 the principle bundle. And then I pulled back my connection omega using these S alphas. So that gives me a bunch of local one forms, which I can also call the connection. So A equals S alpha, A alpha is S star omega. Okay, so what is this term when squared in here give? It gives a contribution to the potential energy, which looks like this. So the Lagrangian equals a half commutator of A with phi squared plus some other terms. But this is the one that we care about. So we just plug those two expressions for phi and A into this Lagrangian. And what we find is my minus two phi squared A mu X squared A mu X squared plus A mu Y squared. So, okay, if you recall, what I said masses look like for free scalar fields, this looks like a mass term for the gauge field. This is sort of remarkable because gauge invariance normally prevents me from adding such a term to my Lagrangian. But I can get this term at low energies, where phi takes this non-zero value. So any questions about why that looks like a mass term? It's like M squared, phi squared. Okay, so what does this mean? Well, it means that at low energies, I can construct a new effective field theory where I'm not able to turn on this AX or AY field anymore. I've got some, so I've just lost some of these gauge fields. It costs too much energy to, you know, this is some contribution to the potential energy in my Lagrangian, so it's just gonna cost too much energy to turn this on. Yeah, just give it a non-zero value. Yeah, or like it'll be extremely small. So for practical purposes, we can neglect them and write down an effective field theory which just involves the light field, A mu z. So what does this mean? We started off with three of these gauge fields, AX, AY, and AZ, and now we only have one. So we say that we've spontaneously broken the SU2 gauge symmetry down to U1. So at high energies, where we're able to explore the whole configuration space of our field theory, we have some SU2 gauge theory, but at low energies, where this phi takes some value, I end up with some effective U1 gauge theory. And what is this U1 here? I mean, it's the group generated by sigma z, but sort of more mathematically it's the carton subgroup that commutes with, that leaves this phi invariant. And that U1 subgroup was not gauge fixed by this gauge choice, because it leaves that invariant. So at low energies, I have some U1 gauge theory. All right, so where are a few applications of this? Well, one, as you may have guessed from the name, is the Higgs boson. The Higgs boson is actually in a slightly different representation of SU2, but I chose this one because it'll be better for application number two. But you might be familiar with the fact that the gauge group of the standard model is SU3 times SU2 times U1. This gauge group is associated to the strong force. What is this? This is called the electroweak force. And the dimension of this lead group is four. So we have four gauge bosons at high energy. But the Higgs mechanism breaks this down to U1, which is just electromagnetism. So this means that three gauge bosons got a mass. And these are what we call the W plus minus and Z bosons. So W plus minus and Z massive. And this is sort of why, while the discovery of the Higgs boson was super exciting, it was extremely expected because we had already seen these W plus minus and Z gauge bosons experimentally. And so this is the only way we have that we figured out for giving gauge bosons masses while being consistent with gauge invariance. No, no, we just have electromagnetism is just U1. So we gave three gauge bosons a mass. So we have four gauge bosons here because this is three dimensional and this is one dimensional. And then we break that down to just U1. No, no, in this case, it's a different representation. So in general, it's whatever group stabilizes the matter fields. Yeah, basically the Higgs field is charged in the two dimensional representation of SU2 and then also under U1. The Higgs field is the scalar field phi. In fact, yeah, the Higgs boson is the only scalar field that we've experimentally seen in nature. Okay, one last application before we move on to quantum field theory. How am I doing on time? Great. So let me just say this rather quickly. This is how physicists think about the relationship between Donaldson theory and cyber-guidant theory. So cyber-guidant theory is some supersymmetric gauge theory. Sorry, Donaldson theory physically is some supersymmetric SU2 gauge theory at high energies and at low energies. And there's some scalar field in the adjoint representation. It takes on some non-zero value just because supersymmetry forbids that field from having any potential. And then I break SU2 down to U1. So we have some high energy theory. This is Donaldson theory. It gets Higgs. And at low energies, I have some U1 gauge theory which is cyber-guidant theory. So cyber-guidant we're concerned with describing the low energy physics of this SU2 gauge theory. Yeah, it's an adjoint field of SU2. Which is just forced to be present by supersymmetry. So if I write down a gauge field for an SU2 gauge field and I just act with supersymmetry, then I get a whole multiple of fields. And one of those is an adjoint scalar. Yeah, it's really present in the high energy theory. Down here, we gauged it away. So it's not dynamical. We completely fixed what it is. Exactly. Okay, yeah, let me say that. I mean, phi there might be dynamical or might not depending on if there's a potential for trace phi squared or not. Okay, so let me move on to quantum mechanics. Here, what I have to say is actually not that much because the important aspects you need from quantum mechanics in order to motivate quantum field theory are somewhat minimal. So okay, let's start a new topic, quantum mechanics. So as you may have heard, the fundamental object here is the path integral. It's not mathematically well-defined. I hope you'll forgive me that, perhaps because it works rather well at CERN. So there must be something there even if it hasn't been properly defined yet. Okay, so I mean, in some special cases it has been well-defined. Like Gromov-Wooden theories, maybe one of the more famous examples. Okay, so with that caveat out of the way, let me just say here's what a path integral looks like. If I have some fields phi, I path integrate, I integrate over all possible values of my fields and then I plug in e to the minus action. And this action is related via a little bit of surgery to the original classical action so that we get a positive definite action. So I define a Euclidean signature time via t becomes minus i Euclidean signature t. And this s, Euclidean s, I'll drop the e signature subscripts after this is minus i of our original s. So this can be justified, but I don't really wanna spend time doing that. So let's just take this as the definition of how we promote a classical theory to a quantum one. And then what are the observables in such a theory? Well, in classical mechanics, classical field theory in any state, I can compute all sorts of functionals of my fields. And they just have some definite value in any state that I might care about. So here, rather than definite values, we have probabilities for what our observables will take. So, okay, let me write O is a functional of my fields. This is what we call an observable. We sometimes call this an operator also. There are more general types of operators, but this is the most common one that we talk about. Yeah, right now, I mean, this is very general. This is just quantum mechanics. We'll move on to quantum field theory very soon. So what is an expectation value of some observable? It's just the path integral d phi. I insert my O, oops. I insert my O inside the path integral, and then I again multiply it by e to the minus s. And then I divide by the path integral without any insertion, which we call the partition function. So you can see here that e to the minus s over z functions as a probability density. And we're just integrating over this infinite dimensional space of fields and computing some expectation value in that probabilistic theory. Yeah, good question. I'll actually say that in a second, but yes. Yeah, field theory books actually often sort of obscure this point, but z often contains some interesting information in itself. Okay, so I told you that in classical field theory, the job of a physicist is to tell you how to time evolve a state. And we can do that here as well. So for simplicity, let me assume that m takes the factorized form n times an interval. Then I need to impose boundary conditions on my fields at zero and t. So let me assume n has no boundary. So let me define two different Hilbert spaces, h zero and h t. H zero is the Hilbert space of square normalizable functionals on the boundary values of my fields at time zero into the complex numbers. So let me write that. We have these square normalizable functionals psi, which are functionals of my fields phi restricted to the boundary time equals zero. And this gives the Hilbert space h zero. Sometimes you'll call these wave functions. And similarly over here, you get a Hilbert space. And then time evolution is supposed to give me a linear map from h zero to h t. And by dualizing, let me say we're gonna give a map from h zero times h t to the complex numbers. And that's just what the path integral is. So we have a map h, h zero, h zero, tensor, h t, bilinear map from h zero times h t to the complex numbers. And you can see how this immediately generalizes to arbitrary numbers of Hilbert spaces where you have lots of boundary components. And what is this map? It's just the path integral d phi restrict, sorry. Yeah, let me write it this way. d phi restricted to time zero. d phi restricted to time t. Then I have these functionals psi of my fields at phi at time zero, my field maybe bar at time t, and then I just do the usual path integral where I've constrained my fields to have these boundary values. So d phi e to the minus s. So this giant path integral gives me a functional where you give me these two linear, these two functionals, I plug it into the path integral and I get some complex number. And that's what we call the path integral. Yeah, sorry, these are all multiplied. Yeah, exactly, maybe psi zero and psi t. Yeah, and so like I said, by dualizing this gives you some linear map from h zero to h t. And if I take t to be infinitesimal, this actually gives me the Schrodinger equation that tells me, it's a differential equation that tells me how to evolve states in time. But this is the finite version of that. Okay, for purposes of time, let me not do the harmonic oscillator. Let me just sort of tell you the answer. So in general, what is the partition function of some one-dimensional theory on a circle? So m is a circle, zero comma beta, a circle of circumference beta. You know, okay, circle. Okay, so say I have some state with some definite value of energy. I claim that these span the Hilbert space. Then, so let me just impose, before I do the sign identification, I have two Hilbert spaces. And so the space of states with some definite value of energy at zero spans the Hilbert space. And so if I wanna evolve those in time, because via Noether's theorem, energy generates time translations, I'm just gonna multiply such a state by e to the minus beta e, where e is the energy of that state. So that linear map is very simple in this case. Now let me identify these two ends. So I'm gonna sum overall possible boundary states where these two states are forced to be the same. So okay, that was a little quick, but the partition function is then gonna be a sum overall energy values of e to the minus beta times the energy of that state. And if you've taken functional analysis, you might have seen the Schrodinger equation for the harmonic oscillator. And you know that the energy eigenvalues take the form some omega times n plus a half, where n is greater than or equal to zero. So the important, and you can reproduce that from the path integral without too much difficulty. So the main reason I wanted to introduce this was both because I wanted to show you how the path integral works, but also because it's very important that these energy levels are quantized. So if I wanna add some energy to the harmonic oscillator, classically I can just give x a little nudge and that'll give it a little bit of energy, but here I have to introduce energy in chunks called quanta. Okay, so now we can go, oh sorry, was there a question? Okay, so now we can move on to quantum field theory finally. So I've just told you, oh sorry, there's one more thing I need to say about quantum mechanics first. How do we recover classical mechanics from the path integral? Well, in classical mechanics, we only look at one single configuration, which is the one that extremizes the action. Here we're pathing over every possible configuration, no matter how badly it violates the classical equations of motion, but if there's some coefficient out in front of the action that's super big, then all of the contributions to that integral are exponentially suppressed if we're violating the classical equations of motion. So this gives you the classical limit. Okay, so now on to quantum field theory. Yeah, that we can violate equations of motion. Yeah, if I have some giant constant out in front of s, so you know, we call this one over h bar, then any contribution that does not minimize s is gonna give an exponentially smaller integrand compared to the classical contribution that does extremize s. Okay, so on to quantum field theory. So this can be motivated by combining two basic observations. One is in quantum mechanics, the classical equations of motion can be violated for short amounts of time. And then observation number two is perhaps the most classical equation of motion, e equals m. In my units, c is one. So I can transfer a particle into mass into energy and vice versa. We're written in a more Lorentz invariant way. I have p mu, p mu equals minus m squared where here p is the four-dimensional momentum. So it's the energy and then the three-dimensional momentum that we're more familiar with. So this classical equation of motion being violated is sort of the heart of quantum field theory because it means that particles no longer have the classical constraints that we thought they did. And so this means that they can take part in interactions that are classically incredibly forbidden. So let me give an example of such a classically forbidden interaction that a particle might take part in. So say I have a photon. This is the particle that carries the electromagnetic force or light. And it's just moving along in time. And all of a sudden it decays to a positron and an electron. I claim this is forbidden by conservation of energy and momentum. I'll leave that as an exercise for you, but it's true. So this will never happen classically. But in quantum field theory it does happen because what I can do is I can say maybe these two guys exist only for some very short amount of time and then they annihilate and they give my photon again. So this happens in quantum field theory and not only does this happen but every possible process happens because I'm required to path integrate over every possible process that can happen. So now we've sort of got a conundrum because particle number is not being conserved and particles can spring in and out of existence anywhere at any time in the universe. So how are we possibly gonna deal with that? And this is where field theory comes to the rescue. We have some fields which indeed extend throughout all of space and time. And so those can give a mathematical formalism for describing these particles springing in and out of existence. So I told you before that we can think of a free scalar field, for example, as infinitely many harmonic oscillators where their frequency, they were labeled by a three-dimensional momentum and their frequency was given by this K squared plus M squared. This is actually precisely the relation for defining the energy of a particle with momentum K and mass M. So this gives us reason to suspect that the quanta of these harmonic oscillators should actually be identified with particles. So I can't inject just a little bit of energy. I have to inject some chunk of energy into each of these Fourier modes and that's what we call a particle. It's an excitation of such a Fourier mode. Similarly, photon is an excitation of the gauge field. And more generally matter particles are excitations of fields. Force carriers are particles which are excitations of fields. And then if I wanna describe interactions, I just draw diagrams of this sort where some electron might shoot some force carrier at some other electron and that's gonna transmit some energy and momentum. Okay, so then the very last thing I wanted to briefly touch on was renormalization. So say this has some momentum P. So by conservation of energy over long times this also has momentum P. But over here we can assign whatever we want. Let me call this Q and this is gonna be P minus Q. Because I'm required to sum over all possible configurations no matter how ridiculous they are, Q is arbitrary, Q is totally unfixed. And so in particular it can naively range up to arbitrarily high values of energy and momentum. And I'm gonna need to probe like extremely high energy physics that I told you I was supposed to be neglecting in effective field theories. So do I need to solve all of quantum gravity in order to describe electromagnetism? The answer is no. And the way we deal with this is in the spirit of effective field theory we introduce some cutoff. So we only integrate over momentum which have some value less than some cutoff lambda. So now part of the data defining a path integral is actually this value lambda, this cutoff lambda. And then if I, the physical meaning of this cutoff can be intuited with this example. So say I have some electron and for reasons of pedantry let me say this is an electron defined in the field theory with some scale lambda. Well, now you're much weaker than I am and you come along and you're not able to probe such short distances, you have a cutoff that only lets you, you can't see within this sphere, this dashed sphere. So you have some cutoff lambda prime which is less than lambda. Then you're gonna call this whole dashed box an electron. Okay, fine, so who cares? Well, I care because this guy is negatively charged and so it's gonna be electrically favorable for positively charged stuff to spring out of the vacuum and actually surround this electron. Cause creating these guys costs some energy equals MC squared but you also lose a whole bunch of energy because of the electrostatic attraction. And so the effective electron that you see has less charge than the one that I saw because it's screened by this cloud. And so we see that the coupling constants defining my effective field theory are functions of the cutoff. And this is called the renormalization group flow which is sort of misleading cause it's not a group, it's not invertible. But you know, said more bluntly, what you call a particle and what I call a particle is gonna differ. It depends on this extra data that you specify. And this is sort of one of the most important ideas in physics. Yeah, okay, let me stop there and ask for questions.