New Revolutions in Particle Physics: Standard Model | Lecture 5
Susskind, Leonard
(February 8, 2010) Professor Leonard Susskind discusses gauge theories.
 Stanford University. All right. We've really studied two different theories a little bit. Quantum electrodynamics and quantum chromodynamics, both of them are gauge theories. In fact, just about all of nature as we know it is one way or another is controlled by gauge theories of different kinds. And so I ought to tell you what a gauge theory is. Now, I'm not going to get into any depth what gauge symmetry is. I'm going to tell you what a gauge theory is in more or less the minimal mathematical way. First and simplest gauge theory is Maxwell's theory. Maxwell's theory of light or Maxwell's theory of electromagnetism, let's not worry about what the word gauge means. We may go through it later, but not now. But what it is, is a theory of fields which have all of the properties of the electromagnetic field. The electromagnetic field has six components, three components of electric field, three components of magnetic field. You can represent those six components in terms of a four vector called the vector potential. I'll just write it down for you. I'm not going to play any big role today anyway. But just to tell you, there's a four vector that's generally called A, and it has an index mu. It has four components, A0, that's the time component of it, and the three space components A1, 2, and 3, or A, X, Y, and Z. Let's not write them out in detail. So it contains a three vector and a fourth component, A0. The significance of A0 is that it's the electrostatic potential. It's the thing whose gradient is the electric force. So for example, the electric force on a charged particle is just gotten by multiplying the electric charge by the gradient or the derivative fx, derivative of A0 with respect to x. So it's just the electrostatic potential that's measured in volts, in particle physics of course it's not measured in volts, volts per unit charge, excuse me, in whatever way you measure electrostatic potential. That's the meaning of A0. Notice first of all, oh, and of course this is the electric field. This is the electric field. The components, derivative with respect to x, y, or z, so it's the gradient of A, and the gradient of A is just called the electric field. So the electric field is determined by the gradient of A0, and the magnetic field is determined by also derivatives, also derivatives of the vector potential, and in fact the magnetic field is the curl of the vector potential. So one of two ways you can study electrodynamics, either in terms of the electric and magnetic fields, of which there are six, or in terms of the vector potential of which there are four, but there are equivalent descriptions of electrodynamics, and we won't go into Maxwell's equations, but let me just remind you that Maxwell's equations give rise to electromagnetic waves. An electromagnetic wave is a wave of electric and magnetic field, and a typical electric magnetic wave has a direction and space that it moves, let's just represent it by an arrow, and it has a polarization. To understand what the polarization means, you'll have to think about the electric and magnetic fields, and the electric and magnetic fields oscillate as you go down the wave. Here's the electric field, for example. The whole thing moves with the speed of light, but at any given instant the electric field might look like that, and the magnetic field is always perpendicular to the electric field. Let's try to draw it, see if I can give it a drawing. I'm imagining the electric field is in the horizontal plane, and it's always perpendicular to the, let's see, is this going to... it's always perpendicular, like so. In appropriate units, the electric and magnetic field are equal to each other and perpendicular. This would be a plane polarized electromagnetic wave. In a plane polarized electromagnetic wave, the direction of polarization, which of course is always perpendicular to the direction of motion, the direction of polarization is determined by the electric field. So the electric field determines the direction of polarization, and that's electromagnetism in a lot of electromagnetic waves in a nutshell. The other thing, of course, about electromagnetic waves is that there are sources. The sources are charges and currents, but let's particularly focus on the sources of the electric field. The sources of the electric field are electric charge, and to just draw a picture, to put it at the level of pictures, every electric charge creates an electric field. The electric field lines never end except out of the charges. A positive convention, the convention is a positive electric charge, puts out an electric field, which is radially outward, falls off as 1 over r squared. You can imagine that the number of electric field lines coming out of a charge is fixed, proportional to the charge, and therefore the number of electric field lines passing through any sphere is the same no matter how far the sphere is. To say that mathematically, the integral of the electric field over the sphere is the same wherever you go, however far out you go. That's Gauss's law. That's Gauss's law, and that Gauss's law has a very strong consequence. If you don't have any other charges in the system, let's say just a plus charge, then you can't get, and the rule is electric lines only end on charges, then there's no way to get rid of a charge. No way to get rid of a charge. If you try to get rid of the charge, either the electric field has to suddenly disappear everywhere. That would violate the rules of the speed of light, that you can't send a signal faster than the speed of light. So instead, if you said that a wave moved outward of missing electric field, that also wouldn't make sense, not unless there were charges at the end points of the electric field. So all this would really be, is this would not be the elimination of the charge, it would just be taking the charge and sending it out as a shell. It would be a shell of charge going out. The charge would not have disappeared. So it's a consequence, a deep consequence of the structure, the mathematical structure, Gauss's law, and other laws that electric charge is conserved. That is the essence of the gauge theory. Essence of the gauge theory, Maxwell-like fields, I'll use the word Maxwell-like because they're not all truly Maxwell's electromagnetic field. Maxwell-like fields consisting of electric-like and magnetic-like fields, if the gauge field is weakly coupled, and I will tell you what that means later, but it basically means that if the interactions between the parts of the electromagnetic field are weak enough that they don't interact with each other seriously, then the motion of a gauge field is exactly the same as a light wave. It will have a polarization. It will move down the axis with the speed of light unless something else, unless it's something in the dynamics that changes that. Naively at least, it's in every way similar to the electromagnetic field. The only difference is, it also has sources. Every gauge field has sources analogous to the electric charge, but it may not be the electric charge. It might be something else. Let me give you an example which is an unphysical example. Not because it's inconsistent, but because it just happens not to be true as far as we know. There is another quantity in nature called the Baryon number. It's simply the number of quarks, but it's a conserved quantity. And you could imagine that Baryon number is the source of... Now, Baryon number is not electric charge. Why not? Well, because protons and neutrons have it. So it is not electric charge, but it could be the source of its own gauge field. If it were, that would mean that there would be forces between protons, neutrons, protons, protons, neutrons, neutrons that would be analogous to the Coulomb forces between charges, except it would have nothing to do with electric charge. They would have only to do with Baryon number. So, for example, two objects with the same Baryon number would repel each other. A proton and a neutron would repel each other because they have the same Baryon number. A proton and a proton would repel. On the other hand, proton and the antiproton having opposite Baryon number would attract. Now, the only rub in this ointment is that there is no such gauge field coupled to the Baryon number. Baryon number does not come with a field like this that surrounds it and no long-range force associated with it. So, Baryon number is not an example. Color is an example and the color forces of quantum chromodynamics also, oh, well, before I do that, one or two other points, one or two other points. The other point, of course, is this is a completely classical description of an electromagnetic wave. The corresponding quantum mechanical description is in terms of quanta, but of course, if we think of quanta as particles, at least that's about the only mental picture that we have to describe these discrete little objects, the discrete little objects which drawing them, of course, is always misleading, but let me draw a photon, there's a photon, a little point. That little photon has a position, moves with a speed of light, but also it has a little flag associated with it and that little flag is its polarization. I won't try to draw a flag on it. In other words, it has a pointer which either points this way, this way, or somewhere in between that indicates the polarization of the electromagnetic wave of which it is a quantum. That's a complicated statement, but I think you get it. So photons come with polarization and that polarization in principle can be rotated. You can rotate it by sending it through a quarter wave plate or in any number of ways. And that's photons. That's the photon theory of electromagnetism. And in the quantum theory, one can think of the Coulomb field as, roughly speaking, a field set up by the emission and absorption of photons, emission and absorption of these photons. Okay, so that's electrodynamics in a nutshell, electrons are charged, protons are charged, and so forth, and they interact in this way with the gauge field. Another aspect of gauge fields, I've already mentioned it, that there's always conservation laws associated with them, namely conservation of the sources. But conservation laws in both quantum mechanics and classical mechanics are always associated with symmetries. Symmetries of some sort or another are always behind conservation laws. The conservation of electric charge, we've talked about that, and I showed you how if you study the quantum mechanics of electrons in terms of charged particles in general, there's a wave field describing those particles, this could be the electron, and the symmetry associated with the conservation of charge is just the multiplication of the field by a phase. So that was the simplest example of a gauge theory, a gauge charge, meaning the electric charge, and a symmetry that goes along with the conservation of that charge. Those things go together. Okay, well, let me just very quickly remind you how it worked in quantum chromodynamics. In quantum chromodynamics, a more complicated structure. You do have Maxwell-like fields. Let's label them. The Maxwell-like fields are labeled by indices. So in quantum, let's continue to call them A. There is no universal symbol for the gauge field of quantum chromodynamics. Sometimes it's called G, sometimes it's called A, sometimes it's called B, sometimes it's called C. I'm going to call it A, but in order to distinguish it from the electromagnetic field, we give it some indices. I and J. It's a matrix. Now, it's a matrix, and that matrix transforms under a group. The group was Su3. So let me just remind you very, very quickly that we started thinking about Su3 not by thinking about gluons. This, of course, would be the gluon field, but by thinking about quarks. So a quark was an object which we called Q. Q could stand for the quantum field of the quark, and it has an index. The index, it's not the up-down index, it's not the charm-strange index. We'll come to that sooner or later. It is the color index, red, green, or blue. I takes on three values. And the symmetry operation is not multiplying by e to the i theta, but multiplying by a special unitary matrix Uij. It gives us Q prime i. That was the symmetry operation, which is a kind of rotation in a kind of three-dimensional complex space. Don't confuse it with ordinary three-dimensional space. This was the symmetry operation, and one would say that the Qs, think of them as particles, if you like, the Qs form a representation of Su3, which is called the fundamental representation. It's called the fundamental representation. It has three entries, red, green, and blue. Sometimes it's called the defining representation. It's the smallest non-trivial representation. And it just can be thought of as a three-component vector, one, two, three. And the unitary matrices can be thought of as matrices, three by three matrices. OK, we've gone through that. The other thing that I told you is that antiquarks, or antiparticles in general, are represented by the complex conjugate fields, fields which are simply the complex conjugates of the original ones. The relation, the mathematical relation between particle and antiparticle, or the wave functions of particles and antiparticles, that relationship is complex conjugation. So we could ask, how do antiparticles transform? Well, we simply realize that if you want to transform the complex conjugate field, you should use the complex conjugate matrix. If you multiply q by u to get q prime, then you must multiply q star by u star to get q prime star. The set of matrices u are called a representation of the group. The set of matrices u star are a distinct and different representation of the group. And the language we would use is that quarks and antiquarks are described by the three-dimensional representation. And we could call it three star, meaning complex conjugate. It's usually indicated by a bar. Two different representations of SU3. So there's seven, we make this. Nine, actually, well, sorry. You asked me how many matrices there are? The number of generators is eight. Yeah, because these matrices are assumed to be special unitary matrices. Now you can ask, and it's interesting to ask, what kind of theory would you make if you didn't insist that they'd be special? And I'll tell you another time, maybe later, but not now. Okay, now what is Aij? Aij has two indices. And the way to think about it is that one of those, it is mathematically got the same group theory structure, the same symmetry structure as having a quark j and an anti-quark i. In other words, this is an object whose indices transform one index as the three-dimensional representation and the other index as the three-bar. Another way to think about it is it has all of the properties with respect to the symmetry, not with respect to all physical properties, but with respect to the way it transforms. It transforms as if it were a quark and an anti-quark sitting on top of each other. Not at spin, just its color, as if it were a quark and an anti-quark, with one special rule, namely the trace of Aij vanishes. The singlet, the singlet that you can make, the piece which has no transformation property under the group at all, that you can kill and forget about. That's why there are eight Aijs instead of nine. So Aij is like a thing with a quark and an anti-quark index. But because it has a quark and an anti-quark index, it also transforms when you under the SU3 transformation group, in that sense it's different than the photon. The photon is completely electrically neutral. Its field, the photon field, does not get a phase when you rotate the electron wave function electrons. Go to e to the i theta times electrons, positrons go to e to the minus i theta times positrons, photons do nothing. That's because they don't have electric charge. But the gluons, they do have color. Now what kind of color do they have? They have the color of a particle and an anti-particle. Like you say, a particle and an anti-particle shouldn't have any color. But they do because you could have a red particle and a blue anti-particle. A red, that would make a thing which was not really neutral, which knew about color. So gluons do have color and they have color in what is called the adjoint representation, which is the representation of the 8th generators. Okay, now, what about the interaction between quarks? Incidentally, there are many other gauge theories of interest. They're all pretty similar to each other. The group might not be SU3, it might be SU2, it might be SU4, it might be SU10, it might be anything. And the objects of the group might not be called quarks. Nevertheless, or the fundamental objects of the group, the analogs of quarks might be called something else. They might have a different name. But they would be objects which had a single index. The anti-objects would also have a single index but of the complex conjugate kind. And the analog of the gluons which are called gauge bosons. Gauge bosons, the Maxwell-like fields always have one index of the particle type and another index of the anti-particle type. So that means, let's see, if we're talking about SUN, that means how many generators altogether. Well, N times N is N squared minus one for the trace. So it's N squared minus one distinct gluons. Now, what is the connection between the quarks or more generally, the particles which transform like quarks perhaps in a more general way? What's the connection between them and the gauge bosons? And I've explained that to you before, but let me just remind you again. If you have a quark labeled by I, where I could be red, green or blue, and another quark labeled by J, that the I-th quark can become the J-th quark if a gauge boson is emitted. What kind of gauge boson? Well, as I showed you before, the way to think about it is just to think about these lines running through the diagram. I runs right through, so that's I, and J also runs right through, except when J is pointing downward, it should be thought of as the analog of an antiparticle. And so an I would become a J by emitting a quantum of the field A, is it J-I or I-J. No, J-I, doesn't matter very much. By emitting a gluon of the I-J-th type, an I-th particle becomes a J-th particle, and that is one of the basic interactions of gauge theories in general. Fundamental objects, analogous to quarks, gauge bosons, analogous to gluons, and a rule about emission and absorption that you have to follow the lines and never lose an index. But now that raises something new. If a gluon itself has the properties of a quark and an antiquark, so let's say this is the I-J-th kind of gluon. Well, the I-J-th kind of gluon can also emit a gluon. How would it do it? The I-J-th gluon could become the I-K-th gluon. I-J, let's say, red-green gluon could become the red-blue gluon, as long as the indices make sense and they're followed, anti-K here. So this would be an interaction in which a gluon of type I-J became a gluon of type I-K, and a gluon of type K-J. The order of I and K matters. A K-J is not the same as a JK. That introduces something absolutely new which is not in electrodynamics, namely that the gluons themselves are charged particles and therefore exert forces on each other. Exert forces in a way that photons do not, exert forces on each other. The dynamics of gluons is much, much more complicated than the dynamics of photons. The last time what I told you was that one of the, and I'm not going to do the mathematics of this, it's not even fully understood it even now, but the effect, one of the important effects is that if you have a source, a color, a bit of color, with field lines coming out of it, again those field lines are not allowed to end, they're not allowed to end, but the effect, let's suppose that's quark, is an anti-quark over here, field lines have to come into it, so the field lines have to be continuous and so forth. I don't want, let me not draw more than that. Now the effect is that the field lines interact with each other in a way that they don't in electrodynamics. And the effect is very simple. The field lines get pulled into bundles which are called flux tubes, the result of which is that no matter how far apart you pull two quarks, they'll never escape from each other because the energy stored in this gooey piece of glue just increases linearly with the distance between them. All right, that's a summary of quantum chromodynamics, but it's also a summary of gauge theory. It's also a summary, very, very quick summary of gauge theory. Maxwell-like fields, sources, Gauss's law, making sure you have conservation and symmetries, symmetries having to do with the conservation laws, but more than that the symmetries telling you how the particles interact with each other in terms of emission and absorption. There is one other thing in a gauge theory, and it's the coupling constant. The coupling constant is the analog of the charge of the electron. The charge of the electron, let's talk about what the charge of the electron means. It is, of course, the coefficient in a Feynman diagram when you emit a photon from a charged particle, the amplitude for that, the quantum mechanical amplitude for that is the electric charge. But if you wanted to think about it operationally, you can imagine an electric charge slamming into the cathode or the anode cathode, the cathode of a cathode ray tube. When it slams into it, it gets accelerated, and the question is what is the probability that an electron which is stopped suddenly emits a photon? The answer is the square of the electric charge. The square of the electric charge, there's some pi's in it and h-bars and things, but basically the square of the electric charge in suitable units is the probability for emitting a photon. So that's what in quantum mechanics and field theory, that is what the meaning of the electric charge is. It's a measure of the square of the electric charge is the probability. Why? Because probabilities are always squares of amplitudes. But that's the fundamental significance of the electric charge, and the electric charge is a dimensionless number, just the probability that when you stop an electron you emit a photon. So it's a dimensionless number. It's a small dimensionless number. With appropriate sets of definitions, the square of the electric charge, really what tends to come into things is not quite the square of the electric charge, it's the square of the electric charge divided by some 4 pi's, 4 pi's are always floating around in there. It's called the fine structure constant, and the fine structure constant, which is pretty much the probability of a photon emission, is a number of about 1%. It's close to being 1 over 137, a famous number, 1 over 137. The 137 has no particular significance. It's just numerically about what the fine structure constant is. And let's just call it 1%, it says there's a 1% probability of emitting a photon. So if you had a picture that when an electron hits a cathode ray tube, it sends out a spray of photons, that's not right. What is right is that if 137 photons hit the cathode ray tube, on the average one of them will give off a photon. In that sense, electromagnetism is a weak force, or a weak process. The probability of emission is weak. All right, that's the quick summary of gauge theory. What I wanted to do is give you very, very briefly a run down on some numbers and orders of magnitude in various parts of particle and atomic physics. And then go on to the weak interactions. Now the reason I'm giving you a run down about numbers is to show you how different the weak interactions. Quantum chromodynamics is the theory of the strong interactions. The strong interactions was the word before quantum chromodynamics was invented for the interactions between subnuclear particles, between hadrons, between protons, neutrons, mesons, and all the things which are made of quarks and gluons. So it's theory of strong interactions. And there are electromagnetic interactions and there are also weak interactions. So I want to spend a little bit of time with numbers to show you what, where the terminology came from, why one is called weak, why one is called strong, why electromagnetism is sort of in the middle between the two of them. But let's just review a couple of numbers. First of all, in terms of units, energy, distance, and time do not have to be thought of as different units. They're all related with each other. First of all, time and distance really should not be distinguished in any sensible set of units that are useful for fundamental physics. We can set the speed of light equal to one, and if we do, if we fix the speed of light equal to one, then time and distance have the same units. And we can use the same units. As far as energy goes, we could also choose to set Planck's constant equal to one. That's a useful thing to do in quantum mechanics. An awful lot of equations get simpler if you set h bar equal to one. Well, they don't get a lot simpler. They just get rid of the h bars, but you can set h bar equal to one. Now, let me just remind you there's a connection between energy, e equals h frequency, right, or h nu, or h bar omega, all the same thing. The units of h bar, well, we've set h bar equal to one, so in units in which h bar are equal to one, energy is equal to frequency, but what's the units of frequency? One over time, number of oscillations per second. So that means that energy and time have inverse units to each other. You don't need to have a separate unit for time and energy. Any time you have a unit of energy, it defines also a unit of time. Now, of course, the unit of energy is inverse to the unit of time, so big energies correspond to small time intervals. And, okay, so let's write down some connections. And since energy and time are connected to each other, and time and space have the same units when C is equal to one, then time and space have units which are just inverse units to the units of energy. So, for example, here's a unit of energy, one electron volt. We've used electron volts before. One electron volt cannot be thought of as a unit of distance, but the inverse, one inverse electron volt, so it's one over an electron. One inverse electron volt is a unit of distance, and just to have an idea of what it is, it's about 10 to the minus seventh meters. One electron volt is about 10 to the minus seventh meters. So an electron volt is a fairly small distance, but on the scale of fundamental physics, it's not a small distance. 10 to the minus seventh meters is what? A atom is 10 to the minus tenth meters, something like that. Yeah, okay, so it's fairly big. An atomic diameter, yeah, an atomic, and we can convert, right, and an atomic diameter, one atom, this is the atomic diameter, that's about 10 to the minus tenth meters, and so you can convert that to electron volts. Is it, I don't know, whatever it is, an inverse of about 1,000 electron volts, a kilo electron volt, an inverse kilo electron volt is an atomic diameter. Okay, another fact, just converting distance, if we know the distance or size of an atom, then we know the time that it takes light to go across it. All right, so the transit time, there's another quantity, the time that it takes transit time, the time for light to cross an atomic diameter, what is that? That's about 10 to the minus 18 seconds. I've used the fact that light goes at about 10 to the eighth meters per second. Order of magnitude, 10 to the minus 18 seconds, these are just some numbers that I just wrote down, but a different time scale. All right, so this is the time scale for light to go across an atom. The different times, and by an atom, I don't mean the nucleus, I mean the whole atom, another time scale is the time scale that it takes for an electron to orbit an atom. Now you might think, why isn't that just about the transit time? Transit time is about 10 to the minus 18 centimeters. The reason is because electromagnetic forces are fairly weak. Because they are fairly weak, the force on, and the ultimate reason that they're weak is because this fine structure constant, this alpha, which is E squared over 4 pi, that's a small number of about 1 over 137. Another way to say it is the electromagnetic force on an electron is weak. Because of that, the acceleration on the electron is not too big, and the orbit is fairly large. The weaker the force on an object, the larger the orbit is going to be. Sorry, the slower the electron will move is what I meant to say. The slower the electron will move, and because the electron moves slowly by comparison with the speed of light, how much slower? About 1 percent slower. About this much slower, the electron moves around with only about 1 percent the speed of light, and that means the orbital time, orbital time, that's about 10 to the minus, let's see, it's bigger. That time is bigger, 10 to the minus 16 seconds. Another quantity sometimes of interest is the decay time for an atom. How long does it take for an atom, an excited atom, let's say a hydrogen atom with its electron, one orbital up from the ground state, one orbit up from the ground state, how long does it take to decay? Now, that is small, doubly small, with two powers of alpha. What are the two powers of alpha? First of all, the acceleration of the electron is small. Remember, charges emit radiation when they're accelerated. The acceleration is small because the force is small. So first of all, the acceleration is small. The electron moves around with a relatively small acceleration compared to what it could have been if alpha were bigger. It moves around with a small acceleration, but on top of that, just like the electron plowing into the cathode ray tube, there's another factor of alpha when the electron is accelerated in the probability that it emits a photon. So as the electron goes around here, first of all, it's moving with a slow acceleration, and second of all, even that acceleration is not very efficient in producing photons just because the fine structure constant is small. The net result is that the time scale, order of magnitude, decay time, how long it takes for an atom to decay. Decay time is about a nanosecond, 10 to the minus 9 seconds, much longer than the orbital time or the transit time. The main thing is there's a variety of different scales in the atom, and they're all related by the fine structure constant. They're related by powers of the fine structure constant, the fine structure constant being a small number. There are fairly large ratios between different time scales, transit, orbital, and decay time. All right, now let's come to Hadrons. Hadrons are like atoms, they're atoms made up out of quarks. We could ask very much the same kind of questions. First of all, we could ask what is the Hadronic diameter? The Hadronic diameter of a typical Hadron, it could be a proton, neutron, meson, they're all more or less the same. Here's Hadrons. Hadrons, Hadrons. The Hadronic diameter is a lot smaller than the electron. Sorry, than the atom. It's a lot smaller than the atom. It's not 10 to the minus 10th meters. It's about five orders of magnitude smaller, 10 to the minus 15 centimeters. That's also more or less the diameter of a nucleus. Nucleus, of course, being a few protons and neutrons could be bigger. Sorry, meters. Hadron diameter is about 10 to the minus 15th meters, five orders of magnitude smaller than an atom. Part of the reason is because there's two reasons. There's two reasons why it's smaller. One of them has to do with the fact that the constituents are heavier and heavier things will sit closer to the center. The other has to do with something else that we'll come to in a minute. But it's about five orders of magnitude smaller. The transit time has just gotten from the size straightforwardly. Transit time, this is not an independent thing. It's just another measure of the size of the thing. Just to get the number straight, it's 10 to the minus 23rd seconds. So that's a typical time scale for, well, it's for light to cross the Hadron. But now we can also talk about the orbital motion of quarks. How long does it take for a quark to swing around its wave function or swing around the interior of a proton? And there the answer is about 10 to the minus 23rd seconds. So how long does it take for it to decay? How long does it take for it to decay? And the answer is about, now I'm talking about particular kinds of decays, a decay where you hit a proton, it starts oscillating, and then emits a pion. Those decays are the same as the quarks. They're different than this hierarchy of scales here. What is the conclusion? The conclusion is the analog of the fine structure constant is much bigger, close to 1. So what is the conclusion? The conclusion is the analog of the fine structure constant is much bigger, close to 1. For Hadrons, in other words, the emission probability, if a quark were to get stopped in a fictitious cathode ray, quarky cathode ray tube of quark flying off, the probability to emit a gluon would be about 1. That's what these things indicate here. The hierarchy here is entirely due to the fine structure constant. The lack of a hierarchy here can only mean that the corresponding quantity in quantum chromodynamics must be close to 1. By now it's been measured in many ways, and it is much closer to 1. It's much larger than the corresponding quantity for electrodynamics. One is a little bit, no. It would seem that just special relativity to say that the case that could be much less than 10 to the minus 23, that's a kind of case like, go for 1. Yeah, I think that's a fair statement. It's about as fast as it could be, in other words. It's about as fast as it could be. Nothing's slowing it down. This is why the strong interactions are called strong. Now, the facts about the numbers here were known before quantum chromodynamics, and it was understood that there was no hierarchy of scales, and that's why it was called strong interactions. We now trace it to the properties of the charge or the fine structure constant. Incidentally, for quantum chromodynamics, the analogous quantity is not really one. It's about a fifth, but much bigger than the 1 over 137 here. All right, so that was just some numbers and some facts about the strong force. It's called alpha QCD. So this is called alpha, and the other one is called alpha QCD, and it's about 0.23 or something like that. Or in context of paper, you write alpha QCD. I've never heard alpha QCD referred to as alpha QCD, but I think if you were writing a paper with both of them, you might call the top one alpha QCD, yes. I've never seen it referred to that, right? Well, particularly if you're talking about shielding or something like that. No, I can't, I don't think I've ever seen it. Oh, oh, oh. It's just called alpha. Sometimes the unshielded one is called the bare alpha. The shielded one is called the renormalized alpha, right? Okay. Well, that brings us to the weak interactions. The weak interactions are called weak. Primarily because the decay times associated with them are very long. They're much longer than the decay times associated with either hadrons, and they even tend to be longer than the decay times associated with atoms. So there are processes in nature, decay processes in nature, in which particles, elementary or otherwise, decay by weak interactions where the time scale, in other words, the half-life of these decays is far longer than any scale that can be accounted for by these, by numbers like this. So as an example, here's some examples. Well, the lifetime of the neutron. The neutron is a hadron. The decay is neutron goes to electron, proton, plus anti-nutrino. And the lifetime for that is about 12 minutes. That's absurdly long on particle physics scales. Part of the reason is easy, but it hardly accounts for the extraordinary stability of the neutron. Part of the reason is the neutron is only a little bit bigger than the combined masses of the electron, the proton, and the neutron, and the neutrino. Let's just see. Somebody have out their pocket? The mass of the neutron is 940 MeV. About 940. So the proton is about 939. I think the actual difference between them is about one and a half MeV. But then there's the electron which has a mass of about half an MeV. All together out of this 940, almost a thousand MeV, the difference of mass of the proton, the amount of available mass left over is tiny. It's 0.1% or less. If the neutron was slightly lighter than the sum of these masses, it could not decay at all. Just energy conservation would not allow it. Remember, mass is energy. It would not have enough energy to decay and still leave over some energy for kinetic energy of these particles. So if the neutron was exactly the same as the sum of the energy of the electron, the proton, and the neutrino, the sum of the mass of them, it wouldn't be able to decay at all. If you gave it a mass, tiny, tiny, tiny, tiny bit above that, it would be able to decay, but it would take a very long time. The decay would be very slow. So part of the slowness or the long lifetime of the neutron is attributable to the fact that there's very little energy available for the decay to take place. But given that, that's hardly the bulk of the story. Twelve minutes is so absurdly long that there's something much, much more than that going on. Another example, let's see, where do I have it? I guess I didn't write it down. Another example of a, this is called a weak decay. For obvious reasons, it's called a weak decay. Another example is a charged pion, let's say a pime minus. A charged pion decays to an electron. Electron has negative charge and an anti-nutrino. Yes, and an anti-nutrino. That's another possible decay. A charged pion goes to an electron and an anti-nutrino. A positively charged pion could become a positron and a neutrino. The lifetime for that, I believe, is about 10 nanoseconds. Longer than these atomic decay times. And atomic decay times are very, very long by comparison with particle physics decay times. Now, there is a decay of the charged pions to these particles. There's another decay involving muons. We're going to come to muons soon enough, similar to electrons. But the point is not the particles that they decay into, but two things. First of all, they also decay very slowly, nanoseconds. But there's plenty of energy available for these decays. These decays are not suppressed by the fact that the energies of the initial and all the masses of the initial and final states are very close to each other. So there's something else going on in weak interactions. Whatever weak interactions are, there's something that makes them proceed very, very slowly and weakly. So that's what we come to next is weak interactions. The primary or oldest example of which is the decay of the neutron. The beta decay of the neutron called beta because electrons are originally called beta waves. And this is called the beta decay of the pion. In fact, it's quite similar. Quite similar physics goes into both of them. So to understand what's going on, we need to understand about some more particles. Particles that we haven't come to yet, the leptons. Let's go back to the quarks for a moment and let me just make a table of the different kinds of quarks. Horizontally, or make a sequence here which will be red, blue and green. These are the colors of quarks. Then along here, let me list the up quark, the down quark, the charmed quark, the strange quark, the top quark, and the bottom quark. Each one of these boxes is filled. There are red, blue, and green up quarks. Let's put an x there. There's an x here. There's an x here. There's an x here. X here, x here, and so forth. The boxes represent all the possible quarks that exist. Now, quantum chromodynamics has to do with symmetries which connect red to green to blue. It has to do with these unitary transformations which mix up the colors of quarks. They do not, those symmetries do not mix horizontally. An SU2-SU3 symmetry, a color symmetry, may rotate an up red quark into an up blue quark or an up green quark. And the nature of the symmetry is to act vertically, to mix up things this way. And also at the same time, we'll mix the different down quarks, the different colors of down quarks. The same symmetry will mix the different colors of charmed quarks and strange quarks and so forth. So the color symmetry acts vertically. It mixes up the different rows here. Is that the unitary matrix? Yes, the unitary matrices mix up red, green, and blue. I might have done better to put the rows where the columns are and the columns where the rows are, but no, it's okay. It's good. No, no, it's good. It's good. Right. The weak interactions are associated with symmetries which act horizontally in this picture. They mix up with down. They mix, and at the same time that they mix up with down, they mix charmed with strange. And they mix top with bottom. Let me just remind you that an up quark has the same properties apart for its mass. Same properties as a charmed quark or a top quark. The down quarks are similar to strange quarks and bottom quarks. Ups have charged two thirds. Downs have charged minus two thirds. Same here and here. So these are symmetries which mix, if you like, upness with downness. At the same time, charm this with strangeness and topness with bottomness. It mixes these up. The symmetries, those symmetries, what would you care to speculate on what group might be involved? SU2, because it acts on things, on doublets. We simply have three doublets here. They don't take up to charm or up to top. They simply act on up and down horizontally among pairs of things. A group SU2 and it is a gauge symmetry. In other words, it also comes together with forces, with gauge bosons, with interactions, and with gauge field. We're going to get to those short enough. But we've left out something from this table here. It's kind of as if there was a fourth color. Now the reason, the fourth row here is not usually identified as a color. And the reason is because the particles do not interact with the gluons. But nevertheless, it is another row here, another row in which there are particles which are filled in here. And in a certain sense, they are also doublets, these particles are also doublets, and also get mixed up under this SU2 symmetry which moves things horizontally. What are they? What are those particles? They're the leptons. So in a sense, the fourth color could be thought of as lepton number. But what are the leptons? The leptons are, first of all, in the first column here, there are analogous to the up, or analogous to the down, there is the electron. The electron is a lepton. But its partner, its partner is the neutrino. But there are different neutrinos. There's not only one neutrino. This is called the electron neutrino. What comes in the next column? The muon. The muon, in every respect except for mass, is the same as an electron. Just as in every respect except for mass, the charmed quark is like an up quark, or the strange quark is like a down quark. And together with it, there's its own neutrino, the muon neutrino. And finally, the last one is called the tau and nu sub tau. All of the charged leptons, incidentally, neutrinos, of course, as you might guess from their name, are electrically neutral. The electrons, muons, and tau's have charged minus one. They are put under the down column here, not because their charge is the same as the down charge, but because the difference between the charges of the two is the same as the difference up here. What's the difference between the charge of an up and a down? Two-thirds minus minus one-third. So the difference between this column and this column is plus one unit of charge. The difference between this column and this column is also one unit of charge. One, sorry, zero minus negative one. So as you go from here to here, you decrease charge by one unit as you go horizontally. Now, should we take a break for five minutes? Let's take a break for five minutes. And then I will tell you about the W bosons. The W bosons are the gauge bosons of the weak interactions. All right, so let's try now to see if we can make a microscopic gauge theory of the weak interactions. As I said, the weak interactions are based on a gauge theory which mixes up to down, neutrino to electron and so forth. And so rather than to get into mathematics for the moment, the mathematics of the group SU2, which we'll come back to, let's think a little more simply. We've seen that in various situations the gauge bosons typically have the quantum numbers or the quantum properties of a particle and an antiparticle. In the case of the quantum-chromodynamic interactions, it is a quark and an antiquark. That's why we had nine gauge bosons and so forth or eight removing the trace. So we could start by saying let's look at the possibilities of gauge bosons which have the properties of a particle and an antiparticle. Let's not distinguish between whether we're talking about quarks, red quarks, blue quarks, green quarks or leptons. Let's just say a particle from this column with an antiparticle of this column. And for simplicity, let's just focus on the leptons. You'll see in a moment that you'll get the same thing if you focused on the quarks. But let's focus on the leptons and talk about objects that we could make with a lepton and an antilepton. First of all, we could have some simple things, electron with a positron. Now I don't really literally mean electrons and positrons. We could have electrons and positrons. Electrons and positrons have no charge at all. So let's drop that for a minute. We'll come back to it. There are things which have properties of electrons and antilelectrons, neutrinos and antinutrinos. But more interesting for the moment is the combination of electron, antinutrino. So what is the charge of an object which has the same conserved quantities, the same properties as an electron with a stuck on top of an antinutrino? Well, let's just focus on its charge. It's electric charge. Its electric charge is minus one. Neutrino is neutral. Electron has negative charge. So whatever this object is, it has negative electric charge. Let's give it a name and let's call it a gauge boson. Let's assume that it is also a gauge boson. Gauge boson means that it behaves like the Maxwell field, the whole same thing. But it has negative charge. Let's call it W. W, that's the traditional name for it. W minus. Why minus? Because it has negative electric charge. We could, that was this one with this one, we could have taken down quark with, I guess, anti-up quark. What happens if you take down quark with anti-up quark? How much charge do you get? Again, minus. Same as this. Down quark with anti-up quark would also have minus charge. So if you like, you could also imagine that the W minus has the same properties as a down quark and an anti-up quark. In other words, it's one from the second column and anti-one from the first column. That's the property that it has. Let's call it W for weak, in fact. I think W was originally the notation for weak. The negatively charged W boson. And of course, there's also a positively charged boson, W boson. Which is like anti-down with an up or like positron E plus with a neutrino. E plus with a neutrino would be a W plus. Let's suppose the pattern is pretty similar to quantum chromodynamics, that the W bosons are like either photons on the one hand or like gluons. Which means that they would be objects which could be emitted from the leptons. They would be objects which would allow a lepton of one kind. Let me not label it. A lepton of one kind to become a lepton of another kind. By emitting a W boson. Let's see if we can find some examples. An electron can emit a W minus to become a neutrino. An electron becomes neutrino by emission of a W minus. You see the action of the gauge bosons is to do the same kind of transition that the symmetry is ultimately going to be imagined to do to mix electron into neutrino in very much the same way that the colors got mixed by emitting gluons. You could follow the lines if you like and say the electron and the anti-neutrino over here can be imagined to be a W minus boson. Same pattern. Let's just draw it by saying a W minus is emitted. But that's not all you can do with a W minus. You can also have a down quark go to an up quark. With a W minus. That also works. In fact, the W minus is an object which can be emitted in a transition from, let's see, from the right column to the left column. A red down quark can become a red up quark by emitting a W. A blue down quark can become a blue up quark by emitting a W. Likewise for green. But you can go further. A strange quark, not the strange quark, the strange quark can become a charmed quark. That's another one. Strange quark, charmed quark, again, W minus. So this new symmetry, this idea of a new symmetry which acts horizontally and only horizontally between neighboring pairs here introduces the possibility of gauge bosons which cause transitions between electron neutrino, down quark and up quark, strange quark and charmed quark. Now a W plus and a W minus are antiparticles of each other. The meaning of that is that you can flip these diagrams, put the W minus down, W here, but when you flip it, it becomes a W plus. So a W plus can be absorbed by an electron, plus a unit of charge, minus a unit of charge to become a neutrino. So a whole family of different processes are codified by this one vertex here. We can turn them upside down, neutrino goes to, you know, what we're allowed to do. We've done these things before. And that's the basic new element of the weak interactions. Actually, that's all, yeah. Well, it's not clear to me why you have W minuses in all three of these cases. There's a lot of charge conservation. It couldn't be W plus. Would you accept I don't know what a W means? W is the name of a particle. It's the name of another gauge. Yeah, but the charge conservation tells me it's going to be a minus. But I haven't seen any reason why it should be the same particle. Well, okay, that's right. But what it comes down to in the end is an empirical fact. It's the empirical fact that the symmetry of the weak interactions acts simultaneously on all of these doublets here. When an up goes to a down, a charm has to go to a strange, a top has to go to a bottom, and the leptons have to mix with each other simultaneously. The implication of that, the implication of that from a practical point of view, is simply it's the same symmetry which acts, it's the same gauge bosons which are there. You could have imagined a theory in which it's quite mathematically consistent to imagine a theory where different W-type bosons would have been emitted by up quarks, charm quarks, or tau quarks, or even leptons. Right? It's a fact of nature. The fact of nature that nature is simpler than it might have been that the same gauge bosons cause transitions between any point, you know what I mean, I'm going to get tongue-tied if I try to say it. Okay, well first of all, what processes can we explain? Good question. Can a W-beg into an electron and a negative neutrino as well? Negative, so say it again. Can you flip the arrows? Yes, yes, yes, yes. You can flip arrows and you can flip lines. Yes, yes, certainly you can flip the arrows. Yeah, absolutely. Absolutely. And you can also flip lines up and down in various ways. For example, another way you could flip the line is you could take this up quark here, instead of going out, instead of an up quark going out, an anti-up quark comes in. Whenever you flip a line from past to future, or from future to past, you'll also change from particle to antiparticle. I'm not sure, but on the right-most side, W-stereo is a strange antitron. Yes. So that's the same as a down antiparticle? Yes. The same thing? Wait a second. Sorry. Yes. That's a strange antitron, right? Yes. That's the same as a down antiparticle. It's not literally a strange antitron, it's just a rule that tells you that whatever these W bosons are, they're associated with transitions from one column to the other. The triton's a charm. Did I? Sorry. Strange. Right. Electron to neutrino. Strange to charm. Is that wrong? No. No. It's not an anti-charm. That's the point. Who said it was an anti-charm? Well, it shouldn't have been. It's different from coming in from the past. Right. If you follow the lines like that, at the point where the line turns around, it has to be thought of as an antiparticle. Right. Okay. Or if you like, you can always think of it as strange anti-charm coming in and making a W boson. Okay. Just keep in mind, whenever you flip from the past to the future, if you want to know what a W boson will decay to, let's read it downward, it will decay to a strange and an anti-charm. And that really can happen. Yeah. So can you create W boson from like electron and a... Anti-neutrino. Anti-neutrino. Yes. And then call the W boson to change into like a strange and a charm. Exactly. That's where some of these processes are going to come from now. So let's think about the various processes. Let's start just for fun. Let's start with the decay of the pi minus. What is a pi minus? A pi minus is a quark and an anti-quark with altogether negative charge. So let's say a quark and an anti-quark with negative charge had better be... Here's the pi minus. That's equal to a down quark and an anti-up quark. Down and anti-up. Now, what can a down and an anti-up do? Let's see if we can find down and anti-up here. Here it is right here. Down and anti-up can become a W minus. So it's a possible transition that is allowed by the physics of W bosons. So here's a W minus. But now a W minus can do something else. A W minus here is... Let's say here is a W plus. Let's put W plus upstairs here. This is W minus now. And let's put the neutrino downstairs, anti-neutrino. And read this as saying that W minus goes to what? Electron. Anti-neutrino. So W minus can go up to electron, anti-neutrino. And here it is. E minus, ordinary electron. So that's the basic underlying process that governs the beta decay of the negatively charged pion into a electron and an anti-neutrino. Now, which anti-neutrino is it? There are three distinct anti-neutrinos. There's an electron anti-neutrino over here. So it's the electron anti-neutrino. But it can also... I didn't draw all the various possibilities, but any place you saw an electron, you can put a muon. Any place... And as long as you carry along the muon neutrino. So another process is muon. Anti-muon neutrino. And just for clarity, this is not strictly a Feynman diagram, right? This is just showing an interaction. It's pretty close to a Feynman diagram. Except for the time direction. Oh, yeah. Right. I don't know why. All of a sudden I change directions. Yeah. For some reason. Yeah, right. Time is now going horizontally. I've become a computer scientist and they measure... Yeah, yeah, they do. Right. Yeah, it's pretty much... Yeah, that's good. We can turn it up. I can't do that easily, but yeah. Yeah, that is essentially a Feynman diagram describing the decay of the pi minus to the muon. As it happens, the... For technical reasons having to do with some complications and Feynman diagrams that are not interesting to us. As it happens, the primary decay... Well, primary, I mean the most probable. The most probable decay of the pi minus is to the muon anti-neutrino, not to the electron anti-neutrino. It's a technical point of no special interest to us. It just happens to be that way. Wouldn't it be consistent instead of considering the Feynman diagrams with a time deal is more like a kinopaly equilibrium deal? Which way you're going? Well, both things can happen. Both things can happen. We're going to have a real equilibrium. We're in empty space. We're in empty space except somebody made a pi minus. A pi minus was created how? A pi minus might be created by some nucleons coming together and creating by strong interactions. By strong interactions, a pi minus might be created. Now that pi minus is just going along and saying, wait a minute, I can decay if I like into a neutrino and a muon. Then of course, the neutrino and the muon have enough kinetic energy to fly off and they won't come back together again because they've departed. They've gone away. Okay, on the other hand, it is true that if you were to take a muon and an anti-neutrino and fire them toward each other, there would be a certain probability that they would make a pi minus. So yes, you can read it both ways. Let's go. A quick question. Like in one dimension, you have an SU3 and the other an SU2. Is it possible to combine them into an SU6 or something? SU5. Well, how great. SU5. Not two times three, two plus three. SU5. Tonight, we want to get to SU2. I'll be happy if we get the basic ideas of SU2. Now, keep in mind, we're doing something a little bit funny because we have, SU2 has how many generators? Three. So how many gauge balls should it have? Three. So far, we only have w plus and w minus. So let's just be another one looking around. And that other one must be associated somehow with something like electron, anti-electron and neutrino, anti-neutrino, but we'll come to it. Just like the, yeah, we'll come to it. There is a third gauge boson looking around. And it does play a big role in the weak interactions, but it didn't play a big role historically in the weak interactions just because for reasons we'll come to. Historically, it was the w boson, which was the first to be conjectured. Speaking of historically, was the weaker the strong force sort of suspected first? Of course. All right. So let's talk about the history. The history of these forces goes back to Becquerel, goes back to radioactivity. Alpha, beta and gamma decay. Alpha decay was what? A mission of a two protons and two neutrons from a nucleus. That's a strong interaction. It's part of the theory of Hadron's and nuclei. Gamma decay, that was this one right here, weak interactions, and gamma decay? Photon. Historically, they happened at the same time in the same experiment. Question? The energy conservation, how it works there? You're assuming some kinetic energy in the world? Yeah, the energy, let's work on it. You never worry about the energy conservation in intermediate states like this. If the energy of the intermediate state doesn't match the energies of the initial and final state, it just means that the intermediate state can only last for a very short time. There's an energy time uncertainty principle. You can violate the energies in a Feynman diagram for short periods of time. You don't have to worry about conservation of energy between here and here. The only energy we have to worry about is the initial energy and the final energy. Okay, so let's worry about it. A pion has an energy of about a mass. Let's suppose the pion is at rest. The mass of the pion is roughly about 140 MeV. The energy of the muon and the neutrino, well first of all it's the rest energy of the muon and the neutrino. And the rest energy, the rest mass of the neutrino, sorry, the mass of the muon, I think is about 100 MeV, I forget exactly, and the mass of a neutrino is for practical purposes zero. It's too small to be important to you. So there's more energy to start with than the energy in the masses of the two final particles. Does that mean it can't happen? No, it just means the two particles will go off and carry some kinetic energy. So the difference between the energy on this side and the sums of the masses on this side is just the kinetic energy of the outgoing particles. That's the energy balance. The momentum also has to be conserved, and that means if the pion was at rest to begin with, the momentum of the muon and the momentum of the neutrino have to be equal and opposite, and they'll go out back to back, so to speak. If you analyze the neutron decay in the same way as you would do that. We're going to do that now. Yeah, let's do the neutron decay right now, see how the neutron works. Oh, let's do something simpler first. Let's do the muon decay. The muon is also something that can decay. So let's take the muon. Here's the muon moving along. Now what can the muon do? The muon can become a neutrino sub muon and emit a W. What kind of W? W minus, right? It can become a W minus and a neutrino. But now one of the things that W minus can do is it can become an electron and an anti-neutrino. So this W minus can now become an electron and an anti-neutrino of the electron variety. Can this happen? The only thing that would, yes it can happen. Not only can it happen, this is the way the muon decays. The muon decays because it is sufficiently much heavier than the electron, about 200 times heavier than the electron. The neutrinos are practically massless. So the energy balance is you have about 100 MeV here, a half an MeV here and the risk goes off in kinetic energy. So that's an example of what's called a purely leptonic process. No quarks in it at all. Is it an anti-neutron? Is it what? An anti-neutron. Yeah, it is. So it's one neutrino, one anti-neutrino and an electron. Okay, so that happens. And that's the primary decay, it's the only decay of the muon. So the muon is an unstable particle because of this. But the reason the muon is unstable and not the electron is just that the muon happens to be heavy enough to be able to decay to the electron. The electron is not heavy enough to decay to the muon. So in a sense it's an accident of the masses. The same is true of the tau, incidentally. The tau can also decay to other leptons. But all right, that's the mu decay. And notice the components are the same. The basic components are the same. A transition from one column to another, followed by a W, and then the W makes another transition from one column to the other, or if you like, a decay from one column to the other. All right, what about the decay of the neutron? Decay of the neutron. Would it be consistent to say the purpose of a linear accelerator, per se, is to effectively increase the mass of the electron so that it can now go the other direction? Absolutely. No. You don't change an answer, just put nothing in it. Absolutely. That's exactly right. Okay, we're going to do neutron decay. So the neutron is, maybe I better switch to vertical. I don't want to, vertical, vertical notation, back to vertical. Neutron is three quarks, two down quarks, and an up quark. Right? Minus four thirds plus two thirds, sorry, minus two thirds, yes, that's a neutron, electrically neutral, neutron coming in. Okay, now what is possible? First of all, we can have the up quark become a down quark, and emit some sort of W, but that'll give us three down quarks. Three down quarks is a fine thing. There's nothing wrong with three down quarks, except that the mass of three down quarks, of a particle with three down quarks, is always too heavy for the neutron to decay. So that's not what happens. What about one of the down quarks? The down quark can emit a W, become an up quark. This is again a W minus. The down quark emits a W minus, becomes an up quark, and now we have two up quarks and a down quark, up, up, down, what is that called? That's a proton. And the W does what the W always does. It decays to an electron and an anti-neutrino. So that's the beta decay of the neutron. Could it decay into a muon and a neutrino? Not enough energy. The muon is too heavy so that there wouldn't be enough energy left over for the muon. But other than that, if it were not for that, it could happen. Another way to say it is if you could jam some more energy into the muon, you could add extra energy into the neutron. How can you do it? You hit it, you give it a shove. You give the neutron a whack, you excite it, and if you excite it enough, it can decay to a proton, a neutrino, anti-neutrino, and a muon. But without somehow providing that extra energy, the neutron will only decay to the proton, electron, and the anti-neutrino. Okay, now let's come, let's see, where are we? I think we'll finish for tonight. But remember, we have a puzzle, and the puzzle is why is it that the weak interactions give such slow decay rates? And I can think of one reason, namely, the coupling constant or the analog of the fine structure constant could be ridiculously small so that the probability of emitting these W bosons would be very, very small. That is not the reason. The fine structure constant for the weak interactions is about the same as for electromagnetism. That is not the reason. The reason is something else, and we'll come to the something else next time. When you say that there are these symmetries, you mean that the Lagrangian is invariant? Lagrangian, the Lagrangian is invariant under those operations. That's exactly what it means. I didn't want to start writing bunches of Lagrangians, so I've used the shorthand of saying there's a symmetry, right? But that's what it means, yeah. For more, please visit us at stanford.edu.
 Stanford University.  All right.  We've really studied two different theories a little bit.  Quantum electrodynamics and quantum chromodynamics,  both of them are gauge theories.  In fact, just about all of nature as we know it  is one way or another is controlled by gauge theories of different kinds.  And so I ought to tell you what a gauge theory is.  Now, I'm not going to get into any depth what gauge symmetry is.  I'm going to tell you what a gauge theory is in more or less the minimal mathematical way.  First and simplest gauge theory is Maxwell's theory.  Maxwell's theory of light or Maxwell's theory of electromagnetism,  let's not worry about what the word gauge means.  We may go through it later, but not now.  But what it is, is a theory of fields which have all of the properties of the electromagnetic field.  The electromagnetic field has six components, three components of electric field,  three components of magnetic field.  You can represent those six components in terms of a four vector called the vector potential.  I'll just write it down for you.  I'm not going to play any big role today anyway.  But just to tell you, there's a four vector that's generally called A,  and it has an index mu.  It has four components, A0, that's the time component of it,  and the three space components A1, 2, and 3, or A, X, Y, and Z.  Let's not write them out in detail.  So it contains a three vector and a fourth component, A0.  The significance of A0 is that it's the electrostatic potential.  It's the thing whose gradient is the electric force.  So for example, the electric force on a charged particle  is just gotten by multiplying the electric charge by the gradient or the derivative fx,  derivative of A0 with respect to x.  So it's just the electrostatic potential that's measured in volts,  in particle physics of course it's not measured in volts,  volts per unit charge, excuse me,  in whatever way you measure electrostatic potential.  That's the meaning of A0.  Notice first of all, oh, and of course this is the electric field.  This is the electric field.  The components, derivative with respect to x, y, or z,  so it's the gradient of A, and the gradient of A is just called the electric field.  So the electric field is determined by the gradient of A0,  and the magnetic field is determined by also derivatives,  also derivatives of the vector potential,  and in fact the magnetic field is the curl of the vector potential.  So one of two ways you can study electrodynamics,  either in terms of the electric and magnetic fields,  of which there are six,  or in terms of the vector potential of which there are four,  but there are equivalent descriptions of electrodynamics,  and we won't go into Maxwell's equations,  but let me just remind you that Maxwell's equations give rise to electromagnetic waves.  An electromagnetic wave is a wave of electric and magnetic field,  and a typical electric magnetic wave has a direction and space that it moves,  let's just represent it by an arrow,  and it has a polarization.  To understand what the polarization means,  you'll have to think about the electric and magnetic fields,  and the electric and magnetic fields oscillate as you go down the wave.  Here's the electric field, for example.  The whole thing moves with the speed of light,  but at any given instant the electric field might look like that,  and the magnetic field is always perpendicular to the electric field.  Let's try to draw it, see if I can give it a drawing.  I'm imagining the electric field is in the horizontal plane,  and it's always perpendicular to the, let's see, is this going to...  it's always perpendicular, like so.  In appropriate units, the electric and magnetic field are equal to each other and perpendicular.  This would be a plane polarized electromagnetic wave.  In a plane polarized electromagnetic wave,  the direction of polarization, which of course is always perpendicular to the direction of motion,  the direction of polarization is determined by the electric field.  So the electric field determines the direction of polarization,  and that's electromagnetism in a lot of electromagnetic waves in a nutshell.  The other thing, of course, about electromagnetic waves is that there are sources.  The sources are charges and currents,  but let's particularly focus on the sources of the electric field.  The sources of the electric field are electric charge,  and to just draw a picture, to put it at the level of pictures,  every electric charge creates an electric field.  The electric field lines never end except out of the charges.  A positive convention, the convention is a positive electric charge,  puts out an electric field, which is radially outward,  falls off as 1 over r squared.  You can imagine that the number of electric field lines coming out of a charge is fixed,  proportional to the charge,  and therefore the number of electric field lines passing through any sphere  is the same no matter how far the sphere is.  To say that mathematically, the integral of the electric field over the sphere  is the same wherever you go, however far out you go.  That's Gauss's law.  That's Gauss's law,  and that Gauss's law has a very strong consequence.  If you don't have any other charges in the system,  let's say just a plus charge,  then you can't get, and the rule is electric lines only end on charges,  then there's no way to get rid of a charge.  No way to get rid of a charge.  If you try to get rid of the charge,  either the electric field has to suddenly disappear everywhere.  That would violate the rules of the speed of light,  that you can't send a signal faster than the speed of light.  So instead, if you said that a wave moved outward of missing electric field,  that also wouldn't make sense,  not unless there were charges at the end points of the electric field.  So all this would really be,  is this would not be the elimination of the charge,  it would just be taking the charge and sending it out as a shell.  It would be a shell of charge going out.  The charge would not have disappeared.  So it's a consequence, a deep consequence of the structure,  the mathematical structure, Gauss's law,  and other laws that electric charge is conserved.  That is the essence of the gauge theory.  Essence of the gauge theory,  Maxwell-like fields,  I'll use the word Maxwell-like because they're not all truly Maxwell's electromagnetic field.  Maxwell-like fields consisting of electric-like and magnetic-like fields,  if the gauge field is weakly coupled,  and I will tell you what that means later,  but it basically means that if the interactions between the parts of the electromagnetic field  are weak enough that they don't interact with each other seriously,  then the motion of a gauge field is exactly the same as a light wave.  It will have a polarization.  It will move down the axis with the speed of light  unless something else, unless it's something in the dynamics that changes that.  Naively at least, it's in every way similar to the electromagnetic field.  The only difference is, it also has sources.  Every gauge field has sources analogous to the electric charge,  but it may not be the electric charge.  It might be something else.  Let me give you an example which is an unphysical example.  Not because it's inconsistent, but because it just happens not to be true as far as we know.  There is another quantity in nature called the Baryon number.  It's simply the number of quarks, but it's a conserved quantity.  And you could imagine that Baryon number is the source of...  Now, Baryon number is not electric charge.  Why not?  Well, because protons and neutrons have it.  So it is not electric charge, but it could be the source of its own gauge field.  If it were, that would mean that there would be forces between protons, neutrons, protons, protons, neutrons, neutrons  that would be analogous to the Coulomb forces between charges,  except it would have nothing to do with electric charge.  They would have only to do with Baryon number.  So, for example, two objects with the same Baryon number would repel each other.  A proton and a neutron would repel each other because they have the same Baryon number.  A proton and a proton would repel.  On the other hand, proton and the antiproton having opposite Baryon number would attract.  Now, the only rub in this ointment is that there is no such gauge field coupled to the Baryon number.  Baryon number does not come with a field like this that surrounds it and no long-range force associated with it.  So, Baryon number is not an example.  Color is an example and the color forces of quantum chromodynamics also,  oh, well, before I do that, one or two other points, one or two other points.  The other point, of course, is this is a completely classical description of an electromagnetic wave.  The corresponding quantum mechanical description is in terms of quanta,  but of course, if we think of quanta as particles, at least that's about the only mental picture that we have  to describe these discrete little objects, the discrete little objects which drawing them, of course, is always misleading,  but let me draw a photon, there's a photon, a little point.  That little photon has a position, moves with a speed of light,  but also it has a little flag associated with it and that little flag is its polarization.  I won't try to draw a flag on it.  In other words, it has a pointer which either points this way, this way, or somewhere in between  that indicates the polarization of the electromagnetic wave of which it is a quantum.  That's a complicated statement, but I think you get it.  So photons come with polarization and that polarization in principle can be rotated.  You can rotate it by sending it through a quarter wave plate or in any number of ways.  And that's photons.  That's the photon theory of electromagnetism.  And in the quantum theory, one can think of the Coulomb field as, roughly speaking, a field set up by the emission  and absorption of photons, emission and absorption of these photons.  Okay, so that's electrodynamics in a nutshell, electrons are charged, protons are charged,  and so forth, and they interact in this way with the gauge field.  Another aspect of gauge fields, I've already mentioned it, that there's always conservation laws  associated with them, namely conservation of the sources.  But conservation laws in both quantum mechanics and classical mechanics are always associated with symmetries.  Symmetries of some sort or another are always behind conservation laws.  The conservation of electric charge, we've talked about that,  and I showed you how if you study the quantum mechanics of electrons in terms of charged particles in general,  there's a wave field describing those particles, this could be the electron,  and the symmetry associated with the conservation of charge is just the multiplication of the field by a phase.  So that was the simplest example of a gauge theory, a gauge charge, meaning the electric charge,  and a symmetry that goes along with the conservation of that charge.  Those things go together.  Okay, well, let me just very quickly remind you how it worked in quantum chromodynamics.  In quantum chromodynamics, a more complicated structure.  You do have Maxwell-like fields.  Let's label them.  The Maxwell-like fields are labeled by indices.  So in quantum, let's continue to call them A.  There is no universal symbol for the gauge field of quantum chromodynamics.  Sometimes it's called G, sometimes it's called A, sometimes it's called B, sometimes it's called C.  I'm going to call it A, but in order to distinguish it from the electromagnetic field, we give it some indices.  I and J. It's a matrix.  Now, it's a matrix, and that matrix transforms under a group.  The group was Su3.  So let me just remind you very, very quickly that we started thinking about Su3 not by thinking about gluons.  This, of course, would be the gluon field, but by thinking about quarks.  So a quark was an object which we called Q. Q could stand for the quantum field of the quark, and it has an index.  The index, it's not the up-down index, it's not the charm-strange index.  We'll come to that sooner or later.  It is the color index, red, green, or blue.  I takes on three values.  And the symmetry operation is not multiplying by e to the i theta, but multiplying by a special unitary matrix Uij.  It gives us Q prime i.  That was the symmetry operation, which is a kind of rotation in a kind of three-dimensional complex space.  Don't confuse it with ordinary three-dimensional space.  This was the symmetry operation, and one would say that the Qs, think of them as particles, if you like,  the Qs form a representation of Su3, which is called the fundamental representation.  It's called the fundamental representation.  It has three entries, red, green, and blue.  Sometimes it's called the defining representation.  It's the smallest non-trivial representation.  And it just can be thought of as a three-component vector, one, two, three.  And the unitary matrices can be thought of as matrices, three by three matrices.  OK, we've gone through that.  The other thing that I told you is that antiquarks, or antiparticles in general, are represented by the complex conjugate fields,  fields which are simply the complex conjugates of the original ones.  The relation, the mathematical relation between particle and antiparticle,  or the wave functions of particles and antiparticles, that relationship is complex conjugation.  So we could ask, how do antiparticles transform?  Well, we simply realize that if you want to transform the complex conjugate field,  you should use the complex conjugate matrix.  If you multiply q by u to get q prime, then you must multiply q star by u star to get q prime star.  The set of matrices u are called a representation of the group.  The set of matrices u star are a distinct and different representation of the group.  And the language we would use is that quarks and antiquarks are described by the three-dimensional representation.  And we could call it three star, meaning complex conjugate.  It's usually indicated by a bar.  Two different representations of SU3.  So there's seven, we make this.  Nine, actually, well, sorry.  You asked me how many matrices there are?  The number of generators is eight.  Yeah, because these matrices are assumed to be special unitary matrices.  Now you can ask, and it's interesting to ask, what kind of theory would you make if you didn't insist that they'd be special?  And I'll tell you another time, maybe later, but not now.  Okay, now what is Aij?  Aij has two indices.  And the way to think about it is that one of those, it is mathematically got the same group theory structure,  the same symmetry structure as having a quark j and an anti-quark i.  In other words, this is an object whose indices transform one index as the three-dimensional representation and the other index as the three-bar.  Another way to think about it is it has all of the properties with respect to the symmetry, not with respect to all physical properties,  but with respect to the way it transforms.  It transforms as if it were a quark and an anti-quark sitting on top of each other.  Not at spin, just its color, as if it were a quark and an anti-quark, with one special rule, namely the trace of Aij vanishes.  The singlet, the singlet that you can make, the piece which has no transformation property under the group at all, that you can kill and forget about.  That's why there are eight Aijs instead of nine.  So Aij is like a thing with a quark and an anti-quark index.  But because it has a quark and an anti-quark index, it also transforms when you under the SU3 transformation group,  in that sense it's different than the photon.  The photon is completely electrically neutral.  Its field, the photon field, does not get a phase when you rotate the electron wave function electrons.  Go to e to the i theta times electrons, positrons go to e to the minus i theta times positrons, photons do nothing.  That's because they don't have electric charge.  But the gluons, they do have color.  Now what kind of color do they have?  They have the color of a particle and an anti-particle.  Like you say, a particle and an anti-particle shouldn't have any color.  But they do because you could have a red particle and a blue anti-particle.  A red, that would make a thing which was not really neutral, which knew about color.  So gluons do have color and they have color in what is called the adjoint representation,  which is the representation of the 8th generators.  Okay, now, what about the interaction between quarks?  Incidentally, there are many other gauge theories of interest.  They're all pretty similar to each other.  The group might not be SU3, it might be SU2, it might be SU4, it might be SU10, it might be anything.  And the objects of the group might not be called quarks.  Nevertheless, or the fundamental objects of the group, the analogs of quarks might be called something else.  They might have a different name.  But they would be objects which had a single index.  The anti-objects would also have a single index but of the complex conjugate kind.  And the analog of the gluons which are called gauge bosons.  Gauge bosons, the Maxwell-like fields always have one index of the particle type  and another index of the anti-particle type.  So that means, let's see, if we're talking about SUN,  that means how many generators altogether.  Well, N times N is N squared minus one for the trace.  So it's N squared minus one distinct gluons.  Now, what is the connection between the quarks or more generally,  the particles which transform like quarks perhaps in a more general way?  What's the connection between them and the gauge bosons?  And I've explained that to you before, but let me just remind you again.  If you have a quark labeled by I, where I could be red, green or blue,  and another quark labeled by J,  that the I-th quark can become the J-th quark if a gauge boson is emitted.  What kind of gauge boson?  Well, as I showed you before, the way to think about it is just to think about these lines running through the diagram.  I runs right through, so that's I, and J also runs right through,  except when J is pointing downward, it should be thought of as the analog of an antiparticle.  And so an I would become a J by emitting a quantum of the field A, is it J-I or I-J.  No, J-I, doesn't matter very much.  By emitting a gluon of the I-J-th type, an I-th particle becomes a J-th particle,  and that is one of the basic interactions of gauge theories in general.  Fundamental objects, analogous to quarks, gauge bosons, analogous to gluons,  and a rule about emission and absorption that you have to follow the lines and never lose an index.  But now that raises something new.  If a gluon itself has the properties of a quark and an antiquark,  so let's say this is the I-J-th kind of gluon.  Well, the I-J-th kind of gluon can also emit a gluon.  How would it do it?  The I-J-th gluon could become the I-K-th gluon.  I-J, let's say, red-green gluon could become the red-blue gluon,  as long as the indices make sense and they're followed, anti-K here.  So this would be an interaction in which a gluon of type I-J became a gluon of type I-K,  and a gluon of type K-J.  The order of I and K matters.  A K-J is not the same as a JK.  That introduces something absolutely new which is not in electrodynamics,  namely that the gluons themselves are charged particles  and therefore exert forces on each other.  Exert forces in a way that photons do not, exert forces on each other.  The dynamics of gluons is much, much more complicated than the dynamics of photons.  The last time what I told you was that one of the,  and I'm not going to do the mathematics of this, it's not even fully understood it even now,  but the effect, one of the important effects is that if you have a source, a color,  a bit of color, with field lines coming out of it,  again those field lines are not allowed to end, they're not allowed to end,  but the effect, let's suppose that's quark, is an anti-quark over here,  field lines have to come into it, so the field lines have to be continuous and so forth.  I don't want, let me not draw more than that.  Now the effect is that the field lines interact with each other  in a way that they don't in electrodynamics.  And the effect is very simple.  The field lines get pulled into bundles which are called flux tubes,  the result of which is that no matter how far apart you pull two quarks,  they'll never escape from each other because the energy stored in this gooey piece of glue  just increases linearly with the distance between them.  All right, that's a summary of quantum chromodynamics,  but it's also a summary of gauge theory.  It's also a summary, very, very quick summary of gauge theory.  Maxwell-like fields, sources, Gauss's law,  making sure you have conservation and symmetries,  symmetries having to do with the conservation laws,  but more than that the symmetries telling you how the particles interact with each other  in terms of emission and absorption.  There is one other thing in a gauge theory, and it's the coupling constant.  The coupling constant is the analog of the charge of the electron.  The charge of the electron, let's talk about what the charge of the electron means.  It is, of course, the coefficient in a Feynman diagram  when you emit a photon from a charged particle, the amplitude for that,  the quantum mechanical amplitude for that is the electric charge.  But if you wanted to think about it operationally,  you can imagine an electric charge slamming into the cathode or the anode cathode,  the cathode of a cathode ray tube.  When it slams into it, it gets accelerated,  and the question is what is the probability that an electron which is stopped suddenly  emits a photon?  The answer is the square of the electric charge.  The square of the electric charge, there's some pi's in it and h-bars and things,  but basically the square of the electric charge in suitable units  is the probability for emitting a photon.  So that's what in quantum mechanics and field theory,  that is what the meaning of the electric charge is.  It's a measure of the square of the electric charge is the probability.  Why?  Because probabilities are always squares of amplitudes.  But that's the fundamental significance of the electric charge,  and the electric charge is a dimensionless number,  just the probability that when you stop an electron you emit a photon.  So it's a dimensionless number.  It's a small dimensionless number.  With appropriate sets of definitions,  the square of the electric charge,  really what tends to come into things is not quite the square of the electric charge,  it's the square of the electric charge divided by some 4 pi's,  4 pi's are always floating around in there.  It's called the fine structure constant,  and the fine structure constant,  which is pretty much the probability of a photon emission,  is a number of about 1%.  It's close to being 1 over 137, a famous number, 1 over 137.  The 137 has no particular significance.  It's just numerically about what the fine structure constant is.  And let's just call it 1%, it says there's a 1% probability of emitting a photon.  So if you had a picture that when an electron hits a cathode ray tube,  it sends out a spray of photons, that's not right.  What is right is that if 137 photons hit the cathode ray tube,  on the average one of them will give off a photon.  In that sense, electromagnetism is a weak force, or a weak process.  The probability of emission is weak.  All right, that's the quick summary of gauge theory.  What I wanted to do is give you very, very briefly a run down on some numbers  and orders of magnitude in various parts of particle and atomic physics.  And then go on to the weak interactions.  Now the reason I'm giving you a run down about numbers  is to show you how different the weak interactions.  Quantum chromodynamics is the theory of the strong interactions.  The strong interactions was the word before quantum chromodynamics was invented  for the interactions between subnuclear particles, between hadrons,  between protons, neutrons, mesons, and all the things which are made of quarks and gluons.  So it's theory of strong interactions.  And there are electromagnetic interactions and there are also weak interactions.  So I want to spend a little bit of time with numbers to show you  what, where the terminology came from, why one is called weak,  why one is called strong,  why electromagnetism is sort of in the middle between the two of them.  But let's just review a couple of numbers.  First of all, in terms of units, energy, distance, and time  do not have to be thought of as different units.  They're all related with each other.  First of all, time and distance really should not be distinguished  in any sensible set of units that are useful for fundamental physics.  We can set the speed of light equal to one, and if we do,  if we fix the speed of light equal to one, then time and distance have the same units.  And we can use the same units.  As far as energy goes, we could also choose to set Planck's constant equal to one.  That's a useful thing to do in quantum mechanics.  An awful lot of equations get simpler if you set h bar equal to one.  Well, they don't get a lot simpler.  They just get rid of the h bars, but you can set h bar equal to one.  Now, let me just remind you there's a connection between energy,  e equals h frequency, right, or h nu, or h bar omega, all the same thing.  The units of h bar, well, we've set h bar equal to one,  so in units in which h bar are equal to one, energy is equal to frequency,  but what's the units of frequency?  One over time, number of oscillations per second.  So that means that energy and time have inverse units to each other.  You don't need to have a separate unit for time and energy.  Any time you have a unit of energy, it defines also a unit of time.  Now, of course, the unit of energy is inverse to the unit of time,  so big energies correspond to small time intervals.  And, okay, so let's write down some connections.  And since energy and time are connected to each other,  and time and space have the same units when C is equal to one,  then time and space have units which are just inverse units to the units of energy.  So, for example, here's a unit of energy, one electron volt.  We've used electron volts before.  One electron volt cannot be thought of as a unit of distance,  but the inverse, one inverse electron volt,  so it's one over an electron.  One inverse electron volt is a unit of distance,  and just to have an idea of what it is,  it's about 10 to the minus seventh meters.  One electron volt is about 10 to the minus seventh meters.  So an electron volt is a fairly small distance,  but on the scale of fundamental physics, it's not a small distance.  10 to the minus seventh meters is what?  A atom is 10 to the minus tenth meters, something like that.  Yeah, okay, so it's fairly big.  An atomic diameter, yeah, an atomic, and we can convert, right,  and an atomic diameter, one atom, this is the atomic diameter,  that's about 10 to the minus tenth meters,  and so you can convert that to electron volts.  Is it, I don't know, whatever it is,  an inverse of about 1,000 electron volts, a kilo electron volt,  an inverse kilo electron volt is an atomic diameter.  Okay, another fact, just converting distance,  if we know the distance or size of an atom,  then we know the time that it takes light to go across it.  All right, so the transit time, there's another quantity,  the time that it takes transit time,  the time for light to cross an atomic diameter, what is that?  That's about 10 to the minus 18 seconds.  I've used the fact that light goes at about 10 to the eighth meters per second.  Order of magnitude, 10 to the minus 18 seconds,  these are just some numbers that I just wrote down,  but a different time scale.  All right, so this is the time scale for light to go across an atom.  The different times, and by an atom, I don't mean the nucleus,  I mean the whole atom, another time scale is the time scale  that it takes for an electron to orbit an atom.  Now you might think, why isn't that just about the transit time?  Transit time is about 10 to the minus 18 centimeters.  The reason is because electromagnetic forces are fairly weak.  Because they are fairly weak, the force on,  and the ultimate reason that they're weak is because this fine structure constant,  this alpha, which is E squared over 4 pi,  that's a small number of about 1 over 137.  Another way to say it is the electromagnetic force on an electron is weak.  Because of that, the acceleration on the electron is not too big,  and the orbit is fairly large.  The weaker the force on an object, the larger the orbit is going to be.  Sorry, the slower the electron will move is what I meant to say.  The slower the electron will move,  and because the electron moves slowly by comparison with the speed of light,  how much slower?  About 1 percent slower.  About this much slower, the electron moves around with only about 1 percent the speed of light,  and that means the orbital time, orbital time,  that's about 10 to the minus, let's see, it's bigger.  That time is bigger, 10 to the minus 16 seconds.  Another quantity sometimes of interest is the decay time for an atom.  How long does it take for an atom, an excited atom,  let's say a hydrogen atom with its electron,  one orbital up from the ground state,  one orbit up from the ground state, how long does it take to decay?  Now, that is small, doubly small, with two powers of alpha.  What are the two powers of alpha?  First of all, the acceleration of the electron is small.  Remember, charges emit radiation when they're accelerated.  The acceleration is small because the force is small.  So first of all, the acceleration is small.  The electron moves around with a relatively small acceleration  compared to what it could have been if alpha were bigger.  It moves around with a small acceleration,  but on top of that, just like the electron plowing into the cathode ray tube,  there's another factor of alpha when the electron is accelerated  in the probability that it emits a photon.  So as the electron goes around here, first of all,  it's moving with a slow acceleration,  and second of all, even that acceleration is not very efficient  in producing photons just because the fine structure constant is small.  The net result is that the time scale, order of magnitude, decay time,  how long it takes for an atom to decay.  Decay time is about a nanosecond, 10 to the minus 9 seconds,  much longer than the orbital time or the transit time.  The main thing is there's a variety of different scales in the atom,  and they're all related by the fine structure constant.  They're related by powers of the fine structure constant,  the fine structure constant being a small number.  There are fairly large ratios between different time scales,  transit, orbital, and decay time.  All right, now let's come to Hadrons.  Hadrons are like atoms, they're atoms made up out of quarks.  We could ask very much the same kind of questions.  First of all, we could ask what is the Hadronic diameter?  The Hadronic diameter of a typical Hadron,  it could be a proton, neutron, meson,  they're all more or less the same.  Here's Hadrons.  Hadrons, Hadrons.  The Hadronic diameter is a lot smaller than the electron.  Sorry, than the atom.  It's a lot smaller than the atom.  It's not 10 to the minus 10th meters.  It's about five orders of magnitude smaller,  10 to the minus 15 centimeters.  That's also more or less the diameter of a nucleus.  Nucleus, of course, being a few protons and neutrons  could be bigger.  Sorry, meters.  Hadron diameter is about 10 to the minus 15th meters,  five orders of magnitude smaller than an atom.  Part of the reason is because there's two reasons.  There's two reasons why it's smaller.  One of them has to do with the fact that the constituents are  heavier and heavier things will sit closer to the center.  The other has to do with something else that we'll come to in a minute.  But it's about five orders of magnitude smaller.  The transit time has just gotten from the size straightforwardly.  Transit time, this is not an independent thing.  It's just another measure of the size of the thing.  Just to get the number straight, it's 10 to the minus 23rd seconds.  So that's a typical time scale for, well,  it's for light to cross the Hadron.  But now we can also talk about the orbital motion of quarks.  How long does it take for a quark to swing around its wave function  or swing around the interior of a proton?  And there the answer is about 10 to the minus 23rd seconds.  So how long does it take for it to decay?  How long does it take for it to decay?  And the answer is about, now I'm talking about particular kinds of  decays, a decay where you hit a proton, it starts oscillating,  and then emits a pion.  Those decays are the same as the quarks.  They're different than this hierarchy of scales here.  What is the conclusion?  The conclusion is the analog of the fine structure constant is  much bigger, close to 1.  So what is the conclusion?  The conclusion is the analog of the fine structure constant is  much bigger, close to 1.  For Hadrons, in other words, the emission probability,  if a quark were to get stopped in a fictitious cathode ray,  quarky cathode ray tube of quark flying off,  the probability to emit a gluon would be about 1.  That's what these things indicate here.  The hierarchy here is entirely due to the fine structure constant.  The lack of a hierarchy here can only mean that the corresponding  quantity in quantum chromodynamics must be close to 1.  By now it's been measured in many ways, and it is much closer to 1.  It's much larger than the corresponding quantity for  electrodynamics.  One is a little bit, no.  It would seem that just special relativity to say that the  case that could be much less than 10 to the minus 23,  that's a kind of case like, go for 1.  Yeah, I think that's a fair statement.  It's about as fast as it could be, in other words.  It's about as fast as it could be.  Nothing's slowing it down.  This is why the strong interactions are called strong.  Now, the facts about the numbers here were known before  quantum chromodynamics, and it was understood that there was  no hierarchy of scales, and that's why it was called  strong interactions.  We now trace it to the properties of the charge or the  fine structure constant.  Incidentally, for quantum chromodynamics, the analogous  quantity is not really one.  It's about a fifth, but much bigger than the 1 over 137 here.  All right, so that was just some numbers and some facts about  the  strong force.  It's called alpha QCD.  So this is called alpha, and the other one is called  alpha QCD, and it's about 0.23 or something like that.  Or in context of paper, you write alpha QCD.  I've never heard alpha QCD referred to as alpha QCD,  but I think if you were writing a paper with both of them,  you might call the top one alpha QCD, yes.  I've never seen it referred to that, right?  Well, particularly if you're talking about shielding or  something like that.  No, I can't, I don't think I've ever seen it.  Oh, oh, oh.  It's just called alpha.  Sometimes the unshielded one is called the bare alpha.  The shielded one is called the renormalized alpha, right?  Okay.  Well, that brings us to the weak interactions.  The weak interactions are called weak.  Primarily because the decay times associated with them are  very long.  They're much longer than the decay times associated with  either hadrons, and they even tend to be longer than the  decay times associated with atoms.  So there are processes in nature, decay processes in nature,  in which particles, elementary or otherwise,  decay by weak interactions where the time scale,  in other words, the half-life of these decays is far longer  than any scale that can be accounted for by these,  by numbers like this.  So as an example, here's some examples.  Well, the lifetime of the neutron.  The neutron is a hadron.  The decay is neutron goes to electron, proton,  plus anti-nutrino.  And the lifetime for that is about 12 minutes.  That's absurdly long on particle physics scales.  Part of the reason is easy, but it hardly accounts for the  extraordinary stability of the neutron.  Part of the reason is the neutron is only a little bit  bigger than the combined masses of the electron,  the proton, and the neutron, and the neutrino.  Let's just see.  Somebody have out their pocket?  The mass of the neutron is 940 MeV.  About 940.  So the proton is about 939.  I think the actual difference between them is about  one and a half MeV.  But then there's the electron which has a mass of about  half an MeV.  All together out of this 940, almost a thousand MeV,  the difference of mass of the proton,  the amount of available mass left over is tiny.  It's 0.1% or less.  If the neutron was slightly lighter than the sum of these  masses, it could not decay at all.  Just energy conservation would not allow it.  Remember, mass is energy.  It would not have enough energy to decay and still leave  over some energy for kinetic energy of these particles.  So if the neutron was exactly the same as the sum of the  energy of the electron, the proton, and the neutrino,  the sum of the mass of them, it wouldn't be able to decay  at all.  If you gave it a mass, tiny, tiny, tiny, tiny bit above that,  it would be able to decay, but it would take a very long time.  The decay would be very slow.  So part of the slowness or the long lifetime of the neutron  is attributable to the fact that there's very little energy  available for the decay to take place.  But given that, that's hardly the bulk of the story.  Twelve minutes is so absurdly long that there's something  much, much more than that going on.  Another example, let's see, where do I have it?  I guess I didn't write it down.  Another example of a, this is called a weak decay.  For obvious reasons, it's called a weak decay.  Another example is a charged pion, let's say a pime minus.  A charged pion decays to an electron.  Electron has negative charge and an anti-nutrino.  Yes, and an anti-nutrino.  That's another possible decay.  A charged pion goes to an electron and an anti-nutrino.  A positively charged pion could become a positron and a neutrino.  The lifetime for that, I believe, is about 10 nanoseconds.  Longer than these atomic decay times.  And atomic decay times are very, very long by comparison  with particle physics decay times.  Now, there is a decay of the charged pions to these particles.  There's another decay involving muons.  We're going to come to muons soon enough, similar to electrons.  But the point is not the particles that they decay into, but two things.  First of all, they also decay very slowly, nanoseconds.  But there's plenty of energy available for these decays.  These decays are not suppressed by the fact that the energies of the initial  and all the masses of the initial and final states are very close to each other.  So there's something else going on in weak interactions.  Whatever weak interactions are, there's something that makes them  proceed very, very slowly and weakly.  So that's what we come to next is weak interactions.  The primary or oldest example of which is the decay of the neutron.  The beta decay of the neutron called beta because electrons  are originally called beta waves.  And this is called the beta decay of the pion.  In fact, it's quite similar.  Quite similar physics goes into both of them.  So to understand what's going on, we need to understand about some more particles.  Particles that we haven't come to yet, the leptons.  Let's go back to the quarks for a moment and let me just make a table  of the different kinds of quarks.  Horizontally, or make a sequence here which will be red,  blue and green.  These are the colors of quarks.  Then along here, let me list the up quark, the down quark,  the charmed quark, the strange quark, the top quark, and the bottom quark.  Each one of these boxes is filled.  There are red, blue, and green up quarks.  Let's put an x there.  There's an x here.  There's an x here.  There's an x here.  X here, x here, and so forth.  The boxes represent all the possible quarks that exist.  Now, quantum chromodynamics has to do with symmetries which connect red to green to blue.  It has to do with these unitary transformations which mix up the colors of quarks.  They do not, those symmetries do not mix horizontally.  An SU2-SU3 symmetry, a color symmetry, may rotate an up red quark into an up blue quark  or an up green quark.  And the nature of the symmetry is to act vertically, to mix up things this way.  And also at the same time, we'll mix the different down quarks, the different colors of down quarks.  The same symmetry will mix the different colors of charmed quarks and strange quarks and so forth.  So the color symmetry acts vertically.  It mixes up the different rows here.  Is that the unitary matrix?  Yes, the unitary matrices mix up red, green, and blue.  I might have done better to put the rows where the columns are and the columns where the rows are,  but no, it's okay.  It's good.  No, no, it's good.  It's good.  Right.  The weak interactions are associated with symmetries which act horizontally in this picture.  They mix up with down.  They mix, and at the same time that they mix up with down, they mix charmed with strange.  And they mix top with bottom.  Let me just remind you that an up quark has the same properties apart for its mass.  Same properties as a charmed quark or a top quark.  The down quarks are similar to strange quarks and bottom quarks.  Ups have charged two thirds.  Downs have charged minus two thirds.  Same here and here.  So these are symmetries which mix, if you like, upness with downness.  At the same time, charm this with strangeness and topness with bottomness.  It mixes these up.  The symmetries, those symmetries, what would you care to speculate on what group might be involved?  SU2, because it acts on things, on doublets.  We simply have three doublets here.  They don't take up to charm or up to top.  They simply act on up and down horizontally among pairs of things.  A group SU2 and it is a gauge symmetry.  In other words, it also comes together with forces, with gauge bosons, with interactions,  and with gauge field.  We're going to get to those short enough.  But we've left out something from this table here.  It's kind of as if there was a fourth color.  Now the reason, the fourth row here is not usually identified as a color.  And the reason is because the particles do not interact with the gluons.  But nevertheless, it is another row here, another row in which there are particles which are filled in here.  And in a certain sense, they are also doublets, these particles are also doublets,  and also get mixed up under this SU2 symmetry which moves things horizontally.  What are they?  What are those particles?  They're the leptons.  So in a sense, the fourth color could be thought of as lepton number.  But what are the leptons?  The leptons are, first of all, in the first column here, there are analogous to the up,  or analogous to the down, there is the electron.  The electron is a lepton.  But its partner, its partner is the neutrino.  But there are different neutrinos.  There's not only one neutrino.  This is called the electron neutrino.  What comes in the next column?  The muon.  The muon, in every respect except for mass, is the same as an electron.  Just as in every respect except for mass, the charmed quark is like an up quark,  or the strange quark is like a down quark.  And together with it, there's its own neutrino, the muon neutrino.  And finally, the last one is called the tau and nu sub tau.  All of the charged leptons, incidentally, neutrinos, of course, as you might guess from their name,  are electrically neutral.  The electrons, muons, and tau's have charged minus one.  They are put under the down column here, not because their charge is the same as the down charge,  but because the difference between the charges of the two is the same as the difference up here.  What's the difference between the charge of an up and a down?  Two-thirds minus minus one-third.  So the difference between this column and this column is plus one unit of charge.  The difference between this column and this column is also one unit of charge.  One, sorry, zero minus negative one.  So as you go from here to here, you decrease charge by one unit as you go horizontally.  Now, should we take a break for five minutes? Let's take a break for five minutes.  And then I will tell you about the W bosons.  The W bosons are the gauge bosons of the weak interactions.  All right, so let's try now to see if we can make a microscopic gauge theory of the weak interactions.  As I said, the weak interactions are based on a gauge theory which mixes up to down,  neutrino to electron and so forth.  And so rather than to get into mathematics for the moment, the mathematics of the group SU2,  which we'll come back to, let's think a little more simply.  We've seen that in various situations the gauge bosons typically have the quantum numbers  or the quantum properties of a particle and an antiparticle.  In the case of the quantum-chromodynamic interactions, it is a quark and an antiquark.  That's why we had nine gauge bosons and so forth or eight removing the trace.  So we could start by saying let's look at the possibilities of gauge bosons which have the properties of a particle and an antiparticle.  Let's not distinguish between whether we're talking about quarks, red quarks, blue quarks, green quarks or leptons.  Let's just say a particle from this column with an antiparticle of this column.  And for simplicity, let's just focus on the leptons.  You'll see in a moment that you'll get the same thing if you focused on the quarks.  But let's focus on the leptons and talk about objects that we could make with a lepton and an antilepton.  First of all, we could have some simple things, electron with a positron.  Now I don't really literally mean electrons and positrons.  We could have electrons and positrons.  Electrons and positrons have no charge at all.  So let's drop that for a minute.  We'll come back to it.  There are things which have properties of electrons and antilelectrons, neutrinos and antinutrinos.  But more interesting for the moment is the combination of electron, antinutrino.  So what is the charge of an object which has the same conserved quantities,  the same properties as an electron with a stuck on top of an antinutrino?  Well, let's just focus on its charge.  It's electric charge.  Its electric charge is minus one.  Neutrino is neutral.  Electron has negative charge.  So whatever this object is, it has negative electric charge.  Let's give it a name and let's call it a gauge boson.  Let's assume that it is also a gauge boson.  Gauge boson means that it behaves like the Maxwell field, the whole same thing.  But it has negative charge.  Let's call it W.  W, that's the traditional name for it.  W minus.  Why minus?  Because it has negative electric charge.  We could, that was this one with this one, we could have taken down quark with, I guess, anti-up quark.  What happens if you take down quark with anti-up quark?  How much charge do you get?  Again, minus.  Same as this.  Down quark with anti-up quark would also have minus charge.  So if you like, you could also imagine that the W minus has the same properties as a down quark and an anti-up quark.  In other words, it's one from the second column and anti-one from the first column.  That's the property that it has.  Let's call it W for weak, in fact.  I think W was originally the notation for weak.  The negatively charged W boson.  And of course, there's also a positively charged boson, W boson.  Which is like anti-down with an up or like positron E plus with a neutrino.  E plus with a neutrino would be a W plus.  Let's suppose the pattern is pretty similar to quantum chromodynamics,  that the W bosons are like either photons on the one hand or like gluons.  Which means that they would be objects which could be emitted from the leptons.  They would be objects which would allow a lepton of one kind.  Let me not label it.  A lepton of one kind to become a lepton of another kind.  By emitting a W boson.  Let's see if we can find some examples.  An electron can emit a W minus to become a neutrino.  An electron becomes neutrino by emission of a W minus.  You see the action of the gauge bosons is to do the same kind of transition  that the symmetry is ultimately going to be imagined to do to mix electron into neutrino  in very much the same way that the colors got mixed by emitting gluons.  You could follow the lines if you like and say the electron and the anti-neutrino over here  can be imagined to be a W minus boson.  Same pattern.  Let's just draw it by saying a W minus is emitted.  But that's not all you can do with a W minus.  You can also have a down quark go to an up quark.  With a W minus.  That also works.  In fact, the W minus is an object which can be emitted in a transition from,  let's see, from the right column to the left column.  A red down quark can become a red up quark by emitting a W.  A blue down quark can become a blue up quark by emitting a W.  Likewise for green.  But you can go further.  A strange quark, not the strange quark, the strange quark can become a charmed quark.  That's another one.  Strange quark, charmed quark, again, W minus.  So this new symmetry, this idea of a new symmetry which acts horizontally  and only horizontally between neighboring pairs here introduces the possibility  of gauge bosons which cause transitions between electron neutrino, down quark and up quark,  strange quark and charmed quark.  Now a W plus and a W minus are antiparticles of each other.  The meaning of that is that you can flip these diagrams,  put the W minus down, W here, but when you flip it, it becomes a W plus.  So a W plus can be absorbed by an electron, plus a unit of charge,  minus a unit of charge to become a neutrino.  So a whole family of different processes are codified by this one vertex here.  We can turn them upside down, neutrino goes to, you know, what we're allowed to do.  We've done these things before.  And that's the basic new element of the weak interactions.  Actually, that's all, yeah.  Well, it's not clear to me why you have W minuses in all three of these cases.  There's a lot of charge conservation.  It couldn't be W plus.  Would you accept I don't know what a W means?  W is the name of a particle.  It's the name of another gauge.  Yeah, but the charge conservation tells me it's going to be a minus.  But I haven't seen any reason why it should be the same particle.  Well, okay, that's right.  But what it comes down to in the end is an empirical fact.  It's the empirical fact that the symmetry of the weak interactions acts simultaneously on all of these doublets here.  When an up goes to a down, a charm has to go to a strange, a top has to go to a bottom,  and the leptons have to mix with each other simultaneously.  The implication of that, the implication of that from a practical point of view,  is simply it's the same symmetry which acts, it's the same gauge bosons which are there.  You could have imagined a theory in which it's quite mathematically consistent  to imagine a theory where different W-type bosons would have been emitted by up quarks,  charm quarks, or tau quarks, or even leptons.  Right?  It's a fact of nature.  The fact of nature that nature is simpler than it might have been  that the same gauge bosons cause transitions between any point,  you know what I mean, I'm going to get tongue-tied if I try to say it.  Okay, well first of all, what processes can we explain?  Good question. Can a W-beg into an electron and a negative neutrino as well?  Negative, so say it again.  Can you flip the arrows?  Yes, yes, yes, yes. You can flip arrows and you can flip lines.  Yes, yes, certainly you can flip the arrows.  Yeah, absolutely. Absolutely.  And you can also flip lines up and down in various ways.  For example, another way you could flip the line is you could take this up quark here,  instead of going out, instead of an up quark going out, an anti-up quark comes in.  Whenever you flip a line from past to future, or from future to past,  you'll also change from particle to antiparticle.  I'm not sure, but on the right-most side, W-stereo is a strange antitron.  Yes.  So that's the same as a down antiparticle?  Yes.  The same thing?  Wait a second.  Sorry.  Yes.  That's a strange antitron, right?  Yes.  That's the same as a down antiparticle.  It's not literally a strange antitron, it's just a rule that tells you  that whatever these W bosons are, they're associated with transitions  from one column to the other.  The triton's a charm.  Did I?  Sorry.  Strange.  Right.  Electron to neutrino.  Strange to charm.  Is that wrong?  No.  No.  It's not an anti-charm.  That's the point.  Who said it was an anti-charm?  Well, it shouldn't have been.  It's different from coming in from the past.  Right.  If you follow the lines like that, at the point where the line turns around,  it has to be thought of as an antiparticle.  Right.  Okay.  Or if you like, you can always think of it as strange anti-charm coming in  and making a W boson.  Okay.  Just keep in mind, whenever you flip from the past to the future,  if you want to know what a W boson will decay to, let's read it downward,  it will decay to a strange and an anti-charm.  And that really can happen.  Yeah.  So can you create W boson from like electron and a...  Anti-neutrino.  Anti-neutrino.  Yes.  And then call the W boson to change into like a strange and a charm.  Exactly.  That's where some of these processes are going to come from now.  So let's think about the various processes.  Let's start just for fun.  Let's start with the decay of the pi minus.  What is a pi minus?  A pi minus is a quark and an anti-quark with altogether negative charge.  So let's say a quark and an anti-quark with negative charge had better be...  Here's the pi minus.  That's equal to a down quark and an anti-up quark.  Down and anti-up.  Now, what can a down and an anti-up do?  Let's see if we can find down and anti-up here.  Here it is right here.  Down and anti-up can become a W minus.  So it's a possible transition that is allowed by the physics of W bosons.  So here's a W minus.  But now a W minus can do something else.  A W minus here is...  Let's say here is a W plus.  Let's put W plus upstairs here.  This is W minus now.  And let's put the neutrino downstairs, anti-neutrino.  And read this as saying that W minus goes to what?  Electron.  Anti-neutrino.  So W minus can go up to electron, anti-neutrino.  And here it is.  E minus, ordinary electron.  So that's the basic underlying process that governs the beta decay of the negatively charged pion into a electron and an anti-neutrino.  Now, which anti-neutrino is it?  There are three distinct anti-neutrinos.  There's an electron anti-neutrino over here.  So it's the electron anti-neutrino.  But it can also...  I didn't draw all the various possibilities,  but any place you saw an electron, you can put a muon.  Any place...  And as long as you carry along the muon neutrino.  So another process is muon.  Anti-muon neutrino.  And just for clarity, this is not strictly a Feynman diagram, right?  This is just showing an interaction.  It's pretty close to a Feynman diagram.  Except for the time direction.  Oh, yeah.  Right.  I don't know why.  All of a sudden I change directions.  Yeah.  For some reason.  Yeah, right.  Time is now going horizontally.  I've become a computer scientist and they measure...  Yeah, yeah, they do.  Right.  Yeah, it's pretty much...  Yeah, that's good.  We can turn it up.  I can't do that easily, but yeah.  Yeah, that is essentially a Feynman diagram describing the decay of the pi minus to the muon.  As it happens, the...  For technical reasons having to do with some complications and Feynman diagrams that are not interesting to us.  As it happens, the primary decay...  Well, primary, I mean the most probable.  The most probable decay of the pi minus is to the muon anti-neutrino, not to the electron anti-neutrino.  It's a technical point of no special interest to us.  It just happens to be that way.  Wouldn't it be consistent instead of considering the Feynman diagrams with a time deal  is more like a kinopaly equilibrium deal?  Which way you're going?  Well, both things can happen.  Both things can happen.  We're going to have a real equilibrium.  We're in empty space.  We're in empty space except somebody made a pi minus.  A pi minus was created how?  A pi minus might be created by some nucleons coming together and creating by strong interactions.  By strong interactions, a pi minus might be created.  Now that pi minus is just going along and saying,  wait a minute, I can decay if I like into a neutrino and a muon.  Then of course, the neutrino and the muon have enough kinetic energy to fly off  and they won't come back together again because they've departed.  They've gone away.  Okay, on the other hand, it is true that if you were to take a muon and an anti-neutrino  and fire them toward each other, there would be a certain probability that they would make a pi minus.  So yes, you can read it both ways.  Let's go.  A quick question. Like in one dimension, you have an SU3 and the other an SU2.  Is it possible to combine them into an SU6 or something?  SU5.  Well, how great.  SU5. Not two times three, two plus three.  SU5.  Tonight, we want to get to SU2. I'll be happy if we get the basic ideas of SU2.  Now, keep in mind, we're doing something a little bit funny because we have,  SU2 has how many generators?  Three.  So how many gauge balls should it have?  Three.  So far, we only have w plus and w minus.  So let's just be another one looking around. And that other one must be associated somehow with something like  electron, anti-electron and neutrino, anti-neutrino, but we'll come to it.  Just like the, yeah, we'll come to it.  There is a third gauge boson looking around.  And it does play a big role in the weak interactions,  but it didn't play a big role historically in the weak interactions just because for reasons we'll come to.  Historically, it was the w boson, which was the first to be conjectured.  Speaking of historically, was the weaker the strong force sort of suspected first?  Of course. All right.  So let's talk about the history.  The history of these forces goes back to Becquerel, goes back to radioactivity.  Alpha, beta and gamma decay.  Alpha decay was what?  A mission of a two protons and two neutrons from a nucleus.  That's a strong interaction.  It's part of the theory of Hadron's and nuclei.  Gamma decay, that was this one right here, weak interactions, and gamma decay?  Photon.  Historically, they happened at the same time in the same experiment.  Question?  The energy conservation, how it works there?  You're assuming some kinetic energy in the world?  Yeah, the energy, let's work on it.  You never worry about the energy conservation in intermediate states like this.  If the energy of the intermediate state doesn't match the energies of the initial and final state,  it just means that the intermediate state can only last for a very short time.  There's an energy time uncertainty principle.  You can violate the energies in a Feynman diagram for short periods of time.  You don't have to worry about conservation of energy between here and here.  The only energy we have to worry about is the initial energy and the final energy.  Okay, so let's worry about it.  A pion has an energy of about a mass.  Let's suppose the pion is at rest.  The mass of the pion is roughly about 140 MeV.  The energy of the muon and the neutrino, well first of all it's the rest energy of the muon and the neutrino.  And the rest energy, the rest mass of the neutrino, sorry, the mass of the muon, I think is about 100 MeV,  I forget exactly, and the mass of a neutrino is for practical purposes zero.  It's too small to be important to you.  So there's more energy to start with than the energy in the masses of the two final particles.  Does that mean it can't happen?  No, it just means the two particles will go off and carry some kinetic energy.  So the difference between the energy on this side and the sums of the masses on this side is just the kinetic energy of the outgoing particles.  That's the energy balance.  The momentum also has to be conserved, and that means if the pion was at rest to begin with,  the momentum of the muon and the momentum of the neutrino have to be equal and opposite,  and they'll go out back to back, so to speak.  If you analyze the neutron decay in the same way as you would do that.  We're going to do that now.  Yeah, let's do the neutron decay right now, see how the neutron works.  Oh, let's do something simpler first.  Let's do the muon decay.  The muon is also something that can decay.  So let's take the muon.  Here's the muon moving along.  Now what can the muon do?  The muon can become a neutrino sub muon and emit a W.  What kind of W?  W minus, right?  It can become a W minus and a neutrino.  But now one of the things that W minus can do is it can become an electron and an anti-neutrino.  So this W minus can now become an electron and an anti-neutrino of the electron variety.  Can this happen?  The only thing that would, yes it can happen.  Not only can it happen, this is the way the muon decays.  The muon decays because it is sufficiently much heavier than the electron,  about 200 times heavier than the electron.  The neutrinos are practically massless.  So the energy balance is you have about 100 MeV here,  a half an MeV here and the risk goes off in kinetic energy.  So that's an example of what's called a purely leptonic process.  No quarks in it at all.  Is it an anti-neutron?  Is it what?  An anti-neutron.  Yeah, it is.  So it's one neutrino, one anti-neutrino and an electron.  Okay, so that happens.  And that's the primary decay, it's the only decay of the muon.  So the muon is an unstable particle because of this.  But the reason the muon is unstable and not the electron is just that the muon happens to be heavy enough to be able to decay to the electron.  The electron is not heavy enough to decay to the muon.  So in a sense it's an accident of the masses.  The same is true of the tau, incidentally.  The tau can also decay to other leptons.  But all right, that's the mu decay.  And notice the components are the same.  The basic components are the same.  A transition from one column to another,  followed by a W,  and then the W makes another transition from one column to the other,  or if you like, a decay from one column to the other.  All right, what about the decay of the neutron?  Decay of the neutron.  Would it be consistent to say the purpose of a linear accelerator,  per se, is to effectively increase the mass of the electron so that it can now go the other direction?  Absolutely.  No.  You don't change an answer, just put nothing in it.  Absolutely.  That's exactly right.  Okay, we're going to do neutron decay.  So the neutron is, maybe I better switch to vertical.  I don't want to, vertical, vertical notation, back to vertical.  Neutron is three quarks, two down quarks, and an up quark.  Right?  Minus four thirds plus two thirds, sorry, minus two thirds, yes, that's a neutron,  electrically neutral, neutron coming in.  Okay, now what is possible?  First of all, we can have the up quark become a down quark,  and emit some sort of W, but that'll give us three down quarks.  Three down quarks is a fine thing.  There's nothing wrong with three down quarks, except that the mass of three down quarks,  of a particle with three down quarks, is always too heavy for the neutron to decay.  So that's not what happens.  What about one of the down quarks?  The down quark can emit a W, become an up quark.  This is again a W minus.  The down quark emits a W minus, becomes an up quark,  and now we have two up quarks and a down quark, up, up, down, what is that called?  That's a proton.  And the W does what the W always does.  It decays to an electron and an anti-neutrino.  So that's the beta decay of the neutron.  Could it decay into a muon and a neutrino?  Not enough energy.  The muon is too heavy so that there wouldn't be enough energy left over for the muon.  But other than that, if it were not for that, it could happen.  Another way to say it is if you could jam some more energy into the muon,  you could add extra energy into the neutron.  How can you do it?  You hit it, you give it a shove.  You give the neutron a whack, you excite it,  and if you excite it enough, it can decay to a proton, a neutrino, anti-neutrino, and a muon.  But without somehow providing that extra energy,  the neutron will only decay to the proton, electron, and the anti-neutrino.  Okay, now let's come, let's see, where are we?  I think we'll finish for tonight.  But remember, we have a puzzle, and the puzzle is why is it that the weak interactions give such slow decay rates?  And I can think of one reason, namely, the coupling constant or the analog of the fine structure constant  could be ridiculously small so that the probability of emitting these W bosons would be very, very small.  That is not the reason.  The fine structure constant for the weak interactions is about the same as for electromagnetism.  That is not the reason.  The reason is something else, and we'll come to the something else next time.  When you say that there are these symmetries, you mean that the Lagrangian is invariant?  Lagrangian, the Lagrangian is invariant under those operations.  That's exactly what it means.  I didn't want to start writing bunches of Lagrangians, so I've used the shorthand of saying there's a symmetry, right?  But that's what it means, yeah.  For more, please visit us at stanford.edu.
Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_0_2207.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_2208_2303.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_2304_2399.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_2400_2603.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_2604_4643.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_4644_5243.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_5244_6131.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_6132_10559.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_10560_11411.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_11412_11735.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_11736_21707.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_21708_21899.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_21900_22199.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_22200_22559.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_22560_23075.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_23076_27671.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_27672_27755.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_27756_28259.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_28260_29159.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_29160_29303.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_29304_29987.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_29988_30215.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_30216_33575.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_33576_33863.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_33864_37415.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_37416_37727.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_37728_38027.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_38028_38579.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_38580_38891.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_38892_39911.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_39912_40379.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_40380_40919.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_40920_41303.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_41304_41435.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_41436_43691.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_43692_43943.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_43944_44711.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_44712_52715.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_52716_55979.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_55980_56135.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_56136_56339.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_56340_57227.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_57228_58523.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_58524_59819.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_59820_61667.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_61668_62303.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_62304_62351.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_62352_62987.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_62988_63755.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_63756_65423.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_65424_67607.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_67608_67931.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_67932_73535.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_73536_77243.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_77244_78215.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_78216_78311.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_78312_78839.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_78840_80735.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_80736_80831.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_80832_80879.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_80880_80939.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_80940_81791.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_81792_82583.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_82584_83207.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_83208_83639.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_83640_84815.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_84816_86519.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_86520_87551.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_87552_88403.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_88404_90323.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_90324_91523.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_91524_92735.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_92736_94475.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_94476_100379.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_100380_104195.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_104196_108107.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_108108_108959.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_108960_112907.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_112908_115607.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_115608_117503.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_117504_118019.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_118020_120563.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_120564_121295.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_121296_123191.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_123192_127355.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_127356_129935.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_129936_131855.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_131856_132935.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_132936_136091.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_136092_136139.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_136140_137051.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_137052_140591.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_140592_140855.png Lecture_5___New_Revolutions_in_Particle_Physics__Standard_Model__480p__flash9_140856_142028.png
Stanford University. All right. We've really studied two different theories a little bit. Quantum electrodynamics and quantum chromodynamics, both of them are gauge theories. In fact, just about all of nature as we know it is one way or another is controlled by gauge theories of different kinds. And so I ought to tell you what a gauge theory is. Now, I'm not going to get into any depth what gauge symmetry is. I'm going to tell you what a gauge theory is in more or less the minimal mathematical way. First and simplest gauge theory is Maxwell's theory. Maxwell's theory of light or Maxwell's theory of electromagnetism, let's not worry about what the word gauge means. We may go through it later, but not now. But what it is, is a theory of fields which have all of the properties of the electromagnetic field. The electromagnetic field has six components, three components of electric field, three components of magnetic field. You can represent those six components in terms of a four vector called the vector potential. I'll just write it down for you. I'm not going to play any big role today anyway. But just to tell you, there's a four vector that's generally called A, and it has an index mu. It has four components, A0, that's the time component of it, and the three space components A1, 2, and 3, or A, X, Y, and Z. Let's not write them out in detail. So it contains a three vector and a fourth component, A0. The significance of A0 is that it's the electrostatic potential. It's the thing whose gradient is the electric force. So for example, the electric force on a charged particle is just gotten by multiplying the electric charge by the gradient or the derivative fx, derivative of A0 with respect to x. So it's just the electrostatic potential that's measured in volts, in particle physics of course it's not measured in volts, volts per unit charge, excuse me, in whatever way you measure electrostatic potential. That's the meaning of A0. Notice first of all, oh, and of course this is the electric field. This is the electric field. The components, derivative with respect to x, y, or z, so it's the gradient of A, and the gradient of A is just called the electric field. So the electric field is determined by the gradient of A0, and the magnetic field is determined by also derivatives, also derivatives of the vector potential, and in fact the magnetic field is the curl of the vector potential. So one of two ways you can study electrodynamics, either in terms of the electric and magnetic fields, of which there are six, or in terms of the vector potential of which there are four, but there are equivalent descriptions of electrodynamics, and we won't go into Maxwell's equations, but let me just remind you that Maxwell's equations give rise to electromagnetic waves. An electromagnetic wave is a wave of electric and magnetic field, and a typical electric magnetic wave has a direction and space that it moves, let's just represent it by an arrow, and it has a polarization. To understand what the polarization means, you'll have to think about the electric and magnetic fields, and the electric and magnetic fields oscillate as you go down the wave. Here's the electric field, for example. The whole thing moves with the speed of light, but at any given instant the electric field might look like that, and the magnetic field is always perpendicular to the electric field. Let's try to draw it, see if I can give it a drawing. I'm imagining the electric field is in the horizontal plane, and it's always perpendicular to the, let's see, is this going to... it's always perpendicular, like so. In appropriate units, the electric and magnetic field are equal to each other and perpendicular. This would be a plane polarized electromagnetic wave. In a plane polarized electromagnetic wave, the direction of polarization, which of course is always perpendicular to the direction of motion, the direction of polarization is determined by the electric field. So the electric field determines the direction of polarization, and that's electromagnetism in a lot of electromagnetic waves in a nutshell. The other thing, of course, about electromagnetic waves is that there are sources. The sources are charges and currents, but let's particularly focus on the sources of the electric field. The sources of the electric field are electric charge, and to just draw a picture, to put it at the level of pictures, every electric charge creates an electric field. The electric field lines never end except out of the charges. A positive convention, the convention is a positive electric charge, puts out an electric field, which is radially outward, falls off as 1 over r squared. You can imagine that the number of electric field lines coming out of a charge is fixed, proportional to the charge, and therefore the number of electric field lines passing through any sphere is the same no matter how far the sphere is. To say that mathematically, the integral of the electric field over the sphere is the same wherever you go, however far out you go. That's Gauss's law. That's Gauss's law, and that Gauss's law has a very strong consequence. If you don't have any other charges in the system, let's say just a plus charge, then you can't get, and the rule is electric lines only end on charges, then there's no way to get rid of a charge. No way to get rid of a charge. If you try to get rid of the charge, either the electric field has to suddenly disappear everywhere. That would violate the rules of the speed of light, that you can't send a signal faster than the speed of light. So instead, if you said that a wave moved outward of missing electric field, that also wouldn't make sense, not unless there were charges at the end points of the electric field. So all this would really be, is this would not be the elimination of the charge, it would just be taking the charge and sending it out as a shell. It would be a shell of charge going out. The charge would not have disappeared. So it's a consequence, a deep consequence of the structure, the mathematical structure, Gauss's law, and other laws that electric charge is conserved. That is the essence of the gauge theory. Essence of the gauge theory, Maxwell-like fields, I'll use the word Maxwell-like because they're not all truly Maxwell's electromagnetic field. Maxwell-like fields consisting of electric-like and magnetic-like fields, if the gauge field is weakly coupled, and I will tell you what that means later, but it basically means that if the interactions between the parts of the electromagnetic field are weak enough that they don't interact with each other seriously, then the motion of a gauge field is exactly the same as a light wave. It will have a polarization. It will move down the axis with the speed of light unless something else, unless it's something in the dynamics that changes that. Naively at least, it's in every way similar to the electromagnetic field. The only difference is, it also has sources. Every gauge field has sources analogous to the electric charge, but it may not be the electric charge. It might be something else. Let me give you an example which is an unphysical example. Not because it's inconsistent, but because it just happens not to be true as far as we know. There is another quantity in nature called the Baryon number. It's simply the number of quarks, but it's a conserved quantity. And you could imagine that Baryon number is the source of... Now, Baryon number is not electric charge. Why not? Well, because protons and neutrons have it. So it is not electric charge, but it could be the source of its own gauge field. If it were, that would mean that there would be forces between protons, neutrons, protons, protons, neutrons, neutrons that would be analogous to the Coulomb forces between charges, except it would have nothing to do with electric charge. They would have only to do with Baryon number. So, for example, two objects with the same Baryon number would repel each other. A proton and a neutron would repel each other because they have the same Baryon number. A proton and a proton would repel. On the other hand, proton and the antiproton having opposite Baryon number would attract. Now, the only rub in this ointment is that there is no such gauge field coupled to the Baryon number. Baryon number does not come with a field like this that surrounds it and no long-range force associated with it. So, Baryon number is not an example. Color is an example and the color forces of quantum chromodynamics also, oh, well, before I do that, one or two other points, one or two other points. The other point, of course, is this is a completely classical description of an electromagnetic wave. The corresponding quantum mechanical description is in terms of quanta, but of course, if we think of quanta as particles, at least that's about the only mental picture that we have to describe these discrete little objects, the discrete little objects which drawing them, of course, is always misleading, but let me draw a photon, there's a photon, a little point. That little photon has a position, moves with a speed of light, but also it has a little flag associated with it and that little flag is its polarization. I won't try to draw a flag on it. In other words, it has a pointer which either points this way, this way, or somewhere in between that indicates the polarization of the electromagnetic wave of which it is a quantum. That's a complicated statement, but I think you get it. So photons come with polarization and that polarization in principle can be rotated. You can rotate it by sending it through a quarter wave plate or in any number of ways. And that's photons. That's the photon theory of electromagnetism. And in the quantum theory, one can think of the Coulomb field as, roughly speaking, a field set up by the emission and absorption of photons, emission and absorption of these photons. Okay, so that's electrodynamics in a nutshell, electrons are charged, protons are charged, and so forth, and they interact in this way with the gauge field. Another aspect of gauge fields, I've already mentioned it, that there's always conservation laws associated with them, namely conservation of the sources. But conservation laws in both quantum mechanics and classical mechanics are always associated with symmetries. Symmetries of some sort or another are always behind conservation laws. The conservation of electric charge, we've talked about that, and I showed you how if you study the quantum mechanics of electrons in terms of charged particles in general, there's a wave field describing those particles, this could be the electron, and the symmetry associated with the conservation of charge is just the multiplication of the field by a phase. So that was the simplest example of a gauge theory, a gauge charge, meaning the electric charge, and a symmetry that goes along with the conservation of that charge. Those things go together. Okay, well, let me just very quickly remind you how it worked in quantum chromodynamics. In quantum chromodynamics, a more complicated structure. You do have Maxwell-like fields. Let's label them. The Maxwell-like fields are labeled by indices. So in quantum, let's continue to call them A. There is no universal symbol for the gauge field of quantum chromodynamics. Sometimes it's called G, sometimes it's called A, sometimes it's called B, sometimes it's called C. I'm going to call it A, but in order to distinguish it from the electromagnetic field, we give it some indices. I and J. It's a matrix. Now, it's a matrix, and that matrix transforms under a group. The group was Su3. So let me just remind you very, very quickly that we started thinking about Su3 not by thinking about gluons. This, of course, would be the gluon field, but by thinking about quarks. So a quark was an object which we called Q. Q could stand for the quantum field of the quark, and it has an index. The index, it's not the up-down index, it's not the charm-strange index. We'll come to that sooner or later. It is the color index, red, green, or blue. I takes on three values. And the symmetry operation is not multiplying by e to the i theta, but multiplying by a special unitary matrix Uij. It gives us Q prime i. That was the symmetry operation, which is a kind of rotation in a kind of three-dimensional complex space. Don't confuse it with ordinary three-dimensional space. This was the symmetry operation, and one would say that the Qs, think of them as particles, if you like, the Qs form a representation of Su3, which is called the fundamental representation. It's called the fundamental representation. It has three entries, red, green, and blue. Sometimes it's called the defining representation. It's the smallest non-trivial representation. And it just can be thought of as a three-component vector, one, two, three. And the unitary matrices can be thought of as matrices, three by three matrices. OK, we've gone through that. The other thing that I told you is that antiquarks, or antiparticles in general, are represented by the complex conjugate fields, fields which are simply the complex conjugates of the original ones. The relation, the mathematical relation between particle and antiparticle, or the wave functions of particles and antiparticles, that relationship is complex conjugation. So we could ask, how do antiparticles transform? Well, we simply realize that if you want to transform the complex conjugate field, you should use the complex conjugate matrix. If you multiply q by u to get q prime, then you must multiply q star by u star to get q prime star. The set of matrices u are called a representation of the group. The set of matrices u star are a distinct and different representation of the group. And the language we would use is that quarks and antiquarks are described by the three-dimensional representation. And we could call it three star, meaning complex conjugate. It's usually indicated by a bar. Two different representations of SU3. So there's seven, we make this. Nine, actually, well, sorry. You asked me how many matrices there are? The number of generators is eight. Yeah, because these matrices are assumed to be special unitary matrices. Now you can ask, and it's interesting to ask, what kind of theory would you make if you didn't insist that they'd be special? And I'll tell you another time, maybe later, but not now. Okay, now what is Aij? Aij has two indices. And the way to think about it is that one of those, it is mathematically got the same group theory structure, the same symmetry structure as having a quark j and an anti-quark i. In other words, this is an object whose indices transform one index as the three-dimensional representation and the other index as the three-bar. Another way to think about it is it has all of the properties with respect to the symmetry, not with respect to all physical properties, but with respect to the way it transforms. It transforms as if it were a quark and an anti-quark sitting on top of each other. Not at spin, just its color, as if it were a quark and an anti-quark, with one special rule, namely the trace of Aij vanishes. The singlet, the singlet that you can make, the piece which has no transformation property under the group at all, that you can kill and forget about. That's why there are eight Aijs instead of nine. So Aij is like a thing with a quark and an anti-quark index. But because it has a quark and an anti-quark index, it also transforms when you under the SU3 transformation group, in that sense it's different than the photon. The photon is completely electrically neutral. Its field, the photon field, does not get a phase when you rotate the electron wave function electrons. Go to e to the i theta times electrons, positrons go to e to the minus i theta times positrons, photons do nothing. That's because they don't have electric charge. But the gluons, they do have color. Now what kind of color do they have? They have the color of a particle and an anti-particle. Like you say, a particle and an anti-particle shouldn't have any color. But they do because you could have a red particle and a blue anti-particle. A red, that would make a thing which was not really neutral, which knew about color. So gluons do have color and they have color in what is called the adjoint representation, which is the representation of the 8th generators. Okay, now, what about the interaction between quarks? Incidentally, there are many other gauge theories of interest. They're all pretty similar to each other. The group might not be SU3, it might be SU2, it might be SU4, it might be SU10, it might be anything. And the objects of the group might not be called quarks. Nevertheless, or the fundamental objects of the group, the analogs of quarks might be called something else. They might have a different name. But they would be objects which had a single index. The anti-objects would also have a single index but of the complex conjugate kind. And the analog of the gluons which are called gauge bosons. Gauge bosons, the Maxwell-like fields always have one index of the particle type and another index of the anti-particle type. So that means, let's see, if we're talking about SUN, that means how many generators altogether. Well, N times N is N squared minus one for the trace. So it's N squared minus one distinct gluons. Now, what is the connection between the quarks or more generally, the particles which transform like quarks perhaps in a more general way? What's the connection between them and the gauge bosons? And I've explained that to you before, but let me just remind you again. If you have a quark labeled by I, where I could be red, green or blue, and another quark labeled by J, that the I-th quark can become the J-th quark if a gauge boson is emitted. What kind of gauge boson? Well, as I showed you before, the way to think about it is just to think about these lines running through the diagram. I runs right through, so that's I, and J also runs right through, except when J is pointing downward, it should be thought of as the analog of an antiparticle. And so an I would become a J by emitting a quantum of the field A, is it J-I or I-J. No, J-I, doesn't matter very much. By emitting a gluon of the I-J-th type, an I-th particle becomes a J-th particle, and that is one of the basic interactions of gauge theories in general. Fundamental objects, analogous to quarks, gauge bosons, analogous to gluons, and a rule about emission and absorption that you have to follow the lines and never lose an index. But now that raises something new. If a gluon itself has the properties of a quark and an antiquark, so let's say this is the I-J-th kind of gluon. Well, the I-J-th kind of gluon can also emit a gluon. How would it do it? The I-J-th gluon could become the I-K-th gluon. I-J, let's say, red-green gluon could become the red-blue gluon, as long as the indices make sense and they're followed, anti-K here. So this would be an interaction in which a gluon of type I-J became a gluon of type I-K, and a gluon of type K-J. The order of I and K matters. A K-J is not the same as a JK. That introduces something absolutely new which is not in electrodynamics, namely that the gluons themselves are charged particles and therefore exert forces on each other. Exert forces in a way that photons do not, exert forces on each other. The dynamics of gluons is much, much more complicated than the dynamics of photons. The last time what I told you was that one of the, and I'm not going to do the mathematics of this, it's not even fully understood it even now, but the effect, one of the important effects is that if you have a source, a color, a bit of color, with field lines coming out of it, again those field lines are not allowed to end, they're not allowed to end, but the effect, let's suppose that's quark, is an anti-quark over here, field lines have to come into it, so the field lines have to be continuous and so forth. I don't want, let me not draw more than that. Now the effect is that the field lines interact with each other in a way that they don't in electrodynamics. And the effect is very simple. The field lines get pulled into bundles which are called flux tubes, the result of which is that no matter how far apart you pull two quarks, they'll never escape from each other because the energy stored in this gooey piece of glue just increases linearly with the distance between them. All right, that's a summary of quantum chromodynamics, but it's also a summary of gauge theory. It's also a summary, very, very quick summary of gauge theory. Maxwell-like fields, sources, Gauss's law, making sure you have conservation and symmetries, symmetries having to do with the conservation laws, but more than that the symmetries telling you how the particles interact with each other in terms of emission and absorption. There is one other thing in a gauge theory, and it's the coupling constant. The coupling constant is the analog of the charge of the electron. The charge of the electron, let's talk about what the charge of the electron means. It is, of course, the coefficient in a Feynman diagram when you emit a photon from a charged particle, the amplitude for that, the quantum mechanical amplitude for that is the electric charge. But if you wanted to think about it operationally, you can imagine an electric charge slamming into the cathode or the anode cathode, the cathode of a cathode ray tube. When it slams into it, it gets accelerated, and the question is what is the probability that an electron which is stopped suddenly emits a photon? The answer is the square of the electric charge. The square of the electric charge, there's some pi's in it and h-bars and things, but basically the square of the electric charge in suitable units is the probability for emitting a photon. So that's what in quantum mechanics and field theory, that is what the meaning of the electric charge is. It's a measure of the square of the electric charge is the probability. Why? Because probabilities are always squares of amplitudes. But that's the fundamental significance of the electric charge, and the electric charge is a dimensionless number, just the probability that when you stop an electron you emit a photon. So it's a dimensionless number. It's a small dimensionless number. With appropriate sets of definitions, the square of the electric charge, really what tends to come into things is not quite the square of the electric charge, it's the square of the electric charge divided by some 4 pi's, 4 pi's are always floating around in there. It's called the fine structure constant, and the fine structure constant, which is pretty much the probability of a photon emission, is a number of about 1%. It's close to being 1 over 137, a famous number, 1 over 137. The 137 has no particular significance. It's just numerically about what the fine structure constant is. And let's just call it 1%, it says there's a 1% probability of emitting a photon. So if you had a picture that when an electron hits a cathode ray tube, it sends out a spray of photons, that's not right. What is right is that if 137 photons hit the cathode ray tube, on the average one of them will give off a photon. In that sense, electromagnetism is a weak force, or a weak process. The probability of emission is weak. All right, that's the quick summary of gauge theory. What I wanted to do is give you very, very briefly a run down on some numbers and orders of magnitude in various parts of particle and atomic physics. And then go on to the weak interactions. Now the reason I'm giving you a run down about numbers is to show you how different the weak interactions. Quantum chromodynamics is the theory of the strong interactions. The strong interactions was the word before quantum chromodynamics was invented for the interactions between subnuclear particles, between hadrons, between protons, neutrons, mesons, and all the things which are made of quarks and gluons. So it's theory of strong interactions. And there are electromagnetic interactions and there are also weak interactions. So I want to spend a little bit of time with numbers to show you what, where the terminology came from, why one is called weak, why one is called strong, why electromagnetism is sort of in the middle between the two of them. But let's just review a couple of numbers. First of all, in terms of units, energy, distance, and time do not have to be thought of as different units. They're all related with each other. First of all, time and distance really should not be distinguished in any sensible set of units that are useful for fundamental physics. We can set the speed of light equal to one, and if we do, if we fix the speed of light equal to one, then time and distance have the same units. And we can use the same units. As far as energy goes, we could also choose to set Planck's constant equal to one. That's a useful thing to do in quantum mechanics. An awful lot of equations get simpler if you set h bar equal to one. Well, they don't get a lot simpler. They just get rid of the h bars, but you can set h bar equal to one. Now, let me just remind you there's a connection between energy, e equals h frequency, right, or h nu, or h bar omega, all the same thing. The units of h bar, well, we've set h bar equal to one, so in units in which h bar are equal to one, energy is equal to frequency, but what's the units of frequency? One over time, number of oscillations per second. So that means that energy and time have inverse units to each other. You don't need to have a separate unit for time and energy. Any time you have a unit of energy, it defines also a unit of time. Now, of course, the unit of energy is inverse to the unit of time, so big energies correspond to small time intervals. And, okay, so let's write down some connections. And since energy and time are connected to each other, and time and space have the same units when C is equal to one, then time and space have units which are just inverse units to the units of energy. So, for example, here's a unit of energy, one electron volt. We've used electron volts before. One electron volt cannot be thought of as a unit of distance, but the inverse, one inverse electron volt, so it's one over an electron. One inverse electron volt is a unit of distance, and just to have an idea of what it is, it's about 10 to the minus seventh meters. One electron volt is about 10 to the minus seventh meters. So an electron volt is a fairly small distance, but on the scale of fundamental physics, it's not a small distance. 10 to the minus seventh meters is what? A atom is 10 to the minus tenth meters, something like that. Yeah, okay, so it's fairly big. An atomic diameter, yeah, an atomic, and we can convert, right, and an atomic diameter, one atom, this is the atomic diameter, that's about 10 to the minus tenth meters, and so you can convert that to electron volts. Is it, I don't know, whatever it is, an inverse of about 1,000 electron volts, a kilo electron volt, an inverse kilo electron volt is an atomic diameter. Okay, another fact, just converting distance, if we know the distance or size of an atom, then we know the time that it takes light to go across it. All right, so the transit time, there's another quantity, the time that it takes transit time, the time for light to cross an atomic diameter, what is that? That's about 10 to the minus 18 seconds. I've used the fact that light goes at about 10 to the eighth meters per second. Order of magnitude, 10 to the minus 18 seconds, these are just some numbers that I just wrote down, but a different time scale. All right, so this is the time scale for light to go across an atom. The different times, and by an atom, I don't mean the nucleus, I mean the whole atom, another time scale is the time scale that it takes for an electron to orbit an atom. Now you might think, why isn't that just about the transit time? Transit time is about 10 to the minus 18 centimeters. The reason is because electromagnetic forces are fairly weak. Because they are fairly weak, the force on, and the ultimate reason that they're weak is because this fine structure constant, this alpha, which is E squared over 4 pi, that's a small number of about 1 over 137. Another way to say it is the electromagnetic force on an electron is weak. Because of that, the acceleration on the electron is not too big, and the orbit is fairly large. The weaker the force on an object, the larger the orbit is going to be. Sorry, the slower the electron will move is what I meant to say. The slower the electron will move, and because the electron moves slowly by comparison with the speed of light, how much slower? About 1 percent slower. About this much slower, the electron moves around with only about 1 percent the speed of light, and that means the orbital time, orbital time, that's about 10 to the minus, let's see, it's bigger. That time is bigger, 10 to the minus 16 seconds. Another quantity sometimes of interest is the decay time for an atom. How long does it take for an atom, an excited atom, let's say a hydrogen atom with its electron, one orbital up from the ground state, one orbit up from the ground state, how long does it take to decay? Now, that is small, doubly small, with two powers of alpha. What are the two powers of alpha? First of all, the acceleration of the electron is small. Remember, charges emit radiation when they're accelerated. The acceleration is small because the force is small. So first of all, the acceleration is small. The electron moves around with a relatively small acceleration compared to what it could have been if alpha were bigger. It moves around with a small acceleration, but on top of that, just like the electron plowing into the cathode ray tube, there's another factor of alpha when the electron is accelerated in the probability that it emits a photon. So as the electron goes around here, first of all, it's moving with a slow acceleration, and second of all, even that acceleration is not very efficient in producing photons just because the fine structure constant is small. The net result is that the time scale, order of magnitude, decay time, how long it takes for an atom to decay. Decay time is about a nanosecond, 10 to the minus 9 seconds, much longer than the orbital time or the transit time. The main thing is there's a variety of different scales in the atom, and they're all related by the fine structure constant. They're related by powers of the fine structure constant, the fine structure constant being a small number. There are fairly large ratios between different time scales, transit, orbital, and decay time. All right, now let's come to Hadrons. Hadrons are like atoms, they're atoms made up out of quarks. We could ask very much the same kind of questions. First of all, we could ask what is the Hadronic diameter? The Hadronic diameter of a typical Hadron, it could be a proton, neutron, meson, they're all more or less the same. Here's Hadrons. Hadrons, Hadrons. The Hadronic diameter is a lot smaller than the electron. Sorry, than the atom. It's a lot smaller than the atom. It's not 10 to the minus 10th meters. It's about five orders of magnitude smaller, 10 to the minus 15 centimeters. That's also more or less the diameter of a nucleus. Nucleus, of course, being a few protons and neutrons could be bigger. Sorry, meters. Hadron diameter is about 10 to the minus 15th meters, five orders of magnitude smaller than an atom. Part of the reason is because there's two reasons. There's two reasons why it's smaller. One of them has to do with the fact that the constituents are heavier and heavier things will sit closer to the center. The other has to do with something else that we'll come to in a minute. But it's about five orders of magnitude smaller. The transit time has just gotten from the size straightforwardly. Transit time, this is not an independent thing. It's just another measure of the size of the thing. Just to get the number straight, it's 10 to the minus 23rd seconds. So that's a typical time scale for, well, it's for light to cross the Hadron. But now we can also talk about the orbital motion of quarks. How long does it take for a quark to swing around its wave function or swing around the interior of a proton? And there the answer is about 10 to the minus 23rd seconds. So how long does it take for it to decay? How long does it take for it to decay? And the answer is about, now I'm talking about particular kinds of decays, a decay where you hit a proton, it starts oscillating, and then emits a pion. Those decays are the same as the quarks. They're different than this hierarchy of scales here. What is the conclusion? The conclusion is the analog of the fine structure constant is much bigger, close to 1. So what is the conclusion? The conclusion is the analog of the fine structure constant is much bigger, close to 1. For Hadrons, in other words, the emission probability, if a quark were to get stopped in a fictitious cathode ray, quarky cathode ray tube of quark flying off, the probability to emit a gluon would be about 1. That's what these things indicate here. The hierarchy here is entirely due to the fine structure constant. The lack of a hierarchy here can only mean that the corresponding quantity in quantum chromodynamics must be close to 1. By now it's been measured in many ways, and it is much closer to 1. It's much larger than the corresponding quantity for electrodynamics. One is a little bit, no. It would seem that just special relativity to say that the case that could be much less than 10 to the minus 23, that's a kind of case like, go for 1. Yeah, I think that's a fair statement. It's about as fast as it could be, in other words. It's about as fast as it could be. Nothing's slowing it down. This is why the strong interactions are called strong. Now, the facts about the numbers here were known before quantum chromodynamics, and it was understood that there was no hierarchy of scales, and that's why it was called strong interactions. We now trace it to the properties of the charge or the fine structure constant. Incidentally, for quantum chromodynamics, the analogous quantity is not really one. It's about a fifth, but much bigger than the 1 over 137 here. All right, so that was just some numbers and some facts about the strong force. It's called alpha QCD. So this is called alpha, and the other one is called alpha QCD, and it's about 0.23 or something like that. Or in context of paper, you write alpha QCD. I've never heard alpha QCD referred to as alpha QCD, but I think if you were writing a paper with both of them, you might call the top one alpha QCD, yes. I've never seen it referred to that, right? Well, particularly if you're talking about shielding or something like that. No, I can't, I don't think I've ever seen it. Oh, oh, oh. It's just called alpha. Sometimes the unshielded one is called the bare alpha. The shielded one is called the renormalized alpha, right? Okay. Well, that brings us to the weak interactions. The weak interactions are called weak. Primarily because the decay times associated with them are very long. They're much longer than the decay times associated with either hadrons, and they even tend to be longer than the decay times associated with atoms. So there are processes in nature, decay processes in nature, in which particles, elementary or otherwise, decay by weak interactions where the time scale, in other words, the half-life of these decays is far longer than any scale that can be accounted for by these, by numbers like this. So as an example, here's some examples. Well, the lifetime of the neutron. The neutron is a hadron. The decay is neutron goes to electron, proton, plus anti-nutrino. And the lifetime for that is about 12 minutes. That's absurdly long on particle physics scales. Part of the reason is easy, but it hardly accounts for the extraordinary stability of the neutron. Part of the reason is the neutron is only a little bit bigger than the combined masses of the electron, the proton, and the neutron, and the neutrino. Let's just see. Somebody have out their pocket? The mass of the neutron is 940 MeV. About 940. So the proton is about 939. I think the actual difference between them is about one and a half MeV. But then there's the electron which has a mass of about half an MeV. All together out of this 940, almost a thousand MeV, the difference of mass of the proton, the amount of available mass left over is tiny. It's 0.1% or less. If the neutron was slightly lighter than the sum of these masses, it could not decay at all. Just energy conservation would not allow it. Remember, mass is energy. It would not have enough energy to decay and still leave over some energy for kinetic energy of these particles. So if the neutron was exactly the same as the sum of the energy of the electron, the proton, and the neutrino, the sum of the mass of them, it wouldn't be able to decay at all. If you gave it a mass, tiny, tiny, tiny, tiny bit above that, it would be able to decay, but it would take a very long time. The decay would be very slow. So part of the slowness or the long lifetime of the neutron is attributable to the fact that there's very little energy available for the decay to take place. But given that, that's hardly the bulk of the story. Twelve minutes is so absurdly long that there's something much, much more than that going on. Another example, let's see, where do I have it? I guess I didn't write it down. Another example of a, this is called a weak decay. For obvious reasons, it's called a weak decay. Another example is a charged pion, let's say a pime minus. A charged pion decays to an electron. Electron has negative charge and an anti-nutrino. Yes, and an anti-nutrino. That's another possible decay. A charged pion goes to an electron and an anti-nutrino. A positively charged pion could become a positron and a neutrino. The lifetime for that, I believe, is about 10 nanoseconds. Longer than these atomic decay times. And atomic decay times are very, very long by comparison with particle physics decay times. Now, there is a decay of the charged pions to these particles. There's another decay involving muons. We're going to come to muons soon enough, similar to electrons. But the point is not the particles that they decay into, but two things. First of all, they also decay very slowly, nanoseconds. But there's plenty of energy available for these decays. These decays are not suppressed by the fact that the energies of the initial and all the masses of the initial and final states are very close to each other. So there's something else going on in weak interactions. Whatever weak interactions are, there's something that makes them proceed very, very slowly and weakly. So that's what we come to next is weak interactions. The primary or oldest example of which is the decay of the neutron. The beta decay of the neutron called beta because electrons are originally called beta waves. And this is called the beta decay of the pion. In fact, it's quite similar. Quite similar physics goes into both of them. So to understand what's going on, we need to understand about some more particles. Particles that we haven't come to yet, the leptons. Let's go back to the quarks for a moment and let me just make a table of the different kinds of quarks. Horizontally, or make a sequence here which will be red, blue and green. These are the colors of quarks. Then along here, let me list the up quark, the down quark, the charmed quark, the strange quark, the top quark, and the bottom quark. Each one of these boxes is filled. There are red, blue, and green up quarks. Let's put an x there. There's an x here. There's an x here. There's an x here. X here, x here, and so forth. The boxes represent all the possible quarks that exist. Now, quantum chromodynamics has to do with symmetries which connect red to green to blue. It has to do with these unitary transformations which mix up the colors of quarks. They do not, those symmetries do not mix horizontally. An SU2-SU3 symmetry, a color symmetry, may rotate an up red quark into an up blue quark or an up green quark. And the nature of the symmetry is to act vertically, to mix up things this way. And also at the same time, we'll mix the different down quarks, the different colors of down quarks. The same symmetry will mix the different colors of charmed quarks and strange quarks and so forth. So the color symmetry acts vertically. It mixes up the different rows here. Is that the unitary matrix? Yes, the unitary matrices mix up red, green, and blue. I might have done better to put the rows where the columns are and the columns where the rows are, but no, it's okay. It's good. No, no, it's good. It's good. Right. The weak interactions are associated with symmetries which act horizontally in this picture. They mix up with down. They mix, and at the same time that they mix up with down, they mix charmed with strange. And they mix top with bottom. Let me just remind you that an up quark has the same properties apart for its mass. Same properties as a charmed quark or a top quark. The down quarks are similar to strange quarks and bottom quarks. Ups have charged two thirds. Downs have charged minus two thirds. Same here and here. So these are symmetries which mix, if you like, upness with downness. At the same time, charm this with strangeness and topness with bottomness. It mixes these up. The symmetries, those symmetries, what would you care to speculate on what group might be involved? SU2, because it acts on things, on doublets. We simply have three doublets here. They don't take up to charm or up to top. They simply act on up and down horizontally among pairs of things. A group SU2 and it is a gauge symmetry. In other words, it also comes together with forces, with gauge bosons, with interactions, and with gauge field. We're going to get to those short enough. But we've left out something from this table here. It's kind of as if there was a fourth color. Now the reason, the fourth row here is not usually identified as a color. And the reason is because the particles do not interact with the gluons. But nevertheless, it is another row here, another row in which there are particles which are filled in here. And in a certain sense, they are also doublets, these particles are also doublets, and also get mixed up under this SU2 symmetry which moves things horizontally. What are they? What are those particles? They're the leptons. So in a sense, the fourth color could be thought of as lepton number. But what are the leptons? The leptons are, first of all, in the first column here, there are analogous to the up, or analogous to the down, there is the electron. The electron is a lepton. But its partner, its partner is the neutrino. But there are different neutrinos. There's not only one neutrino. This is called the electron neutrino. What comes in the next column? The muon. The muon, in every respect except for mass, is the same as an electron. Just as in every respect except for mass, the charmed quark is like an up quark, or the strange quark is like a down quark. And together with it, there's its own neutrino, the muon neutrino. And finally, the last one is called the tau and nu sub tau. All of the charged leptons, incidentally, neutrinos, of course, as you might guess from their name, are electrically neutral. The electrons, muons, and tau's have charged minus one. They are put under the down column here, not because their charge is the same as the down charge, but because the difference between the charges of the two is the same as the difference up here. What's the difference between the charge of an up and a down? Two-thirds minus minus one-third. So the difference between this column and this column is plus one unit of charge. The difference between this column and this column is also one unit of charge. One, sorry, zero minus negative one. So as you go from here to here, you decrease charge by one unit as you go horizontally. Now, should we take a break for five minutes? Let's take a break for five minutes. And then I will tell you about the W bosons. The W bosons are the gauge bosons of the weak interactions. All right, so let's try now to see if we can make a microscopic gauge theory of the weak interactions. As I said, the weak interactions are based on a gauge theory which mixes up to down, neutrino to electron and so forth. And so rather than to get into mathematics for the moment, the mathematics of the group SU2, which we'll come back to, let's think a little more simply. We've seen that in various situations the gauge bosons typically have the quantum numbers or the quantum properties of a particle and an antiparticle. In the case of the quantum-chromodynamic interactions, it is a quark and an antiquark. That's why we had nine gauge bosons and so forth or eight removing the trace. So we could start by saying let's look at the possibilities of gauge bosons which have the properties of a particle and an antiparticle. Let's not distinguish between whether we're talking about quarks, red quarks, blue quarks, green quarks or leptons. Let's just say a particle from this column with an antiparticle of this column. And for simplicity, let's just focus on the leptons. You'll see in a moment that you'll get the same thing if you focused on the quarks. But let's focus on the leptons and talk about objects that we could make with a lepton and an antilepton. First of all, we could have some simple things, electron with a positron. Now I don't really literally mean electrons and positrons. We could have electrons and positrons. Electrons and positrons have no charge at all. So let's drop that for a minute. We'll come back to it. There are things which have properties of electrons and antilelectrons, neutrinos and antinutrinos. But more interesting for the moment is the combination of electron, antinutrino. So what is the charge of an object which has the same conserved quantities, the same properties as an electron with a stuck on top of an antinutrino? Well, let's just focus on its charge. It's electric charge. Its electric charge is minus one. Neutrino is neutral. Electron has negative charge. So whatever this object is, it has negative electric charge. Let's give it a name and let's call it a gauge boson. Let's assume that it is also a gauge boson. Gauge boson means that it behaves like the Maxwell field, the whole same thing. But it has negative charge. Let's call it W. W, that's the traditional name for it. W minus. Why minus? Because it has negative electric charge. We could, that was this one with this one, we could have taken down quark with, I guess, anti-up quark. What happens if you take down quark with anti-up quark? How much charge do you get? Again, minus. Same as this. Down quark with anti-up quark would also have minus charge. So if you like, you could also imagine that the W minus has the same properties as a down quark and an anti-up quark. In other words, it's one from the second column and anti-one from the first column. That's the property that it has. Let's call it W for weak, in fact. I think W was originally the notation for weak. The negatively charged W boson. And of course, there's also a positively charged boson, W boson. Which is like anti-down with an up or like positron E plus with a neutrino. E plus with a neutrino would be a W plus. Let's suppose the pattern is pretty similar to quantum chromodynamics, that the W bosons are like either photons on the one hand or like gluons. Which means that they would be objects which could be emitted from the leptons. They would be objects which would allow a lepton of one kind. Let me not label it. A lepton of one kind to become a lepton of another kind. By emitting a W boson. Let's see if we can find some examples. An electron can emit a W minus to become a neutrino. An electron becomes neutrino by emission of a W minus. You see the action of the gauge bosons is to do the same kind of transition that the symmetry is ultimately going to be imagined to do to mix electron into neutrino in very much the same way that the colors got mixed by emitting gluons. You could follow the lines if you like and say the electron and the anti-neutrino over here can be imagined to be a W minus boson. Same pattern. Let's just draw it by saying a W minus is emitted. But that's not all you can do with a W minus. You can also have a down quark go to an up quark. With a W minus. That also works. In fact, the W minus is an object which can be emitted in a transition from, let's see, from the right column to the left column. A red down quark can become a red up quark by emitting a W. A blue down quark can become a blue up quark by emitting a W. Likewise for green. But you can go further. A strange quark, not the strange quark, the strange quark can become a charmed quark. That's another one. Strange quark, charmed quark, again, W minus. So this new symmetry, this idea of a new symmetry which acts horizontally and only horizontally between neighboring pairs here introduces the possibility of gauge bosons which cause transitions between electron neutrino, down quark and up quark, strange quark and charmed quark. Now a W plus and a W minus are antiparticles of each other. The meaning of that is that you can flip these diagrams, put the W minus down, W here, but when you flip it, it becomes a W plus. So a W plus can be absorbed by an electron, plus a unit of charge, minus a unit of charge to become a neutrino. So a whole family of different processes are codified by this one vertex here. We can turn them upside down, neutrino goes to, you know, what we're allowed to do. We've done these things before. And that's the basic new element of the weak interactions. Actually, that's all, yeah. Well, it's not clear to me why you have W minuses in all three of these cases. There's a lot of charge conservation. It couldn't be W plus. Would you accept I don't know what a W means? W is the name of a particle. It's the name of another gauge. Yeah, but the charge conservation tells me it's going to be a minus. But I haven't seen any reason why it should be the same particle. Well, okay, that's right. But what it comes down to in the end is an empirical fact. It's the empirical fact that the symmetry of the weak interactions acts simultaneously on all of these doublets here. When an up goes to a down, a charm has to go to a strange, a top has to go to a bottom, and the leptons have to mix with each other simultaneously. The implication of that, the implication of that from a practical point of view, is simply it's the same symmetry which acts, it's the same gauge bosons which are there. You could have imagined a theory in which it's quite mathematically consistent to imagine a theory where different W-type bosons would have been emitted by up quarks, charm quarks, or tau quarks, or even leptons. Right? It's a fact of nature. The fact of nature that nature is simpler than it might have been that the same gauge bosons cause transitions between any point, you know what I mean, I'm going to get tongue-tied if I try to say it. Okay, well first of all, what processes can we explain? Good question. Can a W-beg into an electron and a negative neutrino as well? Negative, so say it again. Can you flip the arrows? Yes, yes, yes, yes. You can flip arrows and you can flip lines. Yes, yes, certainly you can flip the arrows. Yeah, absolutely. Absolutely. And you can also flip lines up and down in various ways. For example, another way you could flip the line is you could take this up quark here, instead of going out, instead of an up quark going out, an anti-up quark comes in. Whenever you flip a line from past to future, or from future to past, you'll also change from particle to antiparticle. I'm not sure, but on the right-most side, W-stereo is a strange antitron. Yes. So that's the same as a down antiparticle? Yes. The same thing? Wait a second. Sorry. Yes. That's a strange antitron, right? Yes. That's the same as a down antiparticle. It's not literally a strange antitron, it's just a rule that tells you that whatever these W bosons are, they're associated with transitions from one column to the other. The triton's a charm. Did I? Sorry. Strange. Right. Electron to neutrino. Strange to charm. Is that wrong? No. No. It's not an anti-charm. That's the point. Who said it was an anti-charm? Well, it shouldn't have been. It's different from coming in from the past. Right. If you follow the lines like that, at the point where the line turns around, it has to be thought of as an antiparticle. Right. Okay. Or if you like, you can always think of it as strange anti-charm coming in and making a W boson. Okay. Just keep in mind, whenever you flip from the past to the future, if you want to know what a W boson will decay to, let's read it downward, it will decay to a strange and an anti-charm. And that really can happen. Yeah. So can you create W boson from like electron and a... Anti-neutrino. Anti-neutrino. Yes. And then call the W boson to change into like a strange and a charm. Exactly. That's where some of these processes are going to come from now. So let's think about the various processes. Let's start just for fun. Let's start with the decay of the pi minus. What is a pi minus? A pi minus is a quark and an anti-quark with altogether negative charge. So let's say a quark and an anti-quark with negative charge had better be... Here's the pi minus. That's equal to a down quark and an anti-up quark. Down and anti-up. Now, what can a down and an anti-up do? Let's see if we can find down and anti-up here. Here it is right here. Down and anti-up can become a W minus. So it's a possible transition that is allowed by the physics of W bosons. So here's a W minus. But now a W minus can do something else. A W minus here is... Let's say here is a W plus. Let's put W plus upstairs here. This is W minus now. And let's put the neutrino downstairs, anti-neutrino. And read this as saying that W minus goes to what? Electron. Anti-neutrino. So W minus can go up to electron, anti-neutrino. And here it is. E minus, ordinary electron. So that's the basic underlying process that governs the beta decay of the negatively charged pion into a electron and an anti-neutrino. Now, which anti-neutrino is it? There are three distinct anti-neutrinos. There's an electron anti-neutrino over here. So it's the electron anti-neutrino. But it can also... I didn't draw all the various possibilities, but any place you saw an electron, you can put a muon. Any place... And as long as you carry along the muon neutrino. So another process is muon. Anti-muon neutrino. And just for clarity, this is not strictly a Feynman diagram, right? This is just showing an interaction. It's pretty close to a Feynman diagram. Except for the time direction. Oh, yeah. Right. I don't know why. All of a sudden I change directions. Yeah. For some reason. Yeah, right. Time is now going horizontally. I've become a computer scientist and they measure... Yeah, yeah, they do. Right. Yeah, it's pretty much... Yeah, that's good. We can turn it up. I can't do that easily, but yeah. Yeah, that is essentially a Feynman diagram describing the decay of the pi minus to the muon. As it happens, the... For technical reasons having to do with some complications and Feynman diagrams that are not interesting to us. As it happens, the primary decay... Well, primary, I mean the most probable. The most probable decay of the pi minus is to the muon anti-neutrino, not to the electron anti-neutrino. It's a technical point of no special interest to us. It just happens to be that way. Wouldn't it be consistent instead of considering the Feynman diagrams with a time deal is more like a kinopaly equilibrium deal? Which way you're going? Well, both things can happen. Both things can happen. We're going to have a real equilibrium. We're in empty space. We're in empty space except somebody made a pi minus. A pi minus was created how? A pi minus might be created by some nucleons coming together and creating by strong interactions. By strong interactions, a pi minus might be created. Now that pi minus is just going along and saying, wait a minute, I can decay if I like into a neutrino and a muon. Then of course, the neutrino and the muon have enough kinetic energy to fly off and they won't come back together again because they've departed. They've gone away. Okay, on the other hand, it is true that if you were to take a muon and an anti-neutrino and fire them toward each other, there would be a certain probability that they would make a pi minus. So yes, you can read it both ways. Let's go. A quick question. Like in one dimension, you have an SU3 and the other an SU2. Is it possible to combine them into an SU6 or something? SU5. Well, how great. SU5. Not two times three, two plus three. SU5. Tonight, we want to get to SU2. I'll be happy if we get the basic ideas of SU2. Now, keep in mind, we're doing something a little bit funny because we have, SU2 has how many generators? Three. So how many gauge balls should it have? Three. So far, we only have w plus and w minus. So let's just be another one looking around. And that other one must be associated somehow with something like electron, anti-electron and neutrino, anti-neutrino, but we'll come to it. Just like the, yeah, we'll come to it. There is a third gauge boson looking around. And it does play a big role in the weak interactions, but it didn't play a big role historically in the weak interactions just because for reasons we'll come to. Historically, it was the w boson, which was the first to be conjectured. Speaking of historically, was the weaker the strong force sort of suspected first? Of course. All right. So let's talk about the history. The history of these forces goes back to Becquerel, goes back to radioactivity. Alpha, beta and gamma decay. Alpha decay was what? A mission of a two protons and two neutrons from a nucleus. That's a strong interaction. It's part of the theory of Hadron's and nuclei. Gamma decay, that was this one right here, weak interactions, and gamma decay? Photon. Historically, they happened at the same time in the same experiment. Question? The energy conservation, how it works there? You're assuming some kinetic energy in the world? Yeah, the energy, let's work on it. You never worry about the energy conservation in intermediate states like this. If the energy of the intermediate state doesn't match the energies of the initial and final state, it just means that the intermediate state can only last for a very short time. There's an energy time uncertainty principle. You can violate the energies in a Feynman diagram for short periods of time. You don't have to worry about conservation of energy between here and here. The only energy we have to worry about is the initial energy and the final energy. Okay, so let's worry about it. A pion has an energy of about a mass. Let's suppose the pion is at rest. The mass of the pion is roughly about 140 MeV. The energy of the muon and the neutrino, well first of all it's the rest energy of the muon and the neutrino. And the rest energy, the rest mass of the neutrino, sorry, the mass of the muon, I think is about 100 MeV, I forget exactly, and the mass of a neutrino is for practical purposes zero. It's too small to be important to you. So there's more energy to start with than the energy in the masses of the two final particles. Does that mean it can't happen? No, it just means the two particles will go off and carry some kinetic energy. So the difference between the energy on this side and the sums of the masses on this side is just the kinetic energy of the outgoing particles. That's the energy balance. The momentum also has to be conserved, and that means if the pion was at rest to begin with, the momentum of the muon and the momentum of the neutrino have to be equal and opposite, and they'll go out back to back, so to speak. If you analyze the neutron decay in the same way as you would do that. We're going to do that now. Yeah, let's do the neutron decay right now, see how the neutron works. Oh, let's do something simpler first. Let's do the muon decay. The muon is also something that can decay. So let's take the muon. Here's the muon moving along. Now what can the muon do? The muon can become a neutrino sub muon and emit a W. What kind of W? W minus, right? It can become a W minus and a neutrino. But now one of the things that W minus can do is it can become an electron and an anti-neutrino. So this W minus can now become an electron and an anti-neutrino of the electron variety. Can this happen? The only thing that would, yes it can happen. Not only can it happen, this is the way the muon decays. The muon decays because it is sufficiently much heavier than the electron, about 200 times heavier than the electron. The neutrinos are practically massless. So the energy balance is you have about 100 MeV here, a half an MeV here and the risk goes off in kinetic energy. So that's an example of what's called a purely leptonic process. No quarks in it at all. Is it an anti-neutron? Is it what? An anti-neutron. Yeah, it is. So it's one neutrino, one anti-neutrino and an electron. Okay, so that happens. And that's the primary decay, it's the only decay of the muon. So the muon is an unstable particle because of this. But the reason the muon is unstable and not the electron is just that the muon happens to be heavy enough to be able to decay to the electron. The electron is not heavy enough to decay to the muon. So in a sense it's an accident of the masses. The same is true of the tau, incidentally. The tau can also decay to other leptons. But all right, that's the mu decay. And notice the components are the same. The basic components are the same. A transition from one column to another, followed by a W, and then the W makes another transition from one column to the other, or if you like, a decay from one column to the other. All right, what about the decay of the neutron? Decay of the neutron. Would it be consistent to say the purpose of a linear accelerator, per se, is to effectively increase the mass of the electron so that it can now go the other direction? Absolutely. No. You don't change an answer, just put nothing in it. Absolutely. That's exactly right. Okay, we're going to do neutron decay. So the neutron is, maybe I better switch to vertical. I don't want to, vertical, vertical notation, back to vertical. Neutron is three quarks, two down quarks, and an up quark. Right? Minus four thirds plus two thirds, sorry, minus two thirds, yes, that's a neutron, electrically neutral, neutron coming in. Okay, now what is possible? First of all, we can have the up quark become a down quark, and emit some sort of W, but that'll give us three down quarks. Three down quarks is a fine thing. There's nothing wrong with three down quarks, except that the mass of three down quarks, of a particle with three down quarks, is always too heavy for the neutron to decay. So that's not what happens. What about one of the down quarks? The down quark can emit a W, become an up quark. This is again a W minus. The down quark emits a W minus, becomes an up quark, and now we have two up quarks and a down quark, up, up, down, what is that called? That's a proton. And the W does what the W always does. It decays to an electron and an anti-neutrino. So that's the beta decay of the neutron. Could it decay into a muon and a neutrino? Not enough energy. The muon is too heavy so that there wouldn't be enough energy left over for the muon. But other than that, if it were not for that, it could happen. Another way to say it is if you could jam some more energy into the muon, you could add extra energy into the neutron. How can you do it? You hit it, you give it a shove. You give the neutron a whack, you excite it, and if you excite it enough, it can decay to a proton, a neutrino, anti-neutrino, and a muon. But without somehow providing that extra energy, the neutron will only decay to the proton, electron, and the anti-neutrino. Okay, now let's come, let's see, where are we? I think we'll finish for tonight. But remember, we have a puzzle, and the puzzle is why is it that the weak interactions give such slow decay rates? And I can think of one reason, namely, the coupling constant or the analog of the fine structure constant could be ridiculously small so that the probability of emitting these W bosons would be very, very small. That is not the reason. The fine structure constant for the weak interactions is about the same as for electromagnetism. That is not the reason. The reason is something else, and we'll come to the something else next time. When you say that there are these symmetries, you mean that the Lagrangian is invariant? Lagrangian, the Lagrangian is invariant under those operations. That's exactly what it means. I didn't want to start writing bunches of Lagrangians, so I've used the shorthand of saying there's a symmetry, right? But that's what it means, yeah. For more, please visit us at stanford.edu.
interactions of gauge theories in general.