New Revolutions in Particle Physics: Basic Concepts | Lecture 10
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Susskind, Leonard
(December 3, 2009) Leonard Susskind gives the tenth lecture of a three-quarter sequence of courses that will explore the new revolutions in particle physics. In this lecture he continues on the subject of quantum field theory, including, the diary equation and Higgs Particles.
 Stanford University. Alright, last time I started to tell you something about how quantum field theory gives rise to a theory of the particle interactions through an object called the Lagrangian. I think that was not terribly clear, so I want to go back to it a little bit before starting to discuss the details of particle physics. We're still thinking about quantum field theory. The basic technique that I was alluding to last time is called the path integral method of quantum mechanics. And it is the most direct route to Feynman diagrams, to the theory of particle interactions that are based on Feynman diagrams. So let me just try to briefly go through the basic ideas, which we started to do last time, but I don't think I was sufficiently clear. Anyway, the path integral formalism or method in quantum mechanics due to Feynman is a generalization as the quantum mechanical version of the principle of least action. So I just want to remind you what the principle of least action is. If we're talking about a particle, now I don't mean a particle from the particle physics quantum field theory point of view, just a classical Newtonian particle, the motion of the classical Newtonian particle from one point of space time to another, just very quickly, is determined by a Lagrangian. Lagrangian is a function of the coordinates of the particle, the time derivatives of the particle, and from the Lagrangian, whatever the Lagrangian is, one constructs the action. And the action is an integral along the trajectory of the particle of the Lagrangian. For every trajectory, whether it's the true trajectory or not the true trajectory from here to here, by the true trajectory, I mean the solution of Newton's equations from one space time point to another space time point, whether or not the trajectory is a solution, it has an action, the integral of the Lagrangian along the orbit. The classical principle of least action is that the trajectory followed by the particle through space time minimizes the action. From that, you can derive differential equations, and those differential equations are called Newton's equations. And I will assume that you know a little bit about this idea of action. Now, the same idea applies to classical field theory. And the way the idea works is for a particle, for an ordinary particle motion, the Lagrangian is a function along an orbit, a function of an orbit. For a field theory, and of course it's an orbit which connects some initial configuration to a final configuration. For a field theory, the equations of motion of a field theory, classical equations of motion, are also determined from a Lagrangian. The Lagrangian is a function of whatever fields, I'll call them generically just phi, and not just the time derivatives of the fields, but the space derivatives and the time derivatives, let's say derivative of phi with respect to x mu. That's just a shorthand way of saying that the Lagrangian depends on phi and derivatives of phi. One thing which is required by the theory of relativity is that the Lagrangian be a scalar. That a transform on the Lorentz transformations is a scalar, but that being said, the Lagrangian can be anything built out of phi's and derivatives of phi. What is the action? Imagine we have a region of space time, time as usual flows upward, space is horizontal, and we have a region of space time between an initial time and a final time. Then the action in this region, the action is the action in this region just as the action for a particle trajectory is the action from one end point of the trajectory to another. Here it's the action in a region of space between an initial configuration and a final configuration. The action is equal to the integral over that region, let's write a d4x which means the time and the x and the y and the z of this Lagrangian. What's the total integrated value of the Lagrangian over space and time, whereas for the particle it's just an integral over time? Now what do you do? What's the analog of a starting point for the particle? A value of the field on the initial surface here. Given not just one field, but all the fields in the problem, prescribing the values of the field on the initial surface here and prescribing the values of the fields on the final surface, you can ask is there a solution of the equations of, oh let me go back a step, forget that. Yes, is there a solution of whatever the equations of the theory are which starts with a given initial value of the fields at time t equals zero, let's say, and ends with the fields being something different that's some later time? The answer is the principle of least action again, the equations of the theory are formulated by saying yes, there always doesn't exist something in between which is a correct solution of the theory which has a given initial value of the fields here and a given value of the fields at the end point, and the correct solution of the fields in here is the one which minimizes this action, one which minimizes the action. So the principle of least action gets extended to a kind of space time principle of least action where the degrees of freedom in here are fields. Okay, now let me very quickly remind you what the quantum mechanical use of action is. How exactly you derive the classical principle of least action from the quantum mechanical version of Feynman's quantum mechanical version, we don't really need to get into here, you can go look it up and anything about path integrals, but for a particle, let's come back to the particle motion for a moment, according to Feynman or according to quantum mechanics, a thing that you might want to calculate is the amplitude. Is there anything wrong with the value of a field returning to itself? No, no, no, no more than the possibility of the position of this thing returning to itself. So how do you go from one space time event to another space time event? Why space time events? We're having values of fields here and values of fields here. Well, integrating d4x, so there has to be some d4 trajectory. No, the trajectory is replaced by a history, a history of the field from the beginning to the end. A history means the values of the fields all along here, everywhere is in here. The idea of a trajectory becomes a trajectory in field space, which means- We have to specify actually the coordinates as well as the field values to get an initial point. To get an initial point, what are they? Well, you're specifying the initial boundary condition or whatever as field values. Field values all along some initial surface. You pick the initial surface. In other words, you pick time- It's the whole field at some cross-sectional- Yeah. All right, which is another way of saying the whole field at a given instant of time. And the final configuration is the whole field at some later time. And the question is, is there a solution of the theory, whatever theory is, which interpolates in between these in the same way that a trajectory interpolates from here to here? The rule is minimize the action. That if carried out, we did this number of classes ago, but if that's carried out, the requirement of minimizing the action leads to partial differential equations for the fields and those partial differential equations are things like Maxwell's equations, things like Einstein's equations, and so forth. All right, so let's come back now to the quantum mechanical idea. The quantum mechanical idea has to do with amplitudes. Amplitudes are things out of which you construct probabilities. So for example, you can ask the question, what is the amplitude that if a particle is injected into the world, however it's injected in, at a spacetime point x and t, what is the amplitude that if I look for it at a later time, let's say this is at x and time t, what's the amplitude that if I look at a later time, let's call it t prime, that I will find it at position x prime and t prime. So it's the amplitude that if I start a particle at this point, close my eyes for a while and then turn on a detector which looks for the particle at a certain place, the amplitude for finding it there. That amplitude is a complex number that depends on the initial point, on the final point. It's a complex number whose magnitude squared is the probability. That's the rule. Where the amplitude or multiply by its complex conjugate and that is the probability for the particle to go from here to here. Feynman's rule is the following. It says take the action for any trajectory, take an arbitrary trajectory and write down the expression e to the i minus i, the action of the trajectory. We can just write it out. It's integral of, now we're talking about particles now, integral of the Lagrangian dt from one point to another point. It's a function of or functional of the trajectory. And now, actually there's a factor of h bar in here, a factor of h bar in here, that's where quantum mechanics comes into it. Sum this or integrate it over all possible trajectories. Well, in non-relativistic quantum mechanics, you don't let them go backward in time. Let all possible trajectories which never loop back on themselves in time, sum that over all the possible trajectories. Each trajectory, it's a complex number. Air them all up. Now, how do you add up such a thing in some complicated kind of integral? But we needn't worry too much about the details of how you actually calculate it. That according to Feynman is the amplitude for going from here to here. It's the sum of all possible classical roots. Classical root does not mean a solution of the equations, it just means a possible root. It's the sum of all possible roots whether or not the solutions of the equations, of Newton's equations, of e to the minus i times the action, measured in Planck's constant units. Incidentally, Planck's constant has units of action. So this is simply the action in units of Planck's constant. The e arises because to go a little step involves something very one plus an act term in the action, you have to multiply these both together. Yes, right, exactly. That's exactly right. The other thing here is the i, the i is part of quantum mechanics, what can I say? Yeah? That sum is the probability. That sum is the thing whose square is the probability. By square I mean times its complex conjugate. This sum could hardly be a probability because in general it's complex. It's got an i in it. If you multiply it by its complex conjugate, that is the probability. This is the probability amplitude. The complex thing. So this is Feynman's formulation of quantum mechanics and now it can be extended to quantum field theory. In quantum field theory, the corresponding question, now we're not actually going to formulate it this way because, but this is a correct way to formulate it. We're going to fudge it a little bit and make our way through it without any rigor. But the same idea, you start with an arbitrary configuration of the field. So that's some initial phi of x. X now being just x and not t. An initial configuration of the field. And now you look at some final configuration of the field. Let's call it phi prime of x. These are two different functions. And you can ask, what is the probability amplitude that if I start the field at some value and let the system go and then detect the field at a later time, imagine you had some way of starting an initial condition with a field being prescribed at every point of space. Of course, that's an idealization. You don't have a chance in the world of doing that, really. But you had some sort of apparatus which allowed you to start the field in an arbitrary configuration, let it go, and then you had another apparatus which detects the values of the field at every point. Then you can ask, what is the probability that if you start the field in a given way that at a later time the field will have some value? Again, that probability is determined in terms of a probability amplitude. And the probability amplitude is, again, the sum over all possible ways of interpolating the field between initial and final, all possible, exactly what I said, all possible ways of filling in between the initial state and the final state with a field value at every point of e to the minus i h times the field action. That's the integral dx dt dx dy dz d4x of the Lagrangian, which is a function of phi and the derivatives of phi. So you evaluate the Lagrangian at every point, integrate it up. That gives you the action for a particular, let's call it a trajectory. It's not really a trajectory in any ordinary sense, but it's a history. Let's call it a history, better yet. For each history, each possible history that you can imagine, it doesn't have to be a true history or a real history, for every possible history you can imagine, there is an action. The action is itself an integral, but then you consume this over all possible histories, and that is the amplitude to start with a field value phi and end with a field value phi prime between these two trajectories here. That's the notion of a path integral. I'm not concerned that there is an infinite number of paths, but the action is going to be a finite number for each of these paths. Yeah, but it's a finite complex number, right? Yeah, so how can that converge to anything? Why is that not the infinite number? No, there are many, many integrals. Well, okay, you're asking. Let me phrase it a little more strongly. Aning up an infinite number of numbers, well, that's nothing special. I think what you're pointing out is that each one of these numbers is not only finite, but has a magnitude equal to one. Or something that is not decreasing. It has a magnitude equal to one. It's an exponential of i times something, right? So any number like this is a number which on a complex plane, not minimizing it. You're summing over all. You're just summing over all of these amplitude, okay? These things do converge. They converge because of the oscillate. They all lie on the unit circle, not on the infinite circle, on the unit circle. And yet the integrals do converge. This is definitely a golf box. For example, that's one example, but there are many examples where integrals like this converge. The kind of thing that can't converge, well, if you have an infinite number of numbers, all of which are positive, and you add them up positive and bounded away from zero, all right? You have an infinite number of numbers, all of them positive and bounded away from zero. Of course, that's going to diverge, okay? In this case, you have numbers which can cancel over here as one trajectory, over here as another trajectory, over here as another trajectory, these are the amplitudes for different trajectories, and this one cancels this one. So it's not hard for an integral where the- Well, you would say if they're equally distributed around the unit circle, then you get- Zero. Yeah. In fact, most of them, except for a small fraction of them, they are- That's right. They tend to cancel a lot. The only trajectories which tend not to cancel are the ones near the classical trajectory. We don't have to discuss that now, but that is the way that you go from quantum mechanics to classical theory, at least in this formulation. If you look for the particular trajectories where you have the least cancellation, those are the trajectories of stationary action. The trajectories where the action is minimum. But that's another story. This is the quantum mechanical path integral formulation that much of modern field theory, basically all of modern field theory, quantum field theory is based on. All right. Now, I'm not going to derive the next step. I'm simply going to state the next step, but I wanted to at least explain to you what this quantity is before I show you how in practice it's used. Now when I say in practice, this is a bit of an oversimplification, but not too bad. Not too bad. The last time we talked a little bit about how the Lagrangian is used to calculate processes, particles moving from one place to another, particles interacting. When I say calculate processes, what do I really mean? I really meant calculate the probability for an initial state to go to a final state. But there are two distinct ways to think about relative, well, about quantum field theory. One is in terms of fields and the other is in terms of particles. We know that there is this duality between particles and fields. We could ask a totally different, apparently totally different question. Instead of asking suppose you had a given field value here and you want to know what the amplitude is for a final field value, we could ask supposing you had some incoming particles, quanta of the field. We express things in terms of field quanta rather than in terms of classical field configurations. Supposing I told you that the field initially was in a state which was described by a particular collection of incoming particles. Incidentally when I say the field, I may mean a collection of fields. So I tell you not the value of the field along here, but rather the particle content coming in. And I tell you what the particle content is going out. And I ask you what's the probability that you went from one particle content to another particle content? It's a question of a similar kind of question, but expressed in terms of the particle representation of quantum field theory rather than the classical field configuration. And the answer, not surprisingly, involves exactly this same object. We talked about it a little bit. I think what I told you last time was not incorrect, but it was a real small piece of it. I said what you do is you took one, I'll just remind you of something I said, and then I'm going to say it again, but in a more correct way. You take one plus the Lagrangian, remember what we did. We said let's divide up into space time into lots of little cells. It's hard to give meaning to objects like this, like this path integral directly. The way to give meaning to it is to divide up space into lots of little cells. And instead of thinking about continuous functions or even discontinuous functions, think about the value of the field in each one of these cells. That makes it more concrete. And then in the end, you let the size of the cells go to zero. So let's not let the size of the cells go to zero. Let's keep them finite like this. What I told you is that you take the quantity one plus the Lagrangian in each cell, in the ith cell or it l sub i, that's the value of the Lagrangian in each cell, and you multiply it for all the cells. Remember I said that? If you don't, it doesn't matter because we're going to say it again the right way this time. And then we did something with this to try to calculate, and I tried to show you that there are pieces in here which describe the propagation of particles, the collisions of particles. So let's go back over it again because it really is central. All right. First of all, I missed being tired. I was not terribly clear. It's really one minus i times the Lagrangian in each cell. Now one minus i, one plus a small, this is for the moment, imagine this is a small quantity I shouldn't, actually it's not one plus the Lagrangian. It's one plus the action in each cell. Now the action in each cell is the Lagrangian times the space time volume in each cell. Space time volume is delta x, delta y, delta z times delta t for each cell. Let's call that a small number, let's just call it the space time volume, let's call it a to the fourth, where a is a small number. So each little cell is small, so the action in each cell is itself small because the cell is small. Now one minus or one plus a small quantity, let's call it one plus epsilon, is an approximation to e to the epsilon. e to the epsilon is, this is approximately equal for small epsilon, but the exact formula is a power series in epsilon. That's one plus epsilon plus epsilon squared over two plus epsilon cubed over, what comes next? Three factorial, which is six, three times two times one, and so forth and so on. So one plus epsilon is approximately for small things equal to e to the epsilon, but in fact it's more efficient and correct to really write something different than what I wrote over here, namely this thing. Let's think about what this thing is. In fact, don't you have to put an h bar in the... Here? Yeah. All right. I'll probably wind up setting it h bar equal to one as usual, but right. Okay, what is this object? All right, this exponential here, what's in here, we're going to imagine replacing not by an integral but by a sum. The integral of a space and time here, just imagine that we've replaced it by the sum. And what is it a sum of? It's the sum of the action in every little cell, right? Each when I replace the sum of the integral by a sum up in here, when I replace the integral by a sum up in here, what I'm really doing is just adding up the action in all these little cells. Now, the thing about a exponential is that the exponential of a sum is the product of exponentials. e to the a plus b is e to the a times e to the b. So this can also be written as another way, and another way. You forget the summation here for a minute, that's summing over paths. Before we do that summing over histories, what this object is, it's an exponential of a sum, so it's also the product of e to the minus i over h bar a in the first cell, e to the minus i over h bar action in the second cell, a is action, e to the minus i over h bar a in the third cell. And it's exactly this kind of product, product over all the cells, not of one minus i times the action, but of e to the minus i times the action. They're close to each other. It actually wouldn't matter which we used in practice. But okay, so let's now consider question. Okay. All right, so now having this form here, let's go back to what, let's forget particles for a minute and think about fields and re-express the path integral idea. So here we have a region of space-time that's been chopped up into tiny little cells. Enough. All right, what's the idea of an initial condition? The idea of initial condition is to start on the first row of cells here and give the value of the fields at every point in there. That's an initial condition. That's the analog of an initial condition. A final condition is to specify the fields in the last row. So the question then is what is the probability amplitude that if a field is specified in a certain way on the first row and the field is specified in the last row, what's the probability to go from one to another? Or the amplitude. The amplitude that if you start at a certain way and let the system run, you will later find it in the final state. And the final state now means the field in each one of these cells here. Answer? You take e to the minus i times the sum of the action in all the cells, but we now realize that this is simply the product of e to the minus i of the action in each cell. We multiply together the action in all of these cells for a given history, for a given history, for a given history, which means a given value of the fields in each cell. And we multiply them all together. We compute the exponential of the action and then we sum it over all ways of populating the cells with fields. All possible histories means all possible values of the fields that could be in every cell with the exception of the first row and the last row. The first row and the last row are fixed by the initial and final conditions. In other words, by the initial and final conditions. We don't look at the field in the interior. We simply start the system and then detect it. And the role is sum over all possible field configurations that could exist in between. So that has reduced this idea to a discrete form in which we see that what we have here is this infinite product, or this product over actions. All right, now let's try to formulate some ideas about particles. Instead of asking the question, what if we start with a given field configuration? What if we start with a given particle configuration and end with a given particle configuration? What is the amplitude to go from one to the other? So I'll tell you how you think about it. I know we're not going to derive this. We've talked about it a good deal. We've talked about the ideas a good deal. We've talked about the fact that fields are made up of creation and annihilation operators. And what we're multiplying together here is functions of the fields. Functions of the fields and therefore functions, yeah. I read somewhere that someone thinks that field quanta are not necessarily identified with particles. All right. You would disagree with that? Well, I'm not sure who and what was the context. We've got field theory. No, I'm not sure who said it and what the context was. But you are... Well, give me a little more to go on. The question was, somebody said something. My grandmother probably said it. The field quanta in quantum field theory are not necessarily identifiable as a particle. The only thing I can think of is that he was talking about quarks. And of course, in some sense, quarks are particles. But in some other sense, they're never detectable as particles because they can never escape from one another. That's the only thing I can think of. So quarks are the quanta of the quark field, but they're never directly detected as separate quarks, well separated from other quarks. That's the only thing I can think of that he might be speaking about. Well, I guess during unitary evolution, it doesn't necessarily boil down to observables. And so you may not have particles that are identifiable at intermediate states. I don't know what they're... If you accept the idea of a quark as a particle, it's true that there's not a one-to-one correspondence, necessarily, between the fields in a theory and the particles in the theory. That's an idea which is true as long as the coupling constants in the theory are small. So there is not necessarily a one-to-one correspondence. Yeah, I mean, it is true. There are field theories that for one reason or another, you wouldn't describe their quanta as conventional particles. But at some level, it's just the difference between words. If you define particle to mean quanta, then there's no difference between them. They're indivisible. They carry energy and so forth. So for my money, I would call them particles. But yeah. Can you talk about some of the histories? Is there some limitation on what histories you can talk about? Do they have to sort of continue this in some sense? No. No, in fact, they don't have to be continuous. First of all, of course, the idea of continuity on a discrete space like this doesn't quite mean very much. The field in the neighboring cell is just going to be a different value of the field. The rule is divide the space into cells. And actual practice, this is the way quantum field theory is defined. You divide the theory into cells. You sum over all possible ways of populating the cells with values of the fields, all possible ways. No restriction to things which are approximately continuous. And in fact, quantum fields are not approximately continuous. They jiggle a great deal. And you sum over all the possible values of the fields. And that gives you your input. So in principle, for each field and each derivative, you can assign it all possible values from minus and from minus. Well, you don't assign the derivatives separately from the field. The derivatives, of course, are related to differences of the fields in neighboring boxes. So you populate with field values, and then derivatives are replaced by, yeah. And you can use, there are different rules that you might adopt. You might define the derivative here to be the difference of the field here and here. Right. Yeah, that's a minor detail. Yeah, but in any event, you might take the real line up and break it up into small intervals, but you'd still have a sum over a countable number of possible values of the field. For each field, and if there were two fields, then you'd be doing that over. Yep. By the field here, I mean all of the fields. Right, so you would give a value to each possible field in the theory at each. This looks like- It's a really nasty-looking computation. Absolutely. Well, so since these are complex functions, they're analytic functions, so- Well, the analytic functions of- But the relationships between the values of the field have to conform to what you learn in complex variables. No, no, complex variables is about the theory of analytic functions. Analytic functions are extremely smooth. The smoothest functions you can think of. These functions do not have to be smooth. They're complex valued, but they're not analytic functions. The field as functions of position, definitely not. They're on the average highly discontinuous. If I were to do the computation of the field here, how do I know- That would converge and go all the right way? Yeah, I mean how many different fields I have to consider? How many different assignments have to- So of course this by now is a major industry called lattice gauge theory or lattice quantum field theory lattice because it divides the world into a lattice. And this has been studied to death, how to do these integrals in practice. By now it is a very effective tool, but it took some 30 years to develop the computing technology. I was involved in it in the very, very beginning. I wasn't involved in the computer technology of it, but setting up the rules for lattice gauge theory. So there was a history. The first part of it was setting up the rules for it. And that took about three weeks and then something like 30 years to develop the technology to compute these things. And you're right. I mean, how do you know when you've sampled enough of the space? So there are- and a lot of the wisdom of it came from statistical mechanics where you do very much the same thing. You calculate partition functions or probability distributions for this checkerboard here, not a quantum field theory. It could be a real crystal lattice and you might be summing over configurations of whether there is or isn't an electron at each site. So a lot of the methodology came from the quantitative study of statistical mechanics systems. We have the same question. How do you know when you've sampled enough? This is not the subject of tonight's lecture. So the answer is the experts know. Well, they think they know. And they get good answers. They get good answers and agree with experiments. So it seems to work. Okay. So let's go back and remember that quantum fields are a shorthand for creation and annihilation operators. And if we're talking about, let's say, the product of some fields in this box, times the fields in this box, that can represent the annihilation of a particle in this box and the creation of a particle in a neighboring box. So we might just represent it by a particle moving from one box to another. What kind of things in the action here actually do correspond to a particle moving from one box to another? Let me tell you, there are things in the action here which, strictly speaking, are not associated with one box but with a pair of boxes, namely the derivative terms. The derivative terms, I was a little bit hasty here when I said you multiply all these things together, one for each box, as a little bit of a cheat because there are terms in the Lagrangian which are associated with pairs of boxes. So really you can think of it as summing over the boxes and summing over neighboring pairs of boxes, but that's a detail. Which terms involve pairs of boxes? The terms which involve pairs of boxes are things like the derivative of phi with respect to t squared. The derivative of phi on a lattice becomes the difference of phi in neighboring boxes. So, in here, this derivative of phi with respect to t might really be written as phi at point x, well, phi at point t and x minus phi at point neighboring time and x. Do we actually have second derivatives? No second derivatives. These are first derivatives. I mean, would we? Never. Okay. Never. Bad idea. That was real damage. Now, what about derivatives with respect to spatial coordinates? Same thing. It corresponds to the difference to neighboring points in space. And then the instruction is to take one half of the square of the derivative. Well, one half of the square of the derivative will have in it. I just want to focus now on the terms which multiply phi times phi at a neighboring point. Let me just focus on those. There's also terms in here which multiply by phi by the same, by the value of the same point. But in particular, there are terms in this Lagrangian which multiply phi at one point times phi at a neighboring point. Every time you lay down a term like this, it represents the motion of a particle from here to here. You can think of it that way. Okay. Let's concentrate on these terms. Let's forget the interactions in the Lagrangian. These are simply, these are called the kinetic terms, the quadratic terms in the Lagrangian. They're things which are easy to deal with, and they do correspond to motion of the particle from one point to another. Let's look at this. And let's include only those. We don't even need to write it in this form. Let's just write it as e to the minus i times the sum over all pairs of boxes, over all neighboring pairs of boxes. I said over all boxes, but it's clear that's not quite right. We want to think of it as sums over neighboring pairs of boxes of things like phi in one box times phi in a neighboring box. I'll use the notation x and x prime to represent neighbors on the lattice. That's what goes into the action, or that's what goes into the exponential of the action. There are some coefficients in front of it, of course, but that's of secondary importance. And now we can expand out this exponential. Let's see what's there. As one, minus i times the sum of phi of x, phi of x prime. And then things like, let's see, what's the next term? i times i is minus one, so it looks like it's minus. One of phi of x, phi of x prime squared times another factor of the sum of phi of x, phi of x prime. This could be phi of x prime, x double prime, x triple prime. X double prime and x triple prime are one pair of neighbors. This is another pair of neighbors, and I think I left out a factor. There should be a two factorial downstairs. I'm simply expanding out the exponential here. And what's the next one? Well, it has three powers of the sum, four powers of the sum, and so forth. Let's look at each term here. The first term has something which involves a sum over the lattice of an annihilation of a particle at one point and a creation of another particle at another point, at a neighboring point. The next term, oh, incidentally, the rule that I'm going to tell you again, I mean the business of telling you some rules now, I'm going to tell you some rules now, the amplitude for going from one thing to another, the rule is you've got to close off, you must not have a dangling endpoint. A dangling endpoint can only dangle like that if you've got an operation to put a particle into the system. A dangling endpoint, for example, we're going to have diagrams which have particles going from one point to another. If they end like this, they're illegal unless there's an instruction to put a particle in at this point and to take a particle out at that point. So endpoints like this, we will simply rule out as things unless they correspond to a specific instruction to put a particle in at this point and take it out. Because we're only putting particles in in the initial state and taking them out in the final state, the only endpoints would be allowed on the top and the bottom. Okay, so let's take a particular term here, phi of x and phi of x prime. These are neighboring points on the lattice and they correspond to a motion of a particle from one place to another. There's a sum of terms. There could be a term for a particle from here to here, a term for a particle from here to here, a term for a particle from here to here. But none of these will contribute unless, of course, there would be one situation in which they would contribute. That would be, of course, if this layered structure here was only two layers high and we put in a particle over here and took one out at the neighboring point. Then there would be a contribution to the amplitude of that coming from this term in the product, phi of x times phi of neighboring x. That would contribute and it would contribute to the amplitude, basically the factor minus i. It would tell you that the amplitude to go from one point to another was simply minus i, this coefficient here. Okay, but that's not very good if this, how do we get from here to here? From here to here. How do we get a particle from here to here and calculate the amplitude that if we put in a particle over here, that will detect it over here? For that, we have to find in this sum of products here, we have to have a term which will leave no dangling ends. Okay, let's just take this term here. This has a particle moving from x to x prime and then another particle moving from x double prime to x triple prime. Are you saying the second term contributes only when the initial and final state are adjacent? This term here, yeah, yeah, that's right. This notation means neighboring particles, neighboring boxes. Here we have one neighboring box and here we have two other neighboring boxes. Yeah? Well, now we have the five x prime, the first term is five x five x prime. Yeah. And the second term is five double prime. Yeah, it just means pick two neighboring and sum over all possibilities. Sum, this is a sum over all neighboring pairs. There doesn't have to be adjacent to the first. No, no, no, no, in general not, but it can be. In fact, this could even be the same pair, but it doesn't have to be. Sum over all of them. All right, so what does this one do? It moves a particle from x to x prime and this one moves a particle from x double prime to x triple prime. So one of them moves a particle from here to here. This is x prime to x double, x, from x to x prime. And then the other one moves a particle from here to here. That's x double prime and x triple prime, let's say. Obviously, there's going to be dangling ends unless, well, let's take this case here where we only have three layers. If we only have three layers, the only way to avoid dangling ends is to have the first particle connected to the second particle, sorry, the first box connected to the neighboring box and then this box connected to this box. There won't be any dangling ends except for the dangling ends which correspond to the initial particle and the final particles. So if this was only three layers thick, we would find a contribution here, namely where x prime is the same as x double prime, that transports the particle from the initial position to the intermediate position and then from the intermediate position to the final position. What would be the amplitude then associated with the particle going from here to here, at least corresponding to this term? It would be a one over two factorial and a minus sign. That would be it. We would go off the coefficients and the coefficients tell you the amplitude. Thus far, we don't have a way to get from here to here, not with two steps anyway. Two steps can, however, if this was only two layers thick, we would have now a way to get from here to here, to go across the diagonal. How do we go across the diagonal? We go again to this second order term here, the term which has a two factorial in it, and we find the term which takes us from x to x prime and then from x to x prime and then from x prime to from here to here and then here to here. What's that? There are other paths to get from here. Yeah, there's another path where you jump from here first to here and then up. I don't think there's any other besides that. Yeah, not with just two terms. Right. Okay, but now there was no reason to stop here with only two terms. Let's go on. We can have three terms. Three terms would correspond to going from one box to an adjacent box. Well, it would correspond to three distinct steps. And if those three distinct steps were connected together into a chain, in other words, in this sum of products, we found the term which took us, we could come right back again, that would contribute together with the direct jump from here to here, would contribute to the amplitude to go from this point to this point. So you see, built into this prescription is something like the original particle path integral idea. The amplitude to get from here to here is the sum over the amplitude to go from here to here and the amplitude to go through another root, also this one back here. And eventually, if you expand this out to arbitrary order, there will be a term in there for every possible root that you can take to go from any initial configuration to any final configuration of that particle. So first of all, just thinking about a single particle moving in spacetime, this field Lagrangian contains information about the amplitudes to go from, now let's take an arbitrary, to go from any point to any other point. What you do is you add up all of the possible ways of going there and what is the coefficient for each way of going there? You read it off the coefficient that multiplies that particular term in this sum. One over two factorial, one over three factorial, some of them have eyes, some of them don't have eyes. Remember that I squared is minus one, so some of them have eyes and some of them don't have eyes and so in general, the amplitude to go from one point will be a complex number. That's one of the things that's new here, that they can propagate up and down and that's a feature of relativity. What's that? Oh, yeah. Oh, indeed. Yeah. Yeah. All right. Let's talk about going backward in time. You could think of, let's use one that goes backward in time. You can think of this as either allowing a new rule where particles can go backward in time or you can follow time forward and say what really happened here is the particle moved from here to here and then a particle pair, particle and antiparticle in fact, were created, the particle half of it going to here and the antiparticle combining together with the original particle to annihilate. Think of it either way. But the rule now allows trajectories which go backward. Let's come back to this term here. In this form, since you have n factorials in denominator, does this mean that you sort of have convergence in the nulls? Well, the n factorials certainly help the convergence, but they're not enough to make it converge. Yeah. That's right. No, no, no, no, no, no, but the sum over convergence. Configurations doesn't have to converge. Yeah. This part converges. Yeah. And this is going to be some finite number of total configurations considered. Yeah. Okay. Let's, all right, now let's go on to a slightly different problem. We studied the particle moving from one point to another. Now let's suppose we put in two particles and we want to know what the probability for the two particles to go to two other particles is. In other words, we want to start, and I'm going to take the case where there are only two layers for a moment. There's the case where there are only two layers for a moment. And I want to know the probability that if I start a particle over here, it will get to here. No, if I start two particles. One over here and one over here, that they'll get to here. Okay. Well, come back to this term. Now remember what this term did for me before. Before it allowed me to hop three units. But that same term allows two particles each to go one unit. Look at it. This can create a particle. This can annihilate it. This can create a different particle and annihilate it. This corresponds to a graph where you start a particle in here and goes to here. That's this factor. And the other factor, instead of taking the particle that's already there and moving it over here, it just takes a totally new particle and moves it to here. So this same factor, the same term contributes both to the motion of a single particle, three boxes, and it contributes to two particles each moving one box. Did I say that right? More or less. Three boxes, one, two. Two boxes. Yeah. So there's a lot in here besides just motion of a particle from one place to another. It has information in it about any number of starting particles going to any other number of final particles. Now in fact, particle number doesn't change as long as we just take this into account. Why not? Because every starting, every dangling endpoint here, well, let's see. Yeah, it has to end somewhere. Well, if you start and end with different numbers of particles, then obviously something has to happen. Well, that would mean there has to be a dangling in somewhere. Supposing I wanted to, I mean, I can't, how would you get from one particle to two particles? What if I start with electron and positron, I could end up with no particles? Well, OK, that would be something, that would just be something like this. Two particles. Let's put another layer in here. Yeah. Yes. That's true. That would correspond to an electron and a positron, for example, annihilating each other. The only problem with it is it doesn't conserve energy. So in fact, when you add it up, all these amplitudes, you would get zero. But on the certain circumstances, if there was an electromagnetic field to soak up the energy, yes, you're right. It would have information about an electron and positron annihilating. Right. So there's just a lot in here. There's a lot in here. But not everything. Not everything. To find out the other things that are implicit in the field theory, there are other terms in the Lagrangian. Now what you put in the Lagrangian is determined by experiments. And it's really just a way of codifying the results of experiments. But what else can you have in Lagrangian? We had the derivative terms, let's say, phi dot squared minus the derivative of phi with respect to x squared and y squared and z squared, all that kind of stuff. That was the stuff that I wrote down as phi at one point times phi at another point, at a neighboring point. Then there are possibly other things to just involve phi and not its derivative. So for example, you could have things like phi squared. Remember what the coefficient is? It's usually a 1 half here. That's a convention. Remember what the coefficient of phi squared is? The square of the mass of the particle. We should divide by 2. That's, again, a convention. And if we included this term here, what that would do, it doesn't move a particle from one place to another. It absorbs a particle at one point and emits it from the same point. From the same point. So what would correspond to graphical constructions where we would add a new rule, f of particle enters a region here from one box to another. One of the terms that can act is this mass squared term. It would simply take the particle and do nothing to it. It would leave it in the same spot. Then another term could come and move it, but it would weigh that particular path with a factor of m squared. Every time this term acts, it weighs the path with another factor of m squared in the amplitude. So there would be terms where no m squares act, where only these derivative terms act. In other words, when we're expanding this product out, we only have the derivative terms. That would be the theory of massless particles. The theory of massive particles would have a new rule that one of the possible things that can happen at every step is you may or may not just weigh that path at that point with the m squared. So the mass term is another rule for how you evaluate these paths, these path integrals. Another rule. It's we don't have to get into detail. The main point is that the mass of a particle is codified by another term in the Lagrangian, which is m squared phi squared, which weighs paths in such a way that it knows about their mass. All right, but more interesting is the more complicated terms, things like phi squared. Let's put a g, sorry, phi cubed. Phi cubed can do a number of things. It can annihilate one particle and create two particles. It can annihilate two particles and create one particle. It can create three particles. It can annihilate three particles. All right, so that's a new kind of thing that can happen on this checkerboard here in expanding out the action, expanding out the exponential. There might be additional terms here, for example, g phi cubed. When we expand out, we can have not only these products which take a particle from one point to another, but also processes where particles are created and annihilated in threes, either two in, one out, two out, one in, or three in, or three out. What that looks like on here is, for example, supposing we have several of these quadratic terms which move particles in one cubic term. What can that look like? Well, what it can do is it can take a particle, moving it from here to here, and suddenly in that box, we put a phi cubed. In that box, it can absorb one particle and emit two more. Now, the next thing that could happen is the two particles could move to other boxes. But basically what it does is it takes a particle, absorbs it, and creates two, and those two particles can be on their way. And this would then contribute to a process where one particle came in, two particles went out. In fact, you can make very, very complicated things. You can make things of, let's go back to the one particle going from one place to another. One particle going from one place to another. Let me stop drawing checkerboards. Let's just imagine the checkerboard there. One particle going from one place to another can go directly, just a bunch of small hops, a bunch of these hops. But also something else can happen. On the way, at some point, the cubic term could act. If the cubic term can act, it can take that one particle coming in and split it. Split it into two. And then both particles move along until another cubic term, something quadratic in the cubic term, something with two cubic terms in them, can create an extra particle and reabsorb it. So in fact, the actual amplitude to go from one point to another by one particle is not just the sum of simple trajectories going from one point to another through the space time, but is more complicated. It has infinitely more complicated. It has processes where the particle splits and rejoins. In other words, where the particle emits a second particle which is then reabsorbed. An example of this would be an electron moving along and emitting a photon and reabsorbing it. Part of the amplitude for an electron to go from one place to another is through the process of emission and absorption of a photon. It can get wildly more complicated. Of course, we have to add all of these up. The rule is add up the amplitude for every possible way to go from the initial state to the final state. You could have several of these splitting and joinings. You could have things jumping across the gap between particles. And this can get arbitrarily complicated from the point of view of electrons and photons, things like this could happen. Electron goes through, photon emitted, another photon emitted, but even worse, the photon can break up into an electron and a positron. But then the electron and positron can exchange a photon between them. All of these processes come out of expanding out the action like this and finding the individual terms which do all of these things. If you want the amplitude for any one of these processes, you just go back to what appears here, find the term that you're looking for, and find the coefficient in front of it. It may or may not have an i, depending on whether it's an even power or an odd power, and it will have some factorials and so forth. That will give you the amplitude for a specific process. But then the full amplitude to go from one thing to another is the horrendously complicated sum of all of these things. Right, so you may have certain things that happen in multiple of those histories. In multiple of those histories, you may have similar terms, which then the overall amplitude for that process would be actually from all the different histories. Yeah, from all the different histories going from the initial to the final. So there are two ways to think about field theory. One is to think about fields and think about histories as initial values of the fields, final values of the fields, and you sum over all the values of the fields in between. And the other is this particle way to think about it, where the field is replaced by a distribution of quanta, and again, amplitudes for initial configurations, which means an initial particular set of particles and a final set of particles. The basic processes, the basic underlying processes which happen are governed by this Lagrangian. That is the most important thing. Any time you have more than two fields interacting, you have an interesting interaction process where particles can split and join and so forth. What would happen if you had phi to the fourth here, which is a perfectly good interaction? That would be some place where a particle, for example, could come in and break up at the three other particles. One to absorb the initial state, three to emit the final state, or it could be a scattering in which two particles come along, collide, and two go off. Anything that looks like a vertex with four particles altogether, two in and two out, or two in, or three in, one out, or so forth, or just four out, or four in, they're all governed by this term. And so forth and so on. Let's see, how are we doing? So that's the spontaneous arrival of four particles at the final state? Yeah, except once again, it would violate energy conservation. So if you really worked it out and added up the amplitude from every position where this could happen, every spacetime position where it could happen, you would find out that the constant doesn't happen. It doesn't happen. In other words, they all add up to zero. So processes which violate energy typically all add up to zero in these things, energy and momentum. So the conservation of energy falls out of the new calculation? Yes. And the conservation of momentum, same thing? Yeah. Now, we did some examples. I'll show you how integrating the vertex over all time conserves energy, how integrating it over all space conserves momentum. Same thing is true here. Integrating the position of these vertices and the endpoints of these things over all possible positions in space and time will simply add up to zero unless energy and momentum is conserved. But apart from the conservation laws, everything that you can draw down will happen with some probability. Now, how could it possibly be that you add up this humongous thing and the answer doesn't come out infinite? Well, the answer is that there are coefficients here. There are coefficients here, these are called coupling constants, particularly the coefficients for the higher powers here. We could call this G3, we could call this G4. And the rule is in a diagram, in a Feynman diagram like this, every time you have a vertex, let's say with three particles coming together, the amplitude contains a factor of G, G3 in this case. So the more complicated the diagram gets, the more powers of G that it has. Each vertex gives you a power of G. If G is a small number, then each successive power of G gives you a smaller and smaller contribution and you have a chance that the thing might be able to converge. The technical question of whether it really converges or not is a very difficult one, but you can see that you have a chance if G is a small number. If these coupling constants are small numbers, then the simpler the process is, in other words, the fewer number of vertices, the bigger the contribution and the larger the number of vertices, the more complicated, the smaller the contribution and you have a chance at any rate at some sort of convergence to an answer. If the coupling constants are large, it means these series do not converge and if they don't converge, it doesn't mean that the theory is wrong. It just means you've chosen a naive prescription for trying to work with it. But in practice, most of the quantum field theory, most of the things we do with quantum field theory are based on the assumption of small coupling constants. And in those circumstances, it's possible to evaluate these finite, not just evaluate them, but add them up and find out that the one with four vertices is much smaller than the one with two vertices. So that's the name of the game. Do we do color theory in the same way? Yeah. We're going to talk about color theory. Now the next step is to write down the list of all particles and to write down either in the form of Lagrangians or in the form of Feynman rules. The Feynman rules and the Lagrangians have the same information in them. Three identical particles coming together, let's say three bosons, that's five cubed. Particles going from one point to another, those are the kinetic terms, the quadratic terms here. So the Feynman type rules and the Lagrangian are the same thing. And you can either describe the world by specifying all the vertices incidentally. The lines between one point and another are called propagators. The place where more than one particle, more than two particles come together, those are called vertices, vertices and propagators. But as I said, they're just simply another shorthand way of describing the Lagrangian. The Lagrangian is a shorthand way of describing the graphs. So we would have stated all of particle physics in a kind of dumb way by simply writing down the list of all particles, writing or constructing or even just expressing a quantum field for each particle, and then writing down a Lagrangian for all those particles. We contain all of the interactions or we can write down all of the particles and simply specify what all the possible vertices are. Either way, they come to the same thing. They're kind of a complete list of everything that can happen. Once you know everything that can happen, you can start thinking about calculating amplitudes for particles to go from one place to another. Particles to collide. Particles to collide and scatter. Here's particles colliding and scatter, two electrons colliding and scattering just by a photon jumping from one place to another. Again, Lagrangian tells you how to calculate the amplitude for each of these. As I said, it contains only information, including symmetries, including conservation laws. So I think it's probably, shall we write down the list of all elementary particles? So when you write down a final diagram, basically the bottom is the initial configuration and the top is the initial configuration. That's right. The bottom is the initial configuration of particles. The top is the final configuration of particles. As I said, it is the kind of complementary way to think about quantum field theory, complementary to the field description where you would start with a given field configuration and end with a given field configuration. Incidentally, in some sense, the complementarity between particles and fields is very much like the complementarity between momentum and position. These are two different ways to describe the same reality, and there are uncertainty relations between them. If you know with precision the number of particles, then you know that there's uncertainty in the values of fields. If you know the values of fields, then there's uncertainty in the number of particles. So these are really two complementary ways to describe the same thing. And the Lagrangian is useful in both contexts. Question. The couple of constants, are those predicted by theory or are they perfectly determined? Because under certain circumstances, there are symmetries which relate coupling constants. So this or that coupling constant for this process might, for symmetry reasons, be related to another coupling constant for a different process. So some of them are related to each other by symmetries. But their values in general are simply come from experiment. All right, much of this, if you really go and learn about quantum field theory, has its beauty. It does have elegance and beauty and so forth. Now when we start writing down the list of facts, what particles there are, what their masses are, what their coupling constants are, it's just a mess. It's just a very ugly mess with very little coherence, except the coherence that comes from symmetries. Symmetries tell you relationships between different kinds of particles and their processes. But apart from the symmetries, which are few and not so much, it is a large number of random facts about a somewhat unmotivated collection of different kinds of particles. But still you're pretty confident of the coupling constants because of your… Once you measure them, you measure them. If you change it a little bit, then you get wrong answers. Oh absolutely. No, no, no, I mean, okay, let's put it this way. You have too many particles, which means too many fields, too many for any aesthetic sense. You have a lot of them, a hundred of them or something. A lot of coupling constants, which are just more or less numbers pulled out of experiment, and a lot of masses, which range all over the map. A few relationships between them, but once you know them, once you know them, that's it, you can calculate with great precision anything about those particles, anything about those particles, anything about quarks, electrons and photons and so forth, means atomic physics, or it means particle physics, it means atomic physics, it means nuclear physics, it means chemistry, and maybe it means biology, maybe. So, the amount of input that goes in is perhaps bigger than you might like, but the amount of output is huge. Question for electrons, for example, how many of these G sub n's would be? G2, G3, G4, these be calculated? How many G's are there for electrons and photons? And electrodynamics? One, the electric charge. The electric charge? Well, in a sense, the electric charge is a kind of G3. The electric charge is a kind of G3. The electric charge is the coefficient in the amplitude representing an electron coming in, an electron going out, and a photon being emitted. In terms of symbols, this diagram would be represented by an electron going out, an electron coming in, and the field operator for a photon, which is A. I'm not writing down the details, there are some power, sorry, there are some direct matrices, but then the numerical coefficient in front of it is the electric charge in certain units, certain dimensionless definition. The only processes in quantum electrodynamics are the emission and absorption of photons from electrons. That's all there is. Well, if there are electron collision with a photon, there's just a little bit of electron collision. The simplest process would be photon absorbed. I meant to say proton. Oh, well protons are not usually considered a part of quantum electrodynamics. Now you can think that when you do quantum electrodynamics, you are doing atomic physics. You think of the proton as a point infinitely massive. You don't think of it as something described by quantum field theory. You just think of it as a nailed down, heavy particle which never moves until you open up the theory of protons and neutrons. So the scattering by a photon, a photon by an electron, this is the first diagram. But the electron is moving at almost the speed of light relative to the proton? Proton, yes. Right. Yeah, that's right. Let's draw some diagrams just to get some feel for what kind of things we have to add up. This is the lowest order diagram that goes into the scattering of a photon by an electron. There are two powers of the electric charge. So there's an E squared. But this is the amplitude. The amplitude gets squared to find the probability or the cross section. Cross section is another way of speaking about the probability for the scattering. All right. So that means that the whole process, the probability is proportional to the electric charge to the fourth. The electric charge, at least with the suitable definitions, is a small quantity. The square of the electric charge with, again, with the suitable definition is about one, well, that's the number which is 1 over 137, the fine structure constant. So this is proportional to the square of the fine structure constant. And so scattering in quantum electrodynamics is a unlikely process. A photon comes in and strikes an electron. What is the probability that it goes right through it versus the probability that it really scatters it? The probability that it really scatters it is governed by this e to the fourth, and it's a very small number. Now we have to add all the processes that we can find, add them up to form a real amplitude. Incidentally, there's another one here which looks like this, where the final photon is emitted before the initial photon is absorbed. That's another process. We have to add them all. But then we start adding more complicated things. Let's focus on here, and we can add another photon in many ways. This is not the only way we could have to go from here to here, from here to here, many photons. But how many powers of electric charge does this have? This has four powers of electric charge in the amplitude and eight in the probability. So this is even weaker. But then we can do even more complicated things. We can put an electron-positron pair in here. Each time we complicate it, we add two more powers of electric charge. So in quantum electrodynamics, the expansion is typically an expansion in powers of the square of the electric charge. Every time you add another complicating structure to a Feynman diagram, it's two powers of electric charge, and it means it's about a couple of hundred times smaller in amplitude than the previous one. That's why when you add them up, it looks like it converges, because more and more complicated they get the weaker the probability from that particular configuration. Incidentally, you don't square this and add it to the square of this and add it to the square of this to find the probability. You add them up, and then you square. That's the rule. Add and then square. Add and then square. That means there's cross terms between them in calculating the probabilities, called interference terms. So that's the structure of quantum field theory. That's what we do with it. The next issue, as I said, we're now at the stage where we can really start to sensibly talk about particle physics. We can name all the particles. We can specify their masses. Those are the parameters in Lagrangian here that we've spoken of. We can write down the Lagrangians governing them partly from theory, partly from experiment, and discuss their symmetries. The symmetries can be discussed by looking at the Lagrangian. We'll do that. We can write down the standard model of particle physics and discuss some of its properties. So I think we're off to, I think we're set now to move ahead and really discuss the world of genuine real particles and what can be discovered in the world. Instead of that m squared, you could add the Higgs field and then that would be a quadratic state. We're going to discuss that. Where are the five of the derivatives? Yeah? Where are they at the same time? Oh, they're here, x and x prime could stand for two different times. That's how we manage to move vertically. All right. Now, where does the parameter correspond to the particle just somewhere in the penitentiary? There's also some terms. Now, there are various terms where the particles just stand still. When we wrote down phi of x minus phi of x prime squared, that did have terms in it like phi of x squared and phi of x prime squared. Those are terms where the particles stand still. The cross terms are the ones where it moves. So there are terms where the particles stand still both from this and from this term. Right? Yeah. Okay. For more, please visit us at stanford.edu.
 Stanford University.  Alright, last time I started to tell you something about how quantum field theory gives rise  to a theory of the particle interactions through an object called the Lagrangian.  I think that was not terribly clear, so I want to go back to it a little bit before  starting to discuss the details of particle physics.  We're still thinking about quantum field theory.  The basic technique that I was alluding to last time is called the path integral method  of quantum mechanics.  And it is the most direct route to Feynman diagrams, to the theory of particle interactions  that are based on Feynman diagrams.  So let me just try to briefly go through the basic ideas, which we started to do last time,  but I don't think I was sufficiently clear.  Anyway, the path integral formalism or method in quantum mechanics due to Feynman is a generalization  as the quantum mechanical version of the principle of least action.  So I just want to remind you what the principle of least action is.  If we're talking about a particle, now I don't mean a particle from the particle physics  quantum field theory point of view, just a classical Newtonian particle, the motion of  the classical Newtonian particle from one point of space time to another, just very  quickly, is determined by a Lagrangian.  Lagrangian is a function of the coordinates of the particle, the time derivatives of the  particle, and from the Lagrangian, whatever the Lagrangian is, one constructs the action.  And the action is an integral along the trajectory of the particle of the Lagrangian.  For every trajectory, whether it's the true trajectory or not the true trajectory from  here to here, by the true trajectory, I mean the solution of Newton's equations from one  space time point to another space time point, whether or not the trajectory is a solution,  it has an action, the integral of the Lagrangian along the orbit.  The classical principle of least action is that the trajectory followed by the particle  through space time minimizes the action.  From that, you can derive differential equations, and those differential equations are called  Newton's equations.  And I will assume that you know a little bit about this idea of action.  Now, the same idea applies to classical field theory.  And the way the idea works is for a particle, for an ordinary particle motion, the Lagrangian  is a function along an orbit, a function of an orbit.  For a field theory, and of course it's an orbit which connects some initial configuration  to a final configuration.  For a field theory, the equations of motion of a field theory, classical equations of  motion, are also determined from a Lagrangian.  The Lagrangian is a function of whatever fields, I'll call them generically just phi, and not  just the time derivatives of the fields, but the space derivatives and the time derivatives,  let's say derivative of phi with respect to x mu.  That's just a shorthand way of saying that the Lagrangian depends on phi and derivatives  of phi.  One thing which is required by the theory of relativity is that the Lagrangian be a  scalar.  That a transform on the Lorentz transformations is a scalar, but that being said, the Lagrangian  can be anything built out of phi's and derivatives of phi.  What is the action?  Imagine we have a region of space time, time as usual flows upward, space is horizontal,  and we have a region of space time between an initial time and a final time.  Then the action in this region, the action is the action in this region just as the action  for a particle trajectory is the action from one end point of the trajectory to another.  Here it's the action in a region of space between an initial configuration and a final  configuration.  The action is equal to the integral over that region, let's write a d4x which means the  time and the x and the y and the z of this Lagrangian.  What's the total integrated value of the Lagrangian over space and time, whereas for the particle  it's just an integral over time?  Now what do you do?  What's the analog of a starting point for the particle?  A value of the field on the initial surface here.  Given not just one field, but all the fields in the problem, prescribing the values of  the field on the initial surface here and prescribing the values of the fields on the  final surface, you can ask is there a solution of the equations of, oh let me go back a step,  forget that.  Yes, is there a solution of whatever the equations of the theory are which starts with a given  initial value of the fields at time t equals zero, let's say, and ends with the fields  being something different that's some later time?  The answer is the principle of least action again, the equations of the theory are formulated  by saying yes, there always doesn't exist something in between which is a correct solution  of the theory which has a given initial value of the fields here and a given value of the  fields at the end point, and the correct solution of the fields in here is the one which minimizes  this action, one which minimizes the action.  So the principle of least action gets extended to a kind of space time principle of least  action where the degrees of freedom in here are fields.  Okay, now let me very quickly remind you what the quantum mechanical use of action is.  How exactly you derive the classical principle of least action from the quantum mechanical  version of Feynman's quantum mechanical version, we don't really need to get into here, you  can go look it up and anything about path integrals, but for a particle, let's come  back to the particle motion for a moment, according to Feynman or according to quantum  mechanics, a thing that you might want to calculate is the amplitude.  Is there anything wrong with the value of a field returning to itself?  No, no, no, no more than the possibility of the position of this thing returning to itself.  So how do you go from one space time event to another space time event?  Why space time events?  We're having values of fields here and values of fields here.  Well, integrating d4x, so there has to be some d4 trajectory.  No, the trajectory is replaced by a history, a history of the field from the beginning  to the end.  A history means the values of the fields all along here, everywhere is in here.  The idea of a trajectory becomes a trajectory in field space, which means-  We have to specify actually the coordinates as well as the field values to get an initial  point.  To get an initial point, what are they?  Well, you're specifying the initial boundary condition or whatever as field values.  Field values all along some initial surface.  You pick the initial surface.  In other words, you pick time- It's the whole field at some cross-sectional-  Yeah.  All right, which is another way of saying the whole field at a given instant of time.  And the final configuration is the whole field at some later time.  And the question is, is there a solution of the theory, whatever theory is, which interpolates  in between these in the same way that a trajectory interpolates from here to here?  The rule is minimize the action.  That if carried out, we did this number of classes ago, but if that's carried out, the  requirement of minimizing the action leads to partial differential equations for the  fields and those partial differential equations are things like Maxwell's equations, things  like Einstein's equations, and so forth.  All right, so let's come back now to the quantum mechanical idea.  The quantum mechanical idea has to do with amplitudes.  Amplitudes are things out of which you construct probabilities.  So for example, you can ask the question, what is the amplitude that if a particle is  injected into the world, however it's injected in, at a spacetime point x and t, what is  the amplitude that if I look for it at a later time, let's say this is at x and time t, what's  the amplitude that if I look at a later time, let's call it t prime, that I will find it  at position x prime and t prime.  So it's the amplitude that if I start a particle at this point, close my eyes for a while and  then turn on a detector which looks for the particle at a certain place, the amplitude  for finding it there.  That amplitude is a complex number that depends on the initial point, on the final point.  It's a complex number whose magnitude squared is the probability.  That's the rule.  Where the amplitude or multiply by its complex conjugate and that is the probability for  the particle to go from here to here.  Feynman's rule is the following.  It says take the action for any trajectory, take an arbitrary trajectory and write down  the expression e to the i minus i, the action of the trajectory.  We can just write it out.  It's integral of, now we're talking about particles now, integral of the Lagrangian  dt from one point to another point.  It's a function of or functional of the trajectory.  And now, actually there's a factor of h bar in here, a factor of h bar in here, that's  where quantum mechanics comes into it.  Sum this or integrate it over all possible trajectories.  Well, in non-relativistic quantum mechanics, you don't let them go backward in time.  Let all possible trajectories which never loop back on themselves in time, sum that  over all the possible trajectories.  Each trajectory, it's a complex number.  Air them all up.  Now, how do you add up such a thing in some complicated kind of integral?  But we needn't worry too much about the details of how you actually calculate it.  That according to Feynman is the amplitude for going from here to here.  It's the sum of all possible classical roots.  Classical root does not mean a solution of the equations, it just means a possible root.  It's the sum of all possible roots whether or not the solutions of the equations, of  Newton's equations, of e to the minus i times the action, measured in Planck's constant units.  Incidentally, Planck's constant has units of action.  So this is simply the action in units of Planck's constant.  The e arises because to go a little step involves something very one plus an act term in the  action, you have to multiply these both together.  Yes, right, exactly.  That's exactly right.  The other thing here is the i, the i is part of quantum mechanics, what can I say?  Yeah?  That sum is the probability.  That sum is the thing whose square is the probability.  By square I mean times its complex conjugate.  This sum could hardly be a probability because in general it's complex.  It's got an i in it.  If you multiply it by its complex conjugate, that is the probability.  This is the probability amplitude.  The complex thing.  So this is Feynman's formulation of quantum mechanics and now it can be extended to quantum  field theory.  In quantum field theory, the corresponding question, now we're not actually going to  formulate it this way because, but this is a correct way to formulate it.  We're going to fudge it a little bit and make our way through it without any rigor.  But the same idea, you start with an arbitrary configuration of the field.  So that's some initial phi of x.  X now being just x and not t.  An initial configuration of the field.  And now you look at some final configuration of the field.  Let's call it phi prime of x.  These are two different functions.  And you can ask, what is the probability amplitude that if I start the field at some  value and let the system go and then detect the field at a later time, imagine you had  some way of starting an initial condition with a field being prescribed at every point  of space.  Of course, that's an idealization.  You don't have a chance in the world of doing that, really.  But you had some sort of apparatus which allowed you to start the field in an arbitrary configuration,  let it go, and then you had another apparatus which detects the values of the field at every  point.  Then you can ask, what is the probability that if you start the field in a given way  that at a later time the field will have some value?  Again, that probability is determined in terms of a probability amplitude.  And the probability amplitude is, again, the sum over all possible ways of interpolating  the field between initial and final, all possible, exactly what I said, all possible ways of  filling in between the initial state and the final state with a field value at every point  of e to the minus i h times the field action.  That's the integral dx dt dx dy dz d4x of the Lagrangian, which is a function of phi  and the derivatives of phi.  So you evaluate the Lagrangian at every point, integrate it up.  That gives you the action for a particular, let's call it a trajectory.  It's not really a trajectory in any ordinary sense, but it's a history.  Let's call it a history, better yet.  For each history, each possible history that you can imagine, it doesn't have to be a true  history or a real history, for every possible history you can imagine, there is an action.  The action is itself an integral, but then you consume this over all possible histories,  and that is the amplitude to start with a field value phi and end with a field value phi prime  between these two trajectories here.  That's the notion of a path integral.  I'm not concerned that there is an infinite number of paths, but the action is going to  be a finite number for each of these paths.  Yeah, but it's a finite complex number, right?  Yeah, so how can that converge to anything?  Why is that not the infinite number?  No, there are many, many integrals.  Well, okay, you're asking.  Let me phrase it a little more strongly.  Aning up an infinite number of numbers, well, that's nothing special.  I think what you're pointing out is that each one of these numbers is not only finite, but  has a magnitude equal to one.  Or something that is not decreasing.  It has a magnitude equal to one.  It's an exponential of i times something, right?  So any number like this is a number which on a complex plane, not minimizing it.  You're summing over all.  You're just summing over all of these amplitude, okay?  These things do converge.  They converge because of the oscillate.  They all lie on the unit circle, not on the infinite circle, on the unit circle.  And yet the integrals do converge.  This is definitely a golf box.  For example, that's one example, but there are many examples where integrals like this  converge.  The kind of thing that can't converge, well, if you have an infinite number of numbers,  all of which are positive, and you add them up positive and bounded away from zero, all  right?  You have an infinite number of numbers, all of them positive and bounded away from zero.  Of course, that's going to diverge, okay?  In this case, you have numbers which can cancel over here as one trajectory, over here as  another trajectory, over here as another trajectory, these are the amplitudes for different trajectories,  and this one cancels this one.  So it's not hard for an integral where the-  Well, you would say if they're equally distributed around the unit circle, then you get-  Zero.  Yeah.  In fact, most of them, except for a small fraction of them, they are-  That's right.  They tend to cancel a lot.  The only trajectories which tend not to cancel are the ones near the classical trajectory.  We don't have to discuss that now, but that is the way that you go from quantum mechanics  to classical theory, at least in this formulation.  If you look for the particular trajectories where you have the least cancellation, those  are the trajectories of stationary action.  The trajectories where the action is minimum.  But that's another story.  This is the quantum mechanical path integral formulation that much of modern field theory,  basically all of modern field theory, quantum field theory is based on.  All right.  Now, I'm not going to derive the next step.  I'm simply going to state the next step, but I wanted to at least explain to you what this  quantity is before I show you how in practice it's used.  Now when I say in practice, this is a bit of an oversimplification, but not too bad.  Not too bad.  The last time we talked a little bit about how the Lagrangian is used to calculate processes,  particles moving from one place to another, particles interacting.  When I say calculate processes, what do I really mean?  I really meant calculate the probability for an initial state to go to a final state.  But there are two distinct ways to think about relative, well, about quantum field theory.  One is in terms of fields and the other is in terms of particles.  We know that there is this duality between particles and fields.  We could ask a totally different, apparently totally different question.  Instead of asking suppose you had a given field value here and you want to know what  the amplitude is for a final field value, we could ask supposing you had some incoming  particles, quanta of the field.  We express things in terms of field quanta rather than in terms of classical field configurations.  Supposing I told you that the field initially was in a state which was described by a particular  collection of incoming particles.  Incidentally when I say the field, I may mean a collection of fields.  So I tell you not the value of the field along here, but rather the particle content coming  in.  And I tell you what the particle content is going out.  And I ask you what's the probability that you went from one particle content to another  particle content?  It's a question of a similar kind of question, but expressed in terms of the particle representation  of quantum field theory rather than the classical field configuration.  And the answer, not surprisingly, involves exactly this same object.  We talked about it a little bit.  I think what I told you last time was not incorrect, but it was a real small piece of  it.  I said what you do is you took one, I'll just remind you of something I said, and then I'm  going to say it again, but in a more correct way.  You take one plus the Lagrangian, remember what we did.  We said let's divide up into space time into lots of little cells.  It's hard to give meaning to objects like this, like this path integral directly.  The way to give meaning to it is to divide up space into lots of little cells.  And instead of thinking about continuous functions or even discontinuous functions, think about  the value of the field in each one of these cells.  That makes it more concrete.  And then in the end, you let the size of the cells go to zero.  So let's not let the size of the cells go to zero.  Let's keep them finite like this.  What I told you is that you take the quantity one plus the Lagrangian in each cell, in the  ith cell or it l sub i, that's the value of the Lagrangian in each cell, and you multiply  it for all the cells.  Remember I said that?  If you don't, it doesn't matter because we're going to say it again the right way this time.  And then we did something with this to try to calculate, and I tried to show you that  there are pieces in here which describe the propagation of particles, the collisions of  particles.  So let's go back over it again because it really is central.  All right.  First of all, I missed being tired.  I was not terribly clear.  It's really one minus i times the Lagrangian in each cell.  Now one minus i, one plus a small, this is for the moment, imagine this is a small quantity  I shouldn't, actually it's not one plus the Lagrangian.  It's one plus the action in each cell.  Now the action in each cell is the Lagrangian times the space time volume in each cell.  Space time volume is delta x, delta y, delta z times delta t for each cell.  Let's call that a small number, let's just call it the space time volume, let's call  it a to the fourth, where a is a small number.  So each little cell is small, so the action in each cell is itself small because the cell  is small.  Now one minus or one plus a small quantity, let's call it one plus epsilon, is an approximation  to e to the epsilon.  e to the epsilon is, this is approximately equal for small epsilon, but the exact formula  is a power series in epsilon.  That's one plus epsilon plus epsilon squared over two plus epsilon cubed over, what comes  next?  Three factorial, which is six, three times two times one, and so forth and so on.  So one plus epsilon is approximately for small things equal to e to the epsilon, but in fact  it's more efficient and correct to really write something different than what I wrote  over here, namely this thing.  Let's think about what this thing is.  In fact, don't you have to put an h bar in the...  Here?  Yeah.  All right.  I'll probably wind up setting it h bar equal to one as usual, but right.  Okay, what is this object?  All right, this exponential here, what's in here, we're going to imagine replacing not  by an integral but by a sum.  The integral of a space and time here, just imagine that we've replaced it by the sum.  And what is it a sum of?  It's the sum of the action in every little cell, right?  Each when I replace the sum of the integral by a sum up in here, when I replace the integral  by a sum up in here, what I'm really doing is just adding up the action in all these  little cells.  Now, the thing about a exponential is that the exponential of a sum is the product of  exponentials.  e to the a plus b is e to the a times e to the b.  So this can also be written as another way, and another way.  You forget the summation here for a minute, that's summing over paths.  Before we do that summing over histories, what this object is, it's an exponential of  a sum, so it's also the product of e to the minus i over h bar a in the first cell, e  to the minus i over h bar action in the second cell, a is action, e to the minus i over h  bar a in the third cell.  And it's exactly this kind of product, product over all the cells, not of one minus i times  the action, but of e to the minus i times the action.  They're close to each other.  It actually wouldn't matter which we used in practice.  But okay, so let's now consider question.  Okay.  All right, so now having this form here, let's go back to what, let's forget particles for  a minute and think about fields and re-express the path integral idea.  So here we have a region of space-time that's been chopped up into tiny little cells.  Enough.  All right, what's the idea of an initial condition?  The idea of initial condition is to start on the first row of cells here and give the  value of the fields at every point in there.  That's an initial condition.  That's the analog of an initial condition.  A final condition is to specify the fields in the last row.  So the question then is what is the probability amplitude that if a field is specified in  a certain way on the first row and the field is specified in the last row, what's the probability  to go from one to another?  Or the amplitude.  The amplitude that if you start at a certain way and let the system run, you will later  find it in the final state.  And the final state now means the field in each one of these cells here.  Answer?  You take e to the minus i times the sum of the action in all the cells, but we now realize  that this is simply the product of e to the minus i of the action in each cell.  We multiply together the action in all of these cells for a given history, for a given  history, for a given history, which means a given value of the fields in each cell.  And we multiply them all together.  We compute the exponential of the action and then we sum it over all ways of populating  the cells with fields.  All possible histories means all possible values of the fields that could be in every  cell with the exception of the first row and the last row.  The first row and the last row are fixed by the initial and final conditions.  In other words, by the initial and final conditions.  We don't look at the field in the interior.  We simply start the system and then detect it.  And the role is sum over all possible field configurations that could exist in between.  So that has reduced this idea to a discrete form in which we see that what we have here  is this infinite product, or this product over actions.  All right, now let's try to formulate some ideas about particles.  Instead of asking the question, what if we start with a given field configuration?  What if we start with a given particle configuration and end with a given particle configuration?  What is the amplitude to go from one to the other?  So I'll tell you how you think about it.  I know we're not going to derive this.  We've talked about it a good deal.  We've talked about the ideas a good deal.  We've talked about the fact that fields are made up of creation and annihilation operators.  And what we're multiplying together here is functions of the fields.  Functions of the fields and therefore functions, yeah.  I read somewhere that someone thinks that field quanta are not necessarily identified  with particles.  All right.  You would disagree with that?  Well, I'm not sure who and what was the context.  We've got field theory.  No, I'm not sure who said it and what the context was.  But you are...  Well, give me a little more to go on.  The question was, somebody said something.  My grandmother probably said it.  The field quanta in quantum field theory are not necessarily identifiable as a particle.  The only thing I can think of is that he was talking about quarks.  And of course, in some sense, quarks are particles.  But in some other sense, they're never detectable as particles because they can never escape  from one another.  That's the only thing I can think of.  So quarks are the quanta of the quark field, but they're never directly detected as separate  quarks, well separated from other quarks.  That's the only thing I can think of that he might be speaking about.  Well, I guess during unitary evolution, it doesn't necessarily boil down to observables.  And so you may not have particles that are identifiable at intermediate states.  I don't know what they're...  If you accept the idea of a quark as a particle, it's true that there's not a one-to-one correspondence,  necessarily, between the fields in a theory and the particles in the theory.  That's an idea which is true as long as the coupling constants in the theory are small.  So there is not necessarily a one-to-one correspondence.  Yeah, I mean, it is true.  There are field theories that for one reason or another, you wouldn't describe their quanta  as conventional particles.  But at some level, it's just the difference between words.  If you define particle to mean quanta, then there's no difference between them.  They're indivisible.  They carry energy and so forth.  So for my money, I would call them particles.  But yeah.  Can you talk about some of the histories?  Is there some limitation on what histories you can talk about?  Do they have to sort of continue this in some sense?  No.  No, in fact, they don't have to be continuous.  First of all, of course, the idea of continuity on a discrete space like this doesn't quite  mean very much.  The field in the neighboring cell is just going to be a different value of the field.  The rule is divide the space into cells.  And actual practice, this is the way quantum field theory is defined.  You divide the theory into cells.  You sum over all possible ways of populating the cells with values of the fields, all possible  ways.  No restriction to things which are approximately continuous.  And in fact, quantum fields are not approximately continuous.  They jiggle a great deal.  And you sum over all the possible values of the fields.  And that gives you your input.  So in principle, for each field and each derivative, you can assign it all possible values from  minus and from minus.  Well, you don't assign the derivatives separately from the field.  The derivatives, of course, are related to differences of the fields in neighboring boxes.  So you populate with field values, and then derivatives are replaced by, yeah.  And you can use, there are different rules that you might adopt.  You might define the derivative here to be the difference of the field here and here.  Right.  Yeah, that's a minor detail.  Yeah, but in any event, you might take the real line up and break it up into small intervals,  but you'd still have a sum over a countable number of possible values of the field.  For each field, and if there were two fields, then you'd be doing that over.  Yep.  By the field here, I mean all of the fields.  Right, so you would give a value to each possible field in the theory at each.  This looks like-  It's a really nasty-looking computation.  Absolutely.  Well, so since these are complex functions, they're analytic functions, so-  Well, the analytic functions of-  But the relationships between the values of the field have to conform to what you learn  in complex variables.  No, no, complex variables is about the theory of analytic functions.  Analytic functions are extremely smooth.  The smoothest functions you can think of.  These functions do not have to be smooth.  They're complex valued, but they're not analytic functions.  The field as functions of position, definitely not.  They're on the average highly discontinuous.  If I were to do the computation of the field here, how do I know-  That would converge and go all the right way?  Yeah, I mean how many different fields I have to consider?  How many different assignments have to-  So of course this by now is a major industry called lattice gauge theory or lattice quantum  field theory lattice because it divides the world into a lattice.  And this has been studied to death, how to do these integrals in practice.  By now it is a very effective tool, but it took some 30 years to develop the computing  technology.  I was involved in it in the very, very beginning.  I wasn't involved in the computer technology of it, but setting up the rules for lattice  gauge theory.  So there was a history.  The first part of it was setting up the rules for it.  And that took about three weeks and then something like 30 years to develop the technology to  compute these things.  And you're right.  I mean, how do you know when you've sampled enough of the space?  So there are- and a lot of the wisdom of it came from statistical mechanics where you  do very much the same thing.  You calculate partition functions or probability distributions for this checkerboard here,  not a quantum field theory.  It could be a real crystal lattice and you might be summing over configurations of whether  there is or isn't an electron at each site.  So a lot of the methodology came from the quantitative study of statistical mechanics  systems.  We have the same question.  How do you know when you've sampled enough?  This is not the subject of tonight's lecture.  So the answer is the experts know.  Well, they think they know.  And they get good answers.  They get good answers and agree with experiments.  So it seems to work.  Okay.  So let's go back and remember that quantum fields are a shorthand for creation and annihilation  operators.  And if we're talking about, let's say, the product of some fields in this box, times  the fields in this box, that can represent the annihilation of a particle in this box  and the creation of a particle in a neighboring box.  So we might just represent it by a particle moving from one box to another.  What kind of things in the action here actually do correspond to a particle moving from one  box to another?  Let me tell you, there are things in the action here which, strictly speaking, are not associated  with one box but with a pair of boxes, namely the derivative terms.  The derivative terms, I was a little bit hasty here when I said you multiply all these things  together, one for each box, as a little bit of a cheat because there are terms in the  Lagrangian which are associated with pairs of boxes.  So really you can think of it as summing over the boxes and summing over neighboring pairs  of boxes, but that's a detail.  Which terms involve pairs of boxes?  The terms which involve pairs of boxes are things like the derivative of phi with respect  to t squared.  The derivative of phi on a lattice becomes the difference of phi in neighboring boxes.  So, in here, this derivative of phi with respect to t might really be written as phi at point  x, well, phi at point t and x minus phi at point neighboring time and x.  Do we actually have second derivatives?  No second derivatives.  These are first derivatives.  I mean, would we?  Never.  Okay.  Never.  Bad idea.  That was real damage.  Now, what about derivatives with respect to spatial coordinates?  Same thing.  It corresponds to the difference to neighboring points in space.  And then the instruction is to take one half of the square of the derivative.  Well, one half of the square of the derivative will have in it.  I just want to focus now on the terms which multiply phi times phi at a neighboring point.  Let me just focus on those.  There's also terms in here which multiply by phi by the same, by the value of the same  point.  But in particular, there are terms in this Lagrangian which multiply phi at one point  times phi at a neighboring point.  Every time you lay down a term like this, it represents the motion of a particle from  here to here.  You can think of it that way.  Okay.  Let's concentrate on these terms.  Let's forget the interactions in the Lagrangian.  These are simply, these are called the kinetic terms, the quadratic terms in the Lagrangian.  They're things which are easy to deal with, and they do correspond to motion of the particle  from one point to another.  Let's look at this.  And let's include only those.  We don't even need to write it in this form.  Let's just write it as e to the minus i times the sum over all pairs of boxes, over all  neighboring pairs of boxes.  I said over all boxes, but it's clear that's not quite right.  We want to think of it as sums over neighboring pairs of boxes of things like phi in one box  times phi in a neighboring box.  I'll use the notation x and x prime to represent neighbors on the lattice.  That's what goes into the action, or that's what goes into the exponential of the action.  There are some coefficients in front of it, of course, but that's of secondary importance.  And now we can expand out this exponential.  Let's see what's there.  As one, minus i times the sum of phi of x, phi of x prime.  And then things like, let's see, what's the next term?  i times i is minus one, so it looks like it's minus.  One of phi of x, phi of x prime squared times another factor of the sum of phi of x, phi  of x prime.  This could be phi of x prime, x double prime, x triple prime.  X double prime and x triple prime are one pair of neighbors.  This is another pair of neighbors, and I think I left out a factor.  There should be a two factorial downstairs.  I'm simply expanding out the exponential here.  And what's the next one?  Well, it has three powers of the sum, four powers of the sum, and so forth.  Let's look at each term here.  The first term has something which involves a sum over the lattice of an annihilation of  a particle at one point and a creation of another particle at another point, at a neighboring  point.  The next term, oh, incidentally, the rule that I'm going to tell you again, I mean the  business of telling you some rules now, I'm going to tell you some rules now, the amplitude  for going from one thing to another, the rule is you've got to close off, you must not have  a dangling endpoint.  A dangling endpoint can only dangle like that if you've got an operation to put a particle  into the system.  A dangling endpoint, for example, we're going to have diagrams which have particles going  from one point to another.  If they end like this, they're illegal unless there's an instruction to put a particle in  at this point and to take a particle out at that point.  So endpoints like this, we will simply rule out as things unless they correspond to a  specific instruction to put a particle in at this point and take it out.  Because we're only putting particles in in the initial state and taking them out in the  final state, the only endpoints would be allowed on the top and the bottom.  Okay, so let's take a particular term here, phi of x and phi of x prime.  These are neighboring points on the lattice and they correspond to a motion of a particle  from one place to another.  There's a sum of terms.  There could be a term for a particle from here to here, a term for a particle from here  to here, a term for a particle from here to here.  But none of these will contribute unless, of course, there would be one situation in  which they would contribute.  That would be, of course, if this layered structure here was only two layers high and  we put in a particle over here and took one out at the neighboring point.  Then there would be a contribution to the amplitude of that coming from this term in  the product, phi of x times phi of neighboring x.  That would contribute and it would contribute to the amplitude, basically the factor minus  i.  It would tell you that the amplitude to go from one point to another was simply minus  i, this coefficient here.  Okay, but that's not very good if this, how do we get from here to here?  From here to here.  How do we get a particle from here to here and calculate the amplitude that if we put  in a particle over here, that will detect it over here?  For that, we have to find in this sum of products here, we have to have a term which will leave  no dangling ends.  Okay, let's just take this term here.  This has a particle moving from x to x prime and then another particle moving from x double  prime to x triple prime.  Are you saying the second term contributes only when the initial and final state are  adjacent?  This term here, yeah, yeah, that's right.  This notation means neighboring particles, neighboring boxes.  Here we have one neighboring box and here we have two other neighboring boxes.  Yeah?  Well, now we have the five x prime, the first term is five x five x prime.  Yeah.  And the second term is five double prime.  Yeah, it just means pick two neighboring and sum over all possibilities.  Sum, this is a sum over all neighboring pairs.  There doesn't have to be adjacent to the first.  No, no, no, no, in general not, but it can be.  In fact, this could even be the same pair, but it doesn't have to be.  Sum over all of them.  All right, so what does this one do?  It moves a particle from x to x prime and this one moves a particle from x double prime  to x triple prime.  So one of them moves a particle from here to here.  This is x prime to x double, x, from x to x prime.  And then the other one moves a particle from here to here.  That's x double prime and x triple prime, let's say.  Obviously, there's going to be dangling ends unless, well, let's take this case here where  we only have three layers.  If we only have three layers, the only way to avoid dangling ends is to have the first  particle connected to the second particle, sorry, the first box connected to the neighboring  box and then this box connected to this box.  There won't be any dangling ends except for the dangling ends which correspond to the  initial particle and the final particles.  So if this was only three layers thick, we would find a contribution here, namely where  x prime is the same as x double prime, that transports the particle from the initial position  to the intermediate position and then from the intermediate position to the final position.  What would be the amplitude then associated with the particle going from here to here,  at least corresponding to this term?  It would be a one over two factorial and a minus sign.  That would be it.  We would go off the coefficients and the coefficients tell you the amplitude.  Thus far, we don't have a way to get from here to here, not with two steps anyway.  Two steps can, however, if this was only two layers thick, we would have now a way to get  from here to here, to go across the diagonal.  How do we go across the diagonal?  We go again to this second order term here, the term which has a two factorial in it,  and we find the term which takes us from x to x prime and then from x to x prime and  then from x prime to from here to here and then here to here.  What's that?  There are other paths to get from here.  Yeah, there's another path where you jump from here first to here and then up.  I don't think there's any other besides that.  Yeah, not with just two terms.  Right.  Okay, but now there was no reason to stop here with only two terms.  Let's go on.  We can have three terms.  Three terms would correspond to going from one box to an adjacent box.  Well, it would correspond to three distinct steps.  And if those three distinct steps were connected together into a chain, in other words, in  this sum of products, we found the term which took us, we could come right back again, that  would contribute together with the direct jump from here to here, would contribute to the  amplitude to go from this point to this point.  So you see, built into this prescription is something like the original particle path  integral idea.  The amplitude to get from here to here is the sum over the amplitude to go from here  to here and the amplitude to go through another root, also this one back here.  And eventually, if you expand this out to arbitrary order, there will be a term in there  for every possible root that you can take to go from any initial configuration to any  final configuration of that particle.  So first of all, just thinking about a single particle moving in spacetime, this field Lagrangian  contains information about the amplitudes to go from, now let's take an arbitrary, to  go from any point to any other point.  What you do is you add up all of the possible ways of going there and what is the coefficient  for each way of going there?  You read it off the coefficient that multiplies that particular term in this sum.  One over two factorial, one over three factorial, some of them have eyes, some of them don't  have eyes.  Remember that I squared is minus one, so some of them have eyes and some of them don't have  eyes and so in general, the amplitude to go from one point will be a complex number.  That's one of the things that's new here, that they can propagate up and down and that's  a feature of relativity.  What's that?  Oh, yeah.  Oh, indeed.  Yeah.  Yeah.  All right.  Let's talk about going backward in time.  You could think of, let's use one that goes backward in time.  You can think of this as either allowing a new rule where particles can go backward in  time or you can follow time forward and say what really happened here is the particle moved  from here to here and then a particle pair, particle and antiparticle in fact, were created,  the particle half of it going to here and the antiparticle combining together with the  original particle to annihilate.  Think of it either way.  But the rule now allows trajectories which go backward.  Let's come back to this term here.  In this form, since you have n factorials in denominator, does this mean that you sort  of have convergence in the nulls?  Well, the n factorials certainly help the convergence, but they're not enough to make  it converge.  Yeah.  That's right.  No, no, no, no, no, no, but the sum over convergence.  Configurations doesn't have to converge.  Yeah.  This part converges.  Yeah.  And this is going to be some finite number of total configurations considered.  Yeah.  Okay.  Let's, all right, now let's go on to a slightly different problem.  We studied the particle moving from one point to another.  Now let's suppose we put in two particles and we want to know what the probability for  the two particles to go to two other particles is.  In other words, we want to start, and I'm going to take the case where there are only  two layers for a moment.  There's the case where there are only two layers for a moment.  And I want to know the probability that if I start a particle over here, it will get  to here.  No, if I start two particles.  One over here and one over here, that they'll get to here.  Okay.  Well, come back to this term.  Now remember what this term did for me before.  Before it allowed me to hop three units.  But that same term allows two particles each to go one unit.  Look at it.  This can create a particle.  This can annihilate it.  This can create a different particle and annihilate it.  This corresponds to a graph where you start a particle in here and goes to here.  That's this factor.  And the other factor, instead of taking the particle that's already there and moving it  over here, it just takes a totally new particle and moves it to here.  So this same factor, the same term contributes both to the motion of a single particle, three  boxes, and it contributes to two particles each moving one box.  Did I say that right?  More or less.  Three boxes, one, two.  Two boxes.  Yeah.  So there's a lot in here besides just motion of a particle from one place to another.  It has information in it about any number of starting particles going to any other number  of final particles.  Now in fact, particle number doesn't change as long as we just take this into account.  Why not?  Because every starting, every dangling endpoint here, well, let's see.  Yeah, it has to end somewhere.  Well, if you start and end with different numbers of particles, then obviously something  has to happen.  Well, that would mean there has to be a dangling in somewhere.  Supposing I wanted to, I mean, I can't, how would you get from one particle to two particles?  What if I start with electron and positron, I could end up with no particles?  Well, OK, that would be something, that would just be something like this.  Two particles.  Let's put another layer in here.  Yeah.  Yes.  That's true.  That would correspond to an electron and a positron, for example, annihilating each other.  The only problem with it is it doesn't conserve energy.  So in fact, when you add it up, all these amplitudes, you would get zero.  But on the certain circumstances, if there was an electromagnetic field to soak up the  energy, yes, you're right.  It would have information about an electron and positron annihilating.  Right.  So there's just a lot in here.  There's a lot in here.  But not everything.  Not everything.  To find out the other things that are implicit in the field theory, there are other terms  in the Lagrangian.  Now what you put in the Lagrangian is determined by experiments.  And it's really just a way of codifying the results of experiments.  But what else can you have in Lagrangian?  We had the derivative terms, let's say, phi dot squared minus the derivative of phi with  respect to x squared and y squared and z squared, all that kind of stuff.  That was the stuff that I wrote down as phi at one point times phi at another point, at  a neighboring point.  Then there are possibly other things to just involve phi and not its derivative.  So for example, you could have things like phi squared.  Remember what the coefficient is?  It's usually a 1 half here.  That's a convention.  Remember what the coefficient of phi squared is?  The square of the mass of the particle.  We should divide by 2.  That's, again, a convention.  And if we included this term here, what that would do, it doesn't move a particle from  one place to another.  It absorbs a particle at one point and emits it from the same point.  From the same point.  So what would correspond to graphical constructions where we would add a new rule, f of particle  enters a region here from one box to another.  One of the terms that can act is this mass squared term.  It would simply take the particle and do nothing to it.  It would leave it in the same spot.  Then another term could come and move it, but it would weigh that particular path with  a factor of m squared.  Every time this term acts, it weighs the path with another factor of m squared in the amplitude.  So there would be terms where no m squares act, where only these derivative terms act.  In other words, when we're expanding this product out, we only have the derivative terms.  That would be the theory of massless particles.  The theory of massive particles would have a new rule that one of the possible things  that can happen at every step is you may or may not just weigh that path at that point  with the m squared.  So the mass term is another rule for how you evaluate these paths, these path integrals.  Another rule.  It's we don't have to get into detail.  The main point is that the mass of a particle is codified by another term in the Lagrangian,  which is m squared phi squared, which weighs paths in such a way that it knows about their  mass.  All right, but more interesting is the more complicated terms, things like phi squared.  Let's put a g, sorry, phi cubed.  Phi cubed can do a number of things.  It can annihilate one particle and create two particles.  It can annihilate two particles and create one particle.  It can create three particles.  It can annihilate three particles.  All right, so that's a new kind of thing that can happen on this checkerboard here in expanding  out the action, expanding out the exponential.  There might be additional terms here, for example, g phi cubed.  When we expand out, we can have not only these products which take a particle from one point  to another, but also processes where particles are created and annihilated in threes, either  two in, one out, two out, one in, or three in, or three out.  What that looks like on here is, for example, supposing we have several of these quadratic  terms which move particles in one cubic term.  What can that look like?  Well, what it can do is it can take a particle, moving it from here to here, and suddenly  in that box, we put a phi cubed.  In that box, it can absorb one particle and emit two more.  Now, the next thing that could happen is the two particles could move to other boxes.  But basically what it does is it takes a particle, absorbs it, and creates two, and those two  particles can be on their way.  And this would then contribute to a process where one particle came in, two particles  went out.  In fact, you can make very, very complicated things.  You can make things of, let's go back to the one particle going from one place to another.  One particle going from one place to another.  Let me stop drawing checkerboards.  Let's just imagine the checkerboard there.  One particle going from one place to another can go directly, just a bunch of small hops,  a bunch of these hops.  But also something else can happen.  On the way, at some point, the cubic term could act.  If the cubic term can act, it can take that one particle coming in and split it.  Split it into two.  And then both particles move along until another cubic term, something quadratic in the cubic  term, something with two cubic terms in them, can create an extra particle and reabsorb  it.  So in fact, the actual amplitude to go from one point to another by one particle is not  just the sum of simple trajectories going from one point to another through the space  time, but is more complicated.  It has infinitely more complicated.  It has processes where the particle splits and rejoins.  In other words, where the particle emits a second particle which is then reabsorbed.  An example of this would be an electron moving along and emitting a photon and reabsorbing  it.  Part of the amplitude for an electron to go from one place to another is through the  process of emission and absorption of a photon.  It can get wildly more complicated.  Of course, we have to add all of these up.  The rule is add up the amplitude for every possible way to go from the initial state  to the final state.  You could have several of these splitting and joinings.  You could have things jumping across the gap between particles.  And this can get arbitrarily complicated from the point of view of electrons and photons,  things like this could happen.  Electron goes through, photon emitted, another photon emitted, but even worse, the photon  can break up into an electron and a positron.  But then the electron and positron can exchange a photon between them.  All of these processes come out of expanding out the action like this and finding the individual  terms which do all of these things.  If you want the amplitude for any one of these processes, you just go back to what appears  here, find the term that you're looking for, and find the coefficient in front of it.  It may or may not have an i, depending on whether it's an even power or an odd power,  and it will have some factorials and so forth.  That will give you the amplitude for a specific process.  But then the full amplitude to go from one thing to another is the horrendously complicated  sum of all of these things.  Right, so you may have certain things that happen in multiple of those histories.  In multiple of those histories, you may have similar terms, which then the overall amplitude  for that process would be actually from all the different histories.  Yeah, from all the different histories going from the initial to the final.  So there are two ways to think about field theory.  One is to think about fields and think about histories as initial values of the fields,  final values of the fields, and you sum over all the values of the fields in between.  And the other is this particle way to think about it, where the field is replaced by a  distribution of quanta, and again, amplitudes for initial configurations, which means an  initial particular set of particles and a final set of particles.  The basic processes, the basic underlying processes which happen are governed by this  Lagrangian.  That is the most important thing.  Any time you have more than two fields interacting, you have an interesting interaction process  where particles can split and join and so forth.  What would happen if you had phi to the fourth here, which is a perfectly good interaction?  That would be some place where a particle, for example, could come in and break up at  the three other particles.  One to absorb the initial state, three to emit the final state, or it could be a scattering  in which two particles come along, collide, and two go off.  Anything that looks like a vertex with four particles altogether, two in and two out, or  two in, or three in, one out, or so forth, or just four out, or four in, they're all  governed by this term.  And so forth and so on.  Let's see, how are we doing?  So that's the spontaneous arrival of four particles at the final state?  Yeah, except once again, it would violate energy conservation.  So if you really worked it out and added up the amplitude from every position where this  could happen, every spacetime position where it could happen, you would find out that the  constant doesn't happen.  It doesn't happen.  In other words, they all add up to zero.  So processes which violate energy typically all add up to zero in these things, energy  and momentum.  So the conservation of energy falls out of the new calculation?  Yes.  And the conservation of momentum, same thing?  Yeah.  Now, we did some examples.  I'll show you how integrating the vertex over all time conserves energy, how integrating  it over all space conserves momentum.  Same thing is true here.  Integrating the position of these vertices and the endpoints of these things over all  possible positions in space and time will simply add up to zero unless energy and momentum  is conserved.  But apart from the conservation laws, everything that you can draw down will happen with some  probability.  Now, how could it possibly be that you add up this humongous thing and the answer doesn't  come out infinite?  Well, the answer is that there are coefficients here.  There are coefficients here, these are called coupling constants, particularly the coefficients  for the higher powers here.  We could call this G3, we could call this G4.  And the rule is in a diagram, in a Feynman diagram like this, every time you have a vertex,  let's say with three particles coming together, the amplitude contains a factor of G, G3  in this case.  So the more complicated the diagram gets, the more powers of G that it has.  Each vertex gives you a power of G. If G is a small number, then each successive power  of G gives you a smaller and smaller contribution and you have a chance that the thing might  be able to converge.  The technical question of whether it really converges or not is a very difficult one,  but you can see that you have a chance if G is a small number.  If these coupling constants are small numbers, then the simpler the process is, in other  words, the fewer number of vertices, the bigger the contribution and the larger the number  of vertices, the more complicated, the smaller the contribution and you have a chance at  any rate at some sort of convergence to an answer.  If the coupling constants are large, it means these series do not converge and if they don't  converge, it doesn't mean that the theory is wrong.  It just means you've chosen a naive prescription for trying to work with it.  But in practice, most of the quantum field theory, most of the things we do with quantum  field theory are based on the assumption of small coupling constants.  And in those circumstances, it's possible to evaluate these finite, not just evaluate  them, but add them up and find out that the one with four vertices is much smaller than  the one with two vertices.  So that's the name of the game.  Do we do color theory in the same way?  Yeah.  We're going to talk about color theory.  Now the next step is to write down the list of all particles and to write down either  in the form of Lagrangians or in the form of Feynman rules.  The Feynman rules and the Lagrangians have the same information in them.  Three identical particles coming together, let's say three bosons, that's five cubed.  Particles going from one point to another, those are the kinetic terms, the quadratic  terms here.  So the Feynman type rules and the Lagrangian are the same thing.  And you can either describe the world by specifying all the vertices incidentally.  The lines between one point and another are called propagators.  The place where more than one particle, more than two particles come together, those are  called vertices, vertices and propagators.  But as I said, they're just simply another shorthand way of describing the Lagrangian.  The Lagrangian is a shorthand way of describing the graphs.  So we would have stated all of particle physics in a kind of dumb way by simply writing down  the list of all particles, writing or constructing or even just expressing a quantum field for  each particle, and then writing down a Lagrangian for all those particles.  We contain all of the interactions or we can write down all of the particles and simply  specify what all the possible vertices are.  Either way, they come to the same thing.  They're kind of a complete list of everything that can happen.  Once you know everything that can happen, you can start thinking about calculating amplitudes  for particles to go from one place to another.  Particles to collide.  Particles to collide and scatter.  Here's particles colliding and scatter, two electrons colliding and scattering just by  a photon jumping from one place to another.  Again, Lagrangian tells you how to calculate the amplitude for each of these.  As I said, it contains only information, including symmetries, including conservation  laws.  So I think it's probably, shall we write down the list of all elementary particles?  So when you write down a final diagram, basically the bottom is the initial configuration and  the top is the initial configuration.  That's right.  The bottom is the initial configuration of particles.  The top is the final configuration of particles.  As I said, it is the kind of complementary way to think about quantum field theory, complementary  to the field description where you would start with a given field configuration and end with  a given field configuration.  Incidentally, in some sense, the complementarity between particles and fields is very much like  the complementarity between momentum and position.  These are two different ways to describe the same reality, and there are uncertainty relations  between them.  If you know with precision the number of particles, then you know that there's uncertainty in  the values of fields.  If you know the values of fields, then there's uncertainty in the number of particles.  So these are really two complementary ways to describe the same thing.  And the Lagrangian is useful in both contexts.  Question.  The couple of constants, are those predicted by theory or are they perfectly determined?  Because under certain circumstances, there are symmetries which relate coupling constants.  So this or that coupling constant for this process might, for symmetry reasons, be related  to another coupling constant for a different process.  So some of them are related to each other by symmetries.  But their values in general are simply come from experiment.  All right, much of this, if you really go and learn about quantum field theory, has  its beauty.  It does have elegance and beauty and so forth.  Now when we start writing down the list of facts, what particles there are, what their  masses are, what their coupling constants are, it's just a mess.  It's just a very ugly mess with very little coherence, except the coherence that comes  from symmetries.  Symmetries tell you relationships between different kinds of particles and their processes.  But apart from the symmetries, which are few and not so much, it is a large number of random  facts about a somewhat unmotivated collection of different kinds of particles.  But still you're pretty confident of the coupling constants because of your…  Once you measure them, you measure them.  If you change it a little bit, then you get wrong answers.  Oh absolutely.  No, no, no, I mean, okay, let's put it this way.  You have too many particles, which means too many fields, too many for any aesthetic sense.  You have a lot of them, a hundred of them or something.  A lot of coupling constants, which are just more or less numbers pulled out of experiment,  and a lot of masses, which range all over the map.  A few relationships between them, but once you know them, once you know them, that's  it, you can calculate with great precision anything about those particles, anything about  those particles, anything about quarks, electrons and photons and so forth, means atomic physics,  or it means particle physics, it means atomic physics, it means nuclear physics, it means  chemistry, and maybe it means biology, maybe.  So, the amount of input that goes in is perhaps bigger than you might like, but the amount  of output is huge.  Question for electrons, for example, how many of these G sub n's would be?  G2, G3, G4, these be calculated?  How many G's are there for electrons and photons?  And electrodynamics?  One, the electric charge.  The electric charge?  Well, in a sense, the electric charge is a kind of G3.  The electric charge is a kind of G3.  The electric charge is the coefficient in the amplitude representing an electron coming  in, an electron going out, and a photon being emitted.  In terms of symbols, this diagram would be represented by an electron going out, an electron  coming in, and the field operator for a photon, which is A. I'm not writing down the details,  there are some power, sorry, there are some direct matrices, but then the numerical coefficient  in front of it is the electric charge in certain units, certain dimensionless definition.  The only processes in quantum electrodynamics are the emission and absorption of photons  from electrons.  That's all there is.  Well, if there are electron collision with a photon, there's just a little bit of electron  collision.  The simplest process would be photon absorbed.  I meant to say proton.  Oh, well protons are not usually considered a part of quantum electrodynamics.  Now you can think that when you do quantum electrodynamics, you are doing atomic physics.  You think of the proton as a point infinitely massive.  You don't think of it as something described by quantum field theory.  You just think of it as a nailed down, heavy particle which never moves until you open up  the theory of protons and neutrons.  So the scattering by a photon, a photon by an electron, this is the first diagram.  But the electron is moving at almost the speed of light relative to the proton?  Proton, yes.  Right.  Yeah, that's right.  Let's draw some diagrams just to get some feel for what kind of things we have to add  up.  This is the lowest order diagram that goes into the scattering of a photon by an electron.  There are two powers of the electric charge.  So there's an E squared.  But this is the amplitude.  The amplitude gets squared to find the probability or the cross section.  Cross section is another way of speaking about the probability for the scattering.  All right.  So that means that the whole process, the probability is proportional to the electric  charge to the fourth.  The electric charge, at least with the suitable definitions, is a small quantity.  The square of the electric charge with, again, with the suitable definition is about one,  well, that's the number which is 1 over 137, the fine structure constant.  So this is proportional to the square of the fine structure constant.  And so scattering in quantum electrodynamics is a unlikely process.  A photon comes in and strikes an electron.  What is the probability that it goes right through it versus the probability that it  really scatters it?  The probability that it really scatters it is governed by this e to the fourth, and it's  a very small number.  Now we have to add all the processes that we can find, add them up to form a real amplitude.  Incidentally, there's another one here which looks like this, where the final photon is  emitted before the initial photon is absorbed.  That's another process.  We have to add them all.  But then we start adding more complicated things.  Let's focus on here, and we can add another photon in many ways.  This is not the only way we could have to go from here to here, from here to here, many  photons.  But how many powers of electric charge does this have?  This has four powers of electric charge in the amplitude and eight in the probability.  So this is even weaker.  But then we can do even more complicated things.  We can put an electron-positron pair in here.  Each time we complicate it, we add two more powers of electric charge.  So in quantum electrodynamics, the expansion is typically an expansion in powers of the  square of the electric charge.  Every time you add another complicating structure to a Feynman diagram, it's two powers of electric  charge, and it means it's about a couple of hundred times smaller in amplitude than the  previous one.  That's why when you add them up, it looks like it converges, because more and more complicated  they get the weaker the probability from that particular configuration.  Incidentally, you don't square this and add it to the square of this and add it to the  square of this to find the probability.  You add them up, and then you square.  That's the rule.  Add and then square.  Add and then square.  That means there's cross terms between them in calculating the probabilities, called  interference terms.  So that's the structure of quantum field theory.  That's what we do with it.  The next issue, as I said, we're now at the stage where we can really start to sensibly  talk about particle physics.  We can name all the particles.  We can specify their masses.  Those are the parameters in Lagrangian here that we've spoken of.  We can write down the Lagrangians governing them partly from theory, partly from experiment,  and discuss their symmetries.  The symmetries can be discussed by looking at the Lagrangian.  We'll do that.  We can write down the standard model of particle physics and discuss some of its properties.  So I think we're off to, I think we're set now to move ahead and really discuss the world  of genuine real particles and what can be discovered in the world.  Instead of that m squared, you could add the Higgs field and then that would be a quadratic  state.  We're going to discuss that.  Where are the five of the derivatives?  Yeah?  Where are they at the same time?  Oh, they're here, x and x prime could stand for two different times.  That's how we manage to move vertically.  All right.  Now, where does the parameter correspond to the particle just somewhere in the penitentiary?  There's also some terms.  Now, there are various terms where the particles just stand still.  When we wrote down phi of x minus phi of x prime squared, that did have terms in it like  phi of x squared and phi of x prime squared.  Those are terms where the particles stand still.  The cross terms are the ones where it moves.  So there are terms where the particles stand still both from this and from this term.  Right?  Yeah.  Okay.  For more, please visit us at stanford.edu.
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