 Okay, good morning everybody, we're ready to start QFT 2 and then a standard model. Thank you, good morning everybody, can you hear me good? I know it's, all I need to do is do a little help. I hope everybody's awake, you know, it's this 9am thing. And at Cornell at least I had people literally fall asleep in my class. I don't know what to do, I try my best jokes and they were like sleeping, like totally sleeping. So I hope that you are awake and because we have a lot of fun going on today. So we are going to continue where we started yesterday. And before I go on, let me say that for the, whatever it's called, the question session in the afternoon. I prepared the question, so there's some question I was kind of running on you yesterday and I summarized them in a file and there's some more question that we're going to have today and it's going to be in a file and I'm going to send the file to people and they're going to post it. So then by, when you start having the session you can download it and start looking at the question and definitely I will be around so if you have some question about the questions, I'm here to answer about the answer. Okay, so yesterday we were starting to talk kind of a brief, very brief hand wave introduction to quantum field theory and the way I like to think about it is following. It's a little bit abstract but the idea is it's following. So the way I like to think about field theory is just a generalization of classical mechanics. In classical mechanics the question that we ask is that we have a position of a particle and the only problem we think about a position of a particle is that it's very, it's something that we are very used to. Okay, so it's not abstract enough. But the thing about the position of a particle is also somewhat something abstract and then life would be much simpler. So in mechanics you say I have a position of a particle and there's only one time and in field theory I ask what is the value of a field in four-dimensional time, okay, which is called T mu. Okay, so that's basically the bigger step, the bigger abstraction that we take. Instead of thinking about a position as a function of one time, we think about the value of something as a function of four-variable which we call x mu or T mu. And then we take the Lagrangian instead of x and x dot it's become phi and d mu phi and the action instead of integral over one time it's integral over four times or four x mu. And then just with a little bit of abstraction we just say oh everything we know for classical mechanics we can just take over and we understand. And the next step we did is we always expand around the minimum and we just say just like in classical mechanics in quantum field theory in field theory is the same story we expand around the minimum and when we expand around the minimum everything is harmonic oscillator and then we kind of identify what is a particle and we say oh a particle which is something that we kind of have some intuition what it is it's become more abstract and said oh a particle is just an excitation of a field and it's a little bit more abstract but it's okay that's what we are you know when we do physics and abstraction in physics is just about getting used to it, okay. So after doing quantum mechanics for so many years and I was teaching classical mechanics it looks so weird to me. For example that you know when you have like some matter here and a particle going around and it doesn't feel the whole space all those kind of things or for example when you get used to relativistic notation and you start something and non relativistic is something very weird it's just because we get used to it. So all this idea that a particle is just an excitation of something it's something that we hopefully will get used to it. And when we ended last time we asked the question what's happened with IO order term and then all of you said oh we have to use perturbation theory so that's what we're going to start today. So let's start with perturbation theory and the very basic idea of perturbation theory is the following. So let's consider that we have some Hamiltonian H and we write the Hamiltonian as H0 plus H1 where H1 is much much smaller than H0 and already here it's a little bit surprising what do I mean by this because what is H? Well H is an operator right so what do I mean by an operator it's much smaller than another operator I know something is small when I said oh five is much much smaller than a million that makes sense to me okay the size of the S is much smaller than the size of the galaxy that makes sense to me but what does it mean that I say H1 is much much smaller than H0. Is the eigenvalue of H0 much smaller than the eigenvalue of H1? Is that the correct description? Yes the coupling is small that's what we're actually at the end of the day have that we have some number that the p-order order is small but again it's not very well defined so we have to be kind of careful what we really mean and what we really mean by the eigenvalues usually we mean that the eigenvalue of H of total H minus the eigenvalue of H0 is much smaller than the eigenvalue of H0 okay that's what we really mean kind of formally okay but we have to remember that that but we kind of have the intuition the intuition that we have something very small and we want to see what is the effect of this something that is very small okay and in many ways perturbation theory and that's usually the way we learn perturbation theory we say perturbation theory is that's a mathematical tool to actually use things that we cannot really solve so if I don't know how to solve the full Hamiltonian I use perturbation theory and if I could solve the full Hamiltonian I never care about perturbation theory okay that's kind of the image that we have perturbation theory is just a mathematical tool okay but when we do particle physics perturbation theory in a way the way I like to think about it is much more than a mathematical tool it's really something that really help us to describe the way we see the world okay and let me discuss it in the following sense so in many times we really like to work with the eigenvalue of h of the total Hamiltonian why why would we like to kind of choose a basis where the eigenvalues of the full Hamiltonian is our basis what is special about the basis of the eigenvalue of the Hamiltonian hmm so everything is physical so I can it's a basis choice right so in what way you are you are you are correct I just want you to be a little more precise in what way it's really refer to physical states it is a steady state the eigenvalue of the Hamiltonian are the state that propagate in time without nothing okay and another way that I like to talk about it is that if you're in an eigenvalue of the Hamiltonian you are very boring okay so you know if you have some friend of you that is very boring you say I is an eigenstate of the Hamiltonian nothing happened right an eigenstate of the Hamiltonian there's just a phase and nothing really happened if you want to have some action in your life you better not be an eigenvalue of the Hamiltonian right because only then you have some dynamics okay so actually walking the eigenvalue of the Hamiltonian is very very nice if you just want to understand how a system in a steady state where nothing happened then is very very nice to work in the eigenvalue of the Hamiltonian however when you actually think about dynamics in order to have dynamics you must be in a non eigenstate of the Hamiltonian okay and if we do particle physics and in particle physics we actually really look after the dynamics and we like to ask the question if I have a muon and a muon decay into an electron the muon that I measure better not be an eigenvalue of the Hamiltonian is that clear if the muon were a true eigenvalue of the true Hamiltonian this muon would never decay and would never interact yes so the muon that we all know and love and the one that we keep hitting up from the sky all the time these muons are not eigenvalue of the Hamiltonian they are eigenvalue of h0 they are the eigenvalue of the approximate Hamiltonian and the reason that the muon decay is that because it's not a true eigenvalue of the true Hamiltonian it's only a very close to the full one and it's the h1 is the perturbation that make the the muon decay okay so the way we like to think about in particle physics we think about our particles our particles are excitations of the h0s are not the excitation of the full Hamiltonians are the excitation of the h0 and the decays are what happen because we have perturbation okay so the way I like to think about it when we think about particle physics perturbation theory is a little more than just a mathematical tool it's actually something that bring us into a better understanding of what's going on in the physics okay so let me talk about perturbation theory for two harmonic oscillators and I took the following potential I have two two oscillators x and y and I have some coupling between them alpha and you see that alpha is x cube y so it's a dimension it's a dimension three these two are dimension two x square and y square and this is dimension three and we assume that alpha is small and what's happened classically classically alpha move energy between the two modes and you remember we saw this a couple the swings yesterday so how it's moved the energy between one mode to another when alpha is bigger this the energy flow is is stronger right so now I'm asking you the following question what's really happening quantum mechanics so in classical mechanics we cannot have the following intuition if I have two pendela that are pendela pendulus two two of those two of two of yes if I have two of those of them and there's no coupling at all and I say I only give energy to this one this one would never move now I put a little coupling between them then some energy from this one start moving to the other one okay and the bigger the coupling is the energy flow is stronger okay there's more within if I say oh how much energy I have in the other pendulum after one second the bigger the coupling I have more energy in the other one is that clear now I'm asking what's happening quantum mechanics how in quantum mechanics energy flow from one oscillator to the other do you see the question so what is really the analog the quantum analog of this energy transfer from one pendulum to another from one spring to another yes so the interaction term is what's doing it but what's really what is the dynamics so in in classical mechanics we understand the dynamics again this this example that I gave you I should I should have bring the video that you see the video that you know you have one kid on this swing and then suddenly some energy go to the other kid on the other swing and the time that it takes from one from the energy from one kid go to the other depend on alpha okay the bigger alpha is quicker the energy flow yes yes but now I'm talking just about just about harmonic oscillator what's really going so in in in quantum mechanics what's happening when I make alpha large what's really going on how energy transfer in quantum mechanics how let's the you know something about how to do the calculation and we're actually going to do it soon but I'm asking the description what is really happening the description sorry interactional yes but how it's happening what's really happening okay so let me give you my best example of really what's going on and the answer is following so as you can see there's two faucets here okay I took it from the internet and this one there's a constant flow of water okay that's the classical analog in classical mechanics there's a total flow of energy and if you turn the faucet and you make it more what's happened you have a bigger flow okay so turning the faucet is like making alpha stronger now what's happened when you take a real physical faucet and you start faucet right and you start turning it off and off eventually there's no more flow what you start having you start making this dripping okay one drip at a time yes okay and you know how we call this dripping it's a quanta of water every time there's one quanta of water it's called a drop okay so that's what's really happening quantum mechanics what's happening quantum mechanics that instead of having a constant flow of energy from one to another the flow must be done in quanta okay that's why it's called quantum mechanics every time there's some energy moved from one from one oscillator to another it must go with one quanta what is the quanta by the way h bar omega right so every time I move there's smallest energy that I can move is h bar omega okay and then I ask how much is what is the when I make alpha smaller and smaller the probability of moving becomes smaller and smaller okay so if I say if alpha is some value and then I say on average one I have one quanta moved per second then I make alpha smaller then it will come one one quanta moved per 10 seconds and then one quanta moved per one hour is that clear that's clear the difference and you can do this experiment okay just take whatever go back to your hotel open a little bit don't waste too much water okay and then small and you start seeing the drop okay so by the way this analog is not totally correct what is the thing that is not quantum in this analog of the faucet do you see the analog right what is not not not correct what is not correct in the analog oh that's true but that has to do with statistical mechanics so the thing that is not really correct here that in quantum mechanics we don't know when the next drop going to happen in the faucet you're going to see it's like ta ta ta ta but the quantum mechanics analog would be ta ta ta ta ta something like this it's going to be awesome uh I'll be trying the only thing that we know is that the average going to be the same as this analog that we have here okay so we actually got the understanding what's happening in quantum mechanics so when I take two two oscillators and couple between them there's energy flow from one there could be energy flow from one to the other and the way the energy flows it's not like in a classical constant flow there's some probability that this energy will flow from one to another and when the energy flow it flow in a quanta we okay we all understand this good so then there's the famous Fermi golden rule and as a cool trivia who invented the Fermi golden rule come on I know it's too early okay who invented the Fermi golden rule so the obvious answer would be Fermi why because if it's called the Fermi golden rule you would think that Fermi invented the Fermi golden rule right that's usually the case however it was not Fermi anybody know who invented the Fermi who anybody Dirac very nice so why it is called the Fermi golden rule and not the Dirac golden rule because Fermi went out and tell everybody look this amazing work of Dirac he did so much this is the Dirac thing and everybody call it the Fermi golden rule it's a true story for apparently Dirac was very very quiet person and Fermi was the total opposite he was a very talkative person so what Dirac was saying and Fermi making everybody know about it is that the probability so we asked what is the probability that this one quanta move from one oscillator to the other in order to calculate this probability we have to calculate what we call the transition amplitude the transition amplitude is the f and i are the eigenvalue of h0 so the final state and the initial state and I put in the middle the perturbation so I put the perturbation in the middle and then I calculate the the transition amplitude and the probability which is just a number or I said okay it's not it's not a number it's a how much time it's take between transition on average it's proportional to the amplitude square time phase space okay and phase is just the density of states that we have so this Fermi golden rule which I assume all of you seen before and you studied before it's our starting point it's our starting point to do all calculations all the calculation has to do with this so I have to calculate the transition amplitude then I have to square it and then I add up all the states and I know what is the probability to have the transition from one harmonic oscillator into another harmonic oscillator good oops so let's talk about first order and second order perturbation theory and again I assume you all seen it before so when we have first order perturbation theory basically I just put my perturbation between the final and the initial state that's the transition amplitude now the initial and final state must have the same energy so I can actually have a transition so it's a little different than what you used to so when you do perturbation theory most of the study in quantum mechanics you calculate correction to the spectrum okay and then you just have the same state or if you have the general state you're looking some subspeed of the of the general states what we are doing here is we calculate transition amplitudes which is very similar to calculating correction to the energy but it's not really the same so I have two states that are different i and f but they do have the same energy so I can have transition from one oscillator to the to the other because they have the same energy and this is first order perturbation theory in first order perturbation theory all I care about is just the state that I involve I don't care about the rest of the spectrum so if I ask two states that have an energy three I don't care about the state that have an energy nine okay it's only the state that have the same energy what happened in second order perturbation theory and second order perturbation theory the transition is actually going in the following famous formula that I have the my initial state i coupled to some intermediate state n and then this intermediate state n coupled to my final state f and I have to sum over all the intermediate state n and I sum them but then I pay a price by the by the denominator and the denominator tells me the following things is that when n is farther away from my initial and in the energy of the initial and final state the contribution of this state is less important okay so the way we think about second order perturbation theory is unlike first order perturbation theory that only care about my the state that are with the same energy second order perturbation theory cares about the whole spectrum it's an amazing statement it's the statement that quantum mechanics at the end everything is connected so in second order perturbation theory I actually I know about the state of the theory that has far far away and the farther away they are the contribution is smaller and it's suppressed exactly by one over delta e that the energy difference from the state that I care about okay good so let's come back to and ask what's happened when we have transitions in second order perturbation theory for harmonic oscillators okay and you remember we talked so much about the creation and annihilation operators and why they're important and I said they're important for perturbation theory now it's where they become important so I have the following thing I have my perturbation and I'm asking you the following question if I have a given i what if the amplitude is non-zero okay so this final state f what is the one that this transition matrix element is non-zero do you understand the question so in general I just say I have my initial state i and this f could be anything so there's actually infinite number of states here okay and amazingly enough almost all of them this amplitude is zero only very few this amplitude is non-zero so what I'm asking you is the following looking I would be quiet for one minute and I want you to talk like with your neighbors and make friends and ask the following thing what are the f's that are not zero is the question clear not really so if i let's say i I can say i is a five of x and two of y okay so I can think about some initial state i let's make it specific so my initial state i is nx equal five and ny equal two let's say ny equals nine okay that's i now I'm asking what is f such that this matrix element such that such that f x square y i is not equal to zero so what I'm telling you is that there's only very few f's where this is non-zero for almost all the f's this one is zero so let's take this very specific example five and nine and I want you to tell me what are the ones that are non-zero okay so for example 2020 will be zero yes if you have the answer don't tell it because I want everybody to okay so what I want to do is I want one minute to be quiet please talk to your neighbors and do it just write and I want an answer which are the ones that are non-zero okay please please I know it's early in the morning you can do one two three four and then start writing okay please please try to do it okay so I hope some of you got the answer so can someone tell me what f's are non-zero someone yes you had it before yes nyf to be ny can be either eight or ten and nx five seven or three so let's go three five and seven very nice so that's the right answer and I hope many of you got this answer so let me see how he got the answer I don't know but I assume that that's the way he got the answer and you tell me if we had the telepathic idea and I got you right okay so the way he was doing it he said oh it's very easy it was so easy he just you know I didn't even start talking already had the answer he just said I can write my x x is a ax plus ax dagger right that's what x is and what I have up there I have so let's start with y y is simply y is a y plus a y dagger right so when a when y work on my initial state what my y can do for the initial state so if my initial state have ny equal nine and I have y in the middle what the y can do for my initial state it can either make this nine into an eight by applying a or it can make this nine into a ten by applying the ay okay and therefore my final state only if it's eight or ten will be non-zero because my after I apply it it become either eight or ten and then I have x squared so x is this so x squared go like ax squared plus that's two kind of two a a dagger plus a dagger squared so what the a does a squared what it does it's reduced to a a dagger what is does it do nothing and a dagger squared increase it by two okay so that's why we can either reduce by two stay the same or increase by two okay so what we find we find that out of the infinite possibilities of f almost all of them the matrix element vanish and the only thing that doesn't vanish they do not vanish are only this kind of thing okay so that will be the the case is it clear and can anybody quickly tell me what would be the situation if I have x square y cubed someone else what would be if I have x squared and y y cube what would happen yes so the answer is for x it would be the same so it's either stay the same or plus minus two and for y it's either we plus minus one or plus minus three I hope it's just clear you just take your things do some open open it and I just say how many a's and how many a zegers I have and I work on my final on my initial state with those and I kind of bring them up and down okay so the big thing about this kind of perturbation theory of harmonic oscillators is the fact that why I have some matrix element in second order in particular where I can have infinite number of states many many of them almost all of them vanishes and I have only few that are surviving okay good so let's just then see how using this we can actually calculate transition amplitude so let's say let's consider the initial state to be zero one so I have one y and zero and x and my final state I can do two zero and just I choose my case such that this is have the same energy okay and then I put my perturbation and I see that from zero one I can go into two zero okay and then what I found I found that this amplitude I just plug it in and there's some coefficient in front but what I think about it is the following thing I think about my perturbation this alpha x per y and the following things I start with one y okay and then this y in the perturbation annihilate the the y particle because they have one state of the y particle and then the x squared it creates two x particles so I think about it a transition from one quanta in the y in the y oscillator into two quanta in the x oscillators but in other words another way to say it is that I start with a y particle and this y particle decay into two x particles okay it are not really particles are just excitation of harmonic oscillators but excitation of harmonic oscillators are just particles so we can actually use the same language okay and then we can actually calculate it we square it we do the face space and when we can calculate the lifetime of this state so if I ask what is the lifetime of the state one zero that's how I do the calculation and I calculate the lifetime of the state y zero okay good so now let's do a little bit more complicated perturbation theory and I have the following situation I have three oscillators x y and z and I have the following perturbation and I want to calculate y going to three x using second order perturbation theory yes question no sorry it was above you and then yes yes so there's also the okay when you have the kinetic terms they are also squared right so you also have a and a decor in there but you normally would assume that they don't do anything right yes so yeah so the answer is following I only care about the a and a dagger in terms that are not harmonics and then in the kinetic terms are harmonics this is the p squared and the p square is always second order right so I also don't care about the kx squared I don't care about them because they are quadratic and they are the one that give me the h zero all I care about is the h one so the way we do perturbation theory is I have my h zero and my h zero I use to calculate my zero's order Hamiltonian and my basis that I use is the basis of h zero and all I care about is the h one so I only care about the h one and in particular in the example that I use my h one is only potential term I don't have perturbation to the kinetic term okay there was the other another question down there if how do you recover the continuous foster drop from the discrete one can you have a limit like current states but for interaction yeah so the question is a general question how we recover the classical limit in the from the quantum things and the answer is like the usual answer it's become classical when I actually go to the very very high state so when n goes to very very large so and when say n is 10 to the 20 then the transition becomes so so fast that you just think and then usually also your initial state it's not a true eigenvalue okay so there's the standard answer and that's what we get here and in particle physics most of the time we do not care about the classical limit actually okay so we'll get there okay so let's do this I have three three oscillators x y and z and I use this h one and using this h one you can actually see that I cannot move from here to here so there's no direct transition from this state to this state by the way what I should have had if I wanted to move from this to this what term I should have had I should have I should have write it explicitly so I have this state nx equal to three and my initial state is ny equal to y okay so I'm asking the following question what's kind of a direct interaction I could do in order to move this is like right even worse let's do it like this nx equal to three and ny equal to one what kind of an interaction I need to move from here to here yes x cube y is really good why because I want x cube to create three x's and the y to annihilate one y okay but as you see I don't have it up there and since I don't have it up there I need to use second order perturbation theory and I have to go through some intermediate state in order to calculate the transition amplitude okay what I'm just doing here is classic standard standard standard second order perturbation theory and then I ask what kind of intermediate state I have so you can convince yourself but I'm not going through the exercise that there's actually two passes that I can go and I call them a one and a two so the first one a one I go from a state zero one zero and then I use this beta and I use this beta such that this beta create one x for me and create one z for me and annihilate one y so it's create x annihilate y and create z so I go to one zero one and then I use the alpha term and the alpha term create two x's and annihilate the z to get me into this and I have another way where I first use the alpha and the alpha create two x's and the z for me and then I use a beta and the beta kills the y and z and annihilate an x do you see it so I get two kind of amplitudes and then the total amplitudes will be in both case I have alpha and beta when I put those two things in so both of them are alpha and beta and I have some number over delta e1 and then some number over delta e2 okay so we know how to do it and actually in the homework that I hope that you will do this afternoon I will ask you to do it very carefully with all those numbers some square root of six square root of two all those things everything is known and you will just I will ask you to actually calculate it explicitly so now I want to do a little closer look into this and let's see what come out so the question if you have any question do you understand what I'm trying to do the second order perturbation theory and I just see there's only two terms from the infinite sum only two terms are non-zero okay and I want to actually see what's going on with these two terms okay so I have a chalk and I want to do the following things I want to look at the first amplitude so the way I like to think about it is the following I'm going to use as if this is like kind of time going forward it's not really time but I kind of start with my initial state go into the intermediate state and go to the final state it's not really time it's just as if okay it's quantum mechanics so I start with my initial state and my initial state I can write it like this there's some one line that is y okay and now what's happened when I insert my beta in what's happened then this y disappear and instead two axes one x and one z appear right so let's call this x and let's call it z and what's happened here here I did the alpha sorry the beta thing right you see what I mean so this is my initial state that's my intermediate state so that's my intermediate state and now what's happened after that after that I include the alpha and the alpha kill the z and instead of the z I will have two axes coming in right so I put you an alpha and then I have two axes coming in right and this x keep going okay so this is my initial state that's my intermediate state and that's my final state everybody is with me on this good does it remind you something yes good I don't want to say the word yet but it's just second it's just second order perturbation theory for harmonic oscillators right there's nothing about particle physics yet at this point okay so now let's do the other one let's do the other one and the other one I start with y and then what's happened then I have my alpha and this alpha is creating three particles right it's create an x x and the z right so this is my intermediate state my intermediate state is y z x x yes and then and that's the that's the alpha right that's the alpha and now I have my beta and what the beta does for me in this case it's take a y and a z and annihilate them and create an x instead of them right so I have my x going on okay and that's my beta yes you see so you see what I did here it's just a graphical representation of second order perturbation theory of harmonic oscillators yes and let me just plot those on the side so one of them look like this and the other one look like this yes okay and can you see that those two diagrams I call them diagrams by now okay these two graphical description of second order perturbation theory of harmonic oscillator they are basically topologically the same you see that these two are the same it's just a question what the direction of this line so if this line go forward I have this and when this line go backward I have this you see but topologically they are the same diagram they are both this they both look like this diagram okay yes so they are not the same diagram but now they are the same thank you very much I'm so I'm so happy that one of you is awake the morning okay and not only this also the labels are the same this is y this is x this is z x x y z x x x now they are the same now they are the same all I need to do is put the labels huh cool isn't it it's really cool okay so if I want to do second order perturbation theory all I need to do is just write this one and immediately when I write this I said oh there's also this one and I know that I have to take actually both of them and and put them together okay so and I'm not going to do the the really most impressive fun I'm not going to do because I really want you to do it because it's going to be fun so please try to do it this afternoon I guarantee fun okay once you see the result so what I ask you to do is the following so you do second order perturbation theory you never heard about Feynman okay and you just calculate this diagram and this diagram they are the same right there you know how to do them and you add them up and when you end them up you find the following amazing result you find that when you add them up they denumerate the denominator that you had in the beginning so in the in in second order perturbation theory you have e i minus e n e intermediate okay you have this one and then you add these two amplitudes you have the two amplitudes and when you add the two amplitudes what you find is that this one become one over the energy of the initial state squared minus something that I called q squared and what is q squared q squared is the energy flow inside the z line okay and the energy flow inside the z line in this kind of the case if I start here with some energy y and here I have energy x the energy flow inside the z line is the energy different between the y and the x okay and I mean you all know what I'm going through right you've seen it before but I don't want you in imagine you never seen it before all I want you to do is to do second order perturbation theory and find this is an output that this is what you actually have is e squared minus the energy flow that you have in the in the thing okay and now of course the question where have you seen it before where have you seen it before this is this is ah please please this was the whole thing I was working on it the whole night yesterday for this one point of the lecture yes thank you very much that's just the Feynman propagator that we have and if you ever ask yourself why Feynman propagator have e squared and second order perturbation theory is only linear in energy and Feynman propagator have second order perturbation is quadratic that's the answer that in you really have to add the two amplitude in second order perturbation theory to get one Feynman propagator okay so and in in the homework today if you have time I have another example that you want that I would ask you to do which is this I don't know if you have time so you will do this there's six intermediate states you had all six of them and you see that all six of them are correspond to this one diagram and then you add all six of them and it's all given by one thing that's go there's two of them there's two intermediate this and this and you actually instead of doing adding six diagrams in second order perturbation theory you have only one diagram in terms of this Feynman thing okay so let me kind of sum up what we we did so far so we actually try to ask the following question how we actually have transition of simple harmonic oscillators and the answer is the following we take our h0 let's give my my initial and final state in the and then all the term that have x to the nx something is a vertex it's giving me a vertex in my theory a vertex like this and then all I need to do is to actually start from my in into my out and write all the possible ways in this case there's only one possible path which is this path okay and then I just say the amplitude is the product of the vertices each vertex is just the term that I have in my perturbation and then each of these lines is just this propagator that we call which is basically just come from is the denominator of second order perturbation theory okay there's few more rules and in the homework that I'm going to give you I write them very explicitly they are somewhat simpler than what we know in quantum field theory but you just do them and you can actually prove that that's what it is so then we add the amplitude square them and we get the answer okay so what I was actually doing is that I was actually going through second order perturbation theory in for harmonic oscillators and we find very interesting something that looks like Feynman diagrams something that looks like what we know in quantum field theory and of course this is not an accident it is actually why Feynman diagrams are what we know because fields are just harmonic oscillators the only thing is that here they are somewhat simpler because they are simple harmonic oscillators and you never had to worry about things like fields and Lorentz invariance and all those kind of things that you kind of make the the topic of quantum filter a little heavy mathematically here it's straightforward undergrad quantum mechanics you just do it and you find what we are any questions yes oh so the question is how do we actually know it and the answer is that and I don't have the formal proof for you the only formal proof that I have is just basically taking what Feynman did and just said oh I can do it in one dimension rather in four dimension okay but actually I'm sure that there's actually a proof that you can actually do in quantum mechanics without actually going through quantum field theory all I did is just do it in this case and I did it in this case and you just see wow that's happened but I'm sure and the reason that I'm sure about it is because what when we do Feynman integrals in four dimension I don't care about the number of dimensions so it should also work here okay and that's really this kind of but you can just check it and if you have a cool proof I would love to see it okay and I'm sure people you know you can think about it if you have a proof that this is actually the general thing as I said I didn't I didn't prove it I just you know I know it but it would be really nice any more questions okay so let's move into our a little more complicated thing which Feynman diagrams so basically it's the same story there's nothing when we move from what I was just doing second order perturbation theory for a monic oscillator into quantum filter is the same for a single harmonic oscillator x is just a plus a dagger when I have many of them then for each degree of freedom is a plus a dagger and if I have infinite number then the the i become an a function so I have a of k so a of k is just the a i's of the discrete case and that's how we write a quantum field a quantum field is just the integral of the a's and the a dagger which is just the simple real generalization of a single harmonic oscillator okay just have to do integral because I have many harmonic oscillators okay and then you do perturbation theory is exactly the same what is omega omega is the frequency of the energy so the energy is the time and when I move from one time into four times I have to move from energy into what we call p mu we should have called it emu but we call it p mu for the same reason we call it x mu you see that this is what's going on instead of the energy I have p mu and instead of the energy squared I have the mass square because p mu squared is just the mass so when we have in this kind of situation the thing that flow in the line is the energy square minus q squared and when I move to four times then what is this energy square this energy square is just m squared because it becomes p squared so I have m squared minus q squared okay so you just see how Feynman diagram is just a very straightforward generalization of simple harmonic oscillator perturbation theory okay so here is how we do calculation in Feynman diagrams and I assume many of you did it before and you took some some of you never actually done quantum filtering and you never actually did calculation with Feynman diagrams but it's a well defined procedure of how to do it and this well defined procedure is just a generalization of the simple harmonic oscillator that we just did okay and we usually we care only about one to n or two to n processes why do we usually don't care about like three initial states or four initial states yes experimentally it's hard to set up yeah I think that's a really good answer yes I mean the probability to actually make those kind of collides is very very small and actually Matt yesterday talked about how small is the probability of just to make two proton collides so imagine that you have to put a three a third proton coming in so it's very rare that we do care about but sometimes we do care about it in sometimes when we have a very high density we do care about it but usually we don't and we just have to make sure that we have energy conservation and energy conservation in in particle physics is a little simpler than in harmonic oscillators because we have p mu which is a variable and then we distinguish within two kinds of particles particles that I call on shell and off shell and again it's just a little bit of abstraction of what we started before what is an on shell particle an on shell particle is something that is on the external line is some is a particle that satisfies p squared equal to m squared and this initial line and this initial line you see how we kind of gave it a life in second order perturbation theory just some intermediate state that's what it is in second order perturbation theory and when we move to quantum filter we call this intermediate state an off shell particle and we kind of see it from this picture of the Feynman diagram this is looks like a particle that actually go and kind of transfer in in in time but it's not really there it's intermediate it's only there quantum mechanically it's not really there so something that is not really there and it's only there quantum mechanically we call off shell particle off shell particle is a particle that is square is not equal to p squared plus m squared it's just that off shell particle okay it's just the intermediate state of second order perturbation theory okay and then what we do we write the transition amplitude which is just the product of all the vertices times this one over the energy for second order perturbation theory we add them up we square them we do the phase phase integral and we get the things right okay so I'm not going to get into all the details and if you ever done it you know there's a lot of details and there's so many details that now you put it on a computer and the computer does all the details for you I mean during the course you don't do it on the computer but after the course you never do it yourself anymore right because it's an algorithm so you know how to do it but the physics is this the physics is just the physics of second order perturbation theory so let's do some examples and let's take now I move from just simple harmonic oscillator into a quantum filtering and I'm asking you the following thing is this is the Lagrangian and I'm looking for z going to x y okay and I ask what is the energy conservation condition and what is the diagram and estimate the amplitude so z going to x y can come from this term you see it's annealate and z and create an x y and what is the energy conservation condition so when a z can decay into an x y when the mass of z is larger than the mass of x and y so what we have is the following thing I need that m z is larger than m x plus m y and this is the diagram it's very simple diagram okay and the amplitude is just lambda one okay so that's what we do in Feynman diagrams and we know what the rate so what would the rate would be the rate would be proportional to lambda one squared and here's another example y going to three x's and here I don't have a direct term that does it for me it's just this example that we just have here and you see that this one looks extremely like this one very very similar what is the difference here I have alpha and beta and here I have lambda one and lambda two but otherwise the same and I calculate the diagram and the diagram are proportional to these two vertices lambda one and lambda two and time this intermediate state that's come from second order perturbation theory any question on this I know most of you've seen it before and it's kind of easy and if you've never seen it I really want you to do it so it's really kind of and for your homework I asked you to do this kind of diagram and estimate it okay okay so let me kind of summarize these little parts of what we did we kind of learn Feynman diagrams and the idea of Feynman diagrams is that following that it's really just generalization of second order perturbation theory for harmonic oscillators and the way we think about it is following we have our zeroes order Lagrangian or zeroes order Hamiltonian and the quadratic term described free fields free fields as we say they're boring nothing happened to them they don't interact they don't decay and we use perturbation theory to look for this higher order term higher order terms are what we call interaction and those interaction terms create an annihilate particles and they also create an annihilate offshore particles and these offshore particles give me the tool which is just perturbation theory so it's really really just perturbation theory to do this calculation okay so I want to emphasize the Feynman diagrams is nothing but perturbation theory that we know perturbation theory for harmonic oscillators when we make them into quantum field theory become Feynman diagrams okay Feynman diagrams we have to remember they are perturbation theory it's extremely important because when perturbation theory is not valid we cannot use Feynman diagram okay it's it's obvious when I say it like this right and when QCD becomes strong for example we cannot use Feynman diagram and some people do plot Feynman diagrams on QCD and they always say but it's only a plot it's not really what it is because we cannot use perturbation theory okay any question on all this little thing of Feynman diagram yes sorry can we think about a group can you think about loop diagram in terms of perturbation theory yes of course you know at the end we have to go to loop diagram and in purpose I didn't go into loop into loop because in in simple harmonic oscillators loops are actually never there okay you only get loop when you have more than one time dimension okay but then I definitely didn't go to the full full thing of Feynman diagram basically all I wanted to get from this feeling from this is that Feynman diagram are just a generalization of second order of perturbation theory for harmonic oscillators and there's a little more details that you know you really need to do when you do in the quantum field theory but the basic idea is the same you don't need any quantum field theory in order to understand what Feynman diagrams are any more questions okay so let's move on and we want to move on to this amazing topic of symmetries and symmetries as we know are very important in physics and you are not surprised by this statement but it's never a bad idea to say it again symmetries are very important in physics and since they are so important in physics we actually have to do a little bit more formal explanation of symmetries and once we really understand how we actually use symmetries in physics we have to understand how we use symmetries when we extract when we do particle physics and when we do the standard model so let's come back to the very beginning of what I was telling you yesterday and the question is how do we build Lagrangian it's a little bit weird to say we build Lagrangian okay but this is the idea of how we actually come with Lagrangian that can describe nature so the idea is following we assume what we call if the democratic principle so what is at least ideally the definition of democracy the definition of democracy well there's many definitions but one definition that in democracy everything is allowed unless it's explicitly forbidden that's why the government put these huge books of what is forbidden right because everything else is allowed and they want to make sure that they don't forget anything but in democracy everything is allowed unless it is forbidden which is the opposite of the totalitarian picture the totalitarian picture tells you that everything is forbidden unless this is allowed and therefore in a totalitarian regime the books are very small right there's not much that is allowed but the idea is that in physics we use the democratic principle that means everything is allowed unless it is forbidden and how in physics we forbid things by a symmetry we just say it's violated symmetry therefore it's forbidden okay so I said everything is allowed and I put the symmetry on and I said everything is allowed unless it's forbidden by this symmetry okay so that's the first thing that we do we write the most general one and then we do the thing that we all really love to do and that's expansion okay we start writing it in terms of x squared x cube x to the four x to the five or in quantum field theory let's call it five squared five cube five to the four and then we truncate at some point why we truncate why do we truncate we talked about it why do we truncate why we don't keep going forever ah that's actually the quantum field theory answer you know methodology but actually I don't I want to actually talk about it in a second we truncate because we do perturbation theory and in perturbation theory you have to truncate that's the whole point why because physics is the art of approximation right so we have to truncate at some point and how do we decide where we truncate that's a general question in physics how do we decide where we truncate that's actually it's a really thank you I was I was sure that now we go into two minutes discussion and you immediately got there exactly the right answer experiment the only thing that tells us where to to to stop is experiment if you have an experimental precision of one percent and you first order term give you 10 to the minus five you don't need to expand more if you first order term give you 20 percent and second order give you five percent and only fifth term give you one percent you want to go to fifth term okay so the order that you want to keep expanding depend on how precise is your experiment it's always this answer it's how precise is your experiment yes it is non-renormalizable so let me talk about renormal yes so so the question is as following the question what's happened when we go into non-renormalizable term so let me first even talk about it because as you said I didn't even mention renormalizability and already two of you kind of got me there and I was trying to avoid but now I cannot avoid it anymore so one thing about quantum field theory is the following statement is that if I go above dimension four the theory is not renormalizable what does it mean that the theory is not renormalizable that means that the theory cannot be the full theory it can only be an effective theory is it a big deal no what the big deal it's all the theories are effective so so that's it okay Newtonian mechanics do we ever worry about Newtonian mechanics when you do mechanics the fact that Newtonian mechanics is only an effective theory and at high velocity no all we are saying is that I do Newtonian mechanics when I can when the velocities are small I do Newtonian mechanics do I worry about classical mechanics when I do classical mechanics and I said no it's really quantum no I'm not I only care about it when I say because now I can use you know what happened when I when if I store if I drop it what would happen that's a classical calculation do I worry about quantum effects no I don't why because it's in the classical regime so in it's a good enough approximation right so the point of renormalizability is the point is do I ever care that my theory is an effective theory and the answer is the following if I care then the theory is not good and I have to actually do something else I use the theory only when I don't care about the deep deep uv okay and then I don't care about renormalizability okay and of course I was a little like you know renormalizability is a very important issue but when you do phenomenology you usually don't care about renormalizability okay I think it's very very important all I care about is the fact that I know and appreciate and acknowledge the fact that my theory is not the full theory of nature it's an effective theory and once I acknowledge this fact then I don't care about renormalizability okay it's like when I do classical mechanics I don't care about quantum mechanics as long as I'm in the classical regime so we usually expand and truncate and we usually truncate the dimension four why four because four is renormalizable I just tell you I don't care about it but so historically we really care and it just happened to be that in almost all cases except neutrino masses which we may touch at the end and Andre probably will talk about it in great details when he will start doing his neutrino lectures usually dimension four is all what we need because experiment tell us that is precise enough okay it just happened to be the dimension four which is precise enough it's also where renormalizability come in okay and as I said now neutrino mass give us dimension five okay so that's how we will tell a grandian we write the margin I want that we can and you truncate usually a dimension four okay and what are the inputs the input for writing like grandian is what are the symmetries we impose we impose the symmetry in order to forbid some terms and what are the degrees of freedom I have to say oh I have some system with five degrees of freedom with ten degrees of freedom and in classical mechanic the degrees of freedom are just the number of particles in quantum field theory the degree of freedom are just the number of fields my my building blocks are fields in classical mechanics my building blocks are particles and the position of the particles in quantum field my building blocks are fields okay and then I have to tell you how they transform under the symmetry and I'm going to discuss it in some detail and what is the output the output is a Lagrangian with some finite number of parameters why there's finite number of parameters because I truncate if I didn't truncate the number of parameters were infinite and there was no and we couldn't do physics because I truncate the number of parameters is finite and then what I need to do I need to measure those parameters and after I measure those parameters I can make prediction and I like to make this very clear that a theory a Lagrangian that I have doesn't make any prediction what makes prediction is that I make some measurements those measurements may measure the parameters and only then I can make prediction okay for example there's a very famous prediction of Newtonian gravity is the fact that all particles travel with the same speed right so I don't care if I take two things and I drop them they should drop at the same velocity okay and you see in order to really do it I have to do two experiments I cannot do just one I have to do one to kind of measure g and then the other one I can have a prediction okay because I have one parameter in this theory so in general when I have a theory with 18 parameters why I choose 18 because many times you choose 18 because the standard model has 18 okay if I have a theory with 18 parameters I have to make 18 measurements and then from the 19 measurement on I can make prediction and I can test my theory okay so this is a very important this is kind of a semi philosophical but that's how we actually do physics that's the algorithm to do physics so if you have any question on this I'll be happy to discuss yes yeah yeah so so in principle if I go to higher order terms if I include dimension five say I have more parameters if I have more parameters I need more measurements and then I can start making more and more precise calculations okay so basically the algorithm is the same I go to some up to some order truncated this order make the number of measurements that I need in order to measure all the parameters and then I can start making then I start testing my theory and if I don't expand to high enough order then many times my theory breaks down I say the experiment doesn't really give me what the theory predicts so I have to go to higher order and then I can actually see what the theory does yeah yeah yeah yeah so of course of course of course the first thing you're going to do is you actually start calculating high and higher order perturbation theory using simple harmonic oscillator and Feynman diagram etc etc and only then you say well it's really not that and I know that I need to actually add extra terms and we're going to discuss it with the one example of neutrino mass we're going to get there okay so let me talk now about what I call representation and symmetries and I know some of you study Lie groups and Lie algebras and all this and this is going to be known for you but for some of you I assume you kind of don't know really Lie algebras so I want to get you the very basics of Lie groups of Lie algebra using this idea so what is a representation and how we actually deal with it so let's think about three dimension classical mechanics real space and we have something that may rotate in real space so I want my Lagrangian to be invariant under rotation in real space okay that's a requirement I impose rotational symmetry on my classical Lagrangian okay that's mean that when I rotate my my system nothing happened to my Lagrangian that's the meaning of a symmetry okay and now I look at my degrees of freedom and let's say that my degrees of freedom is some a particle and this particle the position of the particle is a vector what does it mean that is a vector in real space that means that when I rotate the space the components rotate between themselves right so if I have a vector that is in the z direction right a vector in the z direction one second a vector in the z direction is this that's a vector in the z direction and now let's say that I take my system and rotate it into the x direction after I rotate it by 90 degree into the x direction this vector become one zero zero right so in one reference frame it's one it's zero zero one and in another reference frame it's one zero zero so this vector is not invariant under rotation obviously right but there's something that is in still invariant under rotation what is invariant under rotation when I rotate the vector the length of the vector right because what is the length of this vector is one and what is the length of this vector is one so we understand that vectors transform they are not invariant under rotation we say they transform as a vector kind of an obvious thing okay but it's a little deeper and the length of the vector is invariant we say the length of the vector is a scalar it's a scalar because it doesn't change under rotation and the word scalar that we use for scalar filtering has to do with the transformation properties in four-dimensional means Kovsky but the word scalar in general mean it doesn't change under rotation so vector do change under rotation and a scalar doesn't change under rotation okay and actually if I have several vectors I can actually build many things that are invariant many scalars so how do I build this scalar so if I have this vector that I call ri this is let's say this is ri what will be this this will be ri ri this will be the dot product of the vector with itself that the length squared of the vector the length squared of the vector it's invariant okay and a way to see it is just look for the number of indices so here I have one free index and a free index tells me that something is rotate here I sum the two indices okay actually maybe that's the summation convention you can also write it like this with some of i go from one to three okay this is the same but here I do not have a free index ri ri so since I do not have a free index I know that this is invariant under rotation okay so the the point that I want to make is as following is no matter what kind of a system you have you building blocks are seeing that do change when I rotate but I can combine them into combination that do not change when I rotate that's point clear and in vectors I hope it's very clear I take a vector and vector change when I rotate and when I take a dot product of two vectors the dot product is invariant under rotation yes and when I say I require that my Lagrangian would be invariant under rotation that's mean that my Lagrangian can only be a function of those those dot products okay so those those products are invariant and and this is really bad notations this is really not good I should really change this transparency it's really confusing what I just mean is that I take two vectors I dot product then they don't have a free index and then my Lagrangian have to be a function of only all those dot products good so let's be generalized this and that was the example in three-dimensional real space but fields work in some arbitrary mathematical spaces and those arbitrary mathematical spaces are just some abstract space but you want to think about them just like three-dimensional real space and I said I take my fields and I put my fields into representations of those or those mathematical spaces which is exactly the same as we do in three-dimension in three-dimension I say in classical mechanics I said I have three degrees of freedom and I put those three degrees of freedom into a vector of a thing and I call them the x the y and the z component of my particle and now the position of a particle is generalized into fields and then I take many many fields and put those fields into a vector of some representations of some space so if I have a five-dimensional real space I take five fields and I put those five fields into a vector inside this five five-dimensional space is that clear it's just a generalization of positions in three-dimensional space now I use fields instead of positions and it's not three-dimension is some other space okay and what we really care about in terms of going to do physics in particle physics usually we care about SON SON are rotation in n-dimensional real space SUN which is rotation in n-dimensional complex space and U1 which is rotation in one-dimensional complex space or two-dimensional real space the equivalent so let me talk a little bit about all of those and let's talk about how we actually do a how we actually generate singlets in general okay so we know how to do this in in three-dimension real space you just take the two vectors and dot product them we also know how to do it in in spins so when you learn to to combine spins you remember you take spins half and spin half and put them together and you get what do you get from half and half zero and one right so I can there's actually a way to combine the two spins half in order to generate a spin zero and a spin zero is the singlet it seems it doesn't so we already know how to do it with vector in three-dimensional real space we know how to do it with spin half and we know how to do it in and there's actually a generalization and since these are very important for the standard model I want to work on them one one by one okay so the first example I'm going to use is the u1 and u1 is nothing by complex number so when I talk about u1 symmetry it sounds very kind of I don't know fancy I have a u1 symmetry u1 symmetry is nothing but playing with complex number and complex number by now I hope everybody is very familiar with you don't feel they are weird anymore so how do I use u1 all I need is the following thing I telling you that I have a complex number say x is a complex number and I assign to this x some number q q is a real number and now I say I take my number x I take all my numbers in my theory and I rotate all of the of my numbers by some angle theta okay and I rotate them in the angle theta such that for each each value bell I multiply it by some number q is that clear so for example if this theta is 10 degree and this q is 2 then x rotate by 20 degree okay just a complex number that I rotate the phase of the complex number okay so let's take the following example let's say that I have three complex number x y and z and q of of x is 1 q of y is 2 and q of z is 3 okay and I ask how do I get invariance so for example x x star and y y star is an invariant because x x star is an invariant because x star transform with a minus compared to x and I hope you see that x square y is also an invariant y x x star x square y is an invariant because x transform with theta x where transform with 2 theta and y transform also with 2 theta and y star transform with minus 2 theta that was too long okay so let me write it let me write okay so I want to explain why why x per y star is invariant under this rotation so let's write it so x x go to e to the e to the i theta x and y go to e to the 2 i theta y so then what what happened to x squared so x squared go to e to the 2 i theta x squared because I have it twice and what's happened to y star since it's a star y star go to e to the minus 2 i theta y star okay because it's a star and then x times y then this exactly cancel this and this is an invariant okay you're okay with this that we understand how to build invariance okay so these are some more invariants that you can actually write and I hope that you can convince yourself that those are the other invariants that we can build so the whole story of building invariance from complex number become a very easy task the easy task is as following all we need to do is just make sure that the sum of these q's is is zero when I have for each part for each number x I have a number q for for a star the number become minus q and you see that the sum is zero so here it's one minus one plus two minus two here is one plus one it's two minus two here it's one plus two minus three here it's three minus three and here it's four minus one minus three which is also zero okay any question on this I lost you it become late and I was but hopefully we you know we will do more example and hopefully it will be clear so let's move on so that was u1 so u1 I was telling you how we actually built invariance in u1 we just combined the numbers how we go from su2 so su2 is basically just the algebra that we learned and all the things that we learned when we did spin and I assume you all did adding and and subtract and a combining spins in quantum mechanics and basically the same story going for general things when we do su2 okay and the idea is this following is that you actually anything that is leaving su2 we call it spin okay when I have something that has two degrees of freedom in su2 we say that's a doublet of su2 and the way we think about it is like a spin half what is a spin half a spin half is a particle that has two degrees of freedom a spin up and a spin down and of course when I rotate in real space these two components kind of change right in one if I if I have my spin up in the z direction and I measure in the x direction I have 50 chance half and half probability to get in this direction and in this direction so it's depend on the orientation that I have okay so we know how to combine spins we combine for example half and half I can get a I can get a zero so now I'm asking you the following question how can I get a zero if I have spins half and spin three halves how can I get a spin zero from half and three halves so two halves can give me zero how can I get a zero from a three half and half you completely so let me kind of back down and and discuss how we do how we combine spins in quantum mechanics and then we will be able to see how we do a general thing in su2 how we generate environment in su2 so the way we combine spin is this following so if I have two spins some s1 and s2 s1 times s2 I know that I combine them and I combine them from s1 minus s2 and let's assume this s1 is bigger or equal to s2 it's from s1 minus s2 s1 minus s2 plus one s1 minus s2 plus two all the way up to s1 plus s2 let's bring the bell I hope that should be something that I was assuming you know that's something that we learn in quantum mechanics yes that's how we actually combine spins when you combine spins when you make a product of two spins then we have all those kind of things I still lost you so you remember this yes you all remember this yes so why it looks like I could totally lost you okay so that's how we actually know how to combine spins right and now I'm asking you the following question how do I get a spin zero if I keep combining so I take I take this and I can keep combining so now I say if I have a spin half and a spin three halves how can I make this something that make them as into a singlet so I want both of them to be there so for example I can take three half squared three half times three half I can make a singlet yes and if I take half times half I can also make a singlet and someone come with another less trivial example so I can take three half yes thank you very much so I can take this this one also give me a singlet right half cube times three half also give me a singlet okay so in SU2 in order to get a singlet all we need to do is just keep adding them up using spin adding such that at the end I could get a singlet and when I get a singlet that the thing that is invariant under rotation so I already know how to do it in a new one and I know how to do it in SU2 and for the very last thing for today I just want to do it in SU3 and I'm not going to tell you too much about SU3 all I'm just telling you that in SU3 it's very similar to SU2 just a little bit more abstract and I put my thing into the equivalent of spin half in SU3 is something that you call a triplet there are three degrees of freedom and when I take two degrees of freedom and I combine them I can create a singlet and there's something called a three and a three bar and I can create a singlet if you never heard about it it's okay it would be a little bit too much for me to get into the details into here all I want to say is the following that we know that the strong interaction is an SU3 group and we know that there's no free quarks okay there's a statement there's no free quarks and the just statement that there's no free quarks basically tell us that I cannot have anything that I see that is not a singlet under this SU3 and I'm not going to discuss why it is but therefore when I actually see particles that are charged under the strong interaction like bions and mesons they must be singlet under SU3 and in order to build those singlets I need to combine those spin of SU3 and how I combine spin of SU3 I use this kind of seeing the three three times three times three is one and three times three bar is also one and it doesn't really matter what are these two details but the three is a quark a quark is transformed as a SU3 and if I take three quarks I can make a something that is singlet and when I take three quarks how do I call them I call them a bion when I take a quark an anti quark that's three times three bar then I call a meson okay so is this the group theory beyond our understanding of of the spectrum of QCD is the fact that a bion is actually made out of three three three three and there are three quarks and a meson mega those three and a three bar so I want to before we ending I want to do a little game and I call it the building environment games and it's going to be important when we move to the standard model but it's not the standard model so those of you who think it's a standard where I know the answer it's not you have to sing a little bit more and the game is as following I have my symmetry which is SU3 cos SU2 cross U1 which is the symmetry of the standard model and I want to build environment so for the U1 we have to make sure that the number add up to zero for the SU2 we just do a spin and for SU3 all we need to remember is this kind of thing when I have a part a field that I have a bar on it it's become a bar that's it and the fields are Q321 UD and H it looks familiar because it looks like the standard model but it's not everything is there are scalars and I wrote some invariance here and I want to ask the I want to ask you and I be quiet for the usual thing for one minute and I want to ask you which of those things are invariant under this symmetry and which is not okay so the question is that I give you these five things and I want you to tell me if these are invariant under this symmetry or not so please ask me question because I think I lost you a little bit on this last part but if not you know talk to you you people around you or ask me a question and I want to each of those five tell me is it invariant or not okay good please go start talking yes and ask me question if you don't understand what I'm talking about oh I one thing I should have explained one thing I didn't explain so when I tell you these are the field this is the how the field transform on the SU3 this is how the field transform on the SU2 and this is the charge under the U1 so of course all of you saw the standard model familiar but if you never seen it I want to explain this is the U1 charge this is the SU2 and this is the SU3 okay sorry okay keep going tell me when you have it you have it all of you please try to do it try to kind of combine the SU3 combine the SU2 okay so let's start and let's see what we are having this one is it invariant or not this thing is trivial whenever you have a field and the complex conjugate of the field is always invariant because something with the complex conjugate it is always invariant so that's obviously an environment we don't have to worry too much what about h cube what about h cube no okay can you tell me why is it invariant under SU3 it's invariant under SU3 because just is it invariant under SU2 no because three three doublets three spins half cannot make a spin is it in the invariant under the U1 no actually it works very well this game so far okay UDD UDD yes no okay we're already in the quantum regime okay so let's try and see what's happened for the SU3 so actually let's try because the SU3 is a little it's a little thing what's happened to the U1 what's happening the U1 the U1 is 4 minus 2 minus 2 is 0 that's good what happened for the SU2 invariant what's happened for the SU3 invariant why not so it's three times three times three so it's three times three times three and three times three times three is is invariant right yes okay there's actually a subtlety here and I want to tell you this subtlety anybody know what the subtlety here okay I'm just telling you actually this one is totally anti-symmetric and therefore if I have two fields that are the same this one vanishes so assuming that actually these two these have another index then it will be allowed so I the subtlety that I should have think about okay good QUD invariant no what kills it SU2 and U1 very nice HQ U star invariant yes yes okay okay I'm happy I was unhappy 10 minutes ago but actually you know I'm very happy because you all got it so for the homework and that's the last thing we are doing today I will ask you to find more invariants and actually I will I really ask you to find all the invariants to get up to dimension four all the invariant to build out of three or four of those fields so that's end my second lecture and let me just kind of summarize what we did today so what we did today we were actually talking about Feynman diagram and we understand how we do Feynman diagrams and it is just generalization of second order perturbation theory and then we start talking about symmetries and we understand and it looks like at least most of you totally understand how to build invariants and what we're going to do tomorrow and tomorrow I have actually two lectures so I hope that we survive it and the two lectures we first going to keep going a little bit more on on symmetries and then we finally tomorrow is going to be the biggest day of our life if we're going to write the standard model down okay so now we have a break and as usual please please come ask me question as usual okay thank you very much any more questions before the break any question I want to ask you because you say that we assume the symmetries but I was thinking if those symmetries can be accidental and maybe we have to build the Lagrangians with another symmetries that are more fundamental yeah so I don't know yeah yeah yeah so the question is so in this the assumption that I'm making is that I impose the symmetry so the the the rules of the game is the democratic principle I impose the symmetry this is the rule and then I ask is other invariant or not and tomorrow we are going to talk about accidental symmetries and I'm going to actually make some distinguished between imposed symmetry and accidental symmetry okay but at this level I'm just imposing those symmetry one more question there what about the discrete symmetries so in principle I could have done it but you remember the way the the rule of the game was that I decide what symmetry to impose so I decide not to impose them but it's my total decision and if you build your own model you decide to put them that's totally fine just for now I decide not to put them and it's not only me it's the standard model decide not to put them so we are not actually use them but it's in the in the rule of the game you're totally allowed to use them okay okay no more questions thank you very much