 Thank you very much, it's a pleasure to be back here in Trieste, a very pretty place. So my titles are like, you know, I'm going to talk about axioms, the past, the present and the future. So roughly that divides into the three lectures that I'll be talking about in the summer school. The past is quite long and quite distinguished, so it'll actually beat a little bit into the second lecture. The present will mostly just be about bounds and stuff, so rather quick. And the future, of course, that's a third lecture where I'll talk about some new ways to go and search for axioms and why it's so interesting. So this is just like before I get into axioms specifically just to give you an overall idea of why I think axioms are very interesting. So let's step back for a second and look at, you know, what I would like to call as the landscape of particles. I'm a professor at Berkeley and at Berkeley when you start teaching, you're required to use the word landscape at least once in all of your talks. It's part of the contract, so I don't want to violate it. Anyway, so consider something, you know, like as small as a Hubble scale, about 10 of the minus 43 GeV to about 10 of the 19 GeV, that's a Planck scale. So if you have any excitation, anywhere in this roughly 60 orders of magnitude, that is something that is a particle and it can be described a quantum field theory. We have no reason to think that anything breaks anywhere in there, so anything in there is a very legitimate thing for us to think about. Now we know that the standard model emerges around 100 GeV, okay, that's the electric week scale set by the Higgs particle, obviously. And this is where we all live, right? I mean, of course, the mass of the electron is much lighter, but everything fundamentally comes from this one scale, right? And that's where all of the physics we've observed sits. Now, we don't know if there is anything anywhere else. In a sense, naively we don't know, but in fact, we do know that there is stuff out there, right? For example, the standard model cannot explain, observe facts about the universe. It doesn't tell us what dark matter is. It doesn't contain dark matter. It doesn't contain barogenesis, right? We know we exist. That's something that we really know very well. So standard model despite doing a very good job of describing everything we have seen so far fails to explain these essentially fundamental aspects about the world, okay? And in addition, there are obviously these theoretical worries such as the hierarchy problem, which also the standard model doesn't explain. So we are thus very confident that there should be new physics, right? Definitely that aspect is true. However, we don't quite know where in this roughly 60 orders of magnitude this new physics actually exists. We have no idea about that yet. We have several guesses for what it could be, but we don't really know for sure, okay? And that's the large part of the experimental program that we undergo as physicists to figure that out. So the question is, given that we know that this new physics exists, how can we glimpse them? What would we have to do to figure out where it exists, okay? And a natural thing to do when you want to figure out where something is is to make guesses, right? Is to make guesses based upon what we've already observed. So given that the standard model, we've observed, say, a bunch of symmetry. For example, we know standard model contains gauge invariance that are broken gate symmetries and even some approximate global symmetries that are actually broken. It's a reasonable thing to think that there could be such symmetry structures elsewhere in the 60 orders of magnitude from the Hubble scale to the Planck scale. There could be symmetries sitting somewhere, okay? That's a very reasonable thing to assume. In particular, there could be some symmetry structure that exists at some very high scale, FA as I've called it, okay? Could be existing somewhere. And it is also very reasonable that the symmetry could be broken, right? Much like how we have broken symmetries all around us, it may well be that this global symmetry is also broken. That's a reasonable guess for what could exist out there in the world. So suppose you make that guess. Suppose there's a global symmetry that is broken somewhere at some very high scale. One can ask what are its signatures, okay? How would we know? I'm sure you guys all know this, which is that if you have a global symmetry that's broken somewhere, then there's gonna be a Goldstone boson associated with it, right? That is Goldstone's theorem that says that every time you break a global symmetry, there is a massless particle that's associated with that, okay? That's a Goldstone boson. And the important thing about the Goldstone boson is that the mass of the Goldstone boson is much lighter than, I mean, nominally it's got zero mass, is much lighter than the scale where it comes from, right? So even if you had a symmetry that was, let's say, broken at, I don't know, 10 to the 10 GeV, some gigantic scale, we'll never build a particle collider, at least in the foreseeable future, that can figure out what's directly going on at 10 to the 10 GeV. We can't quite do that yet. However, if the symmetry is broken at that high scale, this Goldstone boson will sit at very low energies and therefore gives us a way to access that high scale if we detect it, right? That's why this kind of program is pretty interesting because it allows us to see what's going on at very high scales because it gives a particle that's at very low mass. Now, nominally, of course, if the Goldstone, if this global symmetry is exact, right? If it's exactly preserved by quantum mechanics, this Goldstone boson must be massless, okay? That's a provable fact, okay? But it may well be that this global symmetry is not exact, but it's broken very weakly, okay? There are some, this is not an exact global symmetry. There's something that appears that screws it up a little bit. And if that gets a bit screwed up, then this Goldstone boson will not be massless. It will instead acquire a mass that is proportional to the breaking of this global symmetry, okay? That's, again, a very reasonable thing to think about saying, well, it's not exactly massless, but maybe it's got a tiny mass because the Goldstone boson, I mean, the global symmetry is broken in a very soft way, in a small way. And I would use that as a definition for what particles like axions or axion like particles are. That is, these are just Goldstone bosons that acquire such a small mass because this global symmetry is not exact, okay? It's broken in a very small way, okay? And they go by the name both of axions and axion like particles. And I'll clarify that nomenclature shortly. Then a global symmetry, yeah, I mean, there's no associated gauge invariance or whatever with that, right? Just some global symmetry. You just make rotations over. It's what? It is independent of space. Yeah, I mean, it depends on what, if it's some internal symmetry, for example, it's just some rotation you do in your system, right? Okay, yeah. It's what? That's right, that's right, okay? So in general, it's a good idea to go after these kinds of particles, because the initial context that we talked about is very general. We're just saying there's some symmetry somewhere, it's broken and why not go and search for this, okay? That's why for me, it's really exciting to do this. Now let's clarify some nomenclature. This is just some bad nomenclature in the literature and predates my existence in the world. So there's nothing much I can do about it. But for the purposes of this lecture, I just want to be very clear about what these things are, right? So why do people say there's an axion? And sometimes people use the word axion like particle or an ALP. That's another expression that you would normally hear. What's really the difference, right? The difference is basically the following. It will turn out, as I will show in these lectures, that QCD can give mass to a specific kind of Goldstone boson. That is, we will actually show that there is one particular kind of global symmetry, which gets a bit screwed up by QCD. So QCD ends up giving a mass to that particular Goldstone boson. And that Goldstone boson is called the axion or the QCD axion. Because of its association with QCD, that's the name it gets. And here in this lecture, any time I'm talking about the QCD axion, I will use the letter A to denote what it is. But there's no reason to think that QCD is the only thing that can ruin global symmetries and give masses. There might be other particles, there might be other reasons why a global symmetry gets broken. And therefore, well be Goldstone bosons that get a mass from other mechanisms. Mechanisms that we don't know about, right? It's just something that, yes. Yeah, I don't exactly remember what people define it to be. But an axion would be a pseudo-Nombo-Goldstone boson, because again, it's not a total Nombo-Goldstone boson, it gets a small mass. So it is the same kind of definition that you use, okay? Yeah, that's another bad name that is there in the literature, definitely. Okay, so yeah, if you have some other Goldstone boson that acquired a mass due to something else, some new physics that we don't know about. They're called axion-like particles, or alps. That's a common expression for them. And here in this lecture, we'll refer to them by the letter phi, okay? Since I can create bad notation, I'm gonna use phi. Yes, I mean, it depends on what you want to use them for, okay? So the question is that can axion-like particles be completely massless? I mean, I'm totally fine with them being completely massless. It's just that sometimes you might like these particles to be dark matter, I think so that's all, things that we'll talk about. So if it's completely massless, of course, it can't be dark matter. That's the only difference, yeah. But for much of the score of the phenomenology, it doesn't really matter except for the dark matter, but it is a difference, okay? All right, so we're gonna call them by the letter phi in the effort of alps, or axion-like particles. And of course, all the statements I make about axion-like particles will also apply to the axion because in a sense, the phenomenology is just that they are just Goldstone bosons that are coming from the breaking of some symmetry. So it's the same, the same kind of ideas applied to both. There are, of course, certain very specific things that are true for QCD, okay? So for the QCD axion or the axion gets some properties like it's mass, etc., because of its coupling to QCD. And there's a bunch of phenomenology associated with that, okay? And so those statements that I make about the axion will only refer to the QCD axion, okay? And they don't necessarily apply directly to these axion-like particles. And I think I've been rather careful when I talk about this in the slides, but I may have screwed up, I don't know. We'll find out, okay? So anyway, that's the nomenclature. If your axion, if your Goldstone boson gets a mass from QCD, it's the axion or the QCD axion, otherwise it's an axion-like particle. Okay, so with that overview, let me give you sort of the outline to the sort of three lectures. We'll start off by talking about the past, right? Sort of the historical motivation for the axion. When people actually talked about the axion to start with, they didn't think about it in this very general way in which I described it right now. It was introduced rather specifically to solve a problem with QCD. It's called the strong CP problem. And I will discuss that in this lecture mostly because, you know, in a sense it gives you a very strong motivation for the QCD axion. There's a reason why people really want to go and search for it. It's not just some, you know, it's not like you're going on some fishing expedition and trying to find particles. There's a real problem that the QCD axion solves, and that's something good for you guys to know. So we'll talk about that today. In the next lecture, I'll discuss the phenomenology of these axions, what they can do for you, and discuss constraints and existing ways to actually search for them. And in the final lecture, I'll talk about some new ways that have sort of come up in the last few years to sort of both discover and detect axions. So that's kind of the plan. So for the rest of this lecture, basically we won't even use the word axion again because I'm going to be discussing the QCD issues and something called the strong CP problem. As I said, when people introduced the axions to start with, they were not thinking about go-stone bosons, symmetries broken, all of that stuff. They were just trying to address a very serious problem with QCD and it's called the strong CP problem. And the reason why we want to pursue this historical route is not because we're in Europe and we like history, but rather because it has the virtue of introducing new phenomena, things that are actually cool things to know both in quantum mechanics and quantum field theory. So that makes for a good way to push forward. To that end, this is something boring. So let us consider regular electromagnetism. This is U1 electromagnetism, something that all of you guys have taken courses on and probably spent a lot of time solving problems on. Maxwell's electromagnetism, something very straightforward. And the Lagrangian for electromagnetism that you've always used is this guy, some f mu nu squared, that's a kinetic term, because usually there's a one-quarter sitting in front. By the way, here in these lectures, I will be giving you essentially overall ideas on why these things are important, et cetera. There are a bunch of details of derivations which I'm not going to put up there. They're very well covered in notes that are easily accessible on the internet. So most of the material that will be discussed today, you can find that in Coleman's lectures called the Uses of Instantons. That's basically where it appears. This lecture will help you understand the themes, the big ideas that are out there. The details of the derivation, I'll leave it to you guys to go and look up. So just understand the big concepts of what the paper is trying to do, rather than pesky details about one-quarter and stuff like that. Anyway, so that's the standard Lagrangian that you've always used in electromagnetism. And a very reasonable thing for you to ask is to say, well, if I can write down this term, f mu nu, f mu nu, that thing, why not add this term at theta times f mu nu, f lambda sigma, epsilon mu nu lambda sigma? That's a term that is Lorentz invariant. I can write that down. And just as a matter of notation, instead of calling this f lambda sigma, epsilon mu nu lambda sigma, it's often called f tilde mu nu or f dual. That's just a way of defining it. So I could write down that term as well. So this is the standard Lagrangian that you've always been using in electromagnetism. The kinetic term and some current, and some coupling to electromagnetic currents. But one could have written down this term. And the question to you guys is, why do we never discuss this term proportional to theta? When you've solved your awful Jackson problems, thank God you never included that term. I mean, we've never done that. It's just a parameter. It's just a parameter. Just some constant number. Just like how there's a one quarter or whatever, like an e, I can put a theta in there. So yeah, it is true that you could take theta to be 0. And that's what you've been doing all along, in a sense. But you also put a non-zero value of theta. And the idea is that that doesn't really break anything bad. It's a Lorentz invariant term, so why not put it? So to answer that question, you should ask whether or not this term does anything useful for you. So to start with, just convince yourself that this term f mu nu, f tilde mu nu, that thing. Express that in terms of conventional electromagnetic fields, e and b fields. Just split it up, f mu nu, all the f0i terms are the electric fields. And the terms fij are all magnetic terms. And then compute that thing. You can convince yourself, if you do that exercise, that this term theta ff tilde is just e dot b, the standard e dot b. And of course, now just because you have a term in the Lagrangian, it doesn't mean it does anything useful. For it to do something, it has to really affect the action in some way, right? That's basically how you figure out if a term has dynamical interest or not. So let's calculate that. Let's see what happens. So anytime you have a term in the Lagrangian and you want to compute what it does in the action, you just do the integral d4x of the Lagrangian. And here for now, we only care about this term theta, right? We already know what these terms do. These terms just give you standard Maxwell's equations and all of electrodynamics. So we're simply interested in what this extra term is doing. So just theta ff dual integrated over d4x. And since ff dual is just e dot b, as you can convince yourself, that is just simply theta times e dot b. And a standard thing to do when you have some kind of integral like that is that you can do integration by parts, right? The most useful trick of calculus. So you can write the electric field as the gradient of some potential and then you want to do some integration by parts, right? So integral e dot b is simply integral grad phi dot b and then you do this kind of integral. So that becomes a surface term, right? As you always have when you do integration by parts minus this extra term here which is phi times del dot b, okay? Now the important point to realize here is that if you had a theory where there were no magnetic monopoles, right? So if you look at Maxwell's equations, del dot b is always set to zero, right? And that is a statement that is true in a world where there are no magnetic monopoles. So let us assume that is the kind of world we're living in. So if there are no magnetic monopoles, then by Maxwell's equations, del dot b is zero, right? Identically, always. So that means this term goes away and tells you that the integral e dot b is always equal to just a surface term, okay? And that's only true as long as we're not in the presence of a magnetic monopole. Keep that in mind, okay? So this term theta e dot b that we talked about is always a surface term as long as there are magnetic monopoles. Now I'm not going to have the time to discuss what magnetic monopoles do to this term in this lecture, but it's actually a fascinating story of many cool things that can happen there. I encourage you to look at paper by Witten back in the 1980s that discusses the nontrivial effects of magnetic monopoles on this particular theta term. And it's obvious that it has an effect because in that case this term does not vanish, right? It's a physical term. All right. Okay, so what we have shown is that when there are no magnetic monopoles, this integral e dot b is entirely just equal to a surface term, okay? And I'm sure as you guys know, the surface terms don't contribute to the classical equations of motion, right? They don't contribute. Why do they not contribute? Well, the classical equations of motion are just that, you know, you... I mean, let's think about it in the Euler Lagrange way, right? Some sort of minimum action principle where we basically say I've got some initial field value, there's some final field value, and I'm going to like think about the sum of all paths that go from one to the other. And we say that the classical path is the path where the action is minimized, okay? And the whole point is that if you look at these surface terms, they basically contribute as a constant to these... to that path, and therefore when you're talking about a variation, which is fundamentally asking, let me change things by a small amount, they don't actually affect the minimum. And we will actually see this in a concrete example shortly of how that actually happens. But this is something that you already know, that these kinds of surface terms don't really contribute to the classical equations of motion. So that's a true statement as far as classical electromagnetism is concerned, right? So that is the reason why you've perfectly, legitimately ignored this term through your entire life as classical physicists. This term doesn't do a damn thing if you're thinking about classical field theory. Let's ask what about quantum mechanical, right? After all, we're all now quantum mechanics, so we should ask what does this term do in quantum physics? Now before we discuss something as complicated as quantum field theory and all these terms like theta f of dual, let's ask what this term does for you in standard quantum mechanics, right? Boring old quantum mechanics that you did as an undergrad. So click to that and let's consider a very simple problem, okay? This is just a Lagrangian for a particle undergoing one-dimensional motion along a line in some potential. So this could be, for example, a simple harmonic oscillator. Lagrangian is just x dot square over 2, I've got the 2 right this time, minus v of x and v of x is just some potential. Let's just assume it's a potential which is a single minimum, right? So this could just be your standard harmonic oscillator, one-half x squared or whatever. And let's now add a new term to this, right? So this is just a standard Lagrangian that you've always used. This is a lot like the standard Lagrangian of electromagnetism that you've used, f squared minus, you know, e mu, j mu or whatever. But now we're going to add a term that is similar in spirit to our theta f of dual, very much like that. So let us take this Lagrangian and to this let us add a new term theta x dot, okay? That's a term that I can add. It doesn't violate anything that we know about classical mechanics. And one can ask, what does this term actually do? Does this do anything useful for you? How does the theta term affect physics? Okay. So let's first start with classical physics. You know, since you've never, I'm sure, used this term ever in any of your calculations in classical mechanics. And we can see why immediately, one can calculate the Euler-Lagrange equations, right? The standard things that you do. And see what it does. What does that term do? So the key point to note is that the Euler-Lagrange equation is the motion is that there's some dt times dL dx dot minus dL dx equals 0, right? That's the Euler-Lagrange equation. So you take this Lagrangian and you do a differentiation by x dot. That's going to give you an x dot term here plus a constant. However, when the dt hits the constant, that's going to be 0, right? So basically, you can see that this term does nothing to this part of the calculation and obviously it has no effect on this part of the calculation, right? So if you have this theta term as written here, it does not affect classical physics. It still gives you the same equation of motion. x double dot plus dV dx is 0, right? Yes. Theta is a constant, yes. Well, I just, I mean, that's the term I'm putting in, right? I'm not giving it any more dynamics. Theta is just a constant. It's a constant exactly like a 1 half x squared, right? It's just a number that I'm putting in the Lagrangian and that's my theory. I want to basically understand does a constant theta term have an effect or not? You're absolutely right that if theta is dynamical in the sense that theta has time dependence, it will definitely do non-trivial things. But for now, we are simply asking does a constant value of theta have any effect? Okay. So that's right. So theta is a constant and that's what it is, right? It has no effect. And you can see this on the path integral point of view as well, right? It's a very kind of a reasonable thing to do, which is that let's take the action, right? And then the action is just simply, and just consider what this term does for you. So that's just the integral of dt times x dot. That's a very simple integral to do. Like you don't need any fancy-pants ways to do that. That integral is simply theta times x final minus x initial, right? If you're taking whatever paths, it's always the same parameter theta times x final minus x initial. So when you do your standard variation for all your Lagrange equations of motion, remember that you always keep your initial and your final points fixed, right? That's basically how you're doing your variation. So this tells you that no matter what path you take, this quantity that you're calculating is the same, right? It's a constant contribution to all the paths that you're taking. Therefore, when you try to do some minimization, right, you're trying to say, oh, let me vary this path slightly to see how this parameter affects physics. It's not going to make any contribution because it's a constant contribution to all paths that you're taking. So explicitly, we have shown that this theta term is completely useless for classical physics, right? It doesn't do a damn thing for you. All right. Let's ask the same question in quantum mechanics. What about quantum mechanics? Once again, this is the same Hamiltonian, sorry, the same Lagrangian, right? It's an extra theta term put in there. And one can ask, you know, if you are just doing one-dimensional quantum mechanics, right? Does this actually do anything for you? And you can convince yourself pretty quickly that this doesn't do anything interesting, right? Because you can calculate the Hamiltonian. So there are some, you know, standard things that you do. P is like dL dx dot or whatever. And you can calculate what the Hamiltonian is. And the Hamiltonian turns out to be simply x dot squared over 2 plus v of x because the theta term, again, has dropped out. It doesn't do anything. So, you know, you can go ahead and actually calculate the wave functions, etc. for this particular potential. And again, it turns out to have no effect. Now, this is also evident from the path integral, the same argument that we gave before. Because once again, if you're trying to do one-dimensional quantum mechanics, what do you do? You say, oh, I'm going to go from this point to this point in space. I'm going to take the sum over all paths. And then I do various calculations based upon what paths I'm taking. But once again, we just said that this term basically is the same for all paths. So it doesn't really distinguish one path from another. So it doesn't actually contribute to anything about the problem itself. Good. So why have I spent 30 minutes or whatever telling you guys about a term that has no effect? Maybe Giovanni made a mistake in inviting me. But I will tell you shortly that this is not quite true. This is always true. That's the question. So let's change the problem slightly. So this is pretty cool. Now, instead of considering just a particle in a one-dimensional potential in a 1D line, let us consider this physical system. Let's consider a pendulum that is swinging under gravity. And we will take this to be a solid pendulum. So basically, this is some pendulum. It's got a bob in it. All pendulums have this. And I'm going to assume that this is a stiff rod or whatever. So the pendulum can actually go a full circle and come down, like a thing that can swing the full time. It's not just some string that will crap out half the way. This is a pendulum that can go in a full circle. That's the problem that I'm considering. So that's the potential. It's OK. So that's a standard problem. And this pendulum, you know what is dynamics. It can simply go around in a circle. No problem at all. And one way to think about that is to say there's an angular coordinate where we say, oh, this pendulum is just swinging around. It's going around from theta of 0 to 2 pi and so on and so forth. Another way to say this is to say that, well, I can think about the motion of this pendulum, not just in the angular coordinate, but rather in the x-coordinate. I can say this pendulum is going around theta to 2 pi. And if I'm thinking about that in the x-coordinate, it would basically be making a motion from 0 to, let's say, 2 pi r and then 4 pi r, so on and so forth along the x-direction. And the pendulum is just sitting somewhere here as a function of time over there. So that's the way that you want to view this now, that think about the motion of the pendulum, not in terms of the angular coordinate, but in terms of the one-dimensional x-coordinate. And the way to do that properly is that you identify points that are separated by 2 pi r. That's a sensible thing to do. OK, so why is this interesting? Now, the standard thing you've analyzed all along in your life is to say the Lagrangian is just x-dot squared over 2 minus v of x. But now let me add a term theta x-dot and see what this does for you. Of course, it's not going to do anything for the classical motion because we have explicitly calculated what the classical behavior is. It has no effect at all. But before we talk about quantum mechanics and how that could potentially affect it, let us look at what the potential looks like. So here's the Lagrangian and I've added some theta x-dot. So this is explicitly the story. There's a pendulum and there's gravity that's pulling you down as always. So what does the potential look like? The potential is something periodic like this. Does that make sense? Because you're down at the minimum, there's nothing happening. You're going around like that. That takes you way up the potential and then you start coming down. So the potential looks like one of these funny, funny looking things. It's periodic, the maximum that keeps oscillating back and forth. And as you said, classically, there is no problem. Theta doesn't do anything because the question of motion don't care about it. But what about quantum mechanically? What happens quantum mechanically? Now, unlike the previous problem, the previous problem we just had a single potential. The previous problem had a potential with a single minimum. The standard sort of particle in a box or a simple harmonic oscillator. It has a very simple looking potential. This thing is pretty complicated. This potential has got many, many minimum. It's got this very complicated structure out there. And this, as you all know, has a new phenomenon. So if you have a particle in a box or a simple harmonic oscillator, then the potential, if you look at the ground state of that system, that is a particle that just sits somewhere there. It doesn't do anything for you. It just sits in a single point. But this particular pendulum has something interesting, which is that if you think about it quantum mechanically, the bob can just sit in the minimum. That's just a ground state to start with, let's say. But it can also quantum mechanically tunnel and go a full circle and come down again. Does that make sense? And then it can do that many, many times. So this system intrinsically has features like tunneling that allow it to go many, many times around a circle and come down. So that's something intrinsic about this problem and it's quantum tunneling that you've otherwise calculated before. So you therefore know that if you look at the ground state of the system, or really any state of the system, then it's not just some stupid thing about the particle just sitting locally. The ground state energies of the system get modified by quantum tunneling. The fact that you can tunnel means that there are additional contributions that happen to the wave function and the energies of the system. One way to calculate that is to say, instead of thinking about it as a pendulum going around a circle, you just think of this as some potential that's periodic. And as a periodic potential, then you calculate what the ground state needs to be. You realize, oh look, there is this ground state, this ground state in each individual little bubble there, but all of them can tunnel together and mix up and do something complicated. Does that make sense? So that's basically how the ground state of the system looks like. The true ground state is a superposition of states that are localized at various minimum. So if you had a standard one-dimensional problem with a single minimum, like a harmonic oscillator, the ground state is just this. There's no funny thing happening. But when you have multiple minima, there are superpositions of these things. That is the true ground state of the system. And so that tells you that something interesting could be happening here in the system already, that this is not as trivial as you thought about it. And for now, let's just calculate or at least review how we would go about doing this calculation. Let's ignore theta for now. Let's not talk about what theta does. But let's just simply ask, how do I go about calculating what the ground state of the system looks like? Or the ground state energies? What would I really do to do that? Of course, the sensible thing to do would be to do something like WKB. When you guys calculated tunneling in your undergraduate courses, what you would do is some kind of WKB approximation. You would write some wave function, do matching, things of that sort. And that's the sensible thing to do this problem. But we won't do the sensible thing. We're going to do something more complicated. Basically, because something more complicated is a good way to jump from quantum mechanics to field theory. So that's why we're going to do something more complicated. It's a more easily extendable way of doing it. So let's do that more complicated thing. But ultimately, all we're trying to do is just calculate what the ground state looks like, the ground state energies. That's really what we're trying to do. This is a different way of calculating that. Good. Let's become even a little bit more formal. Because the question we're asking is something very general. We're saying, given some Hamiltonian, what techniques can we use to obtain the ground state energy approximately? We, of course, can't calculate this exactly. But we just want to say, is there an approximation scheme I can use to calculate the ground state energy of some general Hamiltonian or some general Lagrangian? What trick can I use and how can I extend that trick to the problem at hand, which is a problem of this pendulum going around in a circle? So this is just some standard quantum mechanics. You may have already seen it in some advanced course, but it's actually not all that hard to follow. So let's say this is just quantum mechanics, regular or quantum mechanics. I can have localized states like xi or xf. They're just saying the particle is at position xi or particles at position xf. Those are whatever the ket states or something. They're actually called. So let's try to calculate this inner product, x sub f e to the minus ht, now rather xi, in the limit that t is very, very large. So that's a well-defined thing to calculate. I'm just saying I've got this quantum state. I've got another quantum state. And here is some operator e to the minus ht. And I want to stick that in between these two things and calculate this expectation value. That's a sensible thing I can do. Nobody can stop me from doing that. And we're going to calculate this in the limit that t is very, very large. So that's a calculation that I can formally start doing. How do I do that? Well, so what I'm going to do is I'm going to take that state and then I will do the standard trick of inserting a set of complete orthonormal states or whatever it's called. So I'll just stick in some n's and some m's. And then I do a sum over all n and m. And now something interesting happens because these are the eigenstates of the system. Sorry, I'll choose them to be the eigenstates of the system. They're not just some random states. I'll choose these n's and m's to be the orthonormal eigenstates of the system. So I do that. And then this sum becomes something more easier because this guy, e to the minus ht acting on m will only give you some eigenstate. So that gives you an e to the minus ht, say, hits m. It'll give you e to the minus e sub m times t or whatever. And it's non-zero only when n and m are the same. That's what eigenstate is doing for you. So that complicated sum becomes somewhat more simple. It's just some inner product of x sub f times n and this thing times e to the minus nt. Nothing crazy here. This is just some standard inner product. I'm simplifying it using rather standard tricks. Now the cool thing is this. Notice that if you take large n of t for large n of t, this is now some sum of a bunch of exponentials. And it will therefore be dominated by the state of minimum energy. All of the states are going to have higher energy and therefore they will be exponentially smaller than the contribution from the state of lowest energy. So all you have to do, therefore, is to say, take this particular expression and try to calculate what it is at large t. And when you do this at large t, automatically the answer that you're calculating will be dominated by the ground state energy. That is just a trick of exponentials. Now this is, of course, good. We've already said that this inner product is now mostly just this at large t. But we haven't yet done anything very useful yet, basically because we don't actually know what the ground state wave function is. If you kind of already knew what the ground state wave function is, you can then actually go and calculate the energy directly. If you don't know what the ground state wave function is yet, I can't really calculate these kinds of inner products to get anything useful out of it just yet. But still this is pretty good, because we're saying we've already found a nice way to get what the ground state is without doing some hideous calculation. All right, so the important thing to observe now is that this quantity, this x sub f e to the minus h t xi, this thing, can be calculated in a different way. It can be calculated using Feynman's path in a row. So if you look at Feynman's path in a row, what does he say? He says, if you want to understand how a particle, let us say, moves from one point to another point, I do the sum over all histories. And for each path that you take, you sort of cost yourself some action factor, e to the minus i s or something. And that is basically what the standard Feynman calculation is. There's, however, an important difference between those two, between the Feynman calculation and what we're doing here, which is that Feynman's calculation was for the real evolution of a particle in real time. We would say, if you want to go from point A to point B, I'm going to be calculating this in a product, e to the minus i h t times xi. So that's what Feynman is actually telling you to calculate. He's telling you that that's a unitary evolution that is taking you from this point to this point, and that is formally just equal to the integral over all paths times e to the i s. That's Feynman's calculation. But we're not trying to do what Feynman wanted to do. For us, we don't actually have an i. We have a minus sign here. So what we have is a minus sign here rather than an i. That's what we have. So the exact calculation we do is not going to be the same thing as what Feynman did. It is rather for the slightly different object here than what Feynman had before. And that's called the Euclidean action. And this is not something too hard to understand. All we're really going to be doing is to say, oh, I've got some calculation, but I've got time, and I'll replace time by i t. That'll be somewhere in which I can go from one language to another. But we'll sort of describe that shortly in more detail. But this is the general idea. We have this beast that we want to calculate, and we're going to use a Feynman pathological style approach to do that calculation. So what is this Euclidean action? Since this is not the standard Feynman thing that we used, it's not an i. We have a minus sign over there. And you can see that you can basically go from one answer to the other as long as you take your normal action that you had, sorry, the time coordinate that you had there, and send t to i t, i tau. That's just a way for you to do this. So given this, we can see what the Euclidean action looks like. So we have our standard action that we were using, x dot squared over 2 minus v of x plus theta x dot. That's what we were trying to understand. And we're simply going to analytically continue, if you will, take t to go to i tau or something. And that basically does this rather simple change to what the action looks like. So instead of it being x dot squared over 2 minus v of x, that's simply just going to be minus the whole thing, x dot squared over 2 plus v of x. So there's an important sign change between these two parameters. Here there's a lot of the minus sign, and that disappears over here. And very importantly, this term theta x dot now acquires an i times theta x dot. So all of this is going to be important. So here's where you can start seeing that this theta term may have interesting effects. Because this term i, the theta term here enters as a phase. And therefore, if you had something funny with how the paths were changing, et cetera, this is telling you how something may change when you add things together, because this comes in as a phase. And that's basically how it could be important. But as yet, we haven't said anything brilliant. This has all just been some general discussion about Hamiltonians and how you calculate the ground state energy. Everything I've said so far applies both to the pendulum that is going around in a circle, as well as this standard one-dimensional particle in a box-style stories. So as yet, there's no reason why you should think theta is special. Because this is just very general. We haven't said anything yet. But let us persist and see how we can get new effects and how do we do this. So we basically ask how do we calculate the Euclidean action that we already have this. And once we know that, what kind of approximations can we really use? So to do anything useful, I would have to basically know how to calculate this pattern overall. It's all good, looks nice and formally, but to do anything useful, you have to learn how to calculate this. I want to sort of gloss over this technicality here. Well, maybe I'll actually go over it. So as you know, if you look at this kind of integral, the paths that will actually contribute are going to be paths that are near the paths of minimum action. So you've already used this that you try to do like classical calculations. When you try to do a classical field theory calculation or a classical mechanics calculation, you say, ah, the thing that really matters to me is the path of minimum action. That's the one that you always use. And of course, if you just try to do that, you're going to find that that's strictly speaking, a set of measures 0, because it's exactly one set of path. But really what contributes to most of these guys is not the exact path, but rather like perturbations about that. So those are the ones where the coefficient will be small and their set of non-zero measure turns out. Anyway, that's just sort of a technicality. You should sort of understand that when you're doing these path integrals so that you actually understand what you're doing. But really what matters when you're trying to do this kind of computation is that you care about the paths of minimum action for this particular... So, what are the paths that minimize the Euclidean action that is this particular quantity? What are those paths? And we already know the answer to this question. This is very good, because what are the paths that minimize this action? Well, they're just simply the classical trajectories that a particle would undergo if it was moving under this kind of potential. So in the standard story, when you were doing this, you were thinking about a particle that was moving in a... Like the standard Lagrangian that you've always used was a Lagrangian of the form x dot squared over 2 minus v of x. And you calculated your classical paths for that kind of motion. But in this case, there's a relative minus sign. This is x dot squared plus v of x. So what you're saying is that if you look at the paths that actually minimize this action, they're going to be classical paths taken by a particle that moves in a potential minus v of x, rather than plus v of x. That's going to be important as well. Does all of that make sense? So what does it really look like? So the potential minus v of x is simply this guy. So v of x was that funny potential before. You take the negative of it, it looks like this. And now when we say that we want to go from one point to another point, we're simply asking how do particles move in this kind of potential. That's all we really want to do. And for simplicity, we will simply take the initial and the final points to be the same. Xf is an xi in this calculation. It doesn't really do anything for you. It's just some exponential anyway. So just for simplicity, let's simply just do that. So all we're really asking is to say, oh, let me ask what happens for a particle that starts at this point and goes to some other point and comes back, but I'm going to make the initial and the final point to be the same. What kinds of classical paths do I really have? So here's where you clearly see the difference between the standard like 1D problems that you've been using all along and this particular pendulum that you cared about. So with a standard particle in a box that you cared about, your potential was minus v. That's what you're looking at. You're looking at motion of particles over there. What paths, what classical paths can a particle take which starts at this point x1 or xi and comes back to xi? So for your standard issue, the standard problem that you solved, there's only one path. The path that just sits there doesn't do anything. If you go to the left or the right, you can't really come back. You're simply stuck in the same place. For the pendulum, however, you realize there's a class of paths possible. There's one path where it just sits right there. It goes from this point to this point and bounces back. And there's another path where it goes from here to here and then bounces back from there and so on and so forth. So you see that in this case, when you've got this multiple wiggles in your potential like that, there are a large class of paths that contribute to this problem. Yes. Yeah. Yes, but it is good. Keep that thought in mind, but you can see that it's not phenomenologically distinct. We will talk about that shortly. Keep that point in mind. We'll talk about that. Good. You can think about that in that language or you can say there's just something a periodic potential along x and it's the same thing. All right. Okay. So these additional paths that you've got, they get a name. So these additional paths didn't exist for the original problem. So we're going to call them instantons. That's the name that they have. These additional things that are going from one place to another. Okay. Of course, physically what you're thinking about is that in this particular problem, there was no concept of tunneling. There was only one particular minimum I had and I'm just sitting right there and nothing happens. In this particular problem, I intrinsically expect to have tunneling. That's kind of how we even started talking about this. So clearly these additional contributions that exist here must have something to do with these additional instantons that exist here, which are simply the names for these paths. They must clearly have to do something with actual tunneling. Okay. So let's now see what these additional solutions do. These instantons that clearly exist which are obviously important to calculate tunneling and things of that sort. Okay. And in particular, let's ask, what does this data term actually do to these instantons? Okay. Now this is where we actually can address the question that the gentleman there raised. What distinguishes these different paths? Right. In fact, I was indeed saying that I start from this point, I come back to that one point. Is it not the case that somehow I'm coming back to the same point when I do this calculation? But the key thing to note is that there are different kind, each of these paths that exist over there, they're in a sense distinct, they're topologically distinct. Right. Because if I have a path that I'd say that starts from this point and goes there, that basically corresponds to the pendulum going one full circle and coming down there. Right. On the other hand, when it goes all the way here, it sort of goes all the way up there and sort of does it twice and so on and so forth. Right. So if you have this kind of circular issue happening, a path that goes around the pendulum once, that path is topologically distinct from a path that goes around the circle twice. Right. In the sense that you cannot make small changes from one path and get to the other. Right. If you go around the circle once, you've wound around the circle once and if you wound it around twice, you can't really do anything. Okay. So in general, if you look at these kinds of solutions, there is basically a, if you look at each of these instanton paths, or go around, for each such path, I can characterize it by something called the winding number. Okay. Which basically is telling you how many times around the circle is my path actually taking me. Okay. So if you go around the path once, you've got a winding number one, you go to n times, you've got winding number n. Okay. And now we can actually see what theta will really do to this. Right. Because if you look at two different paths that have different winding numbers, right, then this will actually give you slightly different contributions there. Yes. Yeah. Because if you go around once in one direction that's winding number one, other direction will be winding number minus one. So now you clearly see what the theta term is doing for you, which is that if I take one path that goes around the circle once, that's going to give you, I mean say n times, that's going to give you a phase that is i times theta n. Right. So when you've got these multiple paths that exist in your system and you can't really do small changes to go from one set of paths to another set of paths, this theta term is telling you how you should add those different solutions together. Does that make sense? Right. So that's basically how this effect starts becoming physical. Right. So the theta term is a phase that tells you how to add these topologically distinct solutions all together. Okay. So this of course was completely absent in the case of a particle in a box, things that you were used to do before, because there were no instantons, right? They were all just like one trivial solution. So who cared what theta was? If we just reabsorb it in something else, it doesn't really matter. But here in this case, you clearly see that when you do the calculation, you clearly see that depending upon what the value of theta is, I actually count my different paths in a different way. Okay. So that's where theta starts becoming physical, that it counts those different ways that the Bob can tunnel are counted differently in this case. Okay. That's basically how it works. I think I've basically said everything here in this thing. Yeah. Yeah. But they would count the phase would be identical for everything. Yeah. I mean, you count them, but it would be counting zero, right? That's all. Yes. The theta will be reflected at what? Whose vacation motion? Heisenberg's thing? Yeah. Yeah. There it would basically be like you, I mean, you know, there, like you'd have to solve for the actual tunneling, right? You have to do the actual WKB kind of thing. And there you have to see that this tunneling would actually come in that way. It would come in really the way you create boundary conditions for your wave functions. When you actually set, when you solve your problem, you have to choose your boundary conditions, right? And that's where it would actually come in. No, no. But remember, when you're actually defining it properly, you got to pick your correct conjugate momentum, right? I actually wrote down that equation before, but there, if you look at your true definition of the conjugate momentum, it's actually X dot plus theta. So you got to take care of that carefully when you quantize the system correctly. That's where you would actually see it. So I was kind of quick when I said it earlier, saying, oh, it doesn't appear on the Hamiltonian, but that's actually strictly not true. I mean, like, one has to go through that carefully. Okay. Good. All right. So, you know, I've given you the basic idea of how the physics actually works, right? It's telling you that this is what theta is doing for you. That is counting how these circles, I mean, how these different solutions wind around. Okay. Now, there's a bunch of details on how you actually calculate this. If you want to know the full details, I recommend that you look at Coleman's book. This is, you know, uses of instant ons, okay? And essentially what you do is that you take your solution of minimum action and then you perturb about it. As we said earlier, it's these perturbations that actually contribute to your calculation, okay? Anyway, so that's just some detail. I will let you guys look at, you know, Coleman's book or whatever to go through that actual calculation and how you do that. If you, you know, do a bunch of calculation, calculation, calculation, eventually you'll find that if you calculate the ground state energy, right, that will basically have a term that looks like this. So, normally it would just be someone half h bar omega or whatever corresponding to some fluctuation about the minimum, but now you've got some extra term that is, you know, of this particular form. It's got a cosine theta that sits in front. The cosine theta comes about because if you have an e to the i theta and you're, you know, summing over n, for example, that'll give you a cosine theta, right? That's where that appears. And it's got this exponential pre-factor here, which it must because this is a term that is coming from the Euclidean, I mean, sorry, from some kind of tunneling as we talked about, right? It comes from the fact that there are different paths in this circle, okay? That's basically how you see that. This pre-factor k is a pretty complicated pre-factor, okay, that it'll actually be important for us when we discuss the axion, but for now I'm going to ignore how you actually calculate it. It's some, it has to do with the, you know, some sort of functional determinant of these operators here, okay? Anyway, that's just a bunch of details. What we've understood so far, the key thing to understand is that theta is now physical in this particular problem, okay? And fundamentally, this has to do with the actual topology of the system, right? Because when you had a normal 1D potential, it didn't matter because there were not these multiple paths. However, in this particular case where the pendulum is going around in a circle, there are these multiple paths that one can take and therefore theta tells you how to count them in a different way, okay? And that's, yeah. Yeah. Yeah, it's just some tunneling, right? So there's always a ground tunneling that exists, right? That's always true. It is just that there is still a theta dependence on it, that's all. Yeah, because if you set theta to 0, you can still, you know, it's still the potential that you've calculated before. So this tunneling still exists, right? It's just that it's not, yeah. Okay. So the key point that you want to understand here is that theta actually changes the spectrum, right? And this is essentially a, like an important point. Alright. So what does all this have to do with axions, right? I mean we've, I'm supposed to be standing here talking to you guys about axions, but instead I've spent like the last half hour telling you something about one-dimensional quantum mechanics, okay? And all we said was to just say axions, when I defined it, they were just like some goldstone boson of some broken symmetry. That's basically how we defined it. And the point is we were trying to understand why people introduced axions to start with. Why did someone go through the trouble of creating something like that, okay? And to answer that question, we said, oh, let's look at standard electromagnetism, okay? And let me add a term theta f of dual and ask what it does. And then we said it doesn't affect classical electromagnetism. That's an absolutely true statement. And then we were trying to understand whether such terms in field theory have anything interesting. Because as we said this is just a surface term, right? And what we've just seen now is that in one-dimensional quantum mechanics, a surface term of the form theta x dot actually has interesting effects when you sort of treat the problem correctly with all the sort of topological issues coming in there, okay? So we're going to ask therefore, what does this do to field theory? Okay? In particular, what does it give rise to some non-podervative physics like you talked about earlier? Okay. Let me just get some water. All right. So let's in general take, you know, some gate theory. It can be a billion or non-a billion. It doesn't really matter. And well, it will matter. But like for now, let's take it to be something like that. And for now let's consider a world where there are no fermions. Okay? Let's ignore fermions completely. This is a world without fermions. And it's a very simple looking Lagrangian. This is a standard kinetic term. And I add this term theta f of double, right? And we can ask if I take this quantum field theory what does this ground state look like? Right? I mean we were previously calculating the ground state for some pendulum swinging around like that. And now let's ask this question about a quantum field theory that's described by this Lagrangian. So of course, that's a pretty hard thing to calculate in general, you know. So, but the same procedure that we used basically works, right? That is what we did was to say in the case of the particle in the box or this particle in the pendulum, we just calculated some object like this, right? Some integral over all paths, e to the minus Euclidean action, right? And we said that quantity is dominated by the ground state energy of the system. And here we can do something very similar. We take the Euclidean action for this particular theory and we can sum over all particular, all paths in field space. And that's again an answer that should be dominated by the I mean we'll be dominated by the ground state energy of that system, okay? And what are we summing over? In the case of the quantum mechanical problem, we were summing over physical paths that the problem, the particle could take. And here we are summing over different paths in field space that the particle can take, okay? So again, to calculate that thing, we need to know what are the solutions to the classical equations of motion for the Euclidean thing. And as we said, the Euclidean action is something where you just take time to I times time or whatever, okay? So that's what we do. And you know, the Euclidean action, the solutions to the action are simply going to be given by some differential equation, right? As it always says, there are some differential equation that tells you the d mu f mu nu zero or whatever. Those are obviously the classical solutions to that Euclidean equation, okay? And the crucial thing is this, when you solve differential equations and you want to obtain solutions, you always have to impose boundary conditions, right? That's how we solve a problem. So we have to ask what kind of boundary conditions do we really care about in this particular case, okay? What sort of boundary conditions are actually appropriate? Now if you think about it we actually care about quantities that give non-zero answers, right? We don't really want to spend a lot of time calculating something that gives you zero, right? And so there's some Euclidean action that you're actually calculating. It's effectively some kind of energy that you're integrating over, right? And you're obviously doing this over all space, okay? Now if somehow those, if that integral was diverging as you sort of went up, you know, I mean, if you had like finite energy everywhere, like, you know, if you have non-zero energy throughout space, when you do this integral, that will actually give you an infinite answer. And obviously that will not contribute very much to what you're trying to calculate, right? So what you care about are sort of solutions to the Euclidean equation, right? That basically have zero field strength if you will, out at infinity, right? Because if you have zero field strength out at infinity that will not contribute to the energy density and therefore you will get like a finite action out of it, at the end of the day. Therefore, what you want to be doing is that you want to be looking at solutions to these equations of motion that are essentially zero field out at infinity, okay? That's what you want to be looking at. And of course each solution will be something that will appear in your path in a row, just like how our, the rotation of the pendulum bob in the full circle appeared in the path in a row as well. So we care about all of those guys, okay? So obviously for the solution to a finite energy, okay, in this case the field strength must vanish out at infinity. That's something that we require. The central point to note is the following, right? Is that the vanishing of the field strength like an electric field is not the same thing as saying that the potential must vanish, right? As we know, what we're summing over here are various configurations of the gauge potential. That's what we're calculating, right? But the boundary condition is something that is telling you that the field strength f which is equal to da needs to vanish out at infinity, right? So if you had some potential which for whatever reason gave you no field strength out at infinity, that would still count, okay? So you want to not think that just because the field strength vanishes out at infinity, the boundary condition to set is to say that the potential must vanish, right? That's not the right boundary condition. The correct boundary condition is to say the potential could be something which is still the case that the field strength is zero, right? Those are the kinds of boundary conditions one must pick, okay? So this becomes kind of a math problem now, okay? Which is that basically we are saying I want to solve this differential equation. I've got to pick boundary conditions. I'm going to pick boundary conditions so that the field strength is zero out at infinity, right? And that basically tells you that the gauge potential, this value a mu has to be like pure gauge, right? It has to be just a simple gauge out at infinity because that's how it has no field strength even though it's got a non-zero value over there, okay? And that is telling you that you're basically saying that if you look at your space time, right? The boundary of your space time which is where you're putting your boundary condition. So for example, if your space time is r4 the boundary will be s3, okay? And you're saying, oh, given this s3 I'm going to be looking at a map, okay, a function that maps that boundary to the gauge group, right? Because I'm saying I've got my boundary s3 and I've got to pick a boundary condition there where that boundary condition is just purely, just pure gauge, right? So at every point in that space I'm picking some continuous function that maps to a point on your gauge theory, right? So this is a very concrete kind of math question which is simply saying what are the kinds of functions do I have that map me from s3 to the gauge group out at infinity, okay? And every such map will count, right? Because all of those are sort of legitimate solutions that will give rise to a contribution here to this integral over there, okay? And, you know, the issues again the same thing like much like in the case of the pendulum going around in the circle, right? We had a multiple solutions. So there was one solution that went around the circle once and that could never be deformed into a solution that went around the thing twice, right? And that was a property about the actual topology of that system. Similarly over here you're asking a math question which is to say if I've got my boundary of space time which is s3 and I've got a gauge group g are there maps that take you from this space to that space that cannot be continuously deformed to one another? Like is that something you can do? So, I don't want to ask me, but ask your friends in mathematics this is a well-studied topic of what these things can be, right? So, for example here, if you start with let us say an abelian theory like u1, right? Then you can actually show that all maps from s3 to the u1 are trivial that they can always be deformed like to the trivial map, okay? You can continuously change this and go down to 0, okay? So, if you have electromagnetism then this there's only one solution to this problem, right? That is, everything can be trivial and the theta term doesn't matter, right? Because the theta term always came in only when you had multiple solutions that had to be patched together. When you have only one solution doesn't matter, okay? Of course, this conclusion does depend upon the boundary that you chose on the kind of space time that you were in, right? So, in R4, like the world that we live in then it is indeed the case that the boundary of R4 is s3 and mathematically when you map s3 to the u1 of the abelian theory all such maps are trivial. If you had lived in a one plus one dimensional world then the boundary of that space time which is R2 would basically be s1, right? In which case there are now non-trivial maps from s1 to s1 itself, right? You can map a circle and there are multiple paths you can take around it and in that case it would not be the case that there would indeed be multiple possible there, okay? And that is exactly the way in which this solution is identical to the case of the pendulum, right? There would be like multiple gauge paths you can take and each of those paths would contribute differently because of the theta term, right? There would be counter-differently in this calculation, right? So, I'm just going to like breeze through what this final point here is which is that now you can look for, you know, all maps from s3 to g, okay, whatever gauge group you want. Let's take this g to be s1, right? And for any n, I mean other than n equal to 1, so anything above or equal to s2, there are always non-trivial maps that take you from s3 to this gauge group. That's just a math statement, okay? So, each such solution that you take will contribute differently to what this path integral is, right? That's what we said, okay? And formally and much like how in the case of the pendulum, each such winding number contributed to, you know, there's an exact number associated with it, right? Each solution gave rise to some integer that counted how many times you went around it. Similarly, for this more complicated system where you're mapping s3 to this gauge group, there's something called the Pontriagin index which counts how many times if you will that something winds around it, okay? That's just a higher-dimensional mathematical property. And such solutions, these different classes of solutions that are telling you how you wind, you know, the Pontri around this thing, they're called instantons, just like how they were called instantons in the case of the standard pendulum and sort of swinging. And they will indeed, they're sort of like tunneling if you want to think about it that way. They correct the ground state energy of the theory, okay? And now we can truly see what the theta term actually does which is that if you calculate what the theta term is going to do for you, right? It'll once again, when you do the integral in the action, this will just be some integer number. And when theta is non-zero, it will tell you how to add these different instantons together. Okay? That's basically what it's doing. So you can go ahead and calculate what this term looks like. And ultimately, what you will find is that the ground state of the system, exactly like how we calculated this in the case of the pendulum going around a circle, this ground state also gets a contribution times that's, you know, some awful functional determinant times cosine theta times some parameter like that, okay? So if you want the real details of the story, you can go and sort of look at Coleman's paper. But fundamentally, this is the idea, right? As long as you understand this pendulum going around a box, I mean going around a circle, you're exactly mapping that intuition on the field theory, okay? Yes, there's some extra mathematics telling you that there is some map from S3 to some gauge group instead of just maps of things going around a circle, but it's the same fundamental physical point, right? That's really what I want you to take from that particular lecture is not all these details, but really that, you know, understand the pendulum going around a circle and you've got the full story down. Okay. So going back to what we have what the summary of what we have learned so far, okay? Which is that if you want to understand what these theta terms do for you in field theory, right? Much like the case of the initial quantum mechanical pendulum, right? The ground state energy of the system is indeed calculated by using this Euclidean path integral, the same kind of approach that we used. This path integral is dominated by solutions of finite energy density, right? That's what we would need, otherwise it's basically like you're adding zero, it doesn't do anything for you, okay? And the key point is that these finite energy solutions are characterised by the fact that the gauge field strength has to vanish at infinity, but not the gauge field itself, right? So the fact that the gauge field can be non-zero but still give rise to zero field strength, that's what makes all this magic possible. And then all we are just simply saying is to say, oh, given that all I need is to have a configuration whose which is pure gauge out at I mean out in the boundary of space-time, I'm just simply looking at what are the different ways in which that can be possible, right? That's a math question that says, given my boundary of space-time s3, how many ways can I map that onto my gauge group and still get a non-zero answer, okay? And that's some stupid math question, you can ask your math friends what the answer is and for certain groups, the answer is zero, I mean, it's trivial, there's only one possible way and for some other groups like SUN, there are multiple paths that do this. And when you have multiple paths that do this, the surface term theta counts each such path differently, right? And then when you all put them together, it gives you something physical. Okay, great. Therefore, we realize that just because you put a theta gg-dual out there, well I've switched notation, I've gone to QCD now. So we've argued that for electromagnetism, right? Standard U1 electromagnetism this term has no effect, right? Because we just said that for electromagnetism the mathematics is such that there are no non-trivial maps taking you from the boundary of space-time to U1, but this is not true for any SUN, when N is greater than or equal to 2, okay? So both for SU2 and SU3 there are non-trivial instant-down solutions that actually do contribute, okay? But the crucial thing about this whole story here is that to see if this term is important, right? Remember there's an exponential pre-factor sitting here, right? If that exponential is basically 0, right? Then who cares, right? Who cares about theta? It doesn't do anything for you. So I have not actually shown this here but when you go through the correct calculation of this, you know, find the actual solutions, et cetera, et cetera, normalize things correctly, what you will find is that the exponential pre-factor that sits out there is in the form e to the minus 8 pi squared over g squared, where g is the gauge coupling, right? So if you have a theory like SU2 the that will make SU2, then what you would put in there is the G2, like, you know, the gauge strength of SU2 for QCD you would put G3, right? The thing. Now the point is that if you look at our real world in SU2 G2 is very much smaller than 1, it's a tiny number, right? So this parameter e to the minus 8 pi squared over g squared is a super small quantity, right? Because 8 pi squared over g squared is huge and this is e to the minus of that, so that term doesn't matter. So in the standard electroweak theory, okay, in SU2 this term is completely relevant, doesn't do anything for you. That's why nobody talks about theta of SU2, right? Doesn't do anything for you. However, for QCD, we know that G3 basically hits strong coupling, right? This, you know, QCD confines and G3 becomes over 1, right? So for QCD this term is not it's actually important, right? It's not some tiny number. It's an important thing. So the key question that we want to ask is to say what are the effects of theta on QCD? We've already seen that this gives rise to a change in the vacuum energy, for example, right? Like this actually affects the ground state energy of QCD. And that is basically what we will talk about tomorrow. Discussing the strong CP problem. How the theta term actually becomes physical. We've seen how it becomes physical. We're going to discuss how it actually leads to effects on physics, okay? That's for tomorrow, okay? Sorry, could you go back to the slide where the energy is shown in the dependence of theta? So it's proportional to cosine theta? Yes, this one. So if this is an exact result, this means for a particular value of theta by half, tunneling is extremely suppressed. Right. I wonder if this is a real effect or is it at that point approximation breaks down? It is a real effect. Well, I mean it's not that the energy is exactly just this, right? I mean I've only written the terms that are theta dependent. So of course there are other terms out there. Like this is simply the term centered by theta. But for example, for the pendulum, I mean your original motivation was that terms which are not classical solutions are extremely suppressed. Yes. But when you take theta is equal by half, then all the terms with actually finite actions of the classical solutions cancel each other exactly. Right. So it's the other terms which are not classical solutions which will actually contribute. No, but there's always a ground state energy that is big, right? The other terms, yes. So if you really look at the, yeah, so okay, fair enough. So in the true limit of theta equal to pi over 2, there are other things that you would, you know, there's a correction to the classical things that are very, very small that you've not calculated. That's what I meant. So this example cannot be an exact result in the T-teco-pytate. Sure, sure. So it seems that when you have one potential well, you have one energy state, and when you add another one, you have uncountable infinite amount of energy states, right? Isn't that a problem? Why is that a problem? Because this is like the standard solution of block wave, for example, you know, if you take a solid, right? So obviously, you know, if you took the true infinity, you put in a true infinity, so you shouldn't, you know. So if you take a finite, like solid or whatever, right, then you have these sort of band structures on a solid. It's the same kind of idea, right? The fact is you've got these multiple potential wells and into the ground states just being, you know, distinct like that, they split into a sort of a band, right? It is, in fact, the same physics that's happening there as well. So that is a continuum, right? Like we have a band structure, same thing. And of course, if you have a finite box, it won't be truly continuous. Spacings will be very small, but the same thing, yeah. So we've seen that with the periodic potential with the wells, we actually get interesting physics. But why do we know that this system of periodic potentials even exists for QCD, for instance? Why is the correct potential to use? Well, no. So in QCD, all we are saying is let me add this term and ask what does it do, right? So the question we're going to be asking tomorrow, actually, to say, well, we've seen that this term is not irrelevant. It actually has effects. Let me add this and now say what happens to my theory. We've already seen something happen, which is that it changes the energy density. It could do something more, and that's what we'll ask tomorrow. So in the proper space-time calculation, what is the analog of the quantity that you've written on the board for the pendulum, namely XF and Xi bracketed around E to the IHC? So remember, these are actually non-perturbative effects, right? So basically, you can't see the effects of theta in a perturbative language. You can't just say, oh, let me start with something and add some extra term and do it. So you don't quite see it that way. You have to truly see it. This is the way you'd actually calculate it. So you can't just start with a perturbative description of the system. So you can't just start with quantizing in one way and then saying, oh, let me add this theta term and treat it perturbatively, because that will never give you this particular effect. Sure, but when you're doing the calculation, at any point do you write down anything which is analogous to what you've written on the board, namely XF and Xi bracketed around E to the IHT? Sorry, say that again. When you're doing the spacetime calculation, at any point do you write down something that's analogous to what you've written on the board? Are you talking about, is that a question for field theory or are you asking me a question here? I guess I don't understand the gist of your question in field theory. What about it? I don't think I can explain it. Never mind. I think he's trying to ask. You're trying to calculate an expectation value there. Yes, what is the analog in the quantum field theory calculation? Yeah, in the quantum field theory calculation it is simply the fact that you're taking different paths that the field can take in field space. That's exactly what you do in quantum field theory, because in quantum mechanics you take different paths that the field could take in physical space. In field theory you're just saying, I have one classical field, another classical field, what are the different ways I can add them together? Excuse me, is there any particular reason why you say that the surface of R4 should be S3? Couldn't it be a hypertora or something like that? It depends on what you think your topology of space is, right? I think I live in R4, so I'm taking that. Okay, thank you. In the case of SU2, you were saying that the theta parameter that parameterized that term basically just happens to be very small based on the measured value of the gauge coupling for an electric week. I wasn't the impression that there was something a bit more fundamental in a theory that made the theta parameter in the case of SU2 unphysical. Am I mistaken in that? It is unphysical in the case of electromagnetism. You want electromagnetism, okay? Unless you have magnetic monopoles. If you have magnetic monopoles, theta is very much physical. You don't need any of this stuff. It's just directly physical. But for electromagnetism, it is unphysical as we just argued, because it's truly a surface term in that case. If SU2 had been bigger, it would have had more effects. Maybe, no, I think even if SU2 is large, you won't have any effect, because you can rotate it away. Press you too? You have an exact beam of cells. That is the standard model specifically. Here we're just talking specifically about this. There's no fermions in the story, right? Once you add fermions, things are very different. I have a bit of a naive question. You will see an approach that you should write down every possible term. How is theta suppressed in terms of the mass scale? Theta is completely unsuppressed. It's a dimension for operator. That's why there's an interesting question. I should probably emphasize that. Normally, if you're a high-dimensional operator, you'll be like, okay, there's some scale that suppresses it. Here there is your naive expectation of what theta should be, should be order one. Okay. I mean, I was suspicious when I see parameters in the Lagrangian, because they cannot be measured directly. Does this thing get somehow renormalized? When you speak about theta, do you already mean they're renormalized theta? It's always a renormalized theta. It's what we physically measure, of course. The question that is all boiling down to is given that I have this value of theta, what can I physically measure from it? Today, we just talked about how this does something. It changes energies, for example. Tomorrow, we'll talk about how it actually affects what's called the neutron electric dipole moment. It actually gives rise to physically measurable parameters. So you argument crucially dependent on the form of the surface of the topology, what difference would arise if you had another topology of space time? Certainly, if you had one plus one dimensional space, that's something where even electromagnetism would matter. For example, also four dimensional space, but another topology? I don't think it really ultimately ever matters because most of the effects that people really ultimately care about are local effects. So when you do all these instanton calculations for QCD, for example, there's confinement, things like that happening at sort of relatively short distances. It doesn't really matter what happens to space out really, really far away. As long as this instanton solution fits in your region of space time, this calculation would work through. Any more questions? Thank you.