 Can you hear me? Can you hear me? It will be a quiet talk. Can you hear me now? Yes, everybody up there? Can you hear me? Good, thank you. Okay, so I'm lucky enough to give you seven lectures with a total of 90 minutes per lecture, so you know, so it's roughly 10 hours, a little more than 10 hours actually I'm going to spend with you. So I really hope that you're going to enjoy it. And let me start with several little things, okay? So let me say I've been teaching in this school 12 years ago. It was an amazing experience and I still have people I still friend with them since then from students. And I've been doing quite a lot of those kind of schools and what I found the hardest thing to do in this kind of school is just to actually try to measure how much you guys know and where I should start. So what I decided to do based on my experience in the past, I'm going to try to start at the very basic level. And I apologize in advance for those of you who are going to say, what are you talking about? I remember in kindergarten, my teacher was telling me remember when we were going to do some swings and all those kind of things. So please, however, I kind of promise you that if you don't give up till the end, I'm almost sure that you're going to have anybody going to learn something new from what I'm saying today, okay? Please ask questions. I mean, it's amazing how much I love to answer questions. I really love to talk and you can tell. I'm really a talker. But I really like you to ask questions. Here it might be a little bit embarrassing, many people ask me, you know, I'm here just for you. It's an amazing scene. Listen, think about it, okay? Nobody pays me. Well, they pay my flight ticket. But nobody pays me. And I come here for the whole week. All I want to do is just talk to you. I also want to go to the beach. But other than that, I just want to be here for talking to you. And when you see me alone and I'm lonely and sad, come and talk to me, okay? Ask me questions. And I love to talk about anything in physics that I do know about, okay? I don't like to talk about things I don't know. But hopefully, you know, so please, come sit by me when you see me at lunch. Come and sit by me on dinner and breakfast when we are around in the... Come and talk and talk and we'll do some physics. This is my email-free-free to email me with any question if you biased or after the talk. So the plan of what I have to do in the seven lectures, I'm going to start with an introduction to quantum field theory. And then I said, all of you already took quantum field theory, my apologies. But however, what I found that I took quantum field theory like two times and I only understood it when I was teaching it, okay? So it may not be so bad if you kind of hear it again and maybe hear it in a little different way with a different accent, kind of those things so you hopefully it will be still good. So I hope to do it in two lectures. Then I'm going to talk about the standard model for another two lectures and then I go into some, hopefully, a little bit deeper into flavor physics toward the end. Any questions? I knew. It's never... Nobody had questions at this point because it was so clear, right? Okay, good. So let me start with what is high energy physics or another question is what the hell are we doing? So why are we doing high energy physics or why I spend... And now it's like 27 years since I started my master. When I started doing, like, why... What I'm doing, okay? What it is the big picture of what we are doing? What is the part of physics that we are doing is high energy physics? Did you ask yourself this question? You should. You should. Oh, very good, very good. Because otherwise, like, I just go and what do I do? There's some... Oh, let's do high energy physics and you step in the room. So let's... My answer to myself, what is high energy physics? So high energy physics is to find the basic law of nature. And this is one of the things I'm most proud of. Oh, shit. Not this one. This one. Okay. That's the most compact way that I can tell what I am doing, okay? And if someone asked me what I'm doing, I ask, oh, what is the Lagrangian of the standard model? What is the Lagrangian basically of nature? So what we try to do is to find the basic law of nature. And in the way that we are doing it is that I want to find the Lagrangian of nature. And if I know the Lagrangian of nature, then basically I understand nature, okay? That's actually a really important statement, okay? It's important standard because it's just the fact that we can actually do it. I mean, it's really far from trivial that we as human beings can actually see what are the law of nature. And it's even more far from trivial that we actually made so much progress. And we really understand nature at a very, very fundamental level, okay? So we do have a very good answer, okay? It's quite a good answer, and you all know what it is. It's called the standard model, okay? And not only that we have an answer, this answer is extremely elegant. I mean, the standard model and then, as I said, I'm sure you all learn about the standard model to some extent, and I'm going to emphasize why there's so elegancy in this thing. It's based on axioms, just like it's almost like mathematics. You put your axioms very, very, very simple things. You observe some symmetries and you get out something that amazing enough, explain basically almost everything that we saw in nature. There's a few things that we didn't understand yet, and then of course we are really excited about them, but I'm not going to get into them in these lectures, okay? And the way we do it is following, and I'm going to explain it in more details, is it's basically like we did mechanics, and I assumed that all of you took analytical mechanics. That's something that I made, okay? And at least for me, when I took analytical mechanics, that was this wow feeling that I got into the lecture, and I was like this. You had this feeling when you took analytical mechanics, and the professor would start saying, oh, that's the Lagrangian, and every lecture it was this kind of thing. So what we are doing is basically a generalization. In classical mechanics, it's kind of simple. You say, oh, I have generalized coordinates, they call them x and y, and now we do it with fields. The fields are the generalized coordinates of classical mechanics, and we call them phi and psi instead of x and y, but other than that, it's almost the same, okay? And why are we doing particle physics? Why the hell we are, I mean, I kind of include myself in all this, but why we are going and take those amazing protons and accelerate them to such high energy and collide them and start seeing all the crap that come out of this? Why are we doing it? So we are not doing it because we really care so much about what's going on for this particle. We do it because we understand using particle physics, we learn the fundamental law of nature, okay? So if someone asked me, why do you care so much about what is the mass of the electron? Everybody know the mass of the electron? You should. 511. You have this feeling like if you go and you see a car with 511, yes? Yeah, okay, good. Or 137, yes, okay. So the 511, but do I care that there is 511, whatever, KV? Of course I don't. What I really care about is the fact that it's actually part of the big picture of, that it's really explained the fundamental law of nature. Good. So with this philosophical thing, let me start with, again, as I said, I start with very fundamental things and let's go back to under God physics and we ask what is mechanics? So in mechanics class, you go to the class and the professor is telling you, so what is mechanics? Why are we doing mechanics? Because we have a question, the question is how to find X of t. We have to find the trajectory of a particle. Or more generally, we have a system with many, many particles and we have to find the trajectories of all the particles. And if I know this function, X of t, I'm done. Or in more generally, if I know the function, the many function, X i of t, I'm done. I just have to find all those functions. And moreover, once I know the trajectories of all the particles, I know every observable. I can calculate the energy, the momentum, the angular momentum, any observable that you have in mind, I can calculate once I know the trajectories of all the particles, okay? And another very important thing that we learn in classical mechanics is that instead of solving for X, say X1 and X2, I can define two other coordinates, Q1 and Q2, that happen to be, say, the sum and the difference of all those, okay? And in when you wear classical mechanics, it looks like a trivial statement, right? What is more trivial than say, oh, if I know X and Y, and I know X plus Y and X minus Y is the same as knowing X and Y, right? I hope it is a very trivial thing and you are very familiar with it. However, it's a really deep thing and we are going to use it a lot when we start doing particle physics, okay? So how do we solve mechanics? So you remember that was lecture number one in mechanics course, it's a let's find X of t, and then you say, how do we actually find X of t? And the answer is the following, okay? This amazing axiom that come out of nowhere, okay? And it's called the principle of minimal action or the Hamilton principle. And this axiom tells us something like this. It says that there is something that we call the action and the solution is the thing that minimize this action, okay? And that's an axiom. And amazingly enough, this axiom, we never really change it throughout all those years, since the days of Hamilton and Euler and Lagrange, that these all these things, okay? And there's only one action that describes the whole system. So this S, which is some abstract thing that called the action. And it's some integral of something F that is very abstract that we call the Lagrangian. And then the thing that minimize this is the solution. And in order to minimize this, we have to use the Euler-Lagrange equation. You remember all this? Yes? So is it correct what I wrote here? Is the action is the integral of L of X, X dot dt with one particle, one degree of freedom? Is it correct or did I miss something? It should be correct otherwise, you know, I spent so much time thinking about this transparency, right? But maybe it's not really correct. So what about time? Well, I don't know X, X dot and t. So maybe there is a mistake. So I guess, you know, but you're a little too shy. Anybody brave enough to tell me why there's no time? Yes, I assume no explicit time-dependent. Very nice. And is it justified? Why did I do it? Why we usually do it? Because I assume conservative system, very nice. Let me say the way I like to kind of say it is I assume a symmetry. I assume that my Lagrangian is invariant under time translation invariant. I assume that when I take my Lagrangian and do t go to t plus c for an arbitrary number c, this Lagrangian is invariant. And this, the importance of this symmetry is the fact that we have energy conservation or we have conservative system. Okay? So that was actually a very good thing. By the way, why I don't use have X double dot and X triple dot? You remember? Back then, chapter one in Landau-Liffchitz. Yes. So, sorry. Ah, very nice. So this is always the right answer. The right answer is that I do it because it gives me the right answer. So I do it. Okay? And it sounds a little bit, it's called circular reasoning. Okay? So sometimes I choose my Lagrangian to be this because if I choose it with something else, it wouldn't describe nature. So I choose the thing that actually give me the correct answer. However, and that's a very important thing, what we find out when we do deeper and deeper physics, we find out that sometimes I actually have, like, really good reason. For example, for the time, why I didn't include time, there was a really good reason. It was a symmetry reason, right? And it described nature. So of course I could say I didn't put time because it described nature or that's a good answer. But then I give you a different answer which you can take or not and say I impose a symmetry. I say t go to t plus c is a symmetry and therefore there's no time explicitly time dependent in this equation. Okay? Anybody know of a symmetry that forbids X double dot? Not really, actually. You can have it, you can have non-local, you can have a thing that doesn't really have to be a non-local thing when you have double derivative. Have you ever thought about this when we were doing, I want you to think about it, okay? And let's think about it. I'm not going to give you the answer. And it's actually a really important question. So when we start doing mechanics, again, somewhere new under God use, usually that's the way it was presented. And usually we don't ask the question why. And the answer is that if I have X double dot I don't get F equals ma. And I do want to get F equals ma so I set it to zero. Okay? Very good. So what I do after I set my Lagrangian, I use the most famous formula in physics. Right? Maybe there's more, more famous formula like e equal mc squared. But it's a very important formula that all of you probably remember. And you use the Euler-Lagrangian equation. And let's give me X up to initial condition. And when you give me the initial condition, you tell me X zero and V zero, I solve mechanics. Okay? So we did something really nice. We take mechanics and reduce mechanics to the question, what is L? Okay? So you see something really nice the way we do physics. We take a question, which was what is X of t? And I replace this question, what is X of t? But the question, what is L? So if you tell me the Lagrangian, I have a mechanism to get X of t, which was my original question. And that's actually many times how we make progress in science. We take one question, do something and reduce it to another question. Very good. So how we find what is L? So let's take an example, Newtonian mechanics. So Newtonian mechanics, we choose L to be mv squared over two minus V of x. In mechanics, why do we choose this L? So usually the answer is because it's giving me what I want, right? But there's actually a little more into this, okay? So we use the Euler-Lagrange equation, we plug it in, and we find, boom, f equals ma. Wow. And that was, I think, my first wow in my life when I thought this, okay? L is an input. I take L as an input, and I get f equals ma as an output, okay? And most people, I still think most people in the world think that f equals ma is the start of mechanics, right? That's how we teach most people, right? That's how you study in high school. And now we come to this, I know f equals ma is actually an output. But still, how do we find what is this L, okay? And let me give you the following statement, and a following statement is the following, that L is the more general one that is invariant under some symmetries, okay? And I'm going to discuss it a little longer, but again, we rephrase the question. Instead of saying what is L, I said what are the symmetries? And then if you tell me what the symmetry is, write the most general Lagrangian, and then I usually truncate it at some level. I write it to some order in some small parameters, and that's my Lagrangian, okay? So again, let's ask the following question, which you probably asked yourself, what are the symmetry of Newtonian mechanics? So the way we do mechanics, we usually start from the Lagrangian is mv squared over 2 minus v of x, and then we get f equals ma. But can we actually get L equal mv squared over 2 minus v of x from symmetries? Do you know the answer? Maybe let me ask a little simpler answer. If I have a free particle, so let's assume I have one particle, just free, no potential. What is the symmetry? So if I have a particle with just no potential, the Lagrangian is just mv squared over 2, okay? What is the symmetry? And actually L equal mv squared over 2 can get, we can get totally just from symmetry argument. Do you know what symmetry I have to impose? Translation invariant. So translating invariant guaranteed that there's no x in it. Because if I have translation invariant, x goes to x plus c, tell me that the Lagrangian cannot depend on x. Very good. What else? So isotropy state, tell me that it must be proportional to the absolute value of v, not only to v. So it might be important to v squared, okay? No t, we already talked about. So parity actually we do not impose on the Lagrangian. So the parity is come automatically when I say it's a function of v squared. So why I don't have v to the four? What is the symmetry that's guaranteed that I don't have v to the four? And actually let me ask a little different question. So if you see, because the Lagrangian of a free particle is just mv squared over 2, a free particle have only one property, which is a mass, which is an amazing statement. Because if I looked at other things, they have many properties. Just happened to hold this in my hand. It's have a lot of things, it's have mass. But it also have a shape. Do you see? It's have a shape, you look. So why a fundamental particle of only mass? Why doesn't have a shape? Why doesn't have other properties? Or does it? Do we? Are you a little bit confused? I hope so. But I really, you see the question. The question is why a particle of only mass? Why the Lagrangian is mv squared? The over 2 is total normalization, right? I hope you, we always agree. So why the Lagrangian is some number times v squared? And this number is a property of a particle. Anybody? Thank you. Okay. Very nice. Galilean invariance. Galilean invariance is this deep axiom of classical mechanics that tell us that all internal reference frames are equivalent. There's no experiment that we can tell it, tell me if I'm in a different internal frame. And therefore it must be v squared. And if you never done this exercise, please do yourself a favor, please. Okay? Just do this exercise, okay? It's section three in lambda lift sheets, okay? You just go and you just do like, it's two lines of an exercise that you prove to yourself that it must be just some number times v squared. Okay? And what's happening when we give up Galilean invariant? Why do we want to give up Galilean invariant? Because of relativity. So when we give up Galilean invariance, what happened to our Lagrangian? Do we have v to the four terms? When we go to relativity, we have v to the four terms. So everything kind of fits, right? You give up Galilean invariant, you give us the fact that it's only mv squared, you see some v to the four going up. And actually, because of Lorentz invariance, you're going to have a Lagrangian of relativity, which looks a little different. Okay? Good. So I hope we hold on the same page on this how we actually do physics, okay? That's the way we like to think, the way we do physics, okay? We like to do things in a very, very simple way and start on very, very simple axiom, okay? We impose some simple axiom, some symmetries, and we got there. So now that we already, you know, so that's what the whole course in classical mechanics. And now let's move to field theory, okay? So what is field theory? So let me start with the following question. What is a field, okay? So mathematically a field is something that is a function of both x and t. And I call it phi, and there's actually many, many fields that we can think about. And I go with here some examples. For example, a temperature. So why the temperature is a field? Because it changed from place to place, right? So for example, when I come from Itaca in the winter, it's very cold, and I come here in the summer, and it's very hot. So it's both x and t that I change, and I get a different value, okay? And I can actually sit in Itaca and in the summer it becomes warmer, or at the same time when I'm at the same time at different axes, I do have a different value. So temperature is a field, and we call it a scalar field. Why? Because it's only a number, it's just a number at each point. But I can also think about a vector field. For example, a wind. Why a wind is a vector field? Because at each point, I can measure the directionality of the wind. So I say I have a wind, and the winds go in this direction, and it set the following whatever value. So that's a vector field. A mechanical string. So if you take a string, and you take a string that is in rest, and then I start moving the string, I call it a field. Because at any point on the string, it can be either up or down. So it changes with both x and t. How about the density of people? Are the density of people a field or not? It should be because it's in my list. Okay, very nice. Good answer. So do you think it's a day? So let's look, for example, like we take a picture of the world right now, and there's some places that are very dense. Okay? For example, New York City, or actually also we are here in this room, there's a total, it's crazy dense compared to the average number of density of people, right? And two hours ago, this room was actually underdense, right? It was totally, well, I didn't really measure it, I assumed, right? It was empty, right? So the density of people, it's actually, it changes the function of x and t. Makes sense? Is it a vector field or a scalar field? Scalar field, you just say the number of people, okay? But still a little bit surprising that I talk about the density of people as a field, because what is the density of people like like here on my ear or the density of people here on the top of my head? Can I talk about, can I even ask this question? What is the density of people like, again, any other, like on my finger? So what do I actually hinting here? Why I'm asking you, where I'm going? No, it's not the scale, no, it's not, you could, I mean, that's a very interesting thing, because you know, you have this story of Gulliver that's you know that you can, yes, so you say I can actually take this and then, but it's a little, what else? What is, it depends on the volume. So what do I really need in order to make sense of the density of people as a field? A cutter, very nice, thank you Andrew. Andrew took my course in Cornell last semester, but no, you know this before that. Okay, it makes sense, it actually makes sense to talk about density of people, only if I average over a large enough volume, okay, say if I take some volume or surface area of say one kilometer square and I average over it, then it makes sense, right? But if I go to too small of a thing, I cannot make sense of it, okay? I have to average a thing that is clearly, have to be much bigger than the size of a human being. Only then I can make sense of talking about density of people, okay? And how informally in physics we call this size where we have to average, it's called the cut-off. And which kind of cut-off it is? It is there, thank you Andrew, the UV cut-off, right? It's the UV cut-off, it's the cut-off of the small scale of my theory, okay? So actually, basically all the things that I was telling you around have some UV cut-off. So what is the UV cut-off of temperature? At what scale I cannot talk about temperature anymore? So it has to be somewhat the distance between two atoms. So I have to average over large enough distance, okay? So the only way we define temperature, I have to have many molecules within a volume in order to talk about temperature, right? And if I talk about mechanical string, it's the same story, I have to talk about things that are large enough, okay? So actually what I was about to tell you is that fields are seeing the change in X and T, but actually most fields that we know about have a cut-off. They do not they're at any distance. They have a UV cut-off, so it's some small distance I cannot talk about them anymore, and also usually at very very large distance I cannot talk about them anymore. Any fields have some, it's called the IR cut-off. So the UV cut-off is the short distance and the IR cut-off is the large distance. So what's happening? At what scale I cannot talk about things anymore? Large scale. Yes, the infrared, so what is in terms of my, say, my temperature field? I cannot talk about them at what scale when they become too large. The size of the earth, roughly, say if I care about density of people, for example, so I could have actually talked about because there were some people on the moon at some points, right, so we can actually extend it. But the point is that we can never, we never care about things that are bigger than the size of the universe, right? Any field, I never care about things that are bigger than the size of the universe. So any field actually have some UV cut-off and IR cut-off, at least in principle we think there is. So let's bring me to my last example, the electric and magnetic field, which you're all familiar with, you studied them a lot. What is the UV cut-off of the electric and magnetic field? Do they have? Can we talk about the electric and magnetic field at the size of the electron? I don't know, the electron has a size? No, no, no, it's the correct answer. It is, but the electron doesn't really have a size, right? So it's zero, but no it's not really. What? The Compton wavelengths of the electron. So if we talk about empty fields, empty space, just empty without sources. So the answer is, it's really the electron, it's the size of the electron, it's not really the Compton wavelengths. I hope you appreciate those questions, they're really really cool questions and they're kind of simple. And once we understand them, at least for me, it's really helped me to understand quantum field theory, when I really understand those simple questions. What really tells me that somewhat my description of Maxwell equation breaks down? So when you do Maxwell equation in undergrad ENM, and that I really know because I was teaching it now for three years in a row, so I really understand it now. Okay, so when it breaks down, when you cannot use Maxwell equation anymore, you remember that you will calculate the energy that an electron gives is the integral of the, of E squared plus B squared over 8 pi, even remember the 8 pi. You remember this exercise you did? And you get infinity? Yes? And what you did with this infinity? You remember? Not, not. If you never saw this infinity, it's really cool. It's one of the first infinities that you ever see in your life. And many times, so I really want to, so remember the energy, so in electromagnetism for some reason we call it the W, right? So the energy is E squared plus B squared over 8 pi integral over the whole volume of the system. You remember this? There's maybe some units in some, but it doesn't matter. Even the 8 pi, I shouldn't really care, but I'm just happy that I remember it. So you remember that you do this and you ask what is the, what is the energy associated with the electron? And the answer is infinity. You remember this infinity? Good. What do you mean good? If you have an infinity, you cannot just keep going. It is a, it's not, you have to like stop. You can't say that's good and keep going. So what was the thing that cut off this infinity? There's some UV physics. There's something that at very, very short distance tells us that Maxwell equations are not the correct description of nature. Maxwell equations are the correct description of nature only at some large scales. And at some small scale it breaks down. So what is the scale where Maxwell equation cannot describe physics and we need something else to take over? The Planck scale is, is definitely correct, but there's actually a scale that is much bigger than the Planck scale where things are break down, okay? And I don't know if I should give you the answer because I really want you to think about it. The thing that happened is that when you come to a very, very short distance, quantum mechanics become important. And when quantum mechanics become important, you cannot use classical Maxwell equation. You have to use quantum mechanics. And the size when quantum mechanics become important is roughly one over the mass of the electron, okay? And I never know how it is, how much it is in meters. Anybody remember one over m e in meters? Yeah, yeah, we have to remember these numbers. It's amazing I don't remember this number. We really should, okay, so let's do the following. So the Planck scale is 10 to the minus 35 meter, right? And the mass of the electron is MeV, the Planck scale is 10 to the 19, what? Actually, so it's very small. But the point is that it's not zero and it's a cutoff for my theory, okay? So what I wanted to take out of this is that every field theory have a cutoff and we have to understand that every physics that I do actually depend on my cutoff. And that's again a very important lesson that when you start doing quantum field theory and you get all those crazy infinities and you start worrying about them, you have to understand, yes, that's actually because there's a cutoff, okay? And actually what we understand now, we understand that fields are fundamental. In the old days of Maxwell, we say, hey, that's how it works. We have, before Maxwell, we had like electrons and the electron are the source and the fields are some mathematical trick that we use. You remember the way we learned it and that's, I'm sure you learned it somewhere in high school, right? In high school you say the force is qq over r squared and then you say, oh, let's take one q out and let's call it a field and the field is q over r squared and the force is q times the field, such an amazing mathematical trick, right? And it was roughly Maxwell and some other people around his time that say, no, the fields are not just bare mathematics. They are fundamental physics, the real physics in the fields. So fields are something very physical. It's not just some mathematical trick that we use in high school. It's also a mathematical trick to be used in high school but in fundamental physics it's very, very fundamental and since fields are fundamental we really have to try to understand them and that's what we are going to do next. Okay, so let's take an example of a classical field theory and that's the electromagnetism. It's not a simpler example but it's the example that you are familiar with, okay? So you take the electric field and you find that you have a wave equation, okay? And you solve those wave equations and you find that the electric field is given by a cosine omega t minus kx plus phi 0. Phi 0 is some, you have to give me the initial condition and also a, you have to give me the initial condition and then I solve it, okay? And let me emphasize what I mean by solving it because this is clearly not a solution. The solution has to do with all the initial condition that you are giving me. Only then I really know the solution. What I really do when I say I solve Maxwell equation, what I did is it's following. Here I have E is a function of two variables. It's a function of x and t. And on this side you see there's a function of only one variable. It's a function of omega t minus kx. So when I say I solve the field equation, what I do, I take a function that is a function of two parameters and I reduce it to become a function of one parameter. It's clearly not the full solution. The full solution is that I know everything. So in order to know everything I need the initial condition and the initial condition would be A for any omega. So it's not only A. I should divide A of omega. For any omega you have to give me what is the amplitude and then I really solve it. OK? So just to remember that when I say I solve an equation, you just reduce it from two parameters into one parameter. OK? So some very important implication of this result. Each model is on amplitude. So the amplitude depends on the mode. So if I have different modes I have different amplitude. For example, if you look at the light that come here into the room then presumably it have more yellow than blue. Is it correct? I definitely know that this is correct for the sun. For if you look at the sun if you are outside you have more yellow than blue, right? Because the maximum the black body radiation of the sun is centered around yellow. OK? So you have more yellow and you have less red and less blue inside the sun. What does it mean? It means that each mode have its own amplitude. The yellow amplitude is bigger than the red amplitude. OK? The energy in each in each omega is conserved and that's called the superposition principle. What does it mean? It means that if I produce some electromagnetic radiation with some specific frequency it stays forever. In general we know that we have energy conservation in a system. But when we talk about electromagnetic waves we found that actually each mode have its own frequency. If it's on energy conserved there's no energy move from one mode to another and that's called the the superposition principle. So let me ask you the following question. Are the statements are exact? Do we really have superposition principle in in electromagnetic waves? That is if I have a yellow light and another yellow light come in can they actually collide and create like whatever four red photons or something like this? Can we or cannot we? We do. Cool, huh? Do we see it in Maxwell equation? Ah, good. OK. So of course Maxwell equation tell us that that's what we have and we know that Maxwell equation not the end of the story because what's happened because at the UV we have something else going on which calls quantum mechanics and it's actually violates those kind of assumptions. Good. So we know the superposition principle is a classical statement which is extremely good for photons and these energies. OK. But when we have some starts to have higher energy photons they actually have a very large chances to actually interact. Roughly speaking at what energy of photon you think they will have start having a significant probability to interact you see the question? Yes. The mass of the electron roughly at 1 MeV so if I have 1 MeV photon the cross section becomes significant and without doing any calculation it's just the physics that we understand that the cutoff has to give the mass of the electron so that's where we should start seeing some significance. OK. So how do we deal with the with generic field theory? So we all know how to do classical electromagnetism and now I want to actually ask what I do with generic field theories. OK. So generic field theories is basically that we have many, many numbers of degrees of freedom. OK. So the way I like to think about my field is at any point I have like an oscillator. So it's basically like take a harmonic oscillator in classical mechanics and make it many, many degrees of freedom. And I call them infinite although sometimes they are totally finite. OK. So if I talk about the temperature the temperature is a field but clearly it's a finite number of degrees of freedom. How many degrees of freedom do I have when I talk about the temperature of the earth? A lot. Something like 10 to the 50. But it's finite. It's not infinite. But I approximate it by the infinite number. OK. And the way I solve classical field theory is really like we did mechanics. So in mechanics I have my fundamental thing with x and y as a function of t. That's what I wanted. And here is phi of x and t is what I want. OK. So in classical mechanics x is what I want and x is the function of t. In field theory phi is what I want and x and t are my parameters. OK. And it's also kind of cool because in relativity x and t are the same. They are part of Lorentz space of Minkowski space. So let me ask you the following question. So you know when we are studying relativity we usually say the following statement. We say that time is another space dimension and instead of xi we promote xi to become x mu. But there's this other way which is totally equivalent it's just philosophical and I said instead of thinking of time as an extra space dimension why don't I speak about space as extra time dimensions. OK. So one will say well you know there's three space in one time so you know let's take the minority and make them into the majority but maybe there's actually a good reason to think about xi to become the t mu. Do you have an opinion about what is better x mu or t mu? So of course we all you never I assume anybody saw this notation t mu before? No. Yeah so it's kind of obvious because you say three become so going from three to four it's kind of easy going from one to four it's not so easy and the reason it's not only just one to four because one is is really a scalar and four is a vector so why the hell I want to do this instead of doing this right? But if I was you know a lot of things happening in the history of science because of the way it's history but if it was up to me and someone said hey you Val you can rewrite all the notation of the words OK. So first I would not use better for 10, 25 things right there will be only one thing that call better but the other thing that I will do I will not call it x mu I will call it t mu because for me when I think about space time as t mu it make my life so much simpler than I thought about them as x mu and the reason is as following so think about mechanics so in mechanics what I do I say my action is the integral over dt and when I move to to field theory is there is the thing that appear it's x of t and in field theory it's phi of t mu OK so field theory for me is instead in mechanics I have one time and in field theory I have four times OK the the the role of time in classical mechanics is the same role as x mu in field theory do you see it x mu is just the parameter is just the argument of a function so in classical mechanics I have x of t t is just the argument of x and x is what I want and in field theory I have phi of x mu or it better if I could have say phi of t mu and if it's phi of t mu then I just see just extra time and I do the I integrate over there OK so if you're OK with this I still going to call it x mu just because everybody else do it but I really seen that thinking about space time as four time dimension for me it's really much simpler it make my life much simpler when I go to to field theory so that's what I was saying in field theory in mechanics all I need to do is I do s is the integral of l dt and now when I go to field theory I don't have only one time I have say if I have a string in a string I have two times the two times are called t and x OK so I do s instead of the integral of l dt I have integral of l dx dt and l is what a function of what in classical mechanics l is a function of x and x dot but now I have two times the other time is called x and now is a function of phi phi dot and phi prime you see it it's just a generalization of many times system OK and if we have Lorentz invariance so in Lorentz invariance I have integral of l d4x because they have four times OK and the Lagrangian is a function of phi and all the four time derivatives and the four time derivatives are usually d mu phi OK so you see the way I think about field theory is just a generalization of classical mechanics it's the same thing when you go from calculus to multi variable calculus instead of having one time I now have four times and that's it you OK with this I'm sure many of you seen it before it's not new for many of you but I really hope that we understand that this is all what we are doing so just if we really understand classical mechanics moving to field theory it's not such a big deal just have four times rather than one OK OK so what we do we have the Euler Lagrange equation which is the generalization of the Euler Lagrange equation so instead of having just dL to d phi the x dot I also have the dL to the phi one there's also a minus sign that's come from the metric because it's Minkowski space and then that's how I write it in relativistic notation OK so we done so remember when we talked about mechanics I tell you just tell me the Lagrangian and I know everything now is the same with field theory just tell me the Lagrangian and I know everything the only thing that you have to do is well there's two things you have to do so one thing is that actually when you do classical mechanics x has to do with moving in real space and now my what I really care about is not moving in real space is some field which might be some a little bit more abstract than the position of a particle that's one thing that's one abstraction that I'm doing is that instead of asking the position of a particle I ask a value of something at any point in space time and the other thing that I actually instead of working in one time dimension I work in usually four time dimensions or Minkowski space okay so we're done with classical field theory good I really like this course because every 20 minutes I finish a whole course okay so let's take an example for example a free field theory a free field theory is just a generalization of a free particle a free particle only have a kinetic term it's just mv squared over two so now instead of having x derivative I must have also an x derivative because I have two times right so the Lagrangian the kinetic term of a field of a free field is just d mu phi squared okay and that's my Lagrangian and using the Euler-Lagrangian equation I get this very very famous equation called the wave equation so the wave equation of field theory is the generalization of an ma equal to zero of classical mechanics okay it's just the same you just take a free Lagrangian apply the Euler-Lagrangian equation and get the equation so the field theory the the wave equation of electromagnetism is the same as a equals zero in classical mechanics okay very good so that was field theory now I want to start talking about harmonic oscillators and first I want to ask like the following trivial question why the hell we care so much about harmonic oscillators how many times did you learn harmonic oscillators so I try to remember so when I start doing physics I was learning it once then when I was in university I learned it in intro mechanics then I learned it in advanced mechanics then I learned it in a whole thing that's called like fields and waves and then I went to quantum mechanics one and I learned it I went to quantum mechanics two and I still learn harmonic oscillators and then you go to quantum field and you still learn harmonic oscillators and then you know 20 years later you're a professor what you do you touch is harmonic oscillator so that's a really amazing thing at what we do in physics so why the hell we care so much so at one point I said that when I was in in high school I said that's what physicists are doing they have this big pendulum in the room and they watch the pendulum the whole day right why the hell we care so much about harmonic oscillators I mean if you think about how much you learn about harmonic oscillators it looks like totally crazy how much why we care so much about it because it is an approximation for everything that's really nice okay and that's another statement that I'm really proud of myself of saying it I met he was like some really high up scale in the administration of Cornell of my university and you know those things that you have like you have to practice it you have like one minute to meet one of those people and you have to make sure you make the good impression of them because next time when they need to move like three million dollars to the physics department they will have a good thing about it and that's okay so what you are doing is that I'm in the physics department so what is physics for you and I came with this I'm really proud of myself physics is the art of approximation have you heard this before I really really like it I mean physics is the art of approximation okay because actually if you think about it we never care about exact result it's really okay we're proud of the fact that we can never care about exact result we are not mathematical teaching we only care about being sloppy so it's really nice okay so it's the art of approximation and how do we do approximations as a physicist what do you do when you okay you see some system and you say oh I only care about small perturbation around the minimum right have you state this statement before yes good and then sometimes you care about other thing and then you say well I don't know what to do but if you care about small perturbation around the minimum what you do you find the minimum which sometimes is the far from a trivial statement but let's say it's a trivial statement you find the minimum and what you do you expand that's why we love Taylor so much right because he taught us how to expand and then we expand and then what you do you truncate right that's what we do we expand and truncate yes and where would we truncate usually as early as possible right and where is it it's the harmonic terms right and then we are so happy because we can study harmonic oscillator for years and we never have actually to worry about what we neglected right so that's really the answer the answer is that when we have a small parameters and it's not always the case but in many times when we do have a small parameter that's what we do in physics we find the minimum we expand around the minimum and we truncate and the leading order result is the harmonic term okay a generic potential the first result around the minimum is the x squared what about the constant the constant we never care about what about the linear term when we expand in the Taylor expansion it is zero because it's a minimum so that we know that so the first term that is non-vanished is the x squared and not only that in order for this to be a minimum then the coefficient have to be positive right if it's negative then it's a maximum and I don't want to expand around maxima I want to expand around minimum okay very good because almost any function around minimum can be approximated as an harmonic function okay so that's really the reason that we are doing it so it's very important to identify small parameter expand and truncate okay and once we understand that that's what we are doing actually it has an important implication for our understanding of quantum field theory in particle physics okay so let's do the classical harmonic oscillator the potential is just some number time x squared we solve it and we find that the solution is just a cosine omega t where omega is a function of my k and m and we find here a really cool result that the period doesn't depend on the amplitude okay and that the energy is conserved so when I have an harmonic oscillator the energy is conserved you know these two statement yes which of them is exact and which of them is approximate both of them exact both of them approximate in a real pendulum what's going on the first is or not is yes first is exact and second is not so we have actually four options so let's talk let's talk first about energy is energy conserved in general in nature yes good so what can make this statement not correct if actually I neglect a thing like friction but then of course energy is conserved it just get out of my system okay so energy is conserved in the total in the correct absolute way but if I actually take a physical pendulum the energy of the pendulum actually they always some decay because some of the energy do go outside in terms of friction okay but fundamentally energy is conserved what about the second statement that they and that the period doesn't depend on the amplitude one day my son came back from school I think he was in eleventh grade and he took some physics and he said oh my teacher told me today that it doesn't depend on the amplitude and I said and she said it's an approximation no no she said it's exact say no no no come on I mean ask her tomorrow I mean she must and you know he went and say oh it's I'm sure no no no no it's it's exact and I was like you know so what is it what what is teacher neglected that she didn't understand that she neglected large angle very nice what we're doing okay it's a very famous expansion is the expansion of the I learned it I'm very impressed with this is the spine of the sine function all right remember sine of theta is theta anybody remember la no let's talk about the sine first is it plus or minus good what is the nice we all know just from symmetry argument and here's some number right what the number should be three four two six which is a perfect number so that's how I remember it anyway so it's the sine function is the sine function so of course this is the extra term that is there and this term actually make the amplitude and the the period depend on the amplitude okay well let me ask you a little different question which is a little bit of a detour from what I'm having doing here is that when you have a relation like that I did them period doesn't depend on the amplitude it should remind you some kind of degeneracy right they just say that all the all the things have the same frequency independent on the amplitude right I have many many amplitudes and always the same frequency okay so there might be some symmetry that guarantees it you know what is the symmetry that guaranteed that the amplitude doesn't depend on that the the period doesn't depend on the amplitude have you ever asked yourself this question do you see that it should right so if I if energy is conserved if I said the energy doesn't depend on on the movement there's a symmetry that's t go to t plus c what is the symmetry that's guaranteed that the period doesn't depend on the amplitude I leave you with this if you don't know the answer please think about it please please please and good that we have this homework session and I want you to think about it because if I give you the answer you you lose all the fun okay I really want you to think about it and you can talk to your friends please look and say what is the symmetry that guarantees okay so now I want to move to coupled oscillators and what I have here in the picture so this is a picture actually taking from a where I did my undergrad my undergrad study and that's a picture of a coupled oscillator you see here a swing and here's a swing okay and those are connected by another spring in in top okay and when I was a grad student I was going and giving tools to kids coming from school and the way we teach kids from school about coupled oscillator is like this we put one kid on this swing okay and we put another kid on this swing and I asked this kid not to move okay and I take the other kid and I do like this then I said one two three I usually do it in Hebrew okay and I say boom and it lets go and then what's happened can you describe me the system what's really going what's really happening actually it's a YouTube video I should have brought the YouTube video okay so what's happening is the following so one kid start oscillate with some amplitude that depend on my initial condition okay and then because of this spring in the top some energy move to the other kids so this kid go like this and then this kid kind of stop and then the other kids get a maximum amplitude and then this kid kind of stop and then I really should you saw it in maybe you didn't see it with kids maybe you see it with you know some other things that have mass but kids have more than muscle it's only one property okay so they have and so what's really going on what is actually going on is that I have kind of energy in this that move into this so the energy actually move from one to the other because they are coupled okay but then you actually know what are the things that do not move and the two things that do not move are there what energy what things do not move energy from one to the other these are the normal modes right and then actually some of the kids that are advanced enough we do these normal mode things okay it's not exact because this is you know it's not like a system but how I do it I ask a friend to go with me to the same amplitude and I leave them and then the both kids do like this without anything and then we do the other way which is a little harder that I hold the kids and then the other friends drop the game and when the kid is at the maximum then I leave them and we get into this symmetric mode and we go to the anti-symmetric mode and in the anti-symmetric mode the kids just go like this as if there's actually nothing in between okay so we know what it is it's the normal mode it's the famous normal mode that we have each normal mode is not local it's not like a regular pendulum that is here the normal mode is something that is over the whole system it's involved the two pendulum it's this generalized coordinate that we have and the energy in each mode is conserved and of course the fact that the energy in each mode is conserved is as always just an approximation and if I have some higher order terms like this then it's actually move energy from one to the other so what's actually tells me how much energy move from one mode to another what tells me what move from one mode to the other is the coupling between them it's the higher order term if the thing that are not quadratic they're not x squared or y squared the thing that go like say x squared y and the larger this alpha is there's more energy that move between one mode to the other oh yeah yeah yeah yeah she's kind of a very bad picture right so what I want you to think about and I really want you to think about the relation between fields and harmonic oscillators okay and we already talked about it that I see fields as a generalization of harmonic oscillators but I really want you to understand try to think for yourself talk to some of the people that you meet here so too and ask what are really what are how we actually see fields a generalization of harmonic oscillators okay and so the next thing I want to do after we done with there are simple harmonic oscillators I want to actually move to quantum mechanics and once we understand the quantum mechanics of simple harmonic oscillators we finally will be able to really get quantum field theory and we'll be done with quantum field theory course okay so how do we do quantum mechanics so there's many ways to do quantum mechanics and one way that we like to do quantum mechanics is we put a little head above everything right so you take your system and you say my coordinate my classical system has a my degree of freedom is x and I make x into an operator and that's where I do quantum mechanics right and in order to solve quantum mechanics we need to know phi of x and t so you remember when we talked about classical mechanics in order to solve mechanics I need to find x of t in order to solve quantum mechanics I need to find psi of x and t and psi of x and t is actually a field you see that is a field is a function of x and t so it's a field and how many how many wave functions describe a system so when I try to solve classical mechanics I need to find x of all the particles so if I have five degrees of freedom I need to find x1 up to x5 in quantum mechanics do I need how many wave functions do I have one okay so very important in when we have only one wave function to describe the whole system okay so then we move on and do the quantum harmonic oscillator and it's a really really cool we write the Hamiltonian p squared by 2m plus m omega squared x squared we got the spectrum the spectrum is m plus half h by omega and then we use this amazing cool trick and we say oh the Hamiltonian can write as a dagger a time h by omega where a and a dagger are x plus or minus ip there's usually in the book there's a lot of numbers and symbols before that they are very confusing and it's very good not to write them if you really want them you can go to the book and find them but really just ignore them okay of course this must be wrong because they are different dimensions so it must be wrong but this is this sign this sign is an extremely important sign in physics right it's about the sloppiness right so I want you to see but a and a dagger just says x plus ip and x is basically just a and a dagger so that's very abstract right x in classical mechanics is a position of a particle and then in quantum mechanics I write x as a sum of sum a and a dagger which are not even a mission and so just some something okay and then I call this a and a dagger creation and annihilation operator and I say when I apply a on some state I get n minus one and when I apply a day on n I get n n plus one let's look a little abstract why do I even care about those a and a daggers why we even introduce those a and a daggers so when you introduce when you did it what was the professor telling you why are we introducing those a and a daggers because it's really cool to write it like this it's like so much nicer than this why I cannot do a monic oscillator like this I mean I just write my internal like this and I solve it and that's it and I get this and I get the wave function and the wave function are whatever they are the hermit polynomial sorry so this one yes so one one answer is totally nice and it just really cool abstract thing that you can do so you can do it like this and you can do it with other things and use a lot of little really tricks which are really cool mathematically I totally agree but there's some deeper reason that this one is we really like a and a dagger that we want to use it that it's extremely hard to do when I use this Hamiltonian there's one reason that we we really care about a and a dagger much more yes so that's what we say we can actually solve the harmonic oscillator and all that and it's totally correct but there's one big reason that I want yes yes yeah yeah so so you are totally right you can there's so many little really cool things like when you start doing this a and a dagger it's become like so abstract right you know all those yes yeah yeah yeah yeah yeah you kind of you kind of in the direction I want to go they allow us to go from one state to another but can we go from one state to another don't we say that when we have an harmonic oscillator the energy is conserved so if I start with let's say I start with some it's allowed me to actually go from one state to another so it's allowed me to kind of build the Hilbert space very very nice you all going around the one thing that I really want you can find metrics and yes yes yes okay let me come back let me come back to let me come back to quantization of energy yeah we can do it let me come back to the thing that I was telling you from the that I was I'm telling you the whole story that I you know I'm very proud of myself that I was able to make a big contribution to science and I said that physics is the art of approximation you remember this that I was telling you that I'm so proud of myself yes so now that I told you that physics is that why do we really care about a and a dagger so much we really care about a and a dagger so much only when we really go beyond the harmonic approximation in the harmonic approximation to basically if I really have just the harmonic approximation I can do everything without ever mention a and a dagger and you can play with a and a dagger forever and it's very very cool but you don't learn anything new like nothing new learn from them the time that you really learn so much is when we go beyond the leading beyond the harmonic approximation and did you do a perturbation theory with harmonic oscillators back then in undergrad or in grad did you some of you some maybe I don't remember yeah it was one homework but I was sick that day right maybe maybe right okay so now I want to actually go and start doing a perturbation theory and in perturbation theory this a and a dagger become extremely useful and they really really really gonna give us the intuition that we need when we start doing quantum field theory okay so let's talk about coupled oscillators I will come back to the a and a dagger soon and let's think about this kind of thing so I have my potential is this and I have a coupling alpha x y and this is a still harmonic you see it's harmonic in terms that it's have only its second order in x and y and the normal mode are this the x plus minus y and the two frequency are k plus minus alpha and I want to ask you the following question what is the quantum mechanics oh shit energy and spectrum of this system page 25 in my first typo I think it's very impressive no okay what is the quantum mechanics energy and spectrum of this system so please I want to do the following it's already more than one hour that I'm talking I'll be quiet for one minute and I want you and you can talk to your friend around you and if they're not your friend that's a good time to make friends okay and actually try to write the answer so what I want an answer is following I want the answer like this write it write it down I want to answer e with some indices here I don't tell you how many indices equal to something and I want you to write some ket with some indices I'm not going to tell you how many okay so I want an energy and a brine and a ket of this I'll be quiet for a minute please everybody have something to write or a computer no you should when you come to a lecture you should have a pen and a paper oh I see many of you do have it okay so I'm asking and if this question is a little hard let me ask first what is the ground state what is the ground state energy so you know for a simple harmonic oscillator is h bar omega over two what is the ground state energy of this one talk to you first start yes I want to hear I want to hear this background note h bar omega please please I think yes start talking so I can be quiet otherwise it's amazing don't be shy talk ask you prayer do you know the answer so next time I know you will sit alone that's I should have told you in the beginning to make sure you're still right right okay so can someone tell me the ground state energy just the ground state energy h bar nice good start yes h bar time okay let me keep going right anybody someone be brave it's it's believe me it's you have to be brave to talk in front of 150 people someone right now yes alpha over m let's open a parenthesis what is it one minus alpha over m not really good good but you know the fact that you were brave enough to actually talk it's it's good because it's not so trivial anybody else omega plus plus omega minus very nice omega plus plus omega minus very nice so let me explain why it is so basically we know in quantum mechanics we know in in in general in physics when we have two harmonic oscillators we can treat each of them separately that's the big deal about harmonic oscillators that the superposition principle that everything we learn about and if I have only one system with just omega plus this is the ground state energy is h bar omega plus over two if I have only omega minus it's h bar omega minus over two and you have two degrees of freedom the two degrees of freedom are q plus minus so this is the energy the ground state energy of q plus and this is the ground state energy of q minus okay so now how you generalize it to a general e so the e how many indices the e is two let's call them n plus and n minus okay good good and now it's just the generalization so each h bar over two n plus omega plus plus n minus omega minus and what is the state so the state is n plus and minus and what are n plus and n minus these are just the standard harmonic oscillator whatever the hermit polynomial or I don't remember what they are exactly and what is the argument of the hermit polynomial is is it x one or x two what is the argument of this is the q plus and the argument of this is the q minus okay good so we understand how we solve actually a system of capped harmonic oscillators the way we solve the system of capped harmonic oscillator so that's what I was just writing I wrote so this the way we solve a system of capped harmonic oscillators we move to the mode to the eigen mode and then we treat each mode as a separate harmonic oscillator and then we just add their energy and we multiply the wave function to get the wave function of the total system okay any questions on this yes oh that's a really really important part that's section four in Landau-Lichitz and it's really this is the fact that when you have a Lagrangian that is separable then you can actually treat each part each term of the Lagrangian as a separate system such that the total thing is the sum of them and when we go to quantum mechanics the statement is the same if you have a Hamiltonian or a Lagrangian that you can write the Hamiltonian as the sum of two of two Hamiltonians you can solve any Hamiltonian separately and the wave function is the product of each wave function and the energy is the sum and that's a general general statement in quantum mechanics oh so that's of course the whole point is that I neglected the interaction okay I misunderstood the question of course this is a this is in the harmonic approximation of course this is an approximation and we're proud of it right good so how we actually move from this into fields how this is related to field so when I have one degrees of freedom I just say a I just a and a dagger when I have two degrees of freedom I have a plus and a minus when I have seven degrees of freedom I can call them a one up to a seven and when I have a field let's say I have a string with many many many degrees of freedom instead of having an index a i I call it a of k so k is just the the continuous variable there is a generalization of the index do you see what I do here if I have one harmonic oscillator it's just a and a dagger when I have two I have two a's and two a daggers when I have seven I have seven a's and seven a daggers when I have infinite I have infinite a and infinite a daggers but instead of calling them is an index i I put it in a parenthesis and I call it k so this k is just the continuous approximation of the district number i and what happened to the state when I have one harmonic oscillator I have n when I have many my state is n one and two like this and when I have infinite then it's become n as a function of k I just have all of them and what happened to the energy I start with n plus half h by omega then I have to sum over all the a's and when I go to the continuum instead of a sum I have the integral so this will be the integral so you see how I think about a free field a free field is as a set of infinite number of harmonic oscillators and the reason is that I always expand around the minimum and therefore I think about my field as a free field to approximate okay so now let's move to quantum mechanics and in quantum mechanics in when I think about a harmonic oscillator my x become an a and a dagger so the position of the particle is this abstract thing that the position of the particle I can think about it as creation and annihilation of state when I go to quantum field theory my field is an infinite number of harmonic oscillators so my field become infinite number of creation and annihilation operators do you see this statement a field is nothing but many many many harmonic oscillators and see the quantum mechanics harmonic oscillators are just creation and annihilation operators that's what a quantum field is x and t are just the generalization of time so x and t are not are not operators anymore in quantum mechanics x mu become just t mu is just time and the field itself is just generalization of x and the generalization of x is just creation and annihilation I know you heard it before many of you and if you never heard it before you know it's a little bit abstract but that's how I want to think about the field it's very abstract to think about the position of a particle as creation and annihilation operator okay so it's a little more abstract to think about the electric field as creation and annihilation operator and it's not one it just happened to be infinite of them okay but it's just this yes so no no no so that's when I think about a mechanic system when I really care about position of a particle but now I said a quantum field I don't really care about the position of a I care about the electric field at some point and the electric field is the equivalent of x in classical mechanics and x and t so the electric field e is a function of x and t and x and t are actually playing the same role in the electric field right and that's something that you are very familiar when you solve the wave equation you always see that x and t are so similar right so if you think if you solve the wave equation and you take a picture at a given time you see a sign okay and if you do the other way you stand at one point and you just measure the function of time you also see the same function because x and t are so so so much the same so the field is something abstract it's like the electric field is the value of the electric field at this point okay at this x x at this x and t okay and it varies its function of x and its various function of t okay but it's the abstract thing this phi or the electric field and then this electric field that is already kind of abstract you make it more abstract we say oh this electric field is actually creation and annihilation operators of some arbitrary thing but then there's some abstraction in the fact that oh electric field is actually light which is very abstract the moving of the electric field is light so then when we talked about creation and annihilation operator it's a little bit abstract and you have to accept abstraction at this point and then we'll see where it leads us any more questions okay so let's move on so then I asked you two questions so now we try to understand what is the particle so what is the energy that this takes to excited harmonic oscillator by one level and what is the energy of the photon and that's something that we learn many times when you have two things that have the same answer it's much easier to remember them so what is the answer for this so you all know it's the same answer it's H by omega so H by omega is the energy that it takes to excited harmonic oscillator and it's also the energy of the photon and why it is the same answer why the fact that actually the harmonic energy that is taken to excited harmonic oscillator is the same as the energy of the photon and the answer is that basically oops I wanted to do something impressive and it didn't work excitation of simple harmonic oscillator are what we call particles and that's somewhat abstract but I really it's not a very surprising fact okay so you already kind of see it from the fact that both of them H H by omega but if I ask you what is what is the particle what is the photon say what what is the photon the photon is the excitation of the electromagnetic field right that's what it is and when I say I have five photons that means that at this specific omega my quantum mechanics level of this specific omega is at level five that's mean that I have five photons so I hope this is abstract but that's really the way I like to think about the particle so when I think about simple harmonic oscillator and I say my harmonic oscillator is at level nine I think about it as having nine particles in this state okay and then the name creation and annihilation operators actually make a lot of sense because when I create when I go from three to four and I give it an energy of H by omega I create one extra particle so it is abstract but it's no more abstract than what you are familiar with light the fact that we say that light is excitation of the electromagnetic field in a way it is very abstract right this light is you know you really see it you see it in your eye right because our eye is a detector of light but then we say oh this light that we see is just excitation of the electromagnetic field okay and the electromagnetic field to a very good approximation is just given by set of infinite number of harmonic oscillators so light is just excitation of harmonic oscillators right so then if I want to quantize light what do I need to do I said oh it's just a harmonic oscillator so quantize the light it's just quantizing harmonic oscillators and then the photon which is the basic quantity of light is just the excitation one excitation of harmonic oscillators okay so that's the statement and as I said I know that many of you heard before is the fact that particles are basically just excitation of harmonic oscillators or basically the excitation of the field okay so the way I like to think about particles particles are excitations of harmonic oscillators and the harmonic oscillators are my fundamental fields okay we all know what particles are so we can move on but ask me more questions if there's anything so what about masses so when we talk about photons photons have they are massless so now we can actually play the game and I was telling you all there's excitation of simple harmonic oscillators and now we can actually do more things that are more advanced and for example what I want to do I take my Lagrangian I can start adding terms to my Lagrangian I can take my Lagrangian that's the free field Lagrangian that is the one that give me photons and I can add this extra term and when I add this extra term what I find out is that I still have free particle but these free particles have mass yes oh so of course eventually we kind of think about if there's a gauge symmetry okay but at this stage you can think about just 5 go to 5 plus c like just translation so that's the same symmetry that's guaranteed that you have free particles okay but I'm not actually going there yet so I add this this state the m squared plus 5 squared and what you find that this one actually gives you a massive particle how do I know that is a massive particle because for massless particle from this I get omega is equal to k and for massive particle I get omega squared equal to k square plus m squared and what I like you to do I call it homework but of course you know we are not at home here but what I mean is that something to do after the lecture so what I want what I would ask you to do and that we're going to discuss it tomorrow in the solution session is actually to actually prove it that is you take use you guess a solution and by guess a solution it's you know you get you guess the correct solution you take this guess plug it back in and you see that this solution actually satisfy this equation and it's give you this relation omega squared equal k square plus m squared and you know that omega is energy and k is just momentum and this gives you e squared equal p squared p squared plus m squared which describe a particle with mass m okay so please write it down and try to actually do this with the other question that I asked you so what about some other terms so the other terms I can actually add something like lambda five to the four so what's happening you see what I'm doing actually I should I should back up so what I did is the following I start with field theory and I have my Lagrangian for field theory and the first Lagrangian that I did was the free Lagrangian that give me the wave equation that we all know and love that give me the like electromagnetic radiation and then I add this extra term and this extra term is still a free particle in the sense that the energy is conserved but the dispersion relation are such that it's have a mass and now I said I can keep adding terms to my Lagrangian why not actually why not what is the thing that forbid me also actually normalization is the one that tells you five to the six but actually I don't care too much about normalization we're going to talk about it a little bit basically if there's no symmetry that forbid me to add a term I will add this term right so now I can add for example lambda phi to the four and then I plug it into the Euler-Lagrange equation and I get the following relation I get this kind of an equation now if I don't have this term then the solution is very easy and that's what I was just telling you just e to the i something you plug it back in and you see everything and that's what I ask you to do but this I cannot do why I cannot do it because it's non-linear and basically we don't know how to solve non-linear differential equation yes oh so in principle I should add everything right so I just say as an example I add just phi to the four just just for the sake of the example but of course I don't really yet bear how to build the Lagrangian we're going to talk it to too much lengths okay the point I want to make is that I actually I cannot solve this equation because it's non-linear so how do I do what I do when I get stuck we always do the same thing right we do an approximation so what I do in this case how do I solve this equation so I assume that lambda is small so basically I said if lambda is small I know what to do right yes so then I say well let's assume that lambda is small and then if lambda is small what I do I do perturbation here I said let's start with lambda equal to 0 then I solve the the the system and come back so let me end the lecture by a short summary and then tomorrow we continue from this point so what I was kind of doing with you we kind of do a little review of you all three years of undergrad physics and what we did is the following I emphasize to you that fields are generalization of simple harmonic oscillators and I really want this point to get really in us because the the the reason that we care so much about harmonic oscillators to when we do particle physics and energy physics is that our fields are just this they are just harmonic oscillators okay they are much much more abstract harmonic oscillator but that's what they are they are harmonic oscillators okay and then we talked about the fact that particles have this excitation of the harmonic oscillators and what I want I want my Lagrangian to describe the system okay and my Lagrangian is given in terms of fields the fields are the generalization of the coordinate and classical mechanics okay so that's why it's not surprising at all that I can write field as creation and annihilation operators it's nothing but x of classical mechanics and our aim in particle physics is to find the Lagrangian is to find l okay we can only solve the linear case so we only know how to do simple harmonic oscillators and that's why develop all those formalism and now we actually have to start and moving into the perturbation theory so what we're going to do so what I would ask you is that I gave you few little questions throughout the the lecture please try to to think about them right then I should have actually summarized them but I didn't but actually what I I will give this we can post them right we can post my lecture so I will post them and then you see the few questions I ask and we're going to discuss them tomorrow and the last thing I want to say please please please come and talk to me I love to talk much more about everything of this okay so I guess I see you in lunch and tomorrow and all this thanks