 So today is a big pleasure for me to introduce the first lecture by Neymar Kanyamhev from Princeton. I'm sure he doesn't need any introduction, you all know him, he's been the driving force for particle physics and theoretical energy physics in the last, I don't know, 20 years or so. And he's basically an expert of everything, so it's your occasion to get all doubts. I'm done. And with no much more to do, I'll leave the floor to Neymar, please. All right, well, I'm really super delighted to be able to join you guys. I was talking to you earlier, I've lectured at the ACTPP many, many times. It's one of my very favorite places to visit and one of my favorite places to a lecture. And of course, it's too bad that we can't see each other in person, hopefully that will change by next year already. But in any case, I'm really delighted to be able to join virtually. The organizers gave me this topic to talk about that I'm very happy to talk about, of course, the physics of future colliders. But you guys are young people at a summer school, probably mostly in your 20s. I don't know if there are any teenagers out there, but anyway, mostly in your 20s. And very naturally, when you're a young person starting off in theoretical physics, you need to have a sense of urgency and a sense of what's happening sort of now and what you can do right now. And that's of course, in some tension with the topic of this talk, where we're talking about, first of all, future colliders that there's no guarantee whatsoever that they'll actually happen. As you all know, very likely, people around the world are talking about E plus E minus colliders like the linear collider in Japan. E plus E minus colliders as Higgs factories, either at CERN or in China. And more recently, of course, also the possibility of even higher energy, 100 TV proton colliders also being discussed at CERN and in China. And a little more recently, people are starting to, once again, talk about the exciting possibility of muon colliders. These are all things that might happen. There are people are seriously talking about them happening, but whether they happen or not, even if they happen, we're talking about physics on the timescale of at least 10, 20, 30 years from now. And so if you're a kid, as most of you are getting into the subject, you can justifiably wonder why you should care about this, at least right now. Well, there are many answers to this question, and that's really going to be the substance of my lectures to try to get you to think about this very important physics and also to care about it in a deep way. But just to set the context again, and you know, if we're going to talk about future colliders or even the sort of future of particle physics in the sort of traditional sense of exploring physics at higher and higher energies, the very first question to ask is, is there a future of particle physics? Is there some point to continue to go to higher and higher energies? It's more and more difficult to build these machines that take longer and longer. They're very expensive. And of course, famously, so far at the LHC, we've seen the Higgs particle, of course, that was a prime, but we haven't seen any other new particles. And there's unlike the situation with the LHC itself, where beforehand we knew from the physics of electric symmetry breaking that there had to be something new beneath the sort of one TEV scale. Otherwise, we would not have run with physical inconsistencies with WW scattering, for example. So we knew there had to be something at the energy scales that were being probed at the LHC. The only arguments we had before the LHC, why there had to be a lot more beyond the standard model, revolved around mysteries about the Higgs, these famous notions that many of you probably still hear about, but in my experience talking to people of your generation, fewer and fewer people actually understand technically the issues of naturalness and the hierarchy problem. So that's something that I will be talking about in these lectures as well. But anyway, those were the arguments why we should see something beyond just the Higgs at the LHC, and we haven't seen any of that. It's possible those arguments were just slightly wrong, it's possible they're totally wrong, we don't actually, we don't know. But in any case, at the moment, we don't have a concrete argument why we must see new particles beyond the standard model at an energy scale that's anywhere near what we're going to probe with the accelerant. So that's the sort of zero-thorny question, why given that situation, what's the rationale, what's the motivation, why should we sort of keep going and push to higher energies? So that's why I asked the question provocatively, is there a future for particle physics? Never mind. Is there a reason to imagine going for future colliders? And of course, I wouldn't be giving you these lectures if I thought the answer to this question was no. But I think it's really important to pose the question totally openly and honestly, because this is the zero-thorner issue about what is going on today. And to me, there's sort of an irony, because I personally, I mean, this is really speaking for myself, and I think for some fraction of people in the field as well, but certainly not everyone, I view the period we're in right now as intellectually the most exciting phase that sort of fundamental physics has seen in decades and decades, probably going back to the development of relativity and quantum mechanics by the 1930s. I think the issues at stake aren't sort of detailed ones about this or that particle or this or that extension to the standard model. The issues are really profound structural ones about what laws of nature are really about, what they look like, what the deep principles are that are behind them. And so the kind of magnitude and stakes of the questions involved in our field have gone up by an order of magnitude compared to what they were even in the 1960s. A lot of people think of the sort of 1960s as the gold standard, the sort of golden period of what fundamental physics was about with like one particle after another being discovered at colliders and so on. And of course, there was an incredible period, but the real revolutions of the 20th century were those of relativity and quantum mechanics that took place before that. And there, certainly the experimental aspects of some of those discoveries took much, much longer. They didn't have this character of like this rapid fire back and forth, constant discovery, interpretation, constant discovery, interpretation. Part of the reason why the progress was so quick in the 1960s was exactly that the basic framework was sitting there waiting to be filled in as we learned more about the way the world, about the sort of details of the experimental way the world actually worked. But we are basically filling in this paradigm that was handed down to us from earlier in the, from earlier in the century, these big revolutions of relativity quantum mechanics. And where we are today with the fundamental physics is, I believe many other people believe as well, is another one of these sort of bifurcatory periods where the questions at stake are again, as I mentioned before, there are structural ones that go to the heart and sort of challenge some of the foundations of space time and quantum mechanics, these big revolutions that were handed down to us from the early part of the 20th century. We have reasons to expect that we have to go beyond them. Some of those reasons have to do with the mysteries of quantum mechanics and gravity that's just that the notion of space time itself has got to be replaced or extended with something else. Those issues might naively only be relevant at much, much higher energy scales than we'll probe anytime soon, you know, all the way up at the point scale. But also the surprise from the LHC that we've seen the Higgs and nothing else that very much resonates with the surprise of the measurement of a non-zero, well, very possibly the measurement of a non-zero cosmological concept associated with the accelerating universe. These are two major experimental surprises that are perhaps a hint that some of these, some of the, some of the principles that we've taken for granted encoded in the picture of effective field theory, the sort of the Wilsonian view of organizing the way we think about physics, that there's a challenge to these ideas, perhaps not just at the point scale, but at accessible energy scales that we're sort of probing at accelerators today closer to the week scale. And in many ways, the most important experimental character in this drama is the Higgs particle, and that's one of the zero-thorter things that I want to get across in these lectures. People keep whining about there's no new physics, no new physics, no new physics, we don't know what we'll see new physics, well, let's think about other things. And yet, from my point of view, totally backwards because the Higgs is profoundly new physics in a way that's vastly more interesting than had we seen, you know, lots of strong interactions and crazy stuff at the associated electric symmetry at the LAC. That would have been sort of fun for experimentalists and maybe even theorists to try to understand, peel back one more layer of the onion, it would be sort of repeating QCD all over again, but I don't think we would have learned anything profoundly new about the way nature works. Seeing the Higgs and nothing else is something profoundly new because we've never seen anything like the Higgs particle before. We've never seen an elementary spin-zero particle and the very good reasons we haven't seen it anywhere else in nature. And those reasons again go to the heart of some of these fundamental questions about the nature of physical wow, well, Sony defective field theory, the hierarchy problem, the cosmological concept problem, and I want to explain those things. So the Higgs is really special. We've never seen anything like it. It's not just hype. It isn't that it's the latest particle that we discovered. So we hype it up to the media and we tell people how cool it is. No, I mean, people do do that. So that's also true, but the hype is justified in this case because it's really a new thing. We've never seen anything like it before. And so that alone, the fact that we've never seen anything like it before and that its existence is somehow in tension with other principles that we hold dear makes theorists confused. But what you should do experimentally is totally obvious. You take this new thing, you've never seen anything like it before, and you put it on the most powerful microscope you can. So that's the like zero-th order reason for building future colliders, the zero-th order sort of physics motivation. Of course, from the most important motivation of all, which is pure curiosity to see what's going on at the frontier about the direct physics justification, the physics program, and the set of measurements that you're guaranteed to make and guaranteed to learn something important from all revolve around the Higgs particle sort of full stop. So no statements that we'll see is super partners or extra dimensions or compositeness whatever that those arguments were completely reasonable arguments for the LHC. Since we haven't seen any of those things at the LHC, we do not have those solid arguments anymore for what we might see at a collider beyond the LHC. But we have something more precious. We got a big shock to the theoretical system by just seeing the Higgs and nothing else. And so what we should do is just study the Higgs to death. And so that's the zero-th order motivation for future colliders is studying the Higgs to death. The E plus E minus colliders will measure the couplings. So the most dramatic issue about the Higgs is whether it looks point like or not. We'll talk about that in more detail. So from the LHC, we're only going to get a relatively fuzzy picture of what the Higgs looks like. The Higgs could be about as composite as the pion was. And we know there's no huge drama surrounding the theoretical understanding of the pion. In the end, it's a relatively, it's a somewhat point like particle, but it's, but it is composite in the end. And what we'll know about the picture we'll have of the Higgs from the LHC, even if we're done learning everything from the LHC, would be perfectly compatible with it being a composite particle about as composite as the pion. But measurements at a Higgs factory will essentially improve that resolution on the Higgs by an order of magnitude relative to the, to how point like the a pion. So after, and we'll do that by looking at the interaction of the Higgs with other particles with the Higgs to ZZ coupling, for example, or the Higgs to Gamma Gamma and other couplings, the coupling to a fermion, we'll get a picture of how point like the Higgs is. It'll either reveal some substructure as well as the case with the pion or we'll show that it continues to look point like to much shorter distances, in which case we'll have gone one solid order of magnitude beyond anything we've seen before experimentally, as far as seeing a sort of point like elementary scalar. And again, as I'll emphasize it's a sort of point like character of the Higgs, which is where all the theoretical drama revolves around. And ultimately, what you want to see is not only whether the Higgs looks point like the external probes, like in how a couple of the Z particles or the photons or the fermions, but whether it looks point like to itself. And here one of the fascinating facts about the Higgs, which you will also understand a little better after these lectures, is that the very simplest interaction elementary particles can have in nature is when three of them meet at a point. Not two of them, because that two of them would just be considered three propagation. So the first interaction that you can have is when three particles meet with three particles meet at a point, or that's the three particle amplitude. And it's a fascinating fact that we have never seen so far a single elementary particle interact with itself. You might say, well, but we know that in non to be on the engaged theories, the gluons can have a self interaction. Or in general activities, the gravitons can have self interaction. But in none of those cases, there are really a single state that interacts with itself. There's always some quantum number that changes. And it could be a color quantum number or a spin quantum number, but there's always something to change. We'll understand a little why a little better, maybe even by the end of this lecture today, if not by tomorrow. But the Higgs is unique in being a single elementary particle that can interact with itself. We've never seen that interaction before experimentally. And therefore, probing the self interaction of the Higgs tells us both gives us a probe of whether it looks the Higgs looks point like to itself, as well as the sort of first experimental observation of this very simplest possible interaction in nature. And in order to be able to from from the LHC, we'll sort of barely know whether the Higgs interacts with itself at all and get some indication, maybe at the 50% level of whether the strength of this self interaction is compatible with the standard model. But a 100 TV proton-proton collider would produce billions of Higgs particles. And those billions is a large enough number that you'd actually be able to sort of see this triple Higgs interaction and measure it to the sort of percent level, few percent level actually. So that in a snapshot is the sort of zero-author reason why the zero-author sort of physics justification for building the next generation of colliders just given what we know. What we know is we've seen the Higgs, the Higgs is like no other particle that we've seen before. It's special. It's theoretically confusing. Experimentally you need to put it under a microscope and study it to death. And E plus and minus colliders will do that by probing at the sort of percent level, a better than percent level accuracy in many cases, whether the Higgs looks point-like in its couplings to external probes and 100 TV collider. In addition, of course, to going in order of magnitude further into the energy front here, directly for this question of probing the Higgs will allow us to see whether the Higgs interacts with itself, whether the Higgs looks point-like to itself. And in both of these investigations, we'll either see something that deviates from the standard model or we won't. And either one of them, again, it's a sort of moment of bifurcation. If we see some substructure of the Higgs, that's great. And then we'll go, I mean, that's more like the picture of the people of the world people imagine decades ago. Maybe it was just a little bit delayed. There's a reason we didn't see the associated physics at the LAC, people would go off in one direction and try to explore that for another 20 or 30 years. Or we'll see that it doesn't show any signs of sub-substructure. The Higgs continues to look point-like. And that's a much bigger theoretical shock to the system. But right now, theorists are contemplating that that might be the shock, contemplating that that's what's going on. But there's always in the back of your mind the possibility that, well, maybe it's not so dramatic. Maybe we just barely missed something. Such a force is such a huge paradigm shift. You really have to believe that unless we're completely sure experimentally, we have to believe that. We can't know to sort of go marching off in this 90 degree turn compared to the usual trajectory of physics from the last fourth century, unless we get some really big kick from experiment to do so. And so this would give us a really big kick from experiment to try to understand how the heck it came to be that we ended up with this point-like elementary escape. So that's the first set of things that I want to do. So I'm just starting with a very high-level overview of the things that I'm going to talk about. But that's the first set of things that I want to explain in these lectures. Why is the Higgs special? And this has to do with why is the Higgs special? Why is it so surprising that we saw something like an elementary in zero particle? And this has to do with the following tension. And I'd like to explain this tension as simply, but as deeply as I can. So we have the following basic tension that on the one hand, the fundamental laws. Sorry about that. I could be hearing my screen. That's why you're not seeing it. Can you see now? Yes. All right. So our tension is that on the one hand, the fundamental laws of nature appear inevitable. The structure of the fundamental laws appear inevitable. And from here, what I'm going to explain is why if we take quantum mechanics and relativity, so quantum mechanics and spacetime, and so if you imagine that you just handed sufficiently competent theoretical physicists the rules of quantum mechanics, the rules of spacetime, and you lock them in a room, you refuse to let them look at the world outside. Of course, it doesn't really make sense because the room is in the world and is made out of the stuff of the world. But anyway, never mind. Imagine you lock them in a room, refuse to let them see what the world looks like, but you gave them a large supply of food and graduate students and so on. Then it's just an amazing fact that just taking quantum mechanics and spacetime and asking how can you build theories that are compatible with both of these principles of relativity and quantum mechanics. The theorists would at first think that it's actually impossible, but when they worked harder and harder, they would realize it's just barely possible. But it's barely possible in a very restrictive sense. It's a very tight straitjacket on our understanding of how this works. Now in traditional courses, the theoretical framework that unifies quantum mechanics and relativity is quantum field theory and that's what many of you have taken in graduate courses and it's the fundamental language that we all learn in high energy physics. But probably in the way you've learned things, there might be some questions in your mind like, did it have to be this way? Why are we taking these fields and quantizing them, et cetera? At least in some of the ways of discussing quantum field theory, you might be left with some sort of impression that you're making some choices. Maybe the world could have been another way and it happens to be this way and so on. What I want to stress in the first part of these lectures is that there is no choices. I want to describe this physics in a way that's as invariant as possible, that directly connects to principles of quantum mechanics and spacetime to the particles that we see. So we won't be talking about quantum fields and virtual particles and all of that stuff. I want to give you the most direct connection possible with no veil between the fundamental principles of quantum mechanics and spacetime and the properties of elementary particles and what we know about their existence and their interactions. But the upshot is that starting from the principles of quantum mechanics and spacetime and assuming that we are working at high energies, working at high energies, which means that we work in an approximation where the particles are massless. So this is a big assumption, an important assumption. Then you can show that the only possible theories that you can have for massless interacting particles are the cases where the particles are taken from a very tiny menu of possibilities. The particles have spin zero, one-half, one, three-halves, or two. And I'll show you how this follows, as I said, as directly as possible from the underlying principles of spacetime and quantum mechanics. Now, particles have spin a half. Of course, we knew about for a long time the matter particles, electrons, quarks, and so on. Particles have spin one. We have photons. We have, of course, the rest of the gain structure of the standard model of particle physics. But the amazing thing is, again, from pure thought, from pure thought of just beginning, well, not pure thought because we begin with these principles that the world has quantum mechanics and spacetime that we ultimately take from experiment. But starting from that, we can show that the spin one structure must be that of a Yang-Mills theory. And again, it has nothing to do with the elegance or beauty or you like gauge symmetry or anything like that. No, it cannot be any other way. So it's forced on you, the Yang-Mills structure. This guy is interesting. The massless spin two, these people locked up in the room, they haven't looked in the world outside, they don't know about falling apples or elevator thought experiments or anything like that. But you discover that this guy is unique. You can only have one of them. And furthermore, that its interactions are universal. So, of course, this is gravity and you can even go on and discover all of the structure of general activity from this argument. But I want to stress here that this is something else. It's another, that this is the only menu of possibilities you're allowed to have. And you learn this. You learn that there's a universal strength interaction for this massless spin two particle. You would learn that if there happened to be massive particles around, here we're talking about massless particles, but if there happened to be massive ones around that this spin two guy would make them go around in orbits and so on. That it's an attractive force. All of that is just forced on you by the principles of spatial relativity and quantum mechanics. So even if Einstein didn't exist or Einstein's part two didn't exist and we didn't get the GR, just the logical development of thinking about space time and quantum mechanics would force its existence on us as one of the sort of unique possibilities for what nature can do compatible with these grand principles. So these are examples of what we've seen already. So these are like electrons, matter particles and so on and the annual structure in GR. And you can see what's special about the spin zero. So that's what we finally saw with the Higgs. So it's the first really new elementary particle that we've seen. It's the first elementary particle of spin zero. It's the simplest elementary particle in many ways. It has no other properties other than its mass. And here we're working in the approximation where we're ignoring its mass. But it doesn't have charge, it doesn't have spin. So and it's that very simplicity that makes it so perplexed. So again, we'll be talking more about that later. And so these are things that now we've seen nature do. So that's what was exciting about the discovery of the Higgs. There's a menu of possibilities of what nature allows compatible with quantum mechanics and spacetime. We had seen the other examples. Some of the menu filled in already and the Higgs was the first time that we filled something in that we had not seen before. And that leaves the final part of this menu, this spin three half is the last thing nature can do compatible with its grand principles that we have not yet seen it do. And the possibility for having a consistently interacting massless spin three half particles is connected with supersymmetry. Okay. So that's the sort of deepest reason why theorists have been excited about supersymmetry for a long time, because it is the last example of something nature is allowed to do compatible with that sort of grand architecture that we have not yet seen it do. And again, it's nothing to do with elegance and beauty and people like it and it's pretty and aesthetic and that's all crap. So anytime people uses these words elegance and beauty just, you know, they're not saying something in various that sort of beauty and elegance that we talked about in the physics has to do with this. Of course, I'm parroting things that Steve Weinberg has said much more eloquently that I am here. But this is the notion we really care about the rigidity and inevitability. And there's an incredible argument that I'll go through with you, you know, like a half page of algebra, after you get going, that begins with these basic principles and ends after you codify it into some concrete check that you do on the structure of the consistency of scattering amplitudes to this amazing statement that the only possible consistent theories are taken from this menu. And once you know that that's true, then of course, you need to experimentally go look for all of them and to try and see if they're actually realized in nature. Okay, so on the one hand, but this is the attention that I was talking about. On the one hand, we have this incredible fact that that the fundamental laws given the structure of relativity quantum mechanics are inevitable. The attention is that is that exactly is at the same time the same laws appear to make our observed universe, I'll put in quotes impossible, or more specifically unnatural. Okay, so what went into this argument is to begin with that you imagine we have quantum mechanics and the principles of relativity as applies to Minkowski space, you know, so what the space time we're talking about in this argument is flat space time, the space time that to a very good approximation if we ignore questions about cosmology, we appear to live in. So that's the first thing that you have to do. And the second thing is you have to imagine that the elementary particles were ignoring their masses. So we're working in an approximation where we're looking at energies much larger than their masses. So you've had better be able to access that in other words, as we do also an experiment, we have better be able to go to sort of high energies and see a region have an accessible region of energies where we can ignore the masses of the particles. So in order to begin this argument that that makes the structure of the universe inevitable, we have to have a large macroscopic universe to begin with a big enough space time. So there's a flat enough space time so that there's a huge range where it looks like flat space, A, and B, light enough elementary particles that there's a decent range of energies where we can ignore their masses and treat them as approximately masses. So if we have been Kowski space with masses particles, boom, the structure of the laws is locked. And we have to take them from this tiny menu and it's an amazing mind blowing fact. But exactly those same laws because of violent quantum mechanical fluctuations associated with the vacuum seem to want to make exactly that starting point impossible. So for instance, large quantum mechanical fluctuations would naturally give you a big cosmological constant. That big cosmological constant would curl the universe up to a minuscule size. So we wouldn't have this approximately flat space region in which to even begin running this argument. And similarly, the violent quantum mechanical fluctuation. So that that first problem is the famous cosmological constant problem. And the problem associated and it keeps going. The violent quantum fluctuations associated with the Higgs particle would appear to make its existence impossible or unnatural. If the Higgs was much, much heavier or had a much, much larger vacuum expectation value that would also get rid of all the elementary particles. So we wouldn't have we wouldn't have this sort of region of energies in which we could be dealing with approximately massless particles in order to talk about how they scatter. So that's the fascinating tension is that given given the observed fact about the universe that we have quantum mechanics that we have, we have this big flat space time to excellent approximation. And we have some set of elementary particles that we can treat as massless to first approximation. Then then there's this extremely locked and rigid structure. But we have no idea how we could have landed on this zero th order starting point to begin with. So that's the sort of fascinating tension that we find ourselves in is this tension between the rigidity of the laws and the fact that the very same laws seem to want to make their starting points impossible or unnatural. And again, this is associated with the cosmological constant problem and the hierarchy problem that are associated with the cosmological constant and Higgs mass. So so I will be talking about an awful lot of absolute nonsense has been said about these things to my surprise over the past few years. There are even you even hear people say, well, you know, we didn't see we didn't see the hierarchy problem solved with the LHC. The problem was stupid to begin with. It was just an aesthetic problem, just a problem of beauty or whether you like things or not. And this is completely incorrect. It's a very hardcore sort of concrete problem. And I want to explain something about what this what what the problem actually is, because the sort of drama associated with this problem is what's heightened by the fact that it has not been solved in the way people expected it to be solved for 40 years. And it only adds to the motivation for putting the Higgs under microscope, studying it to death and and imagining going for future colliders. Okay, so so so that's my my first plan is to is to explain why the Higgs is special and associated with this tension of the inevitability and impossibility of our laws. Okay. And this, as I said, is probably the most important understanding why the Higgs is special is the most important part of the physics case for future machines. If I have time, I want to talk about something that goes beyond this, which I also think is very important, which are just going beyond the Higgs. I want to talk about some sort of theoretical frameworks that even people talked about before the LHC, you see that there's a sort of a narrative out there that everyone expected before the LHC that I don't know we discover Susie or something like that, you turn on the machine and you know, the detectors would be fried with the super partners. And that didn't happen. And it's only after that that this was a big back to the head of the theorists who said it now theorists have no idea what's going on, a totally blind going into the future. This is incorrect. There was a not even that small a minority, it was a minority, but not a particularly small minority of people before the LHC, certainly including me, definitely not only me, certainly including me though, who were quite loudly saying there's something wrong with these arguments about naturalness. And there was an indication just from experiments that there was something wrong. If these arguments were correct, one would have expected to see new physics before the LHC, we could have seen it at Fermilab, we could have seen it left before the LHC. And there's actually sort of a growing tension, more and more sort of dogs that didn't bark, more and more counter indications to the basic picture already well before the LHC. And yet the main reason for taking things like supersymmetry so seriously were not only that they solved quite beautifully these theoretical puzzles associated with the hierarchy problem, but they had other things going for them. They made other predictions in the context of supersymmetry, particularly the famous picture of the unification of coupling constants at very high energies, and more qualitatively the sort of picture of dark matter where things that sort of fell in your lap without asking for them. And so while people of course knew, even from the 1980s, that there was always some counter indications that there had to be a lot of new stuff at the TEV scale, and the counter indications were that you could have seen indirect evidence for these particles already, early 1980s people realized that. And as time went on, there were more and more indirect things, and even some direct things from colliders that you could have seen that you didn't see. So that was attention in one direction, but there was the counter evidence in the other direction that well, you got these things for free, they just worked spectacularly, the unification of couplings, again more qualitatively the picture of dark matter. So people had in mind that it was a sort of either or that either you took something like the hierarchy problem seriously and then you got all these wonderful things, like the coupling unification and dark matter, or you didn't and then you lost them. And a number of people realized in the early to mid 2000s that this was not necessarily the case, there are other theoretical pictures for what's going on, where you might not expect. In fact, sometimes you definitely would not expect to see the new particles at the LHC. You could still have the successes that you liked before, but lose the sort of justification from the hierarchy problem and naturalness for why you have to see the new particles at the LHC. And I think one of the interesting things about some of these theoretical frameworks, again, many of which were discussed before the LHC, so it's not a matter of cherry picking or moving the goalposts now that the LHC ruled out some of people's favorite theories. These were things that were already talked about well before the LHC. In these pictures, there is a good reason why you might see something at an energy 10 times higher than the LHC. So I just told you in the very beginning that we don't have any guaranteed arguments like we did before the LHC, that we have to see new particles beyond the LHC. And therefore, most people talking about the justification for future colliders are talking about the Higgs with excellent reason. And that's, again, what I will be emphasizing as well. But it is emphatically not the case that we are completely stumbling around blind with no idea what's going on and zero reason to expect we might see some new particles beyond the Higgs just around the corner of a factor of 10 higher in energy. There are pictures of the world where you would expect that. There are not unreasonable pictures that were talked about before. They have some theoretical motivations. They make some interesting predictions. And there are reasons for, and so there are reasons for taking those ideas seriously. And they provide additional motivation for taking the sort of next order of magnitude leap beyond the LHC. Okay, so that's my plan for the lectures. I will start off today talking about these questions about why the Higgs is special. And after doing that, we'll, of course, talk a little bit about, in slightly more detail, what we could learn about this from future colliders, both E plus E minus Higgs factories and proton colliders. And then, again, if I have time, I will talk about question two. On Friday, I will, again, if time permits and we get there, I'll talk about the recent sort of possible excitement with the Muon G minus two and what this might imply for future colliders as well. If the Muon G minus two ends up being real, then one of its most important consequences is it tells us that there's a scale of new physics that is not far away. And in thinking about and studying it in more detail gives even more, gives a more concrete picture for where the new physics might be and can inform really, strongly inform what the kind of next collider is that we should think about building. But so we'll see if we get to this on Friday. Okay, maybe I'll stop right now for a little break. I guess that was the plan, right, Giovanni? All right. Yes, perfect. Yeah. And ask if there are any questions at this point, otherwise we can come back in around five minutes. Yes, perfect. All right, I don't see any question for the moment, so we can have a break and resume in five minutes. Okay. All right. Thanks. Thanks. Hi, Nima. Hey, guys. Hey. I am back. Good. With my copy. Can you see him okay, Giovanni? No. No. Time to unshare and share again. The problem seems to be on my end. Let's see. That's weird. Yeah, I think the problem is on my end. Let me see. We don't see either of the two videos. Okay. I would hope you see, let's see what happened here. Do you see my screen here now? Okay, yeah. Now we can see the screen, but you don't see me still, right? Yeah, I don't see you. All right. Maybe I can try. Let me try leaving it. Okay, now we see. Now you see me. Okay, great. Yes. Very good. By the way, Giovanni, maybe you, I mean, the students can tell me as well, but if you have a sense of whether I'm going too slow or fast or whatever, please as I'm going along. Yeah, I invite all these, yeah, students to interrupt and feedback and ask questions by raising hand. All right. Whenever you're ready. Okay, should we restart? Yes. All right. Very good. So I want to start now talking about why is the heat special? And first, I'm going to sort of talk about things more intuitively, just using ideas that many of you are familiar with already. The first thing to remind you of is that there's a profound difference between massive and massless particles with spin. So spin is a crucial part of this story. This profound difference is in many ways at the heart of all of the sort of deepest conceptual ideas from the 20th century, as we'll just describe, general covariance and the Yang-Mills gauge redundancies are theoretical tools, pigments of the theoretical physicist's imagination, in order to deal with this profound difference between massive and massless particles with spin. And let me just, but let me remind you very simply what this difference is. Like, let's say you have a massive spin one particle, like a W, okay, massive spin one as an example. Okay, so here's a W. Okay. And however it's moving, you can always go to a frame of reference where the W is at rest. And it's a spin one. So however it is the whatever direction it's spinning in, by rotational invariance, you can tilt your head and you can see it spinning in three possible directions. Okay, so a massive W has three spin degrees of freedom, right, just the three degrees of freedom associated with massive spin one. Let's contrast that however with massless spin one, like a photon. So this is three, let me emphasize that it's three spin degrees of freedom. Whereas a massless spin one, like a like a photon. So here's a photon zipping along at the speed of light. And well, you know, even from high school that photons only have polarizations, and they have two polarizations, we can talk about whether they're spinning. For example, whether they have a helicity, whether they have a spin in the direction of motion, or opposite to the direction of motion. So, so a photon has two, not three, has two helicities and not spin. In fact, we normally say this quickly and say that the difference is between three degrees of freedom for a massive spin one particle and two degrees of freedom for a massless spin one particle. So it's spin for mass of spin one and helicity for massless spin one. In fact, really what we know is that there is one degree of freedom, which is the positive helicity photon, a separate degree of freedom, which is a negative helicity photon. And the only reason we call those particles by the same name is that the interactions of the photon to extremely good accuracy have parity invariance. So parity can flip between the plus and the minus helicity particle. But there wasn't even a God given a temporary reason why we have to have both of those. Okay, for example, if we talk about massless spin a half, we know that we don't necessarily have the parity partners of the of the of the of the different helicity particles. We know that we know for an experimental fact that we have left polarized neutrinos, but not that we have left polarized anti-neutrino. Okay, so we don't have to have we don't even have to have parity partners of everyone. But I'm yeah, but I'm just glossing over that point. And so certainly in the case of the photon, we know that it has two helicities. Now, it's really important that three is not equal to two. Okay, so and that's a really a profound difference. You might have thought, and in fact that that and in our previous discussion, I just said this point quickly that, well, when you're going to very high energies, you can just ignore the masses of the particles compared to their energies. So you think that the sort of transition between massless and massive is kind of smooth. But it isn't. You cannot be a little bit massive. It's like being a little bit pregnant. Okay, so so that's not that's not that's not a that's not a real thing. You're either massless or you're massive. There's this profound difference in the number of degrees of freedom between massive and mass. Okay, there's actually a question here. Yeah, hello. So I have a question about what you said before about these two degrees of freedom of the of the masses of the masses has been one field. So because because I've never heard about it, you can also write down a theory for only really that it has only one. So one, you know, I'm wondering, is this as easy like for fermions to just take no actually discover this we'll discover this. I mean, I won't go through every one of these examples. But I will tell you enough about this so that you can work it out for yourself, how we do this really from the bottom up by just thinking about the consistency of scattering. It turns out that while you can have consistent theories of chiral a massless been a half. And of course, that's what we have in nature. You cannot have consistent theories for chiral massless been one. Okay, so in fact, you can't have it. It turns out that you can't have it. But for more detailed reason, it isn't just directly from what the labels of the particles can be. It will come from these other arguments. Okay, so we'll see that the consistency of four particle scattering nails this very restricted set of possibilities. And those possibilities do not include chiral thought, even though they do include chiral massless been a half part. Excellent question. Any other questions? Okay, all right. So this is a familiar anyway, that this is a this is a familiar thing. Probably you all know that there is this difference, but I'm sort of emphasizing again how important it is. So similarly, we can have massive been two, which has five degrees of freedom. But massless been two, again, has two holicities. Okay, so so again, five is not equal to two. All right. So there's a discontinuous jump between massless and massive when we talk about particles with spin. Okay, so now this is a very important consequence. This allows us to understand. So let's say we ask a simple and again, I'm just saying things intuitively right now. Let's say you ask a simple question like, why is the photon massless? Photon massless. Okay. Now, if I was there sort of live and I could see all of you, I would ask you to put up your hand. If you thought that the reason why the photon is massless is because of gauge invariance. Okay, maybe people can can can raise their hand virtually raise their hand. How many people would say that the photon is massless because of gauge invariance? Are they still hands? Probably everyone knows. Yeah, everyone knows I'm going to make fun of them if they raise their hand. They're wisely, they're wisely not raising their hand. But be honest, I'm not judging. They're not honest. I can't even see you. You know, so good, good. The hands are all going up. This is what I'd like to see. All right, now I'm going to make fun of you all. Okay, all the other people are so smart, we're not putting up the hands. These guys, other guys are being super honest, you know, good, good, good for them. Like, all right, good, it keeps going up, just laying a trap for you. But that's fine. All right. Okay, well, I commend, oh, that's interesting. Some people unraised their hand. Anyway, commend me for your honesty. And the rest of you cowards, or the rest of you are either cowards or you know, the right hand. All right, anyway, thank you very much. Well, it's actually completely backwards. It has nothing to do with gait symmetry, why the, why the photon is massless. The deep reason the photon is massless is that two is not equal to three. Okay. So this is the reason why the photon is massless. And let me try to explain this. Let me try to explain this with what was with the picture or the sort of content of the picture. You see, you might think maybe, maybe classically, or at tree level, if we're imagining drawing Feynman diagrams, the photon is massless. But let's look at the sort of possible interactions we could have of the photon, you know, at loop level, we have all these virtual corrections, you know, we can have all kinds of stuff like this going on, maybe more electron loops here and so on. Okay, so this is the photon on the outside. So why don't all of these interactions generate some mass for the photon? Okay, why don't they generate a mass for the photon? And the reason why they can't generate a mass for the photon is simple, but deep. You can't turn on some interaction that discontinuously changes the number of degrees of freedom from two to three. If you just have two degrees of freedom to begin with, there are two degrees of freedom. And just by turning on some interaction, you can't make two discontinuously jump into three. So that's the reason why the photon is massless. That's the reason why the graviton is massless, is that their masslessness is associated with a discontinuous change in the number of degrees of freedom. This, by the way, is also the explanation for why it is that our friends in condensed matter physics can engineer systems in the lab that don't have anything that looked like been one excitation in the microscopic Hamiltonian, you know, they're just talking about some metal or some condensed matter system, but they can arrange it so that the effective field theory that describes a system at long distances has something that looks like an emergent gauge field. Why do they say this is an emergent, but why is that possible? I mean, of course, they have to arrange, they have to find some, you know, cool enough system to do it. But once they do it and they cool it to low temperatures and so on, the emergence of the photon-like excitation is robust. It doesn't depend on very finely adjusting the strength of the interactions in the material or anything like that. It just sort of happens robustly. And the reason it can happen robustly is exactly because of this discontinuous change in the number of degrees of freedom. Once you get something that looks like a massless or a gapless photon excitation, it cannot be given a mass or in the condensed matter language. You can't be gapped by turning on small interactions because of this robust difference in the number of degrees of freedom, because two is not equal to three. There is a question. Yes, there is a question. My question is regarding masslessness. When we are saying it has two degrees of freedom for spin one, when it is massless and three degrees of polarization states for massive particles. So my question is, is that degree of freedom for massless case is hidden or it is present there but not experimentally measurable? Well, so first, there isn't a super invariant difference between absence and not experimentally measurable. So if you take the interactions of something and make it arbitrarily weak at some point when it becomes practically relevant, it is at the same time it becomes conceptually relevant. But if I pursue that argument a little bit more, there are more interesting cases where we have the massless, we have the two degrees of freedom of a massless particle. And then we have another degree of freedom somewhere. And after interactions or other physics is taken into account, they combine together to become the three degrees of freedom of a massive particle. That's exactly what the Higgs mechanism is all about. So we know that we can think of the extra degrees of freedom associated with the massive spin one particle. They have to come from somewhere. And of course, the entire story of the Higgs is finding a source for where those other degrees of freedom are coming from. And there's a way of thinking about it in which we have the two degrees of freedom of the massless spin one. We have some extra degrees of freedom associated with other guys and there's some approximation where you think of them as decoupled but then something happens and they become unified into a single object. But the point is that you have to have those other degrees of freedom around and they cannot be irrelevant to the system. Otherwise, they can't join in with the massless particle that has lots of interactions with other things to become effectively its extra degree of freedom. Okay. So actually, we remove those unnecessary degrees of freedom by gauge conditions because it will not affect our theory. Well, so I'm going to come to this point about gauge redundancy in a moment. So far, I haven't said anything about gauge symmetry, fields, nothing. I've just talked about a very basic fact about particles that there's a difference in the number of degrees of freedom of massless and of massive and massless. In a moment, we'll talk about how we interpret that fact in a particular way of doing thinking about the the physics using fields and how that forces gauge redundancy on us. But I'll talk at some length about the fact that it's not a symmetry, it's a redundancy of description. And yeah, we'll say something about that in just a moment. Okay. Thank you, Professor. No problem. Any other questions? Yeah, there are two. Hey, hello. Am I audible? Yes, you are. Okay. So my question was related to photons being massless. So if they were massive in the first place, then that would mean that they are no longer light like, right? So that would, so isn't that the deeper reason? So otherwise, I would be afraid. I just said it. I said it quickly and we'll be seeing it more technically in just a moment by thinking about the formalism of the so-called little group. But indeed, that the intuitive reason is what we just said already. Why is it different between spin one and spin zero? You can't catch up with the photon. Maybe I didn't see it explicitly. I apologize. Thank you. Why is it intuitively, what's the difference? No matter how the W is moving, I can go to a frame where it's at rest. And then however it's spinning, I can rotate my head and see that it's spinning in three directions. However, I can never go to a frame where the photon is at rest. All I can see is that it's either spinning in the direction of motion or opposite to the direction of motion. I can't catch up with it and use rotations in order to see it spinning in all the three possible directions. And that, in fact, is why really what we know is that we have one spin degree of freedom, either if it's spinning in the direction of motion, there's nothing I can do in any frame or anything to do to see it spinning opposite to the direction of motion. So really, from this argument, I can tell it has one policy degree of freedom and then parity tells me that it has to be paired up to give me two. But indeed, that's the sort of big difference is that in the massive case, you can always catch up with it and do a rotation to see it spinning in three ways. In the massless case, you can't catch up with it and you can't make that argument. So we'll be saying this more formally in a moment when we go back, you know, following Wigner and Weinberg, we'll go back to this question, what is a particle period? And we'll see more technically and more formally what this very intuitive difference is that exactly you just alluded to. But thank you for asking the question. I'm not sure I said it explicitly before that it was because we couldn't catch up with the photon. So thank you. Yes, Dong, please go ahead. Hi, just a silly question. So you said that because the photon has two degrees of freedom, we can observe three. So it's massless. So how about the gluon? The gluon massless is the consequence of the SU2, sorry, SU3 without the spontaneous symmetry breaking. Right. Now, of course, the story of gluons is even more subtle because ultimately all the colored particles are confined. But when we talk about very high energy collisions, the gluons look massive. And in the approximation where we really see individual gluons and very high energy scattering, but we don't see them confined, they really have two holicity degrees of freedom, not three, two. And how can we observe that? Because we never observe the gluon directly. We don't observe the gluons directly, but in the same way we don't observe the quarks directly, we have a picture for what's going on with very high energy collision, when especially when we have, when we look for the sort of point like collisions between the quarks and gluons, you can tell the point like collision has taken place when things come off at large angles relative to the relative to the beam. So that's the way historically, even with deep and elastic scattering, that people first saw the sort of quarks inside the proton. So what we know is that the amplitudes for all those collisions are consistent with the picture of these particles that are ultimately confined later, but the sort of hard collision, the high energy collision is taking place between these point like elementary particles that are massless and the gluons are massless and they have two holicity. Okay, thank you Professor. Okay, any other questions? Yes, so this was what you're basically saying that you kind of turn around the argument so that you say that the reason so that we start off with an initially like totally massless theory, I mean the Higgs particle is the only particle actually has like a mass from the beginning and this is the reason why we can actually construct a gauge invariant theory. Yeah, I'm about to say it the other way around. We use gauge redundancy as a tool to describe massless particles with spit. Okay, and it's not the only tool. There are other tools for describing. It's a very convenient tool. It's a tool that's closely associated with the standard formalism of quantum field theories, but I'm saying it this way in order to emphasize that it's a tool and not a feature of the physics. The physics doesn't know about any language we use to describe it. Gauge symmetry, gauge redundancy, general covariance are language we use to describe the physics. They're not the physics itself and this might seem like a little you know irrelevant academic distinction now, but I promise you if you sort of internalize this way of thinking it'll help you understand things, not get confused about things as you learn more and more things about physics going into the future. So that's why I'm deliberately talking about these things from perhaps a backward steaming upside down language, but it has the advantage that this way of thinking about things is directly tied to the actual observable thing, directly tied to the particles directly tied to their property. And we'll of course connect it to the, and I'll describe in a moment why this way of thinking forces gauge redundancies on us if we want to use fields to describe the physics. Okay, thanks. Any other questions? One more hand. Okay, very good. Okay, yeah, so, okay, so now already here I want to, so, so already here we've already said these words already, but the difference between massless versus massive to its difference in our usual formalism of field theory, usual QFT is associated with gauge redundancies. Okay, so, and I'll explain what I mean by this language in a little bit. So this is, as I've alluded to already, the Yang-Mills gauge redundancy and general covariance or defumorphisms. Okay, but I want to emphasize that these are not features of the real world, these are features of a language that theoretical physicists use to describe the real world, one excellent language, but one language that we use to describe the real world. Okay, and that's the sense in which this, that's the sense in which this distinction, as I said, is at the heart of the sort of these most profound ideas from the 20th century, but already from here we can see why the Higgs is special, why is the Higgs special, because the Higgs has spin zero, so now there is no difference in spin degrees of freedom between massive and massless. Okay, and so what that means is that if we go back, we just said that if I draw this picture for the photon, why is it that all of these corrections don't do anything crazy to give the photon a mass, all of these corrections give me a shift in the mass of the photon equals zero, and that's because two is not equal to three, but the problem with the Higgs, so if I draw analogous diagrams for the Higgs, the most important one is involving the top quart, for example. Here, there's absolutely no reason why delta M squared is not equal to zero because one equals one. There is no discontinuous difference between the number of degrees of freedom of a massive spin zero and a massless spin zero part. Okay, that's the origin, this is the sort of deepest way of talking about the higher fee problem, and we'll talk about this in more detail later, but I'm just sort of already seeing it early at this point, just to emphasize again, how closely tied it is to these basic facts about the relativity and quantum mechanics, as we'll review in a second. Instead, more formally, this two not equal to three is a direct consequence of relativity and quantum mechanics. Okay, we'll see that quite explicitly, but the similarly direct consequence is that one equals one, and that's why that's our difficulty, we don't have any understanding for, there's no difference, there's nothing special about whether a spin zero particle is massive versus massless, and this is also why our friends in condensed matter physics cannot generate a Higgs particle. Okay, they can generate systems that give you emergent photons, emergent chiral fermions, all those ingredients of the standard model are also sort of, you know, they're not, a random system doesn't do it, but you can find robust condensed matter systems that give you as long distance excitations, things like emergent gauge fields, emergent chiral fermions and so on. Once again, all of those objects have their degrees of freedom protected by this discontinuous jump between massive and massless. You can't do it for interest, for interacting massless spin zero particles. There's of course an exception to this, which is if the spin zero particle happens to be non-interacting at low energy, that's what happens with Goldstone boson. Okay, so there is, so if you happen to have no interactions at low energies, then you can be massless, and that of course does arise in condensed matter system as well. Phonons, they're all kinds of examples of Goldstone phenomenon in condensed matter physics too, but interactions spin zero particles that even have interactions at low energies, big interactions at low energies, like the Higgs does, like the Higgs has big interactions, top eukalyle coupling gauge interactions and so on. There, there is no difference between the number of degrees of freedom between massive and massless. These quantum corrections, even if you had a classically massless Higgs would have no reason not to give it a big mass, and that's the content of the hierarchy problem. We don't understand why there's a light Higgs particle. There's nothing special about the Higgs beam, right? And that's exactly why we also can't engineer these things in condensed matter. And in fact, people have not done it. You know, you don't see light scalars, light interacting scalars as a sort of garden variety excitations in condensed matter system. A number of years ago, you know, partially just for fun to try to replicate things that we've seen in particle physics, there are some condensed matter experiments that managed to get a moderately, you know, narrow Higgs-like particle, not with the extra gauge interactions, but just like the sort of Higgs of a, you know, SL3 spontaneously broken global symmetry. But what they have to do is put the system under change the pressure in exactly the right way and fine tune the parameters in order to make the Higgs moderately light, okay? Explicitly, they have to finally adjust the parameters of the system in order to make the Higgs moderately light precisely for this reason, that there's no, there's no spin, there's no difference in the number of degrees of freedom between massive and massless. And therefore you should not expect to find spin zero particles with the decent interactions just lying around. And with very hard work and fine tuning of the parameters, they actually managed to make something that looks a little bit like, like the Higgs. This just further emphasizes the qualitative difference between spin zero and higher spins, okay? So, and that's the, that's the sort of deepest way of talking about what the hierarchy problem is. The existence of massless particles with spin is associated with the, is protected and is associated with this discontinuous difference in the number of degrees of freedom. That difference is not seen for spin, spin zero. And while I'm here, so you see, this is kind of a failure of just symmetry arguments. And we'll talk in more detail about, as I said, why two not equal to three here, why this counts for the number of spins is really coming from an understanding of the, of the, of the, of the Poincare symmetry and the way the, the, and the quantum mechanical representations of Poincare symmetry. But it's really a symmetry argument. So, so the case of the Higgs is an example where, where symmetry arguments do not suffice to explain the phenomenon. Symmetry arguments don't suffice to explain why we have, why we have a light Higgs. And while we're here, I just want to emphasize that this is exactly the same qualitatively as, as with the problem with the cosmological constant. Okay. Because symmetry arguments also do not distinguish between universes that have positive cosmological constant, zero cosmological constant, or negative cosmological constant. Okay. So if you imagine vacuum solutions of Einstein's equations that just have some vacuum energy, cosmological constant is positive. That's the sitter space. If it's zero, it's flat space. If it's a negative, it's empty to sitter space. I'm just drawing a cartoon of what these things look like that the detailed symmetries are different between these spaces. Okay. So, so in flat space, we have the, we have the Poincare symmetry and the Poincare symmetry has, has, has, you know, 10 generators, there are 10 symmetries, which are the four translations, three rotations, three boosts. Okay. So we have a total of sort of 10 symmetries associated with flat space. Well, the detailed symmetry of the sitter space is different. It's not the Poincare symmetry. It's SO 4 comma 1. I want to explain why this is true, but it's SO 4 comma 1, anti-decider space is SO 3 comma 2. But all of these symmetries have 10 generators. Okay. So there's 10 generators here and there's 10 generators here. So that's the problem again. There's no symmetry jump. There's no difference in amount of symmetry as you change the cosmological constant from negative to zero to positive, just like there's no change in the number of degrees of freedom of a spin zero particle as you go from negative to zero to positive. Okay. So, so that's the sort of deep commonality between the problem of the cosmological constant and the Higgs math. We do not have, there's no symmetry argument, unlike the case for, in the case of particle physics, particles of higher spin, there is a difference, but we don't have that difference for either, for things, at the deepest level, we don't have the difference for for excitations that we can turn on that are constant across spacetime. So that's the other distinction between the Higgs and the higher spin particles is that we can talk about the vacuum expectation value of the Higgs. Now I'm not using, I'm using the field language. We'll be mostly talking about the particle language, but I just want to stress this other aspect of things. And that's something which can be constant across all of spacetime. There is no excitation of the photon that's constant across all of spacetime, right? You know, the photon has an electric field, a magnetic field associated with it, it varies. It's very existence involves some variation across spacetime, whereas, whereas the Higgs expectation value was dead constant across spacetime. Similarly, the cosmological constant is dead constant across spacetime. It's exactly these things that are dead constants across spacetime that have the ability to be dead constants across spacetime rather than have variations within spacetime, which are the things that we can't control these symmetry arguments. And that's a sort of a and so exactly the same logic that goes into not understanding why the mass is small because these quantum corrections have no reason to keep it small. Precisely the same argument, this absence of any symmetries, is the reason why we can't understand why the cosmological constant is small. Okay. All right. So that's the first pass on the idea of why the Higgs is special. I guess I'm out of time. Is that right, Giovanni? You still have a few minutes. Okay. Well, okay. If I have a few minutes, maybe I can, maybe I can, do I have 10 minutes or maybe even 15? Or is that, do I have 10? Yeah, 10, 10, then you can have. I have 10. All right. Okay. Let me, okay. So if I have 10 minutes, let me just try to now explain in more detail or at least begin talking about in more detail the issue of the spin of the difference in spin degrees of freedom between massive and the masses. And so maybe I'll just begin, I won't cover it all now, we'll pick it up again tomorrow, but I'll just begin with the question, what is a particle? Okay. At least I'll frame this question and we'll come back and start answering it tomorrow. Okay. So we're going to start with this extremely basic question, what is a particle? And this discussion, again, goes back to Wigner and is most nicely described in Weinberg quantum field theory book volume one. Okay. So I'm going to be giving, okay. So what is the particle? Well, of course, intuitively, we all know what a particle is. It's some localized lump of mass and energy that moves, maybe has a spin. But more conceptually, why do we say that? And what properties are we implicitly assuming when we have that intuitive picture? This is a really good example of the way theoretical physics works because we always begin with very, and it's important to begin with very simple, familiar, intuitive ideas. But often the deeper answer to the question that's actually closely connected to that intuition, it pays to sharpen that intuition to a slightly more abstract statement. And that connection between the very intuitive thing and the slightly more abstract, but more fundamental thing is happens all the time in the development of the subject. And if you're learning it, it's important to sort of be able to go back and forth. Definitely not to always jump to the fancy sounding abstract thing. Actually, it's really important to begin with the simplest, most intuitive, familiar thing you can, but then see the need for finding this more abstract understanding of what's going on. So this question of what is an elementary particle is a great example of that because we just gave what the intuitive picture is. But the more abstract answer is that a particle is an irreducible representation of the Poincare group. So that's definitely a fancy seeming not intuitive abstract answer to this question. But in fact, it's very closely related to the intuitive one. So what's important here is that a particle, even the notion of what a particle is, is inextricably linked to what the symmetries of space and time are. And let me give you some simple examples just to make it clear that these things are related to each other. For example, we just say without thinking about it that an electron that I see over here, I give it the name electron, and it's the same name I give to the electron here or there or somewhere else. Why is that? It's because we've observed the fact that the universe is translational invariant. It's only because of translational invariance that we give the electron the same name everywhere. Why is the universe translational invariant? We don't know. At a deep level, we don't know where the symmetries of space-time come from, but they appear to be there. Also, if I have an electron moving this way, I call it the electron. And if it's moving that way, I call it the electron. I give it the same name. Why? Because of rotational invariance. And for electrons moving at different speeds, again, we call them all the electron. Why? Because we can get from the symmetries of space and time, we can get from one kind of electron to the other kind of electron. To push the point, let's say that for some reason permanently, everywhere in the universe, there was a giant magnetic field turned on in the z-direction. Then it would kind of be dumb to call it the electron, no matter how it moves. Because the electron, when it's moving in the x-y plane, it's a giant magnetic field in z-direction, goes around in little circles, whereas the electron that moves in the z-direction just zips along. They look totally different. They behave totally differently. You would be pointlessly tying your hands behind your back to give them the same name when they do totally different things, when they move in different directions. It's the observed fact in the universe that close to the vacuum, we have these symmetries of space-time that allows us to use the same label to describe all of these particles. That, I hope, makes it clear why the space-time symmetries are related to the sort of labels that we give elementary particles. Now, what was Wigner's argument? Wigner asked the following question. We know that we're going to begin with Poincare symmetry, which is translations and Lorentz transformations. Translations and Lorentz, which are boosts and rotations. Now, we know that we can diagonalize all the translations simultaneously, so we can diagonalize translations. Now, this is actually the first step, and this step is very familiar, but let me just remind you how we do it. First, just most intuitively and non-relativistically, where we already have translation invariance, of course, non-relativistically. The most intuitive thing to do is to take a particle at a point x, some point x. These are sort of position eigenstates, and so if I take a translation t in some direction, let me label this by vector, translation t by an amount a. Well, that's a symmetry, but it should act in some way on the state. What does it do? It acts on the state, and it changes the state, right? Changes the state to a different state where x goes to x plus a. So, while these position eigenstates are very intuitive and close to what we see in the real world for classical macroscopic objects like my copy cup, they're not left invariant by the action of the symmetry, and they're, in fact, they don't transform simply under the action of the symmetry. Under the action of the symmetry, x is changed to x plus a. So, you can ask question, are there states that transform as simply as possible under translations? In other words, they're eigenstates of translations, okay? And the answer is that, of course, these are momentum eigenstates. Yes, and they're momentum eigenstates. So, these states p are related to the states that we're talking about before in the familiar way that you've learned from quantum mechanics, but the important point now is that the action of the translation operator on p just gives you p back. You just get e to the i p dot a times p, right? So, that's what's important, conceptually important about momentum eigenstates, is momentum eigenstates are eigenstates of the translation operator. There are the states under which translations act as simply as possible. And when we get to relativistic systems, this distinction becomes more important. Because again, physically, it is difficult to talk about position eigenstates for single particles. Why? Because we know, we know that especially from this connection, we know that position eigenstates are in turn, linear combination of momentum eigenstates with arbitrarily high momentum. And so, we can't really talk about perfectly localizing, let's say, an electron to a point, because that would involve arbitrarily high momenta, and those arbitrarily high momenta would allow you to produce electron-positron pairs. So, we can't talk about a single electron in a position eigenstate in a useful way. So, the notion of position eigenstate that's familiar from non-relativistic quantum mechanics is not a useful one in relativistic particle physics. However, the notion of a momentum eigenstate is perfectly sensible, even for a single electron. So, that's why even though these two ideas are very simply connected to each other in non-relativistic physics, when we talk about relativistic particle physics, it's better to imagine labeling a single particle as being an eigenstate of translations. So, we think about it as a momentum eigenstate. So, now I didn't have to say anything about position eigenstates at all. I could have just jumped in to say that what's important about using momenta is that momenta diagonalized translations, but I wanted to go through this little detour to sort of explain why this is a more fundamental picture, even still in the context of relativistic particles. Okay, so to begin with, Wigner asked this question. So, let's diagonalize translations. So, if we diagonalize translations, then any state will be labeled by its momentum and maybe some other labels. So, any state is momentum and some other labels. So, this is a momentum p that satisfies p squared equals m squared. And these are some other labels. And so, Wigner wanted to know what could these other labels be and how does this state transform now under the full Poincare group? So, we've already seen that it's the diagonalizes translations. So, all we have to do is talk about what the action of a general Lorentz transformation is on. And so, Wigner asked this question. Let's say you have a general Lorentz transformation, capital lambda. There should be a unitary operator, U of lambda, okay, that acts on Hilbert space. And so, how does U of lambda on p and sigma, what does U of lambda do on p and sigma? So, equals what? Okay. All right. Well, the most obvious guess for what it does is that U of lambda on p and sigma gives me, well, just the Lorentz transformation on p. Well, on sigma goes along for the right. Okay. So, this is the guess. This is the most obvious guess one. And indeed, this is what we would get for spin zero particles with no charge, with no other quantum numbers. And sigma would just label if you had five spin zero particles, sigma would run through one through five. Okay. But it's not the most general answer. And what Wigner found is the correct solution to this problem is more interesting. And he found that in general Lorentz transformation on p and sigma actually mixes up the sigma indices. Okay. And what you get is, of course, you Lorentz transform p, but you get other indices, sigma prime. And so, you get a sum of a representation of the so-called Wigner little group. Okay. So, W of lambda. And here I'm using Weinberg notation sigma sigma prime. Okay. So, in general, Lorentz transformation mixes up these other labels. And what these other labels are, are therefore a representation of this thing called the Wigner little group. Very intuitively, we know what these other degrees are free tomorrow. Right. For instance, if I have a photon, if I have a photon moving in the z direction, then we knew ahead of time that this U of lambda couldn't just rotate p because I could do a boost. Let's say the photon is moving in the z direction. Let me, let me do a Lorentz transformation that doesn't change the momentum of the photon at all. I do a Lorentz transformation that's just a rotation in the z direction. Right. Well, if I do that, then we know that I have to pick up a phase associated with the spin of the photon. So, even though p doesn't change, I have to change something else. Okay. And that's something else is the, has to represent the holicity of the photon. Or if I have a massive spin one particle, just at rest, then I could do the Lorentz transformation that's no boost at all, but it's just a rotation. So that's something that doesn't leave them, that again, doesn't change the momentum of the particle, but must change the spin labels and mix up the spin labels with each other. Okay. So this intuition makes it clear that the story of the, of this, of what these extra spin degrees are free tomorrow, what these sigma prime degrees are free tomorrow has a lot to do with those Lorentz transformations that do not change the momentum of the particle. That's the sort of most obviously, because, because a lambda p would equal p would not change. And so by focusing on those Lorentz transformations that don't change the momentum of the particle, you get access to just whatever these other degrees of freedom are. Okay. So that's what we'll start with tomorrow. Okay. So we will begin and carefully set this problem up, as Wigner did following Weinberg, and we will learn, and we'll learn why there's this slightly more formally why there's this profound difference between massive and massless that's reflected in the difference of the little groups between massive and massive particles. Then I will quickly talk about how we account for this difference in the usual way of doing field theory. And I'll explain it goes all the way back to very basic questions about even representing the physical particle states using polarization vectors. And I'll talk about why that forces us into thinking about gauge redundancy in the standard language of doing things. And I'll even, and I'll review some of the early indications going back to the 60s from this more standard language of why it is that there's such a strong constraint on massless particles with spin, why we're forced to have the equivalence principle for gravity, why we're forced to have the Yang-Mills structure for massless spin one particle. And then I will come back to the discussion even more invariantly, and we'll just talk directly about the particles directly about what the, what the, how to think about the amplitudes for the scattering of massless particles and go over the whole argument again at an even deeper level where you can see how the inevitability of the possible particles and then their interactions comes out of a very simple but deep physical argument that we can go through in like a page or so of algebra in order to achieve the, combat the conclusion I advertised them at the end of the last lecture for why the only possible menu of particles we can have are spin zero, one-half, one-three-halves, and two, why the spin one has to be associated with Yang-Mills, why the spin two has to be associated with the general covariance in the usual language, and why the spin three apples got to be associated with Susie. Okay, so those are the things that we'll talk about tomorrow. And I think by the end of that you'll have a, I hope a much better appreciation for why the spin zero case of the Higgs is so special. And after that setup we'll spend some time talking about probing the properties of the Higgs at the future collide. All right, that's it for today again. Thank you, Onima. So now we can move to the Q&A session. So invite anybody with a question to raise their hand. Yes, the first, please call, please. Yeah, so in various points you're talking about