 Okay, so before I can get to what quantum field theory is, I need to spend a bit of time explaining the background. So let's start off with classical field theory. So why should we care about this? Well, I'll give you two reasons. One is that it's necessary to understand classical field theory in order to promote it to a quantum field theory. A lot of the classical ingredients play an important role. So need for QFT. And it's also just a good idea to, before you study a quantum theory, you should understand the classical limit to get a lot of intuition. But secondly, it works. You may have used electromagnetism in your intro electromagnetism courses, for example, or more generally classical physics. It works rather well. And this gives me an excuse to introduce an idea that I'll need throughout this talk, which is effective field theory, which is why classical physics works. So the idea is very simple. If I wanna tell you how a football, well, the trajectory of a football, I don't need to know all of special relativity in order to do that. And so there's some set of physics that is relevant at the particular energy scales that you care about. And you can neglect all of the other physics above those energy scales in some self-consistent way. So this is theory of physics that gives correct answers as long as you're careful to only ask questions that are within its domain of validity. Answers to low energy questions. So a sort of cartoon that you should have in mind for effective field theory is if I've got some very complicated hills and landscape and something else, and I have a ball sitting here with very little energy, so I can't possibly escape this hill, then I don't need to know the details of all of this. I can just focus on the geometry here and I have some nice quadratic potential. So this is in fact the source, for example, of why the harmonic oscillator is so important in physics is, it gives the universal description of this low energy physics, where you don't have to worry about all these details. Okay, so let me start off now with a scalar field theory. So in general, a quantum field theory is, sorry, a classical field theory. The basic ingredients you need are some fields. So what do the fields live on? They live on some manifold M. I'll call this space time. I'll explain the reason for the quotations in a second, but I mean, it's because what we call M might not always be what we think of as all of space and time. And I'll equip this with a Lorentzian metric for now. G Lorentzian. So the negative signature part corresponds to the time direction. And then fields are just maps from M into some target space. And this is very general. So for example, this can include sections of a tangent bundle, for example. So that would be functions from a manifold to the fibers over each point in that manifold. And we would call those vector fields. If y is not a non-trivial bundle over our manifold, it's just, this is a genuine function, then we call this a scalar field. So more generally, we have scalars, we have spinner fields, vectors, tensors, et cetera. Okay, so let me start off just with the simplest field theories. These are scalar field theories. So as an example, let's look at the harmonic oscillator. So this is a particle which feels a linear restoring force, which pulls it back to the origin. So mathematically, what do we mean by a particle? We mean that it has no spatial extent. So it exists for all of time, but it has no spatial extent. So this means that M here is just gonna be the real line equipped with the Lorentzian metric, ds squared is minus dt squared. If you're not happy with that notation, we have like, so this is why I put space timing quotes over there is because M is really gonna be the manifold over which our fields are allowed to vary. Okay, and then our field here is just a function x from the real line parameterized by time to the real line. And we think of x of t as giving the position of this particle at some time t on the real line. Okay, so that's not all I need in order to specify the harmonic oscillator. An unglamorous description of the point of physics is if I give you some state at some time, then I'm supposed to tell you how that state will evolve over time. And in order to do that, I need to provide what's called an action. So from the action, we extract the dynamical principles of the theory. That is the rules that govern time evolution of the theory. And the way we do that is by extremizing the action. So, you know, delta s delta x equals zero. And these give what's called the Euler Lagrange equations. Okay, so in this particular example of the harmonic oscillator, oh, so in general, by locality, the action will always be the integral of some Lagrangian. So s equals integral over M, your volume form times, yeah, it's the integral of the Lagrangian is some function from M to the real numbers. And importantly, L of x only depends on the fields and their derivatives at that point x. That's what I mean by locality. Okay, so let's look at this for the harmonic oscillator. No, it's a function on M. I'm integrating it over M. It does depend on the fields though. So it is a functional of the fields. That is true. Yeah, I mean, the fields depend on M. So in that way, you get a function from M. Okay, so the action of the harmonic oscillator, I'm gonna write it in a sort of obnoxious way just to emphasize that metrics matter in physics. So, you know, in general, physics cares about lengths and angles. So here, g inverse is the bilinear form on the cotangent space. And okay, I can write this in a less obnoxious way as just, and if I extremize this action with respect to x, then I will indeed find the Euler Lagrange equation, which is, and we can recognize this as Newton's law, F equals MA, where here we have some linear restoring force. Okay, so that's probably the simplest possible quantum field theory, sorry, classical field theory. So now let's upgrade this a bit to higher dimensions. So let's now consider M to be R N minus one comma N, sorry, comma one. And our field will again be a real scalar field. So by that, I mean x is a map from M to the real numbers. And the Lagrangian of this theory is minus a half D mu phi, sorry, D mu x, D mu x, where here I'm implicitly summing over these mu indices from zero up to N minus one and raising and lowering with the metric. So, sorry, explicitly this looks like a half, DT x squared minus sum over the spatial indices of DI phi, DI x squared. And so just like in the harmonic oscillator, this takes the form of kinetic energy minus some potential energy. And we sometimes just call the potential energy the potential, but here, so here the form of the potential energy is some gradient energy rather than explicit x squared, M squared, x squared. Okay, so I can upgrade this theory a little bit by adding such a term to the Lagrangian. So let me call this L sub M. And this is the same theory just with an extra term added to the potential, minus a half M squared, x squared. Okay, so both of these theories are called free field theories. And the reason is just if you compute the Euler-Lagrange equations, you get the de Lombardian, this is the Laplacian in Minkowski space, minus M acting on your field equals zero. And this is a linear equation that hopefully we can all solve in our sleep with separation of variables. Box, yes, it's the Laplacian in Minkowski space. Sorry? Oh, I mean it's like minus dT squared plus sum over spatial components dI squared. Here? Yeah, thank you. Okay, so I just told you that we could solve this equation using separation of variables. Let me actually rewrite this action in a sort of strange looking way because it'll prove very useful when we turn to the quantum theory. So we're just gonna rewrite this in terms of a Fourier transform but only in the spatial directions. So let me define x of T, this is equal to integral d3k over 2 pi cubed, either the ikx and then x of tk. So if we rewrite the action in terms of this Fourier transform, the action will look almost identical to the harmonic oscillator. Just we'll have continuously many harmonic oscillators, one for each choice of k. So let me write this out. Yeah, sorry. So if I just focus on one particular value of k, this looks like the exact same action that I had before for the harmonic oscillator, where here the frequency of the harmonic oscillator is just k squared, this is magnitude squared of k plus m squared. So for every single point in k space, I have a harmonic oscillator and they're not interacting with each other. I can solve for each Fourier coefficient independently. And that's again just saying what you already know about separation of variables just in a sort of slightly different way. Okay, let me generalize this example even further. So now let's consider a target space y with a Riemannian metric h. And our field will be a map x from space time into y. So in the examples we've been studying, this generalizes the examples we've been studying where y is either the real line or real line over here also. So more generally, I can write a Lagrangian of this form minus a half. T is just some positive coefficient. So in the harmonic oscillator, this is the mass. And then let me write it like this. G inverse tensor h applied to dx, dx. And here dx is the matrix of partial derivatives of x. So explicitly, explicitly dx is a map from space time to cotangent space to space time, tensor the tangent space to y. So that's just the matrix of partial derivatives of x. And yeah, this gives me a diffeomorphism and variant Lagrangian. Sorry, T is a positive coefficient. So in the harmonic oscillator, it played the role of the mass. And indeed, okay, so let me give this theory a name. It's called the nonlinear sigma model for historical reasons. There are no sigmas. And physically, this is describing the dynamics of maps from space time into some target space y. So if space time is one dimensional, it's just time, this is describing a particle moving around in y. If m is two dimensional, so it's one space and one time dimension, this is describing the dynamics of a string exploring y. And more generally, this is describing some higher dimensional version of a membrane probing some space y. So let me just say physics of a dim m. Dimensional object exploring y. And then you can make this even fancier. You can add potential terms like we did for the harmonic oscillator that sort of prefers certain places in y relative to other places. So yeah, h is a Riemannian metric on y. So okay, let me write out this Lagrangian in physics language, because it's better. Okay, so if you like the first version, there's no saving you. All right, and nonlinear sigma models are very ubiquitous in supersymmetric theories. Why is that? Well, generally there's no reason that you shouldn't add a potential to your theory. And sort of in the philosophy of effective field theory, if there's no reason that you shouldn't add some term to your Lagrangian, it's not prevented by symmetries, then you should probably expect it to be there with some coefficient. But in supersymmetric theories, you very frequently have such strong constraints on your Lagrangian from supersymmetry that you just can't possibly add any potential for your scalars. And so you get these nonlinear sigma models at low energies. And a very concrete example that I'll show up next week is for certain four-dimensional field theories wrapped on a circle, this y will be the moduli space, the Hitchin moduli space. Okay, so let me now turn to gauge theories. So gauge. So the setting here, as we're probably familiar, is principal bundle P over M. And our field here is a connection on this bundle, is a principal connection on this bundle. So, okay, I should say P is a principal G bundle for some lead group G. And then we can add to this certain other fields, which are sections of associated bundles. So let me choose some representation, row from G to some vector space V, then I can form an associated vector bundle P times V, row, which is P times V, mod G, where G acts from the right on P and acts via this row on V. Okay, and we can form a field connection, sorry, a curvature, which physicists call field strength from this connection. So we get some F, which is D omega plus, this I is probably not gonna make you happy, but it's so that my connection is gonna be her mission rather than anti-her mission. Okay, so once I have this curvature, this genuinely transforms in the adjoint representation, and so I can form the gauge invariant combination, trace F wedge star F, star is the hodge star, and this guy gives me a gauge invariant Lagrangian for Yang-Mills theory. So let me write that out more explicitly. The Lagrangian for Yang-Mills theory is one over two E squared star trace F wedge star F, which again in human language is trace F mu nu F mu nu. Okay, so that's fine mathematically, physically. What is the point of this theory? And as we're all familiar, at least when G is U one, this is the mathematical description we use for electromagnetism. So here F is a two tensor whose components are the electric and magnetic fields. And okay, so let me say this. Okay, and then we're familiar with the potential energy in electromagnetism takes this form. This is some electrostatic potential energy between two charges. And here this E is the same E that showed up over there. So E is what we call the coupling constant, and it determines the strength of the force. So E is the coupling constant. And I've written this in such a way that Q one and Q two are both integers. So this is maybe a fact that's not taught a lot in electromagnetism courses, but this is just an experimental fact. Every charge that we've ever observed in nature, it has an integer multiple of the electrons charge. So that might sound incredibly bizarre, but that actually has a natural explanation in terms of gauge theory. So in order to explain that, let me explain what it means to have some charged matter, like electrons, for example. So charged matter, by the way, the Euler Lagrange equations from the Yang-Mills equations with gauge group U one are Maxwell's equations. Okay, so if I have some charged matter, this is, like I said, just a section of some associated bundle. So this means informally I can say I have some field in some row representation. Rho is a representation. And we'll take row to be irreducible just because if we have some direct sum of irreducible represent, sorry, if we have some direct sum of irreducible representations, we'll just call those each different fields. Okay, so row is irreducible. Well, what are the irreducible representations of U one? They're just labeled by integers. Your vector maps to E to the IQ theta times your vector, where Q is an integer. So right away we see why charge is quantized in electromagnetism. It's because it's a U one gauge theory and irreducible representations of U one are labeled by these integers. So more generally I can, for an arbitrary gauge group, I can have matter charged under some representation. When you hear charged, you should think irreducible representation of the gauge group. Yeah, it's another field that you added to your Lagrangian. That's a section of some associated bundle. An electron field is a section of the bundle associated to Q equals minus one. Yeah, you can have Q equals zero. That's a good representation. Yeah. We can have fancy nonlinear actions of gauge symmetries, but in most applications, we care about they're just linear. But yeah, I mean, what you just hinted at is the existence of like gauge nonlinear sigma models where you can gauge some isometry of the target space. I mean, that is important in math. Like we can describe symplectic and hypercaler quotients in that way, for example. Okay, so let me give a cool mechanism that is both of interest to you and physicists. So this is called spontaneous symmetry, breaking of gauge symmetries. Or in a shorter way, we just call this the Higgs mechanism. So let's say that I have, I'll start a new board. I have an SU2 gauge three. So my principle bundle is a principle SU2 bundle. And furthermore, I add to this theory some matter, which is in the adjoint representation of SU2. So the three-dimensional representation. Okay, so let me write that down. We have omega, which is a connection on a SU2 principle bundle. And we have a field phi, which is an element of the, it's a section of P times adjoint representation. G here is SU2, the algebra. Okay, so let's suppose that, for whatever reason, my gauge field, sorry, my scalar field phi, takes on some non-zero value. So there are good reasons why this might happen. Let me give you two. One is in supersymmetric theories, I told you that you can't have a potential oftentimes. And so this field is just allowed to wander off. And so it can take any value that it wants. And so it's not impossible for it to take some non-zero value. Another way that I can say that it has some non-zero value is I can actually force it to have some non-zero value. So I can add to my Lagrangian some potential which depends on trace of phi squared. And if I can choose my potential, so it sort of looks like this. So it forces the trace of phi squared to be non-zero at low energies. Yeah, sorry, this is trace of phi squared. And this is the potential energy. So I can just add some potential to my Lagrangian that forces it to be non-zero at low energies. But for whatever reason, let's say that our field has some non-zero value. Well, then I can choose a gauge where it takes the following form, phi equals little phi times sigma z, where little phi is positive. So it's a field from m to r plus. So this might look funny because we're used to gauge fixing by imposing some conditions on the connection, but this is a perfectly sensible way to fix a gauge too. Physicists call this unitary gauge. Yeah, you know, I have some non-zero field phi. I know nothing about it other than it's non-zero. I can act on it by conjugation to put it in this form. The poly z matrix, you know, the S-U-2 Lie Algebra is the span of the poly matrices. Okay, and similarly, I can write my connection I'll write it in some gauge. So, you know, a mu equals a mu x sigma x plus a mu y sigma y plus a mu z sigma z. So this is just the most general, you know, element in the adjoint representation of S-U-2 that I can write down. And the mu is, because it's a one form locally. Okay, so let's now look at what the kinetic term for phi looks like. So we have the gauge invariant version of the kinetic term that we had before. So d mu phi squared. That's the kinetic term for phi in the Slagrangian where here d mu is the covariant derivative. So d mu big phi equals partial mu phi plus i commutator of a mu and phi. So I can plug these two expressions for phi and a into this kinetic term. Yeah, so what I did is I covered my manifold with some open sets u alpha and I chose a gauge meaning I picked some sections, some local sections, S alpha from u alpha into.