 interrupt and they will raise hands and then it's dead. Yeah, that's great. Okay, we're ready. Okay, yeah, maybe just a second. Let me put in all the slides just to make sure that I can share them because that would be nice. Just, okay, I think now you can see my slides. Okay, great, then everything's ready. Sorry for the delay. Would you like to go in play mode, in presentation mode? Sure, I can, here it is. Thank you. All right, then we are ready to start. It's a pleasure for me to introduce the last lecturer of the day, which is going to start with the first lectures on the Standard Model, which is Stefania Gorey from University of California, Santa Cruz. She has been working in several topics of energy physics. She's an expert of Higgs physics, flavor physics, beyond the Standard Model, and she will talk, give an introduction and go more in details about topics of Standard Model, flavor, and effect. Please. Okay, thanks a lot for the introduction. Thanks for inviting me to this nice summer school. So here I have the, so first welcome everybody and here I have the first slide where I introduce a little bit to myself. Well, that's my name and this is the University where I am, University of California, Santa Cruz. And yeah, I placed it here a couple of pictures of our beautiful campus in the middle of the Redwood Forest. So a little bit of advertisement. And then here you can find my contact. So the email address and everyone is invited to send me an email with questions or if you want to have some more references, just welcome to contact me. So as Giovanni was mentioning, this course is going to be about the Standard Model of particle physics. And here in this first reference, what I've done is to collect a couple of references. That could be useful for our study of the Standard Model. As we see the Standard Model is obviously very much based on quantum theory and here record three beautiful books about quantum theory. And by Schwarz, Peskin, Schröder and Mangiore. Some books about the Standard Model, as you can see here. And then also some, some radio paper about the Standard Model. So these are general references and then throughout the course, I will show you more specific references on specific topics, okay? But this is, so these references are here for completeness and I would invite you to take a look. But obviously, you know, they are, most of them are books. So they contain a lot of information, much, much more, of course, than what we will discuss during this course, okay? So this is just to start. Now let me switch to my iPad. So I will stop sharing the slides. And here is my iPad, okay. So let's start with the goals, goals of these lectures. So what goals? Also please interrupt me if you don't read in my handwriting because there is something not clear. That's great to know. So what are the goals? So we want to learn these three different things. So the first thing we would like to learn is how to build the Standard Model at Grantsson. So how to build the Standard Model. I will always have a separate abbreviate with the SM, so the Standard Model at Grantsson. So this is a, as we'll see, we have some what are called rules. So we have to follow the rules of a quantum field theory. We have to apply gauge invariance. We have symmetries to use. And then we have a set of ingredients that are our particles, the particles that we have discovered. So this is obviously quite theoretical and this, as I said, is basically learning QFT for the Standard Model. But then also we would like to understand how to test the Standard Model at Grantsson. So we will link what we have learned about the Standard Model at Grantsson to experiments. And we see indeed that things work, right? That we have quite a lot of tests, quite a lot of measurements that are confirming this framework of the Standard Model that we studied this week. But then last topic that we leave quite a bit of room for going beyond the Standard Model and being creative beyond the Grantsson of the Standard Model. We'll discuss if we want to understand what are the anomalies in data. So we see that there are some measurements that don't fully agree with the Standard Model prediction. And then we want to understand a little bit what can be wrong and especially if these anomalies are staying. And the focus here will be on something that is pretty recent, that is a new measurement of g minus toward the neon that is probably quite a few of you have heard about it and this is quite an exciting new measurement. So we would like to understand it a little bit. And then I will introduce why we want to go or we would like to go beyond the Standard Model. So what are the questions that we would like to address and why Particle Physicists study also beyond the Standard Model Physics. Okay, so these are the goals and this is what we would like to learn this week. So I've seen maybe there is a question in the chat. Let's see. No, don't worry, I will answer. Okay, great. Thank you. And so this is for the goals and then for the topics that we will discuss. So for today, what we'll do is mainly to introduce tools. So we'll talk a little bit about Lawrence's invariance. I will recall some notions about the QFT, spoiler about the natural units and a little bit about symmetries. This topic about symmetries, we'll see how much we do today or we'll finish them tomorrow. And these are the tools that then we'll use to start building the several pieces of the Standard Model Lagrangian. So for tomorrow, for Tuesday, our focus will be on quantum electrodynamics and the weak interactions. Then Wednesday, we'll talk about electric week, symmetry breaking and the Higgs. So we'll take what we have learned on Tuesday and we see how to break the symmetries that are associated to the weak interactions. On Thursday, our class on Thursday will be dedicated to clever physics. So the clever sector of the Standard Model. And then finally on Friday, we'll do a little bit of introduction to neutrino physics and we'll link neutrino physics to what we have learned about clever physics the day before. We'll speak about, as I mentioned already, experimental anomalies and why beyond the Standard Model physics. So that's the plan for these lectures. And yeah, so if there are no questions about the overall organization and the goals, we are ready to go into the details. And then please just ask questions and interact me as much as you can. That's more fun. Okay, so let's start with our first chapter. So this will be some overview plus tools. So this is the chapter of today. So the first thing I want to discuss with you are the fundamental principles of the Standard Model. Okay, so first of all, let's start very, let's go back in time. Let's think a little bit again about Newtonian physics. So in, oops, brown color. In Newtonian physics, say that we want to describe the system of many particles. And what we can do is to introduce the set of generalized coordinates. So introduce a set of generalize coordinates. Let me call them XI and the time derivative of XI. So this will be the derivative with respect to time. And then if you want to describe the system, what we would like to understand is what is the time evolution of these coordinates. And to do that, what we do is to minimize the action S that is defined as the integral of the Lagrangian of the system that does depend on these generalized coordinates with respect to time. So if you want to study the time evolution between T1 and T2, you need to solve this integral here. And once you minimize, you will find some paths. And so basically what you will need is to solve the Euler Lagrangian equation of motion. So solve the Euler Lagrangian equation of motion. They tell us that there is a relation between the derivative of the Lagrangian with respect to X dot and the derivative of the Lagrangian with respect to X. Namely, what we know is that if you take the time derivative of this object, this is equal to this. So what we see from here is that once we know the Lagrangian, L, we can find the path of the several particles X of T. And this, of course, is up to initial conditions. So in the case of above, what we'll need to define are the initial conditions at T equal to T1. Now this is what we know from Newtonian physics. And then we can generalize this to 50 or is. So what happened to a field theories? So in field theories, the message is the same, namely that we need to know the Lagrangian in such a way to understand the evolution of the several fields. And in field theories, what we do is to define the action again as an integral of the Lagrangian that now does depends on the fields. So we call this field phi and the derivative of all the fields. And this time we do the integral in four dimensions, so D4X. And then the Lagrangian equation of motion can be written as, will be written as in this way, again relating the derivative of the Lagrangian with respect to the derivative of the field. And this will be equal to the derivative of the Lagrangian with respect to the field. Notice that I'm using here the relativistic notation that we'll introduce briefly. But here, before I enter into those type of details, what I want to really to highlight here is that the conclusion, so the conclusion is that we need to know the Lagrangian. And then hopefully we can solve this differential equation in such a way to find the evolution of the several fields. Okay, I say hopefully because obviously depending on the nature of the Lagrangian, this differential equation here can be quite complicated. And actually comment for the expertise that we might want to use perturbation theory for this differential equation is too complicated to solve otherwise. Okay, but then following this message, this is our motivation to try to learn how to write down the standard model Lagrangian, okay? And then we can understand the phenomenology of the several particles of the standard model. So I see there is a question. Yes, there is one question by Sonali Pappanik. Please. Hello, hi, Stefania. Hello. Is it audible? Yes. Yeah, here I want to ask that what is the motivation of introducing action, like integral of Lagrangian? What is the physical interpretation? So in Newtonian physics, we know that the, so what the action is, is the, so we introduce the action as, in quantum mechanics as the quantum of, of, so let me back up a little bit. So if you think about the Planck constant, so the Planck constant is the quantum of action. And then basically you can define all, or your physics using this quantity that is the action. That is basically, I mean, you see from these relations here that you can build, you can study your physics using the action or the Lagrangian or Hamiltonian in a similar manner. Just that the way that usually we start is in a axiomatic way saying that you build this action here and what your physics does is to follow path that are minimizing this action. So this is the axiomatic way of defining your physics that describe whatever system that you like basically. So the physics starts with the action? Yeah, that's the way I would say it. In this, as I said, this is mainly an axiom, right? An axiom. Sorry, not an axiom. Namely that your physics will, so your system will minimize the action. And then from there, you learn all your dynamics. Okay, okay. And again, by minimizing what do you mean? So you have this quantity that is defined here, you know, for Newtonian physics, but then we said that this is the same for field theory. And then you can, so this quantity s will be a function of x and x dot and then you minimize your function, right? You find the minimum of this function as a function of x and x dot. And if you do so, you can demonstrate that you end up obtaining this Euler Lagrangian Equation of Motion that is relating x and x dot. Okay. Okay. Thank you. Yeah, sure. Any other questions? Yeah, thanks for the question. Okay. So let's introduce a little bit the fundamental principle, so the standard model Lagrangian. So what are the rules that we would need to follow in such a way to be able to write down the Lagrangian? So fundamentally, we would need to be able to write down the Lagrangian so fundamental principles of the standard model. And so as we discussed quite a bit in details, but this is, you know, for now just an overview. So the Lagrangian, actually, you know, not only of the standard model, but this is the focus of these lectures. So the Lagrangian is based on the following principles. So we'll have a set of symmetries that we'll discuss a bit today and the rest throughout the course. So we'll have symmetries. Then what we do is to apply the principle, to apply unitarity. So we need to, we need to write down a Hamiltonian that is a mission. Then what we also impose is renormalizability. So the reason is that we want to have finite predictions for the physical observables that we want to measure. And therefore we need to have a Lagrangian that is renormalizable. So we need to have only a finite number of amplitudes that are diverging in the ultraviolet. And then finally, typically what we say is that the Lagrangian of the standard model is also based on minimality, namely that we use only a minimal set of objects to write down our Lagrangian. This if you want, so as you see I put this word in quotation marks. And the reason being that the particle content of the standard model as we'll see is highly non-minimal. So the particle content of the standard model is non-minimal. Since as we'll discuss quite a bit we have a set of gauge bosons. We have a lot of fermions and one hex. But then you might wonder what we mean with minimality. We mean that we don't have particles that we have not at least observed. But still there is in mind this idea of having a Lagrangian that describes a system that is as minimal as possible. And in fact in this context I can show you maybe his slide so I will switch for a few minutes to my slides again. So let me stop this sharing here and we'll go back to my slides. I guess you see my slides, right? So this is the particle content of the standard model and we'll study this quite in detail this week. We have the gauge bosons that I was mentioning. We have the glue on the photon and the weak gauge bosons. The W and the Z, one hex and then all these fermions. We have quarks and leptons with different electric charge. And what is quite cute and that's why I like to present this slide is that the next slide it took us quite a bit of time to discover all these particles of this cartoon that actually I took from the economist that shows when we discover all these particles if you're curious. So this is a nice summary. Starting from the electron we have here the experimental discovery by Thompson and then we have here in 2012 the discovery of the hex boson at CERN. And also what is interesting of this cartoon is that each particle has two dates. So there is the first date you see here and here and here they correspond to the time that the particle was first theorized by someone and then afterwards you see the time of the experimental discovery and sometimes you see that it takes quite a bit from the theoretical idea to the experimental discovery especially as we see here for the hex boson that took us as we'll discuss quite a bit in detail took us quite a bit of time and all the opposite there are also particles that are discovered by chance as the muon in the sense that we didn't expect them theoretically and that's why there is the famous sentence by Ravi we didn't expect the muon discovery but I see that there is another question I have a question related to the unitarity of Lagrangian is it only related to the hermiticity of that Hamiltonian or is there something else which is also related to so you can think about unitarity in many different ways so what I mean with unitarity is simply that the Hamiltonian is a mission so that you have your time evolution operator that is unitary then you can depending on the research that you do you can think about perturbative unitarity and other type of unitarity issues so this is simply that the Hamiltonian is a mission so here we are back to my iPad so this is what I wanted to say about minimality and particle content of the standard model and then for symmetries so will I see another question maybe before symmetries go ahead so when it comes to question of minimality does it also tie into the fact as to why we don't use higher order Lagrangian in our theories no so what I mean here with minimality is that we use just the minimal set of objects of particles or fields that we need to describe nature higher order I mean this is for perturbation theory and this is a complete different thing I mean as we'll see a little bit about perturbation theory this is a sort of a tool that you need in such a way to make predictions and to do calculation using your standard model Lagrangian but this is different from minimality the minimality that I mentioned here so maybe one more question and then go ahead I want to know while constructing the Lagrangian under certain symmetry why do we want the terms in the Lagrangian to be singlet under that particular symmetry why don't we discuss this when we discuss the gauge symmetries so this is absolutely so at the end of the day we'll introduce some symmetry or some symmetry and then especially you know when we discuss the gauge symmetries we see that several terms of the Lagrangian need to be singlet under this gauge transformation and yeah I would delay this question to when we discuss gauge symmetries I think it's much better but if it is still not clear I would invite you to ask again okay thank you okay so if there are no more questions so here let me do a couple of brief comments about symmetries and then this is something that we study quite a bit more later so for symmetries so let's set to symmetries that we'll discuss this week so we'll introduce a little bit the Lorentz symmetry and we see how to use it to build a standard model Lagrangian terms then as I just mentioned we'll discuss gauge symmetries and we see that the standard model has a quite large gauge symmetry so let's see SU3 times SU2 times U1 we will see and let me put this maybe in parentheses that we do not impose global symmetries but they are accidental meaning that for the standard model Lagrangian once that we follow the other set of rules that will give you during these classes and in particular we see that in terms of global symmetries at least approximated symmetries the standard model Lagrangian has quite a large set we see that we have as examples we have SU3 to the fifth power flavor symmetry we'll discuss this on Thursday when we'll discuss flavor physics then we see that we have a lepton and varion numbers that are again global symmetries of the standard model we'll discuss a tiny bit about those not too much and then we see that we have another symmetry that is what we call the custodial symmetry just to introduce a little bit to the names that is again an approximated global symmetry of the standard model and we see indeed that how these symmetries are broken even though they are indeed as I said already approximated symmetries of the standard model Lagrangian so this is just a few comments about the symmetries that we need to understand a little bit in order to be able to build the standard model Lagrangian for the last few minutes before the break what I wanted to do is a few reminders about Lorenz transformations and relativistic notation so let's introduce a little bit Lorenz connected to these different symmetries that I mentioned here so the first reminder that I wanted to mention in this context is so how do we relate two observers that are moving with respect to each other with the velocity that is not too far away from the speed of light so the classic example is that we have two observers O and O prime and the system of coordinates of these observers is the following so we have here the first one with some time T this is O the observer O and then we have the second one and then this second one is moving at a speed that I call B0 with respect to the first one and then we have a Lorenz transformation that is relating the coordinates of an object as seen by the first observer to the coordinates of the same object as seen by the second observer so the Lorenz transformation in this case that I am moving on the X axis is given by the following equation so instead of the speed of light times time X, Y, Z what we have is CT prime that is given by gamma CT minus beta X this is minus beta CT plus X and then this so as you see we don't have any transformation for Y and Z and here I define this beta as B0 over C so B0 again this was the speed that we see here and then the gamma factor is equal to 1 minus B0 squared over C squared this is our Lorenz transformation we can obviously generalize it if we have some motion in a generic direction that is not necessarily X and what is interesting here is that we can build invariance so if we write down this delta square that is defined as C squared T squared minus X, Y and Z so this quantity is invariant under Lorenz transformations so I can write down my delta prime as C squared, T prime squared and this is simply again equal to delta squared this is what we need with invariant no matter what type of Lorenz transformation I do and this delta has actually name so this is the proper space time distance and actually we can write it in a little bit more compact form as let me write it here so this delta square is equal to eta mu nu so this is a quantity that we use quite a bit that's why I introduce it here times X mu, X nu sorry so this delta mu nu is our metric that is given by 1, minus 1, minus 1, minus 1 and all the rest is 0 of diagonal and obviously this is not the only invariant that we can write down and we see that the way that we can write down Lagrangian terms is to impose that we have invariant objects another object that we use quite a bit that is also invariant is given by this combination here so if you take say a particle with energy E and momentum given by Px, P1, Pz in several directions then we can build the following invariant as E squared over C squared minus Px, Py and Pz and this is again an invariant and I'll ask you what invariant is this I don't know if someone can answer maybe if anybody knows one can say it so this is actually a physical quantity that is given by what is this someone so the mass squared the so this will be the mass squared of the particle and this doesn't depend on the Lorentz frame that we are considering so we have learned that indeed we have this invariant and this other invariant and then the last thing that I wanted to mention concerning Lorentz is just a little bit of notation that will be useful for later namely that we can relate the four vectors with the index up or down in the following manner using this eta mu nu namely that x mu with the index down is equal to eta mu nu x nu and this will be given by Ct minus x minus y minus c and then we can define the derivative with respect to this four vector namely so the derivative with an index mu that is given by derivative with respect to x mu and then in terms of Lorentz transformation we have the x mu that goes into x prime mu that is given by this matrix two indexes x nu so this was the matrix that we brought down explicitly just above in the case of a transformation on the x direction here you see what I'm putting this red arrow and a useful relation is that if you apply two of these Lorentz transforms then you get again your metric and you can check this relation for the Lorentz transform of above so you can check that so the Lorentz transform of above in a matrix form can be written in the following manner and then you can apply to the metric minus one and then you have the same object here all of these are zeroes and then you have a one here on the diagonal and then indeed you get the same metric this is just an example if you take Lorentz transform on the x axis this relation will be actually useful for later that's why we have to keep this in mind so these are the little reminders that I wanted to mention for concerning Lorentz and now we are ready to take a break but of course I'm happy to take questions in the meantime okay thank you so we'll have a break for five minutes so since there is one there is time will you teach leptogenesis in your course? sorry can you repeat I didn't will you teach leptogenesis in your course? no no yeah that's something that goes beyond what I'm planning to do okay fine I can give you references if you are interested but it is something that we want to discuss okay fine thank you there's another one hi so I want to ask again that what difference it would make if we introduce Hamiltonian and not Lagrangian initially yeah so you can work both with Hamiltonian formalism or Lagrangian formalism yeah that's so you can formulate your field theory both in terms of the Hamiltonian or Lagrangian typically people what they do is to use the Hamiltonian formalism and then you can write down your time evolution using Hamiltonian but you can also use simply Lagrangian formalism yeah okay so the dynamics would be same sorry I didn't hear what you said the dynamics would be same thank you okay I think we can start okay excellent so now let's see how to apply what we have learned on simple Lagrangians so let's let's build simple Lagrangians that will bring two examples so one is for a real scalar field phi and then the second one is for a fermion field so a Dirac fermion field sorry okay so let's start with one so what I can write down in my Lagrangian so since this is a scalar field let me try to write down these two terms and then we see if they work so the first term that I can write down is the term of this type so this yeah this is a term that we'll discuss a little bit and then let me put maybe a factor of two in front okay and then the other thing that I want to write down in my Lagrangian is a function of the field and this will be my potential and you see that you know I'm not obviously putting any Lorentz index here this is just a scalar function of this real scalar phi okay now this field phi is the field that will describe a spin zero particle or quantized and we see that actually the Lagrangian that we brought down here is very similar to the Lagrangian that we can write down for the Higgs particle okay but this yeah for later now if we write down a Lagrangian like this so first of all we can check if it is Lorentz invariant or not so the first question is is it Lorentz invariant now for the transformation of the field since this is a scalar field so this this field will depend on the coordinate x so here to be precise actually so this will be you know I have four components so I have time x minus z and actually since this is a scalar field what we know is that we have that the transformed field completed at x prime is equal to phi of x so using this transformation it's it's trivial to demonstrate that the potential v of phi is a trivially invariant under Lorentz transformations a little bit less trivial is the transformation of the first term that I wrote down here in the Lagrangian so let's see what happens what happens to it so what we have learned is that x mu transforms in the following way and therefore if we take a the derivative of the field phi this will transform as Lorentz transformation then we have the derivative prime and phi prime where with derivative prime what I mean is simply the derivative with respect to x prime as the Lorentz transformed of x and therefore the first term of the Lagrangian the mu phi will transform in the following way so I have a few indexes here that I have to take care of then I have my metric at mu nu and then I have the transformed derivatives or the transformed field prime and prime so what I've done is simply to have two transformations each corresponding to one of these derivatives and then you see that I have this combination here that we already saw above so let me highlight it so this object here is this thing here that we saw so these tell us that indeed this is given by eta mu prime nu prime derivative of phi prime and then this is ok is equal to two old steps to mu prime prime prime and this is indeed invariant in the sense that this is equal to the Lagrangian term that we were starting with so using these Lorentz transformations that we introduced 10 minutes ago we can show that indeed this would be a term of the Lagrangian that is allowed by Lorentz so we are happy that we can write it down ok so that is just a simple example of how we can see how to use Lorentz transformations to see what we are allowed to write down in the Lagrangian and what we are not ok I see some questions and some hands raised yes the first is by Enga Me hi I just wanted to ask has there even been an instance of the Lorentz invariance symmetry breaking has the Lorentz symmetry ever been broken basically so there are tests meaning experimental tests to see if the Lorentz symmetry is an exact symmetry and so there are bounds on the possibility of breaking Lorentz but so far there is no evidence really for experimental evidence for breaking Lorentz symmetry but yeah there are experimental searches for that measurements thank you ma'am I have a question you have said that for the Lagrangian in order to check the Lagrangian is Lorentz invariant the V of phi has to be really invariant but it is seen that usually it is seen that when we make a model we check for the minimization of the potential part under the symmetry that has been considered so why is it done and how is it different from this concept sorry can you repeat the question I am not sure I got this ma'am you have written that in order the Lagrangian to be Lorentz invariant the V of phi should be trivially invariant but it is usually seen that while constructing models under particular symmetry we check for the potential minimization part under the symmetries that has been considered so how is this different from this concept V of phi is trivially invariant go ahead sorry so when I said that it is trivially invariant simply this is a consequence of the way that scalar fields transform under Lorentz so I mentioned that a scalar field transform in this way so it doesn't transform under Lorentz and that is why if you write down a scalar potential of this type then also that potential doesn't transform so it is invariant but this is a if you want a consequence of the fact that I am taking a real scalar field with those property of transformations under Lorentz and then I see what happens to the potential this is not the you know this is a consequence of having a scalar field okay thank you okay so this is for scalar fields and then for fermion fields oh sure sorry yeah hi sorry I was just thinking about what you said about global and local symmetries and I was wondering did we consider Lorentz symmetry a global symmetry yeah so we don't yes so that's right that's right I didn't so I'm putting it on a different footing so the global symmetries let's go back so the global symmetries I was mentioning here global symmetries that are accidental so they come out once that I impose you know my Lorentz symmetry my gauge symmetries and so on so I was putting them in a different footing but definitely it's so that the Lorentz symmetry that we are considering is not is not gauged so this is a also a symmetry of that type even though I was putting that on a different footing but would that change if the metric well if you're not talking about the Minkowski metric and change the metric yeah so it's a bit different than so when I speak about gauge symmetries the idea is that or local symmetries so the gauge symmetries I was putting here for the standard model these are locally in the sense that the transformation will depend on the specific point in space time and yeah so this is sort of different than what you are thinking about having a non-Minkowski metric sorry I just asked that because I usually deal with I guess like cosmology and particle physics a lot and I was just wondering like in the case of having a non-Minkowski metric then I guess like the Lorentz symmetry in that case is not really like a global symmetry it will depend on space time coordinates that's right but it's a bit different because you don't I mean you don't have the same way of treating it right as you know introducing as we see gauge bosons that are connected to these local gauge symmetries and so on but yeah I agree that here what you can do is instead of using this Minkowski metric you can use you can take a metric that does depend on your space time on your x thank you okay very good but then of course I mean just to connect what we have seen at the very beginning of this class you can use this Lagrangian to write down your equation of motion and the equation of motion that you get is your Klein Gordon equation so what you get is that the box operator applied to the file field is equal to minus the derivative of your potential just to connect to what we have seen at the very beginning of this class so this is all I wanted to say about scalar fields and then the second example that I want to mention is the Lagrangian for a fermion field so what I take here is a psi field that is a in this case a four component Dirac spinor and so this field will describe the spin one particle when it is quantized and then again the Lagrangian that I might try to use is a Lagrangian of this type so here let me write it down and then I will discuss minus m psi where this psi bar is defined as psi dagger gamma zero so these are the gamma matrixes that do have these anti-communication properties namely plus gamma mu is equal to two times the metric so this is our clipboard algebra and what one can do and I want to discuss this because it will take me quite a bit of time but what one can check is that indeed this Lagrangian that I brought above this LF is again Lorentz invariant so I find maggiore a nice book explaining and showing details of this aspect so for the people interested I would suggest you to take a look at the book of QFT by maggiore so this is one of the references that I showed you at the very beginning and then again so this is the first thing that we can check and then the second thing is that using the equational motion and using this clipboard algebra telling us how the gamma matrix is anti-communic we can show that we obtain the Dirac equation so namely that i, gamma mu, the derivative minus m applied to psi is equal to zero and we see that indeed this type of Lagrangian is the Lagrangian that is at least part of the Lagrangian that is describing the quarks and leptons of the standard model now since you are speaking about fermions let me take this as an opportunity to define two quantities that we'll use quite a bit tomorrow when we'll discuss the weak interactions we can define the the helicity of a particle so let's maybe open a small parenthesis so what is the helicity of a particle so this is defined as the product between the spin and the momentum so S is the spin and P is the momentum so these are three-dimensional vectors and then we are used to normalize this with the absolute value of the momentum it's easy to demonstrate that this quantity is actually not Lorentz invariant so if you choose a different Lorentz frame you will get a different answer and we can use the helicity to define right-handed and left-handed particles so we'll have a right-handed particle that is a particle that has a spin that is in the same direction as the momentum and then we'll have left-handed particles where this spin is instead in the opposite direction so we can use this property of particles to do actually our calculations but there is another property of particles that is connected to a helicity that is Lorentz invariant that is what we call the carality of a particle so what is the carality so the carality for a four-dimensional spinor is defined by the gamma-5 operator so what we have is that if we take a massless spinor what we can show is that actually the carality and the helicity are describing the same property so carality is the same thing as helicity again we should not forget that this is only in the limit in the sense that we have the energy of the particle that is much bigger than the mass and so in this limit what we can demonstrate is that if we take the gamma-5 matrix and we apply it to the right-handed particle then we get the same sorry if we apply it to the left-handed particle we get a minus sign okay so this is a spinor for a particle so a four-dimensional spinor and then for completeness we can also ask what happens if I apply this operator to an antiparticle and then we have sort of the opposite behaviour in the sense that if we take the spinor V for an antiparticle we have that this is equal to minus V and then similarly we have this with a palace okay I see a question in there hello Stephanie I have a couple of questions from Helicity in the context of Fermion to distinguish between left-handed and right-handed we needed this Helicity quantity but my question is from Photon that in Photon we have also defined polarisation and because it is massless we already know that there are two polarisation states we called left-handed and right-handed polarisation states of Photon so how it is different from polarisation and if it is not different I mean Helicity and polarisation why we called it in a different name Helicity yeah I mean the physics that is going to describe this similar if you want typically people use this notion I mean the word of Helicity to describe fermions but if you want I mean this is a just to understand what is the direction of the spin and with respect to momentum and what is but I mean in terms of physics there is not much more to it yeah so because we are talking about fermions we are using new kind of what instead of polarisation that is Helicity sorry say it again yeah in case of Photon we are talking about polarisation there also Photon can move and it may have a component along spin or to the opposite of spin but here it is Helicity and you are talking about because we are talking about fermions so we are using this new word Helicity yeah but I mean you can obviously you can apply this the same definition also to a Photon or to anything that has spin right and then you can compute your Helicity also for Photons is just a matter of what is the you know the observable that is most convenient to describe the specific system but no one is telling you that you cannot compute the you know this quantity here also for particles that are not fermions yeah this is what I want you to say okay and another question I am not too much aware about the chirality means other than these matrices and other things so why we need this chirality or chiral quantum number other than Helicity how is Helicity and chirality different yeah so the way I presented this here I mentioned that in the massless limit they are the same thing but this is only in the massless limit so if you take a particle that has an energy that is comparable to the mass this is not anymore the case so this is the first thing to keep in mind the second thing that is very important to to have in mind and this relates to the lecture of tomorrow is that the chirality is a property that is very important when we study the weak interactions namely that we see that only so let me write here for tomorrow so what we will discuss is that only left-handed fermion fields couple to the W bosons the weak interactions so when we study weak interactions we'll see that only a specific class of fermions do couple to the weak interactions so in this sense it's very important to use this idea of chirality to describe and to identify different type of fermions okay and this will develop this much more tomorrow but that's why I'm introducing that here because this is a property that is differentiating particles those communicating to the W bosons and those that are not other questions one question by Gengameh I'm not sure if this scope allows us to talk about this stuff but I guess I was wondering when we write down representations of the Lorenz or the Poincare group I thought that the left handedness or right handedness corresponds to whether the representation of the fermion is 1-0 or 0-1 I guess I always get I guess if you want to talk about chirality or the handedness of a particle in terms of the representations of the Poincare group how would that correspond yeah so basically this is obviously a long discussion but this is maybe let me think if I can say I'm also happy to wait and we can discuss this later yeah I would be happy to discuss it later but maybe a couple of words so when I mentioned here that we have this fermion Lagrangian and I point you to the major this is exactly this type of discussion so basically the idea is that you split your Dirac in two parts you have your left and right part so these are while spinors that do transform in a different manner as you say under the Lorenz transformations and using this basically you can demonstrate that this is the Lagrangian in its invariant and this is connected precisely to what you are asking but yeah it's quite a bit of a long discussion that I was not planning to but yeah thanks hi my question was just as we said that we can define chirality for all the other vector fields or any other particle just here we have the gamma-5 how would just an idea of definition of chirality for I don't know for W ? ? ? ?