and that there's no violation of charge conservation. That's totally obvious that an electron can emit an electrically neutral thing.

But nevertheless, let's just draw the emission of a photon by thinking of the photon as a composite of an electron and positron. It's not useful for electrodynamics. The analog is quite useful for quantum chromodynamics.

Okay, now let's come to quarks. Electrons, we're finished with. And the important thing in quantum chromodynamics is the color. So let's begin with quarks. Quarks can be red, green, or blue.

Let's make a column vector out of them and use the language of quantum mechanics. A quark can either be in the red state, the green state, or the blue state and make a column out of it. An antiquark, let's represent antiquarks by red bar,

green bar, and blue bar. A gluon, first of all, a gluon is an object that can be emitted by a quark. If this is a quark and a quark goes off, a gluon can be emitted.

But the interesting thing is the gluon behaves in some respects like a quark and an antiquark. It's not a quark and an antiquark. Well, I'll tell you precisely in what sense it behaves

like a quark and an antiquark. But let's think of the gluon as a quark and an antiquark in the same way. So how do we label, if we label each quark by a color, this now becomes a quark and an antiquark. A quark and an antiquark.

How many different kinds of quarks and antiquarks are there? Well, did I hear nine? Yeah, okay. You're almost right. Right, the logic was what I was looking for. I wanted you to say nine.

There's a subtlety which we'll come to. There are really only eight. But let's say nine to begin with. What are those nine quarks? Sorry, what are those nine gluons? They're the nine combinations that you can make by taking a quark and an antiquark. In other words, they make a matrix.

If quarks are like a column, then, and we might think of, just for fun, we might think of antiquarks as making a row. Red bar, green bar, blue bar. Then gluons would fill out a matrix.

All right, in other words, to label a gluon, you would label it with two indices, two colors. And what would they be? They would be red, red bar. Red, green bar. Red, blue bar.

Green, red bar. Green, green bar. Green, blue bar. What else? B R bar, B G bar, B B bar. So the diagonals are indistinguishable.

Say it again? The diagonal elements are indistinguishable. Well, the diagonal elements are not really indistinguishable, but the point is that the sum of the diagonal elements is not an independent quantum state.

All right, so it would seem like there are nine gluons. Later on, we will talk about a particular subtlety, which tells us the quantum mechanical superposition of red, red bar, plus green, green bar, plus blue, blue bar is a nothing, doesn't exist. But for now, for our purposes right now,

let's neglect that. There are really only eight gluons, but it really does look from this pattern here that there should be nine. So let's play with it as if there were nine. Then what kind of Feynman diagrams can exist? Well, a quark can become another kind of quark

and emit a gluon. All right, so let's draw the diagram for that. Let's take the case of a red quark becoming a green quark and emitting a gluon. What kind of gluon gets emitted?

Red, green bar. Where's red, green bar here? Red, green bar. And if you like, you can draw that with a neat notation.

The notation is just a bookkeeping device, really. And here it's useful. Here it really is useful. Just think of the red is going through and the green also is going through, except when you flip the green line over, it becomes an anti-green and a red. So the gluon that's emitted is as if the red just went through

and the green annihilated a green bar. That's the basic vertex of quantum chromodynamics. There's not just one of them. There are nine of them, or actually only eight, but the pattern is quark goes to another kind of quark

and a gluon is emitted. If a red quark goes to a red quark, then it's a red, red bar. If a red quark goes to a blue quark, then it's a red, blue bar, and so forth and so on.

Now, that's, as I said, the basic phenomenon or the basic primitive building block of quantum chromodynamics. You can build all sorts of, sorry, that's half,

that's one of the building blocks. There is another building block that isn't there for quantum electrodynamics. As I said, photons don't interact with each other. In particular, a photon can't emit a photon and another photon. Electron can emit a photon and stay an electron.

Photons don't emit photons. So photons don't interact with each other in any way except in materials. Excuse me. Gluons are only emitted with a change of quark.

No, they can, no, you can have, they don't have a change of quark. No, I mean, about this process, and whether it changes or whether it does not change, nevertheless, this is the only method in which one emits a gluon. That's right. This is the only way in which gluons are emitted.

But there is something new that makes quantum chromodynamics, first of all, far more complicated and far more interesting than electrodynamics. Let's take a gluon moving along. Here's a gluon. Let's see.

Let's take that gluon to be a red blue bar gluon. That's a red blue bar gluon. Now, can this happen? I'm going to show you something that, yes, is really part of quantum chromodynamics.

Maybe it's not too surprising once I draw it. This is not really a quark, but just imagine it's a fictitious quark making up the gluon. All right, fictitious quark goes off. Well, if it's a fictitious quark, it better not really go off, but we'll see in a moment what happens to it.

And the blue bar keeps going. But now, if these really were quarks, a quark and an anti-quark could form. What kind of quark and anti-quark would you like to put there? Green? Let's put green there. This would be green going this way, and this would be anti-green going this way,

and this would be red. So red goes through, blue bar goes through, and green becomes anti-green going that way. Well, what do we have now? Now we have a basic vertex in which three gluons come together.

Let's draw gluons as wavy lines similar to photons. We now have a vertex in which a red-blue bar becomes a red-green bar and a green-blue bar.

We don't really have to remember what combinations are possible. All we have to do is figure out which diagrams we can draw where all of the lines go through without being interrupted. You can figure out what the various couplings or what the various possible

fundamental diagrams there are connecting gluons. This is the new thing. What it means is that gluons interact with each other. Gluons exert forces on gluons in a way that would be unthinkable for photons. Photons cannot exchange photons between them.

Gluons can exchange gluons between them. So now let's come to forces. The gluon behaves in a way which is similar to the photon and does something which is similar to what the photon does. It can be exchanged.

Here's a diagram. It's easy to draw diagrams where quarks interact with each other. Let's draw a diagram in which an antiquark interacts with a quark. Here's a blue bar antiquark which emits a blue bar green gluon,

becomes a green gluon, and now the green gluon, the green, sorry let's see,

this would be a green bar. This would be a blue green bar gluon. His green bar is blue, but let's say that blue goes right through like that.

And what do we have here? We have here now an exchange of a gluon between a blue bar and a blue, making a green bar and a green. That's a kind of force between quarks. That creates a force between quarks in very much the same way that photons

exchanged between electrons create forces. You might want to make a force between a blue bar and a blue bar. Supposing we wanted a force between a blue bar and a blue, then we would make this blue bar here.

All right, so we can make all sorts of forces this way, but what about forces between gluons? All we have to do to this diagram is add an extra two lines. Let's, what did I have this originally? Blue bar, let's make it blue bar.

Green bar up here, green up here. Let's put another line in here. It's not another quark now. It's really going to be representing a gluon. And this one, let's take to be red. Red goes right through.

Let's put another line over here. I don't know, what did I take that one to be? I think I took that one to be blue bar, blue bar. This is now a diagram which represents the exchange of a gluon between two gluons. This is really something new.

This is something very, very different than electrodynamics. What does it lead to? It leads to forces between gluons. Quarks can bind together because of the exchange of gluons and make hadrons, but gluons can bind together and make objects.

There are objects in quantum chromodynamics which contain no quarks. There are no bound objects, there are no bound states, composite objects in quantum electrodynamics just made up out of photons. There are objects which are just made up out of gluons. And how do they happen? They happen because gluons can exchange gluons back and forth.

We could just summarize this by saying, there's a force due to the exchange of gluons between gluons. This would also mean, for example,

that if you had two waves of gluons going past each other, they would interact with each other. They would deform each other. They wouldn't just go through each other just like two electromagnetic waves. In fact, even if you had a single gluon wave moving along, the different parts of it would exert forces on each other

and cause it to deform or do whatever it might do. So, the watchword is that the dynamics of gluons is non-linear. Do we have to have the same number of reds, greens, and blues?

You just have to make sure the lines follow through the diagram uninterrupted. Well, that's what following the lines means. Did I draw this wrong? I might have drawn this wrong.

Oh, sorry, this should be green bar. This should be green bar, green bar. Yeah, sorry. The rule is follow the lines. When a line turns around in time like that, change it to an antiparticle. And that's basically the only rule of quantum chromodynamics,

that there are interactions between quarks and gluons. And they satisfy the, let's call it the follow the line rule. And there are interactions between gluons and gluons, and they also follow the line rule.

We still have an anti-color problem with RG. Let's see where. RG. Oh, sorry, RG. RG. RB bar, this should be B bar.

This should be B bar going this way. This one should be G, and this one should be G bar. Okay, let's draw, let's do it over. I messed it up badly enough, but I should do it over.

Okay, so I think I had blue bar over here, red if I remember. I don't remember. Red goes through, so it stays red.

Blue bar goes over here, and then turns around, so this must be blue. This one was blue bar. Has to be a quark and an anti-quark. Blue bar, straight through, blue bar here. And now we have our choice what we want to put over here.

So I think I put green going this way. Green, green, green bar. Green goes right through the diagram. Okay, now I think it makes sense.

Every line just goes straight through. Red goes straight through that way. Blue bar goes straight through this way. Well, we can put the arrow the other way to indicate anti-particle.

Gluons always have the properties of quark, anti-quarks. Earlier, by analogy with the photons, you said the gluons were massless. Yeah. Are these diagrams with all these still convincing the massless?

They are massless. We're going to come to what the meaning of the mass of a quark and a gluon are. All right. Yes, they are massless in a technical and special sense. I'll tell you right now what the special sense is.

We're going to quit in a minute or two, in fact, right now. I'll give you an example. We're going to come, we're going to study this theory one more week, and we're going to talk about the confinement of quarks, and we're going to talk about the structure of hadrons and so forth. But let me just tell you in what sense a quark or a gluon does or does not have a mass,

or does or does not have the mass that we ascribe to it. Let me imagine that I have a object of mass m, a small object of mass m.

And the small object of mass m has attached to it a, I don't know what it is. It's whatever you want it to be. Some sort of wiggly soft piece of chewing gum or something that dangles off it. Now I want to move this object.

If I move the object with a very small force, the force being so small that it doesn't deform, that the acceleration is so slow that the whole thing moves off together,

what kind of mass does it have? The answer is it has the mass of whatever you put here, plus the mass of the wad of chewing gum or whatever it happens to be. We're thinking now purely non-relativistically, just to give you an analogy. On the other hand, supposing I shake this thing with a very high frequency and I ask,

what are the properties of the motion of the core of it over here when it's been subjected to a very, very high frequency force of some sort? What kind of mass does it have over here?

Just the mass of this object alone, right? The rest of it doesn't have time to adjust to the forces. It just stands still. Now maybe this sends out a wave all right, but the very rapid oscillation here, the response of this end here would be the response of an object of mass m.

So what is the mass of it? Is it the mass of the sum of them or is it the mass of the thing at the end? The answer is ill-defined. The answer depends on the frequency of the force that you exert on it. Masses of objects are frequency dependent or, well, in this sense,

in this sense, the mass of this object, the observed mass of it that it would respond with would depend on the frequency of the motion. In the same sense, the mass of a quark,

is frequency dependent. If you shake a quark as some kind of object inside a hadron, there are three of them and there's a bunch of mushy gluon stuff in there holding them together. If you were to try to move the quark by itself,

if you were trying to move the quark slowly, if you grab the whole of that quark, if you could do so, and you moved it very slowly, the whole thing would drag along and what would be the mass of it? What would be the mass you would experience? The mass of the whole thing, which could be the mass of the proton, which is 930 something or others.

What about if you were to exert a very high frequency force on it? Well, I'm not going to tell you what, well, the answer would be what we normally call the mass of the quark. That's this 5 or 10 MEV for an up or down quark.

So the mass of the quark, when it's subjected to a very, very high frequency, or when it's hit very hard and you try to see how it flies off, dragging this other stuff behind it, the initial impulse that it gets and the initial velocity that it goes off with, will be sensitive to one value of the mass.

On the other hand, if you hit it very slowly with a very gentle, low frequency force, the whole thing would move off. So the concept of what is the mass of a quark is somewhat ambiguous. And to keep the discussion simple, let's say the mass that we usually ascribe to the quark

are these high frequency responses, analogous to the mass at the end of the wall of chewing gum here. Isn't that a restatement of delta x, delta t? Not really. This is a completely, in this case, this is a completely classical phenomenon that you have a small little nut

at the end of a wiggly something or other. You shake it very rapidly. You see it accelerates with one kind of acceleration. You accelerate it with a low frequency.

It accelerates different. That's a classical phenomenon. It does not have to do with the... In the last lecture, we described the electric charge, the charge of the electron, but it's similar. Yes, yes, yes. Lesson, that's exactly right. The lesson is the parameters that we describe particles with

are dependent on frequencies and wavelengths of the interactions that they engage in. They're called running constants. So we say the charge of a quark is the same as the constant theory. Is that the same story applies as the case in the lecture? It does, it does. We have to sort out exactly what we mean by the charge of a quark,

but let's just put it this way. It's one third the charge of a proton, but yes, you're right. We do have to define it carefully, and that's our whole story into itself. Yes, why do you always have to have a quark and an anti-quark?

Why can't two quarks form a part of it? That's a great question. That's what we're going to talk about. This might be not the right time for this question,

but all this stuff with the quarks here, it's so odd with the fractional charges and color. And my question is, because you can't look at individual quarks, how did people figure this out? They took time.

Yeah, well, there was a number of clues. There were a number of clues. By the time I came into physics, the idea of quarks was already established.

I was a graduate student in 1963 when Murray Gell-Mann announced the idea of quarks. And I do know how he came to it, but there were clues. There were a lot of clues.

There were a lot of clues, but there were also a lot of inconsistencies. So it was a pattern of suggestive facts together with apparent inconsistencies, such as the violation of Fermi statistics.

And there was another fact, which was very peculiar. The other fact we'll come to next time had to do with the fact that quarks are produced in the laboratory, that they're always permanently confined inside protons, neutrons, mesons. That was another fact.

And it wasn't one person who put it together. It was a whole variety of people who put the whole thing together. Nambu had the right idea in the early 60s. He had the right idea. But the whole thing got put together and nailed in place, and the whole structure was put together over a period of 10 years.

Yeah, we will talk about that. From the start, this whole theory is ignoring electrons. Yes, this theory is ignoring electrons now.

Just like in quantum electrodynamics, we ignore quarks. Then we have to put them together. We have to put them together into some coherent thing in which quarks and electrons and photons and all of them form one bigger structure,

some of which has been done and some of it has not been. So yeah, it is a process of isolating. I mean, physics always works that way. You isolate, you divide and conquer, and then you have to put it all together. Okay, good.

For more, please visit us at stanford.edu.