 When I moved to America and I had to learn a lot of American Expression there's an expression called doubleheader. Anybody knows who is not American what a doubleheader means That means some I was thinking it's an animal with two heads, but it's not anybody knows what the doubleheader means Okay, so here's your cultural thing that you know so in baseball It's a you know that the game baseball. Yeah, it's a game that most of the time nothing happened That's why it's so exciting Okay, and if only because nothing happens they sometimes can play play two games in a day A usual game is like five hours, so they have five hour little break and do the second one so let's call it doubleheader when they have two baseball games and that's how I feel today I should have a doubleheader and I was actually preparing my After yesterday talk I stopped preparing the today talks and then in the evening We got into some interesting discussion about hiking that took away and not know to blame anybody But of course then I couldn't continue and then this morning when I was about to finish this lecture I was listening to Andre which was so interesting so much so many things that was really exciting So I hope that the lecture will be okay And if you find typos and usually the number of typos increase with the number of transparency in some monotonic function Partly that's my item and we'll go on. So we start talking about neutrinos Symmetry I don't gonna talk much about neutrinos So what we were doing yesterday we start talking about symmetries and we keep going to talk about symmetry and We made this say how to build inviant games And I was very impressed and happy to see that most of you got it very easily and we know how to build inviant So whatever symmetry and representation I'm gonna throw on you and you have 10 seconds to build inviant I know you can do it. Okay, so I'm not worried about it and The main idea of how we built Lagrangian as you said the Lagrangian have to be inviant under some symmetry And we know how to build those inviant based on our building blocks and the building blocks are our fields and The way we built inviant will make a distinguish between you and symmetry you want symmetry is just rotation on a plane And actually I forgot to mention a very important fact that rotation on a plane is Commutative that is I don't really care about the order that I do my rotations Unlike rotation in bigger planes in in bigger spaces like in in space where rotations are not commutative and that's this thing that we again very much familiar with and These are called non-abillion symmetries and a billion symmetries although they commute where I don't care about the order So when I have you one the thing that really tells me how things transform How things rotate under the symmetry operation are called just simply charge and of course We all familiar with electromagnetic charge and today we're going to discuss a little more about And some subtleties about electromagnetic charge and when we talk about non-abillion symmetries We say that we put things into a representation and we just say it's like the similar to the spin representation or angular momentum representation that we have in quantum mechanics if we have a state with a J equal one we know that it's have three degrees of freedom and they transform like a spin one And if I have a spin to it's a five degrees of freedom and we say it transform like a spin to okay And if you never learn about it, then you hopefully you will have a chance to learn more of the details But that's all what we need for today of how to build Lagrangian and what we like to talk about today I want to start with discussing more aspects of symmetries. It's going to be a little Technical and I hope I will be able to actually get the the message across for those of you who didn't see it before And I want to talk about several things. I want to talk about the different between impose symmetry and accidental symmetry I want to talk about the different between Lorentz symmetry and internal symmetry And I want to talk about the different in global and local symmetries Okay, so we'll get in and hopefully till the end of the lecture We're going to start doing finally real model building and in the second lecture We're going to get into some the standard model finally so let's start with some more on symmetries and The first thing I want to talk about I want to talk about the different between an impose symmetry and accidental symmetry so the way that we think about doing model building is the following we We tell what is the symmetry and we impose it That is we say everything is allowed unless it is forbidden and it is forbidden by a symmetry And it's a symmetry that I choose to put or you choose to put anybody who do model building That's the power that you have you have the power to impose any symmetry that you like And when you impose a symmetry the Lagrangian must be invariant under the symmetry Why because that the symmetry that you impose, okay? Then there's a different kind of symmetry a symmetry that is actually an output of Your theory. It's not a symmetry that you impose, but then you write your Lagrangian and It's like a little miracle happened and you see that you Lagrangian Actually a bigger symmetry that the symmetry that you impose and the reason that you see a bigger symmetry is because you Truncated your expansion if you keep going and add more and more terms at the end You cannot get extra symmetry at the end all the symmetries that you have are only the symmetry that you put in but if you truncate you can get the output theory has more symmetries than the symmetry that you impose and The simple example. I know about is the example There's one that we already discussed to a great details is the one that we have with the pendulum Okay, so when we impose say we have One-dimensional theory and I put a pendulum inside the theory I have say like energy conservation that has to do with the T go to t plus c symmetry that I impose But then if I truncate at the harmonic term Boom another symmetry emerge the symmetry is the dilation symmetry that we discussed But if I keep going if I add the x to the fourth term then the dilation symmetry is not there anymore Okay, so this the dilation symmetry is an output if I truncate at x squared okay Good, and let me give you here another example I take a same a Theory with you one and I give them two charges equal one is one charge and the other is minus four Okay, and then what I can write all the terms that I can write will be like x x star and y y star Okay, and there we actually have two symmetries We have you one x and you one y the x rotates simply by itself There's a you one that the x rotate under and a you one that they y rotate under okay But then there's a dimension 5 operator a dimension 5 stem x 4 to y and this term is Inviant under the you one everybody see that this one is in violent under this you one because it's 4 minus 4 equals 0 very nice Okay, but this one it's not there if I go up to dimension 4 if I go to dimension 5 I have this symmetry and then the x and y cannot rotate independently So if I if I go up to dimension 4 the x can rotate independently of the y and both x rotation and y rotation are a symmetry when I put the dimension 5 I don't have it anymore So this become very important when we start talking about the standard model and we're going to see that in the standard model There are some symmetries that are accidental in particular the symmetry that are accidental in the standard model our bion and lepton number Okay, and just telling you that they are accidental You should have a feeling to them that they may not be really really Conserving nature why because they are just there because we truncate the CS So so far we see bion and lepton number seems to be conserved We didn't see any experimental direct evidences are not conserved But just the fact that they are accidental in the standard model and we believe that standard model is correct We kind of feel that they might be broken. Okay, and I Really hope that I will that we soon gonna see some really really experimental evidence that they are broken It would be really really cool Any question about accidental symmetry so let me go on and The next symmetry I'm going to talk about I want to talk about Lorentz invariance or more generally about the symmetries of spacetime and We really distinguish within these two kinds of symmetry one symmetry is the symmetry of spacetime. Yes So the question is is it generic that if I have accidental symmetry that it will be broken by non perturbative effect of the theory So what I would say is the following It's general it will be broken by higher dimension operator. Okay, and It doesn't have to be broken by higher dimension operator. Okay, it broke by higher dimension operator Assuming that I know nothing about the higher dimension operators if I write the most general higher dimension operators, they will be broken Okay, but maybe there's some actually UV physics that is actually more fundamental than my theory that make them actually Exact for some reason. Okay, so this accidental symmetry may come from some deeper things and There is actually a reason for them to be exact All I'm saying is that if I am agnostic and I'm just saying writing them the most general higher dimension operators Eventually all my accidental symmetry has to be broken by higher dimension operators Okay, but we don't know if writing the higher dimension operator is the correct explanation So let me move to Lorentz invariance and I want to distinguish within those kind of two symmetry There's a symmetry of spacetime and the symmetry of my fields. Okay the fields They live in this mathematical space Okay, and we say there's some mathematical space that I do some rotation within the internal fields The field space. Okay, so I take if say I have 10 degrees of freedom I say these 10 degrees of freedom behave as a vector in some 10 dimensional mathematical space and I move between the fields Is the equivalent of moving between? The different x if I have x1 and x2 in classical mechanics and I can move into x1 plus x2 and x1 minus x2 Is rotation on my generalized coordinate then we have rotation of spacetime itself in classical mechanics it has to do with rotation of t because t is not a it's not a Generized coordinate t is just my integration variable in the Lagrangian, right? And when we move to four-time dimensions then of course we talk about their rotation is that this four-time dimension Well in in classical mechanics, we have only one time dimension So all I can think about is t go to t plus c or t go to lambda c But when I go to four-time dimension then I have a much bigger symmetry and there's the Lorentz symmetry Which is rotation in four-dimension Minkovsky space now the point is that basically Everything that we're doing in in particle physics. We always impose Lorentz invariance. Okay, and from this Philosophical idea of how we built Lagrangian. This is our choice We choose to impose Lorentz invariant of course historically The reason is that we see that everything is Lorentz invariant, so we want it to be Lorentz invariant Okay, but the philosophical idea is that we impose it Okay, and then just like we do with the other thing that we talked about representation if we talk about SU 2 and we So in SU 2 we can have a spin 0 spin half spin 1 etc when we go to Lorentz invariant the story is very similar all the Building blocks that we have must be in a specific representation of this Lorentz invariant Okay, coming back to classical mechanics You can think oh everything that I have should be either a scalar of rotation a vector of rotation and tensor of rotation So then we conclude that all the fields that I have in my theory must have a very specific representation under the Lorentz in a symmetry and that says to do in a somewhat non-trivial way Should it tells me the spin of the particle and I said it's really not a trivial way And I'm not going to tell you to get into the details but all I'm saying is that I know that all the fields have a very defined spin and We really distinguish between only we only care about those three things a singlet is a spin zero Which is a scalar field and that's what all I did so far was about scalar fields and it's turned out that those scalar fields are Kind of important, but there's are not the most important in a way of what we're doing And then we have fermionic field and fermionic field as we know have spin off And it's turned out that when I actually look for the Lorentz transformation It's turned out that there's two building blocks of fermionic field The two building block of fermionic field has to do with what we call a left-handed field and the right-handed field What is a left-handed field a left-handed field is the one that excitation are purely left-handed What does mean that it's purely left-handed when it's traveled at the speed of light the polarization is only is always left-handed So it should be this one because that's left. Okay, and if I have a Field that is right-handed when it's traveled of the speed of light then the polarization is this Okay, so of course, it's totally arbitrary if I use this or this what is not arbitrary is the fact that I have two of them I have this one or this one and the surprising fact is that Actually, these are the fundamental building block of of fermionic field and This came a little bit of a surprise through the years because when we actually look for the electron and the electron is massive them The massive electron have four degrees of freedom because have the electron and the positron and each of them have spin up and spin down And it's down that the most fundamental things are what we call while fermions and while fermions have a well Very well defined Chirality, which is either left-handed or right-handed and what we're going to see soon is that we actually we take those Fundamental building block and from this building block we're going to make to get actually the particles that we're going to see and The big step from using this to see the particle is the following fact So you remember that when we talked about particles I make a big deal about the fact that particles are nothing but excitation of harmonic oscillators or excitation of fields Okay, what we are going to see soon that particles can be actually excitation of more than one field They can be some superposition of fields So I can have some fundamental fields and the particles that I see are not correspond to excitation of one field They can correspond to excitation of two fields So I have two particles that correspond to excitation of two fields But none of them corresponds to one by one corresponding to one field Okay, again if you think about it, it's not a big surprise. It's always what we have in physics Okay, in the beginning you have one-to-one and then you start having some matrices and we have to diagonalize those matrices Okay, so that's what we have we have actually that our Electron the physical electron that we see is not just an excitation of the simple electron field It's actually an excitation of two fundamental field the left and the electron the right and the electron And I cannot tell you oh this electron is just an excitation of a left and electron any physical electron that we see is an excitation of two Fundamental fields, okay If you like to say the electron and the positron each of them are excitation of the two fundamental fields that makes sense Somehow yes, okay good. It was nice. It was like a field all of you was together. No thing I was very happy to see it. Okay And then you know are much more complicated than scalars and Not only they are more complicated also the notation become extremely Awkward, okay, I don't know who invented this. I think it's fine. I invented this notation with some slashes, okay And the slush that you usually it means that you want to erase something become very very important Okay, so if you see it before you know what I'm talking about you get to see it again If you didn't see it before you're also not eager to see it for the first time, but we're still gonna see it for the first time Okay, so here it is. Okay Look look Isn't it like it's so amazingly nice. No, you just look at it and you have no idea what I'm talking about, right? So the idea is as following so The fermionic field as I say it's much more complicated than the scaler and the comb and it must be a complex field by Construction it must be a complex field think about it again just like a number if a if a scalar field is just a simple number a fermionic field must be a complex number and Being a complex number mean that it can create Particle and antiparticles we didn't talk about it when we talked about like the photon There's only photons. There's no anti photons when we go to fermions we also have an electron and a positron so it's become more complicated and One important thing about the kinetic term and if you've seen it before you're familiar if you know These gamma matrices are some generalization of the Pauli matrices But the important thing about the kinetic term for the fermion is that it have only one derivative so I have two fields and one derivative and that's very surprising in a sense because we are so familiar with the fact that the kinetic term is p squared is the Momentum squared and then we move from from classical mechanics into Filtering and we see phi squared the second derivative d mu phi squared is again the second derivative And somehow for fermion is only is only single derivative And again, it's kind of important if you know why it's good. If not, let's not go much into the details Now how do I get masses for fermions? So for the scalar It was kind of easy to get mass you remember how we get mass We add this term m squared phi squared and then you had this little homework to actually verify that This is mass and all of you did it and we're all very happy You remember and now we are maybe not so happy because you try to look for the same thing for the fermion And what you find out that actually fermion masses are much more complicated Okay, and they are much more complicated in the following way that in order to get a mass to the fermion I need a combination of a left-ended field and the right-ended field Okay, and I'm just stating this as of now But I want to ask you why it is so why it is that in order to get a Fermion mass I need a combination of left-ended field and right-ended field well for the scalar. There was no problem I just have a scalar I add the mass term and that's it. Okay, and This is a very hand-wavy argument. So I'm gonna do like this. Okay, and the point is that as Much as hand-wavy is it's there actually a fundamental physics in it And the point is as following for a scalar field There's no spin and it doesn't matter in what frame I'm walking Okay, for a fermion because it's ever spin and spin is not Lorentz invariant I have the following really interesting property if I have a massless fermion and the massless fermion is a left-handed No matter what frame I am it's always keep going at the speed of light and therefore it's a it's it's left-handed Now let's assume that the fermion have some mass. It can have a really really tiny mass. Okay, say a neutrino Okay, and in my frame it looks totally left-handed We just learned from and read it they did this amazing experiment and found that they left-handed all the experiment that we did so far Totally agree with both totally left-handed. However, if it's even a tiniest mass That implies that there must be a frame where the neutrino actually going the other direction Yes, so I'm here. I'm left-handed and then there's another frame That then it's go to the other direction when it's go to the other direction it start looking right-handed. Why because the direction When I do Lorentz boost the direction Change, but the spin is invariant under the Lorentz moves. It's always going in this direction So if in this direction I go in like this and then I start going like this It's become right-handed Okay, another way to say it is that also since if a particle is a mass It's also have a rest frame and in a rest frame. I cannot tell left-handed from right-handed Why because left-handed right-handed has to do with velocity and in a rest frame you spin can be in any direction There's no way you can tell the direction. So in a rest frame you always 50% left-handed and 50% right-handed Okay, so what we conclude is that if we have a massive particle a massive fermion this massive fermion We cannot really tell if it's a left-handed or right-handed So if my building block have a well-defined spin either left-handed or right-handed They must be much less and if I want to have something that is massive I have to combine the left and the right so I can have both of them Okay, so that's kind of the hand-wave argument. Why a master for new tree for neutrino You killed me after saying you three of us alone cannot say any other thing a fermion a fermion for example a new tree No, okay must have a left-handed and the right-handed component. Okay Now this left-handed and right-handed component can be two different fundamental fields and when there are two different Fundamental field we call this master a Dirac master because that's the master that we know from the electron And that was the idea of Dirac and very nicely. It's called Dirac not Fermi because he kept the name. Okay, so that the Dirac master We can also have a very cool Things that the particle can be the right-handed is the same particle And if you take a left-handed particle and this left-handed particle You apply some charge conjugation and become A right-handed particle and they it can combine to itself. It's called my orana mass. I'm not going to talk about it I just want you to hear about it. Are you going to talk about it? Oh great So I don't want to talk about it because andre gonna tell everything about it But I just want you to hear about it that this is the a possibility Okay, so I want to ask you the following question. What are the conditions to have a master? So for a For scaler, I just say if I impose a symmetry there a master and actually there's no way to forbid a master There are some ways but in general it's very very hard to forbid a master for a scaler For fermion is actually the other way around you have to be very specific in order to be able to write a master So I'm asking you the question What are the symmetry properties of phi left and phi right should be in order for them to be able for us to be able to Write a master for the fermion That's a question Yes They should transform as the same u1 or let me generalize it They should transform in a way that I can make a singlet out of everything Okay, and since this one is a psi oops Since this one is a psi left bar and this is a psi r Bar is like a complex conjugate. So all the things become conjugate So that's mean that psi left and psi right must transform the same under the symmetry So they must have the same charge under any u1 and they must have the same representation under an su2 or su3 Okay, so actually in order to have fermion masses It's become a non trivial thing You must put both left and then the right-handed things with totally the same quantum number Oops, that's a new with totally the same representation and we have to put them together only then we can have masses Okay, so that's an important lesson that we learn here Masses for fermions. I kind of natural the masses of scalars are natural. They are there once you just Introduce a scalar field. It says a mass When you introduce a fermion field, you have to kind of walk You have to make sure that you can actually make master and let me give you a a piece of vocabulary The piece of vocabulary that's following if I have a theory that the left-handed particle and the right-handed particles Are actually total mirror image of each other So I have every left-handed field have a right-handed field with the same quantum number We call this theory a vectorial theory It's a little bit Frustration name because the name vector used for so many other things So it's also used for this theory a theory where the left-handed and right-handed fermions are all matched up in pairs I'll call vectorial theories And theories that are not vectorial that there's theories that I have some fermions that do not have a partner with exactly the same Representation they call chiral theories and in general in vectorial theories I have masses and in chiral theories because I don't have this match Not every left-handed fermion marry a right-handed fermion and I have some single people people I have some Field that are not matched up those fields do not have mass so in general chiral theories You we should have massless fermions Good So we know this word vector and chiral and we might use them in the future okay Next again, there's some technical details that either you heard about it or not But we kind of use them in the future and now I want to talk a little bit about the dimension of of the field So when we were talking about Classical mechanics things were rather easy I said I have my degrees of freedom x and I expand I do tell expanded and expanding x And it was very clear that if I have x to the 6 I said that's dimension 6 because I have x to the 6 If I have x To the 4 it was dimension 4 very easy. How do we actually count dimension when we do field theory? So when we were dealing with the Scaler it was very easy because the scalar was just the generalization of the x And I just say if I have 5 to the 4 that's dimension 4 5 to the 6 is dimension 6 When we put fermion into the game the situation become a little bit different So let me actually discuss what is the dimension of a fermion field Okay, and it's another dimension of a fermion field is different than the dimension of a scalar field So the generalization for classical mechanics a little breaks down. So how do we use it? How do we do it? We use this System of Reference we our system is that we use h by equal c equal 1 which of course we can always bring them back And then we say that the dimension of energy is 1 or must have 1 dimension And the dimension of x in this unit is 1 over its minus 1 And now we can actually start building up and ask what is the dimension of all those things So let's start with asking. What is the dimension of the action? Do you understand what i'm saying if I have a potential in classical mechanics the time What is the dimension of a potential in classical mechanics? Potential is energy. So it's dimension 1 yes Yes Good. What is the dimension of x? Minus 1 okay good. So now i'm asking you what is the dimension of Oh, let me start first. What is the dimension of h bar? Zero because it's a number just the number 1, okay So now let me ask what is the dimension of the action s It was nice at the beginning there was some count and then start to consider the zero. How do I know that it's zero because s Is just s dt It's just l dt And l has dimension of one because l in classical mechanics l is just energy. So energy is dimension one t is dimension one minus one and it's Zero another way to think about it that we know that the action has dimension of angular momentum And the action is dimension of angular momentum and h bar is dimension of angular momentum. So s has dimension zero Very nice So now what is the dimension of l? one One minus one t Nice I like the the fact that you don't give up and you keep telling me the right idea. It's one Okay, and if I have l in a fourth in some time number of dimension. So this is in one time dimension if I have four time dimension Nice and you can generalize it to d time dimensions It will be d. Okay, and how do we know it because we know that s is the integral of l d to the d of t Okay, so that's the number of dimension. Tell me that l has dimension Of the number of dimensions. Okay, very nice. So based on this we can understand how what is the dimension of a of a scalar field So a scalar field. We have something like this d mu phi squared l Is d mu phi squared So I know that this one has dimension of four d mu is derivative with respect to x So if x is dimension one is one a derivative with respect to it is dimension plus one Okay, so therefore this second two derivative give me plus two and therefore phi must have be also one Okay, so then from here I conclude that phi Is dimension one. Okay, because this is one. This is one squared It's give me four and now what's happened for fermion. So for fermion. It's something like this And a derivative as dimension of one and therefore the fermions the two fermions have to have Dimensions three and from here we get this weird result that the dimension of a fermion field is three half Okay, which you probably most of you seen before But what is so let me say few let me say two things one thing is that it's Really weird for me and I have no intuition why it is and I'm happy to hear if anybody Have some deep intuition why the dimension of a fermion field is different than the dimension of a and At the beginning I was thinking. Okay. It's it's not a big deal. It's just a dimension But then when we think about the whole big deal of women we expand and truncate We really really care about the dimension because when we expand and truncate Then the number of fermion fields that I have at a given order is less than the number of fermions Scaler field that I have to a given order So the the the number of dimension of the dimension of the field is very important when we do Physics, okay, it's not just a number that I can rescale it I don't know I don't have any any intuition and I'd be love to hear if somebody had been thinking about this question And have an intuition. You know, what is the fundamental thing? But what it's really implied for us is that when I expand And I think about say up to dimension four I cannot actually have I can only have two fermions. Why because fermion must appear in pairs. Why fermion must appear in pairs because because of lorenz environment very nice One way to think about it is that I have spin half and The Lagrangian must be spin zero. It's have to be invariant under rotation So spin half times spin half give me zero and any odd numbers of fermions always give me Something that is a half so it cannot be invariant. So lorenz fermion must appear in pairs And any pairs of fermion is dimension three Okay, so if I need four fermions, it's become dimension six Okay, well four scalars are dimension four. So when I go up to dimension four, I can have terms that have Two two scalars three scalars and four scalars But for the fermions, I can only have two fermions and that's it Okay, so let's make a very big difference in the phenomenology between scalars and fermions That somehow I don't see why it's come from the mere fact that fermion have a spin And it's nothing to do with spin statistics or all this Okay, so we understood this and I want to talk about one other Let's say there's a lot of little details that I start with and I'm sorry if it's a little boring But I really hope that we just get it over and we know it And I want to talk about the discrete symmetries of spacetime and here it's a again something that I'm assuming most of you heard It's about charge conjugation parity And time so time is the one that take t to minus t parity is the one that takes x comma x x And charge is something that take the charge of the particle and make it a minus the charge of the particle. Okay and We say that they are related to to spacetime because I Say p take x to minus x the vector x the three dimensional x so it's related to spacetime and There's something called the cpt theorem and the cpt theorem tells us that any Lorentz invariant local Lorentz invariant to field theory Must be invariant under the combination of all of them if I take the charge and make take it opposite Take the parity and opposite and time is opposite. I Get the same theory and in terms of everyday language The the pattern is following that if I have an electron going forward in time It's actually a equivalent of a positron going back in time. Okay, because I also take The electron when I apply cp on the electron it becomes a positron So the electron going forward in time is the same as a positron going backward in time. Okay now c and p are Taking left-handed fermion into right-handed fermion. It's kind of understandable if you think about p So p I take a left-handed fermion and I apply p. So I change the direction of motion but the spin is A pseudo vector. Okay, it's angular momentum an angular momentum Do not change shine under x go to minus x So when I apply p on a particle of the left-handed particle, it's changed the direction But do not change the spin So it's going from this direction. I changed the direction, but the spin kept going that way So it's become right-handed. So c And that's happened to also for for c so In a way, it's kind of very easy to break parity. It's very easy in a sense that It's very easy in a model building way to break parity. So what is the easy way to break parity? The characterity means that I have some left-handed field that do not have Match up on right-handed field or the other way around and when I apply parity the left-handed become right-handed and since there's no right-handed The symmetry is not symmetric under p So the statement that I'm saying here is that any chiral theory Must violate parity. Okay I didn't say the other way around vectorial theory can also violate parity, but it's not it's not if and only if And now we move to cp and in cp the situation is a little bit different because cp You take a left-handed field into a left-handed field So chiral theory do not have to violate parity cp And i'm just stating now the fact that in order to see to have cp violation in our theory I must have that my Lagrangian is not real and we're going to actually discuss it in great details Hopefully tomorrow here. I'm just mentioning it Any question on this? Okay So now I want to talk about local symmetries. Yes Oh, I said that vector theories could break parity. Okay, so you don't have to actually break parity and The point is that What i'm just saying is that if I applied parity I take you from left to right But actually also other things happened. Okay, so you can actually write vectorial theories that break parity We can talk about it later And actually it's not important for the standard model. Okay So now I want to move and talk about local symmetries Also known as gauge symmetries And you all learn about it or you should have learned about it in your undergrad electromagnetism And in undergrad electromagnetism We actually discuss it in great detail that we show that you have a gauge transformation And it's a symmetry it's kind of a symmetry of our things and you keep going and there's a lot of cool stuff and When we move on to do particle physics We actually I really like the way andres said everything is electromagnetism You just kind of promote it in many different directions. So that's the same story We just kind of understand what's happening electromagnetism and and go up So we talked about the symmetry operator operation and the symmetry operation where that you take my phi field And rotate it with some theta And how much if I rotate by theta how much my field rotates to do with the charge q Okay, and now I say Let's assume that this theta Which is just a number in the symmetry become actually a function of space time itself Okay, so what did I do mathematically? I take phi I take theta that was a number and I promote this number to a to a Sorry To a function. Yes to a function and I want to hear another word to a field You remember a field is nothing but something that is dependent x and t something that depend on x and mu a specific function A specific function that depend on x and mu we call the field So we promote the theta from a number into a field, okay and There's a lot of hand-wavy argument why this is logical So let me say the following. I'm going to explain you why it is logical But before I explain you why it is logical Let me say that the way I like to think about it is just When you do model building and we talked about symmetries You just want to use all the things that you have in your toolbox of symmetries So I have a way to do symmetries where theta is a number And now I'm going to discuss how I can do it when theta is a field. So that's kind of the argument and The people say that This makes a lot of sense that things are local in the following sense So oh, I should have said that this is also the depending global symmetry and a local symmetry It's a global because theta is the same all over space time and local symmetry is when theta is different at different point of space time And why people wants to do things locally? Because physics is local Nice, okay, and it's kind of amazing that we were always kind of surprising that you know when when Newton discovered this action on a Distance principle that looks very weird and then we got used to the idea that actually the sun Affect us here in terms of the gravity the gravity of the sun affect us here. How can it be? It's so far out It's amazing. It's magical and then it took whatever how many years until we realized that physics actually is local What it's happening is that the sun generate And a gravitational field the gravitational field kind of propagate into the earth And then here we feel the gravitational field that is Produced by the sun but everything is local And the field is the one that is propagated and any time that it seems like it's not really local at the end of the day We find that physics is local So we like it to keep it local So why don't we also promote our symmetry into local symmetry and amazingly enough when you look at the standard model Somehow we don't need global symmetries only need local symmetry And again this idea that physics is local is kind of coming back to us Okay, and we never see any physics now that is actually non-local So we kind of like the idea that things are local Very good. So then we actually Look at our at my if I have some Lagrangian that is invariant under a global symmetry I look at the at the terms and I find that any term that depend only on the fields Doesn't really care if I have if this theta depend or doesn't depend on the angle because Because it is invariant. It should be independent on theta So I don't care if that is is a global or a local thing However, the kinetic term the derivative is depend on the fact that is local because When I take a derivative of a field and I take the derivative of this theta I give extra term So the kinetic term is not invariant Under rotation if I if it was invariant under this it's not invariant under this So according to the rules of the game, we just say if it's not invariant under rotation I shouldn't write it you remember what we said we said if it's forbidden by a symmetry it should not be there Okay, so that's the answer. We should just drop the kinetic term and we're done However, we really want a kinetic term Why do we want kinetic term? Right because if I don't have kinetic term, I don't have dynamics, right? And if I don't have dynamics, it's cannot describe the world Kind of nice, right? Yes, so any theory that I write I want to make sure that I have a kinetic term So I have to fix the kinetic term and I'm not going through all the algebra In a way, you can actually see it from electromagnetism from your undergrad electromagnetism But in order to fix it what you need to do you need to add a massless field This massless field must be a spin one Amazingly enough, how do we call spin one particle? We call them vectors Okay, as if there's no other words and we use vector for any words that they just around you just call it a vector So we call it a vector field because it's a spin one which is different from the vectorial theories Which is different than the vector and the rotation Okay And not only this we know how this spin how this new particle Have what is the presentation under the symmetries? So under u1 it's have q equals zero so it's not charged under the u1 And that's what we know and love from the photon the photon has electric charge zero Our for su2 is a triplet this field that I add for su2 It's not a singlet of su2. It's actually a spin one under su2 also known as a vector of su2 Okay, nice, uh, we use one word and I use the whole lecture with this one word vector Okay, and for su3 it's turned out to be an octet of su3 Okay So it's kind of a fundamental very important different between the abelian and non abelian in the abelian It's q equals zero which is the singlet for the non abelian This must this this particle is a triplet or an octet so it is charged under it. Okay Before I go on let me do a little detour here to the side and I was talking to you about scalar fields I was talking to you about A fermion fields and when I say fermion I really mean spin half or those For the general public fermion mean anything that have half spins But in particle physics when we say fermion we usually mean spin half And then I talk about the vector which is spin one. Okay So naturally you would expect me to start talking about spin three half spring two spring five halves, etc However, I'm not going to do this. Okay, and the question is why I'm not going to keep going. Okay And the answer is kind of very interesting answer And The point is it's following that why classical field theory you can write for any spin that you like When you try to quantize the theory it's turned out that you can quantize a spin zero theory quite easy Spin half in order to quantize it become more complicated and you need all those Things but at the end of the day we know how to quantize spin half Spin one basically you cannot quantize the theory Unless this spin one is this gauge boson So actually not only that I told you so here I was telling you one Seeing like this that I said oh if I want to make the the symmetry local it must involve a spin one But there's another route to come to the same conclusion and the same conclusion is to do the following things If I want a quantized theory of spin one This spin one must be the gauge boson of a local symmetry If I just try to write up a spin one thing without a gauge symmetry this theory is not self consistent Okay, so the only way to write a spin one quantum theory is if it's a gauge theory And if you go to spin three half Basically from spin three half and above you cannot quantize any theory With some subtleties if you have gravity you have also the spin two and it must be a massless spin two for gravity And if you have super symmetry you can also have a gravitina which is spin three half But forget about those kind of gravity stuff. Okay, if we forget about gravity then Just from mathematical point of view it turns out that the only thing that we can quantize is spin half It's been zero spin half and a gauge boson and it's amazing It's really amazing that that's what we see in nature So maybe there's some deep connection between the fact that mathematically we can only do spin zero half and one And that's also what we see in nature, but I don't know there's just Kind of a fact Okay, so I'm not going to talk more because we don't know how to do it Okay So now let's talk a little bit about the gauge boson kinetic term So I say I add this gauge boson. I need have a kinetic term for it And it's kind of a generalization of the scalar But it's a many indices because the scalar do not have indices and here we have and here we really love the kinetic term from Electromagnetism by the way, what is the standard normalization for this kinetic term? Okay, just the fact that all of you say I know you've seen it before it's minus a quarter and f mu nu is This and f mu nu is basically just e and b the the derivative of the Vector potential is give you e and b in in electromagnetism We start from e and b and build The the the vector potential a when we do particle physics would like to go the other way around We start from a a is our starting point and then the electric and magnetic field will be a derived quantity by defining f mu nu and The kinetic term is just this now for non abelian case There's one very important difference between the kinetic term So in the abelian case we just have this d mu a mu For the non abelian case and I call here for to distinguish from the abelian instead of a I call it a g And I call it g because it will be there. We soon will see them. There are gluons There's an extra term that go like this There's Derivative of g derivative of g and then another term that there's no derivatives. It's have only g's Okay, and a little bit surprising that I say it's part of the kinetic term because kinetic terms must be a derivative That's why we call them kinetic term, right? But we already abuse the notion kinetic term so many times, right? We start with kinetic terms just something they have time derivative And then we actually have also an x derivative because we think about x as an extra time derivative And now we totally Say oh, that's a kinetic term although it's have some terms. They do not have any derivative at all Okay, but it is part of this f mu nu. So when I write a kinetic term for a su 2 and su 3 I have f mu nu f mu nu And if you see that I take this and square it I have the kinetic term which is second derivative of of g The d mu g squared But I also have seen that look good like g cube which is then the product of this time this And also seen that looks like g to the 4 which is the product of those guys, okay? And these are extremely important. This is something that make the phenomenology of non-habilian Theories and a billion theories very very different So it's like it's looked like a small deviation from electromagnetism, but the phenomenology is very very different So The last thing I want to talk about almost the last thing about the gauge symmetry is how do we copy it to the fermions? Okay, and again, let me come back to Do your old when you were young and you were taking undergrad classes and you were discussing how do you actually Put the electromagnetic potential in you into your Hamiltonian. You remember those days? Yes So then you talk about how you do it for electron for the electric field and for the electric field We just say oh I have my Hamiltonian. I have my kinetic term Which is p squared over 2m And then I add my potential that would be the electric potential And then I said how can I add my vector potential the magnetic field and The answer was that the magnetic field I add by this I do p minus a squared Over 2m Plus five and that's the Hamiltonian of a particle in an electromagnetic field Okay, and I don't know if you were kind of worried about the fact that this looks a little weird because A magnetic field give you a magnetic force Okay, and our intuition from working with Hamiltonian is that this one is a kinetic term And this one is a potential term and forces come from potential. They don't come from kinetic terms, right? And then suddenly we have the magnetic field and we have the magnetic force And then you put the magnetic force in the wrong way It's like excuse me. This is not this is put it in another drawer. This is the drawer for kinetic term This is the drawer for potentials, right? So why did we do it? What was kind of the reason that we did it? You remember why you did it back then? It was a very hot day. Yeah, you remember this day. It was lecture number 37 three days before the end of the semester Okay, and you were already like have this other exam on your day But then you came to class and suddenly this hits you you remember Yeah, I cannot forget this day. Okay So why why it was that we put it in the kinetic term what was kind of the reason Yes, so that's always how we do it in undergrad We just say let's do the formula that give me the correct nature, which is always a good idea I don't I don't like yeah, but we are trying to do something a little deeper But this is always the correct thing the correct things to do is I want to build my theory to give me nature Okay, and only later on 50 years later. We say oh have this philosophical idea I built Lagrangian symmetry tata. So now we are 50 years later or whatever five years later for you And we say what is the symmetry property that yes Yes, yes, yes So do one way to do it is actually to understand what is really the conserved momentum And that's so yes It says something about the conservation of momentum and it's really interesting that when we go to A magnetic field you find that what is really conservative It's not the mechanical momentum. It's not just mv But there's actually something else that has to do with qa and you actually go through this and say wow That's amazing. It's not regular momentum That's because there's something else that is conserved and this has to do with this and we put it here But there's something a simpler reason that I put it here. Yes Yes, yes Yes, so you're almost there. Let me try to say it my my way, but I think that's what you meant it just has to do with the properties of Of what I have so if I if I have a scalar I put it in a scale if I have a vector I need to combine it with another vector In order to make a scalar the only way that I can make a scalar out of a vector Is that I take the scalar squared? So this one is a vector times a vector and this one is a scalar And if I give you a vector into the theory say oh the vector should go where the vector is Okay, so actually you don't have to label your drawer as kinetic versus Potentially have to label your drawers Vectors squared time scalar to begin with so then I just have a vector and the vector must be combining Into the vector that I have right because a is a vector Huh, that's nice. I'm so happy right now. We see what's going on Yes, because a is a vector it has to come together with another vector, which is a So you are not surprised by the fact that actually magnetic field the interaction actually come from the kinetic term Okay, so the intuition that we kind of grew up when we started classical mechanics that the potential It's what give me forces when we go to electromagnetism you find that this is kind of already breaking down It's a it come a little shaky and this potential is only the electric field and this one is the magnetic field And now what's happened when you promote The three vector of the magnetic field into a four vector when you go to relativity and become a mu What it's better have to happen It's better come together with the p mu. Okay, so when I actually promote the Three the three dimensions placed into four dimension and I promote my a mu into a four vector a mu The a i into a mu then the a mu all come together in the kinetic term And then we come to this really interesting phenomena that all of electromagnetism All the electromagnetic interaction are part of the kinetic term Okay, you see what we did for the kinetic term It started with just some Time derivative is just x dot squared And then when we move to filter it's become also x prime dot And then when we went to gauge fields for non-habilian gauge field We already have some g g cubed and g four coming from the kinetic term And now when we actually look for how electromagnetism is coupled to fermions We find that all the coupling is actually coming from the kinetic term. Okay So the kinetic terms is Lost its original idea that is a kinetic term the kinetic term now is the whole city full of a lot of things It's also have a kinetic term, but they also have a lot of interruptions. Okay, so in particular what we have We find that in order to do it we have to take my Derivative and promote it to something that called the covariant derivative The name covariant derivative comes from general relativity And it's somewhat related And this covariant derivative is not only the d8 also have the amu which is nothing by this Okay, so if this looks a little bit abstract to you Then come back to what you know What I just did d mu is just the momentum and amu is just this and there's actually also a q here that I didn't Write so this one is nothing, but you familiar Hamiltonian for particle in a magnetic field Is just this Okay, and this is the kinetic term which is just this Okay, and the important point that in the kinetic term I have the interaction and the interaction here has to do with the photon coupled to my To my field so if I have field phi and I couple it to the photon I have such an interaction okay, and So I was telling you the fact that when I have a gauge boson it's coupled to the fermions From the kinetic term and the coupling is proponent to the q to the charge That's for the electromagnetism and I'm not going to get into the details But I'm telling you that when I have a non-habilian symmetry the coupling is proportional to some matrices And these matrices are the generalization of charge in a non-habilian case so in In an abelian case I said charge is just the number that tells me how strong I am interacting in a non-habilian case It's not a number anymore. It's a matrix in a way That's what you expect if I tell you I have some A theory that is commuting and it's have numbers and then you take your theory and make it non-commutative What should happen to the numbers? They should become matrices No Yeah, right you've seen it before in quantum mechanics. Now we see it again But it's a general thing when things are non-commutative numbers become matrices Okay, so you are not surprised that the generalization of the notion of charge in electromagnetism become matrices in non-habilian symmetry, okay And you kind of know what are those matrices in some cases So what are the matrices? So if I have a particle that is a spin half What is the charge that the generalization of charge for this spin half particle? What matrices they are? Pauli, what else if I tell you spin half you say pauli We talked a lot about this guy today, huh? He did a lot of things He wrote some three matrices. He went to the party. It was quite good. Okay And if you move on to su3, then we have some other matrices and they are called the gelman matrices And he was the one who actually first took the su3 matrices. It's a little story that I just heard A few days ago, so he passed away a month ago less than a month ago. He just passed away, okay, and I was back in israel and nati zeiburg, which you probably heard He just came and started talking about him and what he told me I think he told me this story that actually the gelman matrices, which are the generalization of the pauli matrices were first written by gelman Despite the fact that mathematician did su3 for like 50 years before gelman. So why nobody wrote them down? Okay, okay any idea And I think the answer is that's because they are mathematicians Okay, all they care about is that they can prove that they exist Okay, and then like why write them down and then suddenly we actually need them They need the physicists to actually take what the mathematicians do and write them down And the gelman matrices in this language for me is the generalization of a choir of a charge instead of a charge I say what is the What is the Strong charge of a quark the strong charge of a quark is the gelman matrices What is the weak charge of the electron the weak charge of the electron is the pauli matrices Okay, I know the word charge is very it kind of looks like a number and at the end of the day when I measure A strength of interaction the strength if if the charge is a matrix and I want to measure some number This number should be proportional to what? kind of the intuition So the strength of the interaction, okay, let me do it like this So if I have some scattering electron electron positron scattering, okay, just something like this some electron positron And here I have some photon, okay The amplitude is proportional to q squared Yes, because I have q here and q here Okay, now if instead of this photon I have a gluon something that is non-Abelian Okay, so here instead of a q I have some lambda and lambda these are the gelman matrices So then at the end of the day I have to take this this matrix square it and Took the phase space integral. We talked about it, right? So what would be the instead of the q square what should appear in my Cross-second, yes So it should be somewhat proportional to the matrices, but it should be a number So how you take matrices and get numbers out of those matrices? Ah So nice you can also do the terminate, but you already got the right answer So the equivalent is actually something that go like trace trace of lambda lambda Okay, and that's kind of the equivalent of q squared, okay So q squared is kind of a scalar which is a trace and then to If it was a number then taking the square root is easy But they're taking a square root of a trace of a matrix It's any more complicated and that's just a matrix And that's again something that we see many many times in physics that things are squared are the invariants and The square root is actually a matrix Okay, let me Talk one extra piece about electromagnetism and the way we think about charge in electromagnetism And again, that's something that you learn back then as an undergrad We say charge actually have two very different Kind of characteristics one thing is that charge is conserved And charges conserved of course in electromagnetism is trivial because you cannot create an annihilate particle But in particle physics the fact that charge is conserved is a non-trivial statement It's the fact that whenever I create an annihilate particle the total charge of the interaction is stay the same Charge conservation in quantum field theory Or in in in general in Relativistic mechanics is the same as energy conservation. It's move around charge is floating around you create an annihilate particle But the total charge is conserved. So one aspect of charge is of charge is a conserved quantity Another aspect of charge is the fact that it actually tells us the strength of the interaction If I have a charge a particle that has more charge, it's coupled stronger to the electric field Okay So do you see that there are two different aspects of the definition of charge and most of them are very familiar to you, right? So now my question is these two aspects I'm telling you that one of them come from the global nature of the interaction It's just there because I don't care if it's a local or global interaction And one of them is there only because the interaction is local So I want you to tell me which one is which so there was There's only two options because I tell you this one and one, okay So which one is which so you have 50 chance to to guess just without saying anything So I hope that there is I will be more than 50% anybody Any brave person to tell me which one is which Yes Very nice. So it's the global nature of the symmetry that give us conservation laws Okay, and that's if you like is the net earth here, etc It's the local nature of the symmetry that give us the coupling to the photon If you like the local symmetry is just the fact that we actually do have a photon to start with But charge conservation, we don't really need the photon to start with charge conservation is there without even having the photon Okay So let me say the following if I have a global symmetry global symmetry have conservation law But no gauge bosons and local symmetry of gauge boson and it's also the charge is also responsible for the strength of the interaction So let me kind of before I move on I want to kind of summarize One thing about masses because mass is going to be a very important part when we start talking about standard model We discussed about three types of particles. We discussed about scalars fermions and gauge bosons And scalars basically they must have masses It's to you just write a scalar you write a mass term. There's basically no symmetry that can forbid it Okay, there is some cool cases that you can do it, but I'm not going to get there For fermions, it's very easy to forbid masses. All I need is to make a chiral symmetry What do I mean by a chiral symmetry? I just mean that I don't have the same Number of left hand and right handed fermion to meet each other and gauge symmetry Gauge symmetry guaranteed that the gauge boson is massless Okay, so we kind of go from one to another Scalars are must be massive fermions. It's up to us and gauge boson must be massless Okay So that's kind of where we are going and of course we know that in nature the w and z are massive And we need spontaneous symmetry breaking and all that but before that that's where we are So That's really nice because we just mentioned spontaneous symmetry breaking. It just happened to be the next transparency Okay, so I want to talk about spontaneous symmetry breaking And you probably all know about spontaneous symmetry breaking and I like to tell you some little stories about spontaneous symmetry breaking And the first one time symmetry breaking story has to do with the Mathematical and physical donkey. So how many of you know about the story about the mathematical physical donkey? Okay, so that's a a cool story that really helped me to understand them Oh, there's actually a third one. It's a biological donkey. There's three of them. Okay, so Let's say that you There's a donkey and the donkey is very hungry and the donkey come into a room and there's two Piles of hay donkey is supposed to eat hay. Is that's correct? Anybody have a donkey? No, okay, so you come in and there's two things and the donkey come exactly in the middle of the room And you see the two Hayes and the donkey is very hungry and it's so hungry and the donkey, of course, know the principle of minimum election So he wants to go to the one that is closer to him, right? However, the two Hayes are exactly at the same distance. Okay, so what the donkey will do? Okay So now it's really depend of what kind of a donkey you are. Okay So if you are a biological donkey, what you're going to do You're going to just choose one and you don't care which one. Okay, and you go and eat and you Totally fine. And maybe you so biological you don't even understand that you have such a big philosophical question to start with You just choose one of them. Yes, okay If you're a mathematical donkey, what would you do? Okay A mathematical donkey will stand there and it said it's totally symmetric. So I cannot make my mind So what would happen to the mathematical donkey? He will die of starvation, right? Okay, and that's actually one of the reasons that we've seen that they extinct today. Okay, there's no mathematical donkeys in nature anymore Okay And what happened to the physical donkey? Whatever to the physical donkeys or physical donkey, of course, he totally understand the the the deepness of the situation But he still choose one of them. Why? Almost quantum fluctuation, right So he just choose one of them. Okay But it's the point of spontaneity and tree breaking So the story of spontaneity breaking has to do with this the point is that any physical systems behave more like that The biological donkey you come and although you have totally two symmetric places You kind of have to go and choose one of them. Okay, so in particular You always want to go into one of the minima and it doesn't matter which one You have to go you you choose, but you do choose. Okay, let's say if you say, why do I choose the answer can always be quantum fluctuation and that's another example of Fondant symmetry breaking and You've probably been in similar situation. That's happened to be a wedding table And it was down from google So I don't know who got married, but And then you sit on a table And you you know you talk and everything and then you want a drink And you sit let's say you sit here. Okay And you see that those glass these glasses and these glasses are exactly the same distance from your hand Okay, so the question is what which glass do you choose? Okay, so I know the same story with the donkey. Tata you choose one glass The second you choose one glass. Let's say you are the first person and you choose this one glass Then everybody else on the table know which glass to choose So you choose the glass to the right and everybody should choose the glass to the right if you choose the glass to the left Everybody should choose the glass to the left. Yes Okay, and that's fontan symmetry breaking It starts with a symmetric situation that all the glasses were the same distance You can pick left or right and you choose one of them. Okay Sometimes actually since can very interesting thing can happen Sometimes you have very long table and you choose right and someone in the other side of the table choose left You know this feeling you had it. Yeah, you had it, right? You want to all these restaurants with a big group and then what's happened every day start here Left right right left left left and then one person end up with two glasses and one person gets with no glasses Right, you know you you've been there. I'm sure right And something with the fork or something. So what do we do? What do we do in real situations? What's happened with the two glasses? That's the one glasses person Very nice. So you give the extra glass to the other side. Okay, very nice. So that's that's also has a name I'm not going to talk much about it But this phenomenon is called a domain wall phenomena A domain wall phenomena has to do with the fact that you have spontaneous symmetry breaking on this side Which is different on the other side and then they clash in the middle And when they clash in the middle, they can annihilate each other. They move the glass to the other side. Okay So you're all being in this situation and by the way again Being in america, I learned quite a lot of Interesting things and one of them are interesting what so society actually decide for us which glass should we take Okay, anybody know which glass should you take? How do you know it? Okay, that's a very good answer No, no Very israeli answer. Okay, but okay. It's like yes, that's the right answer. It's to the right. Okay, but that's how I learn it in america Okay, so if you don't remember now, you all remember because so but you bring up your hands and you do like this Okay, and then you look at your hand and this one looks like a D D stands for drink. That's where you cup should be And B stands for bread. I mean here they don't have it, but sometimes you have this little You know the little plate for the bread that's also in the middle That's the one you go for the left So you do like this and you never forget it and since this, you know, I always I'm I'm so good at this now I never have these mistakes Good, so let's talk about some physics out of this wedding table And what is the idea? Okay So the idea of spontaneity signature baking is the fact that The vacuum state is degenerate. That's always the case. So if I have a vacuum state that is Non-degenerate, I'm there. Okay. If I have a vacuum state that is degenerate I have to choose and why do I have to choose? I have to choose because of this very very very big principle that we expand around the minimum You remember we talked about this principle and you remember that I tell you it's a very important one I didn't cheat. It's a very important one. Okay, so if I have a system I need to go and go to the minimum and start expanding Okay, so if I have one minimum, it's very easy. I go to the minimum and I expand But what's happened when I have two minima? I have to go and expand around the minima So what I have to do I have to choose one minimum when I choose one minimum It doesn't matter which one I choose. I just make a choice when I make the choice That's that's what's happened when we have spontaneous symmetry breaking. Okay, and then we use perturbation theory around the minimum and we get Fields and particles and everything around this minimum. Okay So now Let's we understand what is the basic idea of spontaneous symmetry breaking. Let me ask the following question What is the fundamental difference between spontaneous symmetry breaking and no symmetry? So in spontaneous symmetry breaking once I go to the say, you know The donkey when the donkey go to the right and start eating the right. There's no symmetry anymore, right? So what is the difference between spontaneous symmetry breaking that I have to choose as a A minimum versus no symmetry at all So the Lagrangian totally has a symmetry So the fact that I actually go and choose my I choose where to expand around so I kind of choose my ground state I have a degenerate ground state and I choose to sit in one ground state So of course the Lagrangian itself still have the symmetry But locally I don't know about it because locally I'm only here I don't know the fact that there's another minimum far far far out. Okay, and the standard standard example of the Of spontaneous symmetry breaking in physics is the double well potential and the double well potential look like this Right, so if I go and I start expanding here, I have no idea that there's another minimum Here, right? So locally, I don't know that I actually have this symmetry So how I would know That I have spontaneous symmetry breaking versus no symmetry Okay, and spontaneous symmetry breaking is such a huge concept in particle physics and in quantum filtering So what is the fundamental thing about but spontaneous symmetry breaking versus the non-symmetry? non-symmetry at all and this is this statement that I made here in Box and yellow and all this Spontaneously symmetry breaking implies relations between parameters That's the important thing So even though i'm here and I expand around the minimum and I kind of don't know that I have it The fact that my Lagrangian still have this symmetry and this symmetry is hidden because I don't really see it I can still see indication that is there kind of indirect evidence that I have the symmetry And indirect evidence has to do with the fact that I do have relation between parameters If I if I'm just here if this one was not there There will be some kind of expansion going on here and there will be no relation between parameters Then suddenly I have amazing relation between parameters. Okay I'm going to give you an example in the next transparency But the bottom line is that's what I say is this one Spontaneously symmetry breaking theories imply relation between parameters And that will be very important when we're going to discuss the standard model Okay, we're going to see those really cool relations that looks like coming out of nowhere and then we want to remember this Okay, so today in the afternoon when I go and say do you remember what's happening in the morning? You have to remember this exact moment. Just don't forget this Okay, so let me give you the example of how we do we get a relation between parameters So let's take the following potential some function f of x and this this function is basically this double well potential Okay, you see it a and b are positive and the minimum are in these two points are plus minus b over a Here and here and I have to choose one one minima Let's say that I choose this minimum Okay, and I expand around this minimum and when I expand around this minimum I use my new variable u and u is the variable that vanishes and the minimum and then my function f of x become this um This function So you see here I have x to the four x cube x x squared and x to the four and here I also have u cubed So there's no u going to minus u symmetry. That's the meaning that I break the symmetry spontaneous Here it's x squared and x to the four so x to minus x is a symmetry And here since I have u cubed I don't have u go to minus u symmetry Of course, I still have x go to minus x symmetry But I don't care about the value of x anymore because u is my value bell U is the small parameter that I expand around you going to be the field that's going to have excitation and create my particles, etc Etc. So all I care about is just u I don't know that there's actually x somewhere to start with because x doesn't give me all the particles and everything Okay, but how do I still know that I have this function that there's still this symmetry like hiding? So if I just take this function f of u and I write the general one the general one would be c2 u squared plus c cube three three u cubed plus c4 u to the four And here in general I have three parameters. So if I just sit around a minimum that is not symmetric I should measure three numbers c2 c3 and c4 And amazingly enough when I look into this function It still have only two parameters why it's have only two parameters because my original theory have only two parameters So I must still have the same number of parameters, which is only two So how do I measure these two parameters that I start with into the three parameters here? There's one relation c3 squared equal to four c2 c4 So what you see is that I expand around this minimum And it looks as if I have a theory with three parameters because it's not symmetric under u go to minus u It looks as if I have three parameters But then magically I have two parameters And then I said, ah, I do know why there's two parameters. Why there's two parameters? Because this is a spontaneous symmetry broken theory and not just a theory without a symmetry Good Yes Any questions? The point is that I'm not going to guess them what I'm going to actually find out I actually do the calculation and I find them and I don't know there's actually description to do it without All I know is that I can tell you how many Relation I should find that I can count and we're going to count it to some degree And the way I know how to do it is just right there You're you just start expanding and you find it you see you expand and you just look at it and say wow This one is actually dependent on these two. Okay, and We're going to see some examples Okay Now I want to talk a little bit about the fact that we can have also partial Spontaneous symmetry breaking so the example I gave you so far is that I have a symmetry and the symmetry is broken For example, it was broken when I have an The double will potential and when I choose this X go to minus X is broken But sometimes I can have a situation when I break only part of the symmetry Okay, so I want to talk about the case that I have Something that is symmetric and the three dimension. Okay For example, think about the hydrogen atom and then I can break it. I can break it spontaneously by Choosing a magnetic field and I apply magnetic field into it. Okay In which direction I apply the magnetic field In which direction in all you undergrad classes you apply the magnetic field The z direction Why why you didn't apply it in the x direction? Oh, because it's easier because it's spontaneous symmetry breaking. There was a symmetry you can choose You can choose that's the whole point and we as a society choose the z direction Okay, no idea why but we could choose anything. Okay, but we choose the z direction You did spontaneous symmetry breaking. Do you see that? That's what we did any direction is the same you can do everything with Choose it in the x direction or in the y direction or in some arbitrary direction It will be the same but we choose it to be in the z direction Okay, very nice. So after we choose it to be in the z direction What's happened to the symmetry? So the symmetry was three dimensional and now I choose a vector in the z direction What is the symmetry after I choose it? It is It's s o 2 it is rotation in the plane So if I have rotation in the general Three dimension and I choose a vector then I start having Symmetry only rotation in the plane perpendicular to this vector So in particular the symmetry is always broken down from rotation in a In a space into rotation in a plane or in group theory language is from s o 3 to s o 2 Okay And it doesn't matter which direction I could choose it in this in this way And then the symmetry will be in this plane The symmetry is always there. It's perpendicular to the direction and the direction is an arbitrary choice It's spontaneous symmetry breaking. So let me General summarize what I wanted to say in this transparency that when I have spontaneous symmetry breaking I can break the symmetry not completely I have a symmetry that have many generators Rotation in three dimension I can rotate around the x y and z plane And then when I break it I can break only rotation around the x and y axis But not around the z axis. This still be a symmetry Okay Now let's talk a little bit about what's happened when we have spontaneous symmetry breaking in in quantum field theory And in quantum field theory the idea of spontaneous symmetry breaking is the following I have my potential and It's exactly the same story right here. It was a function of x now It's a function of phi and I hope you got used to the fact that phi is just a generalization of x It's just Like any number so I have a potential for it And then I look for the minimum of this potential and I expand around the minimum of this potential Okay And when I expand around the minimum of this potential I write it explicitly phi of x mu that means that's a field And it's equal to the number a number is the minimum plus another field Okay, just like when I expand x when I say u is equal to some number plus You know x is some number plus u x and u were Valuable and the number was a number. It's still a number. It didn't change. Okay So what these things do that for us? It's break the symmetries That phi is not charged under so if it's just a z2 just a just a parity is break parity If it's a bigger symmetry it breaks all the thing that is not charged under. Okay And then we can have actually important implication For masses and that's very important. So you remember about half an hour ago We talked about which can what can get mass etc Okay, and we talked about masses to fermions and masses to gauge bosons And we said that fermions could have Could be massless if the theory is chiral, right? However now when I put my when my scalar acquires a verb Oh, I should use this when I say acquires a verb That's mean that it's actually has a non-trivial minimum and this is a verb. That's that's vocabulary. Okay When I have a situation when I have spontaneous Symmetry breaking and I expand around the minimum expanding around the minimum can give fermion a mass Why I can give a fermion a mass So I can actually here I'm going to see it explicitly But fundamentally what's going on? There's actually a symmetry and if my fermions are chiral under this symmetry They do not have the same charge the left hand and the right hand that fermion do not have the same charge Then this fermion is massless Now I break this symmetry and I break the symmetry into another symmetry Such that under this other symmetry this fermion do have the same charge And since they do have the same charge then they could have a mass Okay, so I break my chiral symmetry into a Into a Thank you You really and into a vectorial symmetry You should have guessed because everything today we use the word vector, right? So I have a chiral symmetry I spontaneously break it into a chiral a vectorial symmetry so the the fermion can get a mass and Technically we see it in the following sense. Why is the number phi is my scalar field psi by left phi right is my fermion field So this is an interaction term. It's a three field phi psi bar and psi. There are three fields It's an interaction term now after this phi acquires a wave after this phi Find its minimum then phi become v plus h and in particular v become a number So a field become a number plus a field and the number the minimum Then I have a term that's go like y v psi by psi and psi by psi It's only two fields. It's a psi by psi by left psi right, which is nothing but a mass term for the fermion Okay So that the technical way that we see that fermions get acquires a mass and the mass is proportional to the coupling Times the value of the minimum So the bigger the value of the minimum is the mass of the fermion is is larger Okay, and it has to do with the coupling between the my fermion and the fields that actually gets a minimum Okay So we learn one thing fermions who could be massless in the full theory can acquire the mass after spontaneous symmetry breaking Point number one point number two the gauge bosons can also acquire the become massive and this has to do with the A covariant with the derivative with the kinetic term for my scalar fields and how this is come about Philosophically is the same idea if the symmetry is Unbroken I told you it must be massless when the symmetry is broken Then there's no symmetry that forbids it to have a mass so it acquires the mass Of course, there's a lot of Satellites in in this statement But that's the overall picture when I break the symmetry the thing that protect the vector boson to be massive Is not there anymore so the thing can be become massive and technically we see it from the kinetic term for the for the scalar The kinetic term involves the gauge bosons. Okay, this is the coupling and then when this phi Acquires the wave when this phi I replace phi by v plus h then the terned phi that become v This a square phi squared that come from here Then this phi squared I replace it by v. I have v squared a squared and That's that's a master because it's a number time A squared that's a master for the gauge bosons. So we see that the gauge boson acquire mass From spontane symmetry breaking Okay, so what we learn and that's the important part of spontane symmetry breaking is the following when I spontaneous symmetry breaking Massless fermion could become massive and masters gauge bosons Become massive because of the breaking. Okay And they because of the breaking I should expect to have some relation between the parameters four minutes. Okay Let me go on to this last part and we do the model building in the afternoon. So We almost there and We are finally have all the tools Maybe not we don't fully cover them, but we have all the tools to start doing model buildings. Okay So now we actually can go and start building. Okay, and how do we do model building? And I already mentioned a few times throughout those lectures We put some input and we get an output. So what are the input the input is the generalized coordinate? I tell you what are the fields that are the input I tell you what the symmetry are and then I write the more general a grandeur and I truncate it And then how can actually start writing models? Okay And let me say the following whenever we write such a model and we truncate it a dimension four Always the Lagrangian can be written in the following form. There's always four terms in the Lagrangian The kinetic term that involves all the kinetic term for the fermions the scalars and the gauge bosons And remember there's so much a lot of things in the kinetic term give me interactions So it's not only kinetic thing. It's also interaction. Then I have something I call L psi The psi is only thing that involve only fermions. So how many fermions it could involve either zero or two Because I cannot have four fermions. So this is only mass term for the fermions Then I have something that I call L yukawa And I don't know. Do you know maybe you know why it's called yukawa? It's the same yukawa with the pion story and it looks totally irrelevant But maybe Andrea, you know and you can tell me in the break But it's called the yukawa interaction and that's a term that involves both scalars and fermions So it's two fermions and one scalar So this one is two fermions. This one is two fermions and one scalar and then we'll l phi that involve only scalars Okay, so what we are going to do in the afternoon is to actually start doing this and I'm going to Give you input and then we start writing a Lagrangian and we're going to see what's come out of this Lagrangian Okay, so we're going to do qed. We're going to do qcd. We're going to do standard model We're going to do more standard model and we're going to do more standard model. Okay, so and that will be end for the morning Thank you