Lecture 1: Classical Field Theories and Principle of Locality（原声文稿）

Yes, I'm Hong, so you can just call me Hong.

I am a theoretical physicist.

I work on a high energy theory, including string theory, quantum gravity, non -equivalent statistical physics, et cetera, and many different topics.

So now let me say a few words regarding the QFT, key at the quantum field theory itself.

So the goal of this quantum field theory class is to develop concepts, formalism, and techniques for quantum dynamics of fields. So I'm sure all of you have studied Markzware equations.

So Markzware equations describe fully classical dynamics say, of electric and the magnetic fields.

But we live in the quantum world.

And so we should also treat electric and the magnetic fields using quantum language.

And then you find that once you do that, and then the concept of photon, once you treat the electric field and the magnetic field quantum mechanically, and then you you get the concept of the photon.

And the photon now is the fundamental particle which mediates, say, electromagnetic interactions.

So you actually get a completely different physical picture from what you get classically.

And yeah, so one goal of the class is for you to appreciate, say, to understand the quantum electrical dynamics.

And that's actually where we will end this class, and this we will discuss the quantum electrodynamics.

And quantum field theory is also very important for many other, for other interactions in nature.

So among the four fundamental interactions in nature, three of them are described by quantum field theory, completely.

And you can also use quantum field theory to describe gravity, to describe quantum gravity, But at the moment, quantum field theory does not offer a complete description of quantum gravity.

But still you can use the quantum field theory techniques for understanding certain questions of quantum gravity.

And also quantum field theory over the years, even though it's initially developed for particle physics, but over the years has been also found many applications in many, many other branches of physics, in condensed matter, statistical physics, et cetera.

And it's fair to say, nowadays quantum field theory has become a universal language for theoretical physics, essentially in all different fields. And so if you are a theorist, certainly you need to master this language.

And even if you are only interested, say, in atomic physics or statistical physics, and the field theory concept will be very important.

And for experimentalists to know basic concept of quantum field theory and to have some basic understanding of it should also go a long way, help you to appreciate say the most recent theoretical development and also for help you to communicate with theorists and yeah.

And so this class is the first of a series master sequence.

And so this semester we mostly developed the fundamental concept and the quantum field theory 2 and 3 will be more about technical development.

And say, if you're an experimentalist, and then you can view this quantum field theory 1 as a standalone class, and which, yeah, just enough for you to to get the basic idea of quantum fields, quantum serial fields, which you don't, yeah, you may not depend on your lead, and whether you have two or three or not, yeah, may depend on your lead.

Okay, so, under the main topics I plan to cover are listed in the outline, this document of the outline, which is already on the website.

And so I should emphasize that outline is a rough. It's only a rough roadmap. It may change, it depends on the pace.

It may change things along the way. And sometimes I change mind, say, half through the course.

Somehow I feel, yeah, anyway. So don't treat it too literally.

And any questions about this subject or quantum field theory so far?

And also, let me say a few words that the quantum field theory has a reputation of being a very difficult subject.

Actually, indeed, myself have suffered a lot when I landed myself.

But with 2020 hindsight and also from interacting with a large number of students through teaching various level of quantum field theory classes in many years, I can assure you now that actually quantum field theory is actually not difficult at all.

If you lend it the right way.

And of Of course, anything is not difficult if you lend it in the right way.

So in the sense that this is the empty words, but keep it in mind.

Whenever you find this, you think it's too hard, and the reason might be that you have to change your perspective. You have to change your perspective. you.

And so quantum field theory, one thing people complain about is that quantum field theory often involves a lot of calculations.

And that's true. That's just the fact of life. You cannot avoid it. But that's not what it makes it difficult. Okay.

Complicated calculations, you can just go through them from one line to the next line to the next line. If you're careful enough, patients enough you can go through.

So the difficulty I think for most people of quantum field theory is more at the conceptual level.

It's because this subject is not a very intuitive subject.

It's not something you can just understand just by thinking, okay.

So that's why I emphasized earlier the exercise and really working through it is very important.

It's was developed through examples, but working through many examples, you slowly develop intuition about quantum mechanics, and if you learn all the lessons, then you get good feeling about quantum mechanics. And quantum field theory is the same thing.

It's that you have to, some kind of intuition has to be developed, okay?

It's not something I think you can easily, yeah, just like mechanics, which you can, in some other subject, maybe if you have very good intuition, you can just imagine it, okay.

Yeah, so to help you develop good intuition about quantum field theory, I can offer you three pieces of the devices, okay?

So, so first, yeah, so the first piece of advice is that the quantum field theory is essentially quantum mechanics, but dealing with infinite number of degrees of freedom, okay? So, so in your quantum mechanics class, you always treat with finite number of degrees of freedom.

But quantum field theory, the difference for quantum field theory is now you treat infinite number of degrees of freedom.

It turns out that this treating infinite number somebody with freedom makes a difference. So more is different.

And actually, sometimes make conceptual differences.

And so that's why sometimes the quantum field theory is unintuitive.

But that said, I found for many people, including myself when I learned it, many difficulties you encounter in learning quantum field theory is not due to the difficulty in quantum field theory itself, it's actually due to your gap in understanding of quantum mechanics.

So whenever you encounter something you don't quite understand in quantum field theory, try to step it back to say, can I formulate this difficulty in terms of quantum mechanics with only finite number somebody who's freedom.

And often you find actually your difficulty can already be formulated in quantum mechanics.

And then that way, then you should be able to just settle it yourself, because we are supposed already to be a master of quantum mechanics.

And certainly, when I learned quantum field theory, I was stuck at certain points for a long time.

And then later I realized just because I didn't understand a certain Heisenberg picture of quantum mechanics very well.

When you understood, somehow I realized when I understood the Heisenberg picture of quantum mechanics well, and those difficulties just went away, and have nothing to do with quantum field theory itself.

So that's why in your first p -set you will get familiar with Heisenberg picture of quantum mechanics. That came from my own experience.

And also, the second point is that quantum field theory deal with formalisms.

And sometimes this subject seems very formal, okay, you have a lot of formalisms, okay.

But just keep in mind, any formalism in physics, no matter how abstract it is, it was always designed to solve some concrete physical problems and physical questions.

very concrete physical questions and if you understand what kind of concrete physical questions quantum field theory was designed to solve then that can give you a very good perspective on those formalism and why people do this why people do that why people do this trick why people do that trick and because they were invented to solve certain concrete problems okay and once you understand the questions understand the problems then the formalism become are much easier to understand.

And the third thing we already said is that in quantum field theory, as in quantum mechanics, intuition was built through experiences, through examples.

So when you do your PSET, when you look at the examples in the class, you should always ask yourself afterwards, say after you have done your PSET problems, always look back at that problem, say, what did I learn from this problem, okay, and just think through it again, think through, think through what you learned from that problem again, and that is a very good way to help you to learn from your experiences and to help you develop intuitions, okay.

And so, yeah, so, so also a very important thing you should keep in mind is that in the graduate to a course like this.

And most of the things should be learned outside the class.

So inside the class, the purpose to give you a guide is to emphasize the conceptual picture and the physical intuition, et cetera. And so sometimes I will leave some details for you to finish in the p -set.

And sometimes the p -set will involve problems which I did not fully discuss in lecture, but I want you to work out yourself, okay?

And so PSET is an important part of the learning, even new things, okay?

Not just to practice something, but it's also a very important part for learning new things.

Also, finally, I would like to make a apology.

So you will soon find that the lotations are used in the lecture are different from the lotations in the recommended reading books, okay?

So I recommended reading Paskin and Wemberg, and you will find that my lotations are actually different from them.

Also the order of presentations are also different from them.

I know this is very annoying, but there are just no perfect textbooks, and there's no perfect set of notations everybody use, and we all use the notations which we find the most convenient to use.

Even though I realize this problem, but I don't have a good resolution.

OK, so just keep in mind that the rotations in my lecture can be different from the rotations in those textbooks.

Good, so do you have any other questions?

Good, OK. So if you don't have any other questions, so let's start.

So the chapter 1 will be about why we can see the quantum field theory, OK?

So first, we talk a little bit about the classical field series to stand the stage for quantum field series.

And the first important concept is called the principle of locality.

So if you remember from your, say, high school days, Newtonian mechanics, so in Newtonian mechanics, you have action at a distance.

For example, if you look at gravity, and gravity is exerted by the sun on the earth, but they are very far away.

And the same thing with the Coulomb interactions between the charged particles.

But then in the 19th century, they came from this principle of locality, and that's formulated by Faraday around 1830, OK?

So the principle of locality says all points, you actually don't have action at the distance.

You set all points in space, participate in the physical process, OK? OK?

And the effect, so if you have interactions, OK, so effect propagates from points to a labeling point, OK?

So in this principle of locality, you don't have action at a distance. So action at a distance is always conveyed.

The action is always conveyed from one point to another point through the propagating in the space, OK?

And the fields, the concept of the fields is the mathematical device or vehicle that the principle of locality is at work.

So this is essentially the device we need to use to realize that this principle of locality.

And so the main idea of the field is that we associate each point with each point in space a dynamical variable or dynamical variables.

So for example, if you have an electric field, the electric field is defined for all space.

So at each point x, we can introduce an electric field. OK?

And then this electric field can also depend on time.

And yeah. So of course, normally we write it this way, e is xt.

And the reason I write it this way is to emphasize that in the definition of the electric field, the space and time actually play a very different role.

the space plays a role of a label, okay, so at each point we have an electric field, okay, at each point x we have an electric field, and so x here is just a label, okay, and the t is used to describe the evolution, the change of the electric field, okay, so the x and t plays a very different role, and similarly you can do it for magnetic field, it's the same thing, OK.

And here, you should always view x as labels. OK. So the spatial point, we should denote as vector x is the labels.

And we know that the evolution of electrical in my left field, I described by Maxwell equations.

Let me just write them down to remind you.

So we have dot b, zero dot e, four pi rho.

So this is so -called differential form of the Markzware equations.

So the reason I'm taking trouble to write them down is to emphasize in the following point.

So this set of equations exemplifies perfectly the principle of locality.

It's because you see those equations only involve the value of the electric field and the magnetic field at a single point.

So if here is at x -point, so here is also at x -point. You level say x here and here is some other point of y.

And so this is reflected the principle of locality that everything, so the effect is propagates from point to point. You just have the derivative of the same points.

It's never, you don't involve separated points.

So these are local equations.

So these are what we call the local equations, and they contain only E and B at the same point, and also the charge density and the current density at the same point with finite number derivatives.

So the derivatives are the ones to help you to propagate, because it relates the point to the laboring point.

And so that's how it propagates it through the derivatives. So the derivatives are key. The derivatives are key.

And another example, which examples locality, which I will not go into here, which some of you may know, is the Einstein gravity. So Einstein's general relativity.

So in Einstein gravity, the dynamical variables are so -called spacetime metric.

So they are object with two indices, two space time indices.

And then the Einstein equations are equations for this kind of object.

And again, the equations are local equations in the sense that they only depend on the g evaluated at the same point with the finite number of derivatives.

So in Einstein's gravity, you no longer have action at the distance.

OK, so the effect of the gravity is propagated through the spacetime, through spacetime.

So in fact, so here we say these two examples exemplifies the principle of locality.

In fact, the principle of locality played a very important role in formulating those equations.

Because as we will very soon see, when you have principle of locality, you can significantly to be constrained, the theory you can write down. So that's a very, very powerful principle.

Yeah, you can have non -local equations, but it's believed that the fundamental equations in nature, they're all local.

Yeah, and so far, the equations govern all fundamental interactions, all four different interactions in nature, the equations are local.

Other questions? Yes?

Say it again. I don't quite understand the question. Yeah.

Oh Here I'm just one for size again That the of course in Einstein's theory this these two are treated the same way But but if you think in terms of the fields and they actually have very different physical interpretation Yeah, yeah, so that's why I write this way. Yeah.

Yeah, it's the same Same way I write it like here, yeah, yeah, yeah, same reason.

Just here to emphasize for this, to emphasize the different row played by X and T. Yeah, but then, but I think you're asking a very good question.

So I'm going to mention nature in the relativistic series. Then, of course, then these two become the same. Yeah, become, play the very equal row, yeah.

Does that answer your question? Yeah, okay, good.

So let me also very quickly mention the different types of fields.

Okay? So you can have what's called a scalar field.

So these are quantities which at given point, given point there's only one value, okay? Say, for example, the temperature.

So it's just a single quantity defined at the point.

For example, the temperature, maybe other quantities. And then you also have a vector field.

And E and B are the examples of a vector field because at each point, you have a vector.

So here, I'm using the three -dimensional rotations. But in the relativistic series, is you can also use the four vector rotations.

Say, for example, the vector potential, we have the following form. Say, a mu will be a four factor, and then the spacetime, I combine them together into a four vector.

And so at each point, and then you have a four vector. And so, yeah.

And also, you have a tensor field. So this metric is an example of a tensor field.

here you have two components to have two indices okay so so you have many many different components that depend on the so again this is right to be stick notation I can write it as a right to be stick notation okay so at each point and now you have a now you have some object with two indices okay and you can also have later we will see your you also have something called spinner field out alpha. And this alpha is some other indices which we will define later.

So we can also have so -called spinor and alpha some other indices.

And as our convention, you said mu is always from 0 to 3.

Then sometimes I just write x. When I write x, it just means a four vector. It means x mu equal to ct and x.

So x always denotes a, x vector always denotes a spatial vector, okay?

And yeah, and then this is the same as ct xi, okay? So the spatial indices always denoted by i, and yeah.

And so partial mu will be the same as 1 over c partial t.

and then the derivative on the gradient on the on spatial directions and say if you have a full vector then again you we have the convention that a0 then the a vector it's the same also as a0 and ai I use the i to load the spatial components and and this is the last time I will write speed of light, so the c will always take to be 1, and h bar will take to be 1.

That's for notational convenience.

Do you have questions?

Okay, so now, so with this preparation, then we can talk about action principle for classical fields.

So first recall, in your classical mechanics, so we introduce the action, which is the integral of your Lagrangian, and Lagrangian is the function of x is your variable and x dot and the time.

And so this is a one -dimensional, yeah, this is 8 over 1, one -dimensional particle motion. motion.

And you can also introduce some momentum, canonical momentum, which is defined by partial l, partial x dot.

And then you can also define a Hamiltonian. And Hamiltonian is related to the Lagrangian by Px dot minus l, from a Legendre transform.

And the equation of motion is obtained defined by extremize S.

So S is considered to be a functional of your trajectory.

So whatever trajectory you have, and you extremize this S, and then you get equation motion.

So now you can generalize to field theory.

So now from principle of locality, so for field theory from principle of locality, so the form of the Lagrangian, so to the form of the Lagrangian L.

So again, we hear, for field theory, we can again define S as a time integral of a Lagrangian.

And the form L is significantly constrained to have the following form.

So you must have the form L equal to a spatial integral. So this is integration of all spatial directions. OK, so d3x.

So let me first write down the notation, and then I will explain the notation.

OK. Yeah, maybe just write it to partial i.

OK, so let me now just explain the notation a little bit.

Well, first, so here I use a short -hand notation into the load of the fields.

So phi A, so it will be a function of spatial direction on the time, is general field.

And A, label different fields.

So this index, A, labels different fields.

So for example, A can label different scalar fields. If you have multiple scalar fields, then they can label them.

And they can also refer to indices, space time indices, like a mu, and also, yeah, etc. Okay. So a just label whatever field you have. Okay.

And then the second point is that this L, this script L, is a function.

The emphasize here is a function, okay, of phi a and these derivatives.

And these derivatives.

So in other words, the a l, so this is a key point.

So that is a l only depends on the value of phi a and these derivatives at a single point, say x, and then you integrate over all x.

OK, so L is called the Lagrangian density.

Say it again? Yeah, yeah, yeah, yeah. Yes?

Yeah, yeah, we'll mention that. Yeah, we'll mention that.

So here I just explained the notation, OK?

So now let me just make some remarks on why the Lagrangian we must have this form.

So first, the principle of locality implies here must only involve a single integral, because the locality does not allow something like this.

Say, for example, does not allow a term like this.

Does not allow a term like this, which involves the phi at different points.

It's because if you have terms like this in Lagrangian, OK, so later you will see, oh, sorry, L, not L script.

So if your Lagrangian have these kind of terms, and then so as we will describe the equation motion later, you will see from the equation motion, then your equation notion will not be local.

OK? So equation notion will involve the behavior of your field at one point, and then influenced by point at some point far away.

And then you will not have local. So the locality significantly constrained.

Because if you give away locality, and then in principle your Lagrangian can be arbitrarily complicated. You can have as many integrals as you want.

But because of locality, you only allow to have such a simple integral, a y integral of a function. So this is the key.

So that's not allowed. Here is key point.

So the second point is that we only allow first derivative in time.

We only allow first derivative in time. We don't allow the second derivative in time.

So the reason is that, again, as you will see equation motion, if you involve the second derivative in time in your action, or in your Lagrangian, and then when you get the equation motion, you will get equation motions involving more than two derivatives in time, OK?

So So this will lead to, so this implies the equation motion only contains two derivatives, two time derivatives.

So this constraint comes from our experiences.

It's the same reason here. We only include the first derivative in time.

It's because in real life, all the experiment is determined by the initial condition. the initial condition only need to specify the location and the velocity, okay?

You don't need to specify more. If you're in quotient motion, you're moving more than two derivatives, then you need to specify more general initial conditions.

And so, yeah, so here is the same thing. We only allow the first derivative in time.

But you can, in principle, allow arbitrary normal derivative in spatial direction, okay?

But for simplicity, for the most of the time, as we will see, we will restrict to quantum field theory in spatial relativity, means that there will be a relativistic invariant, maybe Lorentz invariant. And in Lorentz invariant theory, space and time, they can transform to each other, play equal role.

So if you only have single derivative in time, you example we will see they will all have only single derivative in spatial directions okay but certainly non relativistic systems you can have a higher a number of spatial derivatives good any questions on this okay good so now so as in classical mechanics allowing we can introduce the canonical momentum Hamiltonian etc okay so so here we can so so here we can introduce so called that the canonical momentum density, the rhythmical density will be clear.

So we can remember, for this phi, x is just a label. So you can just view this theory.

It essentially has infinite number of such kind of, yeah, it's an unfortunate notation. Here we use x as a dynamical variable.

But here the x is only a label. OK? Here, x is a label.

So here, you can just imagine you have infinite number of degrees of freedom just each one is labeled by x, OK?

So now, just imagine here, you have many, many x, just you have some labels for it.

And so for each such one, we can introduce this momentum, OK?

So each phi a, we can introduce this canonical momentum, OK?

Phi a dot derivative, OK? OK? Yeah, maybe just.

So defined as the derivative of the Lagrangian density divided by time derivative of phi a.

So this is just the direct generation of here.

So remember, so this is evaluated at each point in spatial direction.

And each, yeah, so this thing will also depend on x and t. OK?

Oh, this is just the capital pi. Yeah, capital pi.

OK, and this phi A dot, just a time derivative of phi A.

And remember, x are the labels. OK, so x does not do anything here.

So the reason we call this Lagrangian density is because the A is a Lagrangian density. density.

And then we can also define the Hamiltonian density the same way.

So we have defined the script edge, which is defined by phi A dot pi A.

Again, this is all defined on the same spatial point minus L, okay, so this is not going to density.

And then the then the Hamiltonian, you just integrate over all space, essentially sum over all the good freedom, okay?

Just give him that. Yes?

Yeah, yeah, it's just like you sum, so if you treat the, yeah, so yeah, yeah, yeah, so so the A is summed.

So I will always, yeah, I forgot to mention, good, this is a great question.

So I always assume in this Einstein convention, in the sense that all the repeated indices are assumed to be summed. So no matter how many components are there, you just sum over A. OK, so A is summed. Here, yeah.

Good. Other questions?

So now we can talk about equation motion.

So it's the same thing as classical mechanics.

So classical field theory, just like classical mechanics, you generalize your infinite number of degrees of freedom.

So now with each point in space, you associate with some degrees of freedom.

So now let's look at equation motion.

So again, you just do the variation of the action, you extremize the action to be 0. integral.

So the action is again just defined by the dt over Lagrangian.

So the s is the dtl and then can be written as the four integral dt and dx of the density.

So now let me just write, let me not just write it in the more right basic notation, just assuming there's one derivative, one time and one spatial derivatives.

Okay, so the mu now, yeah, I combine the space and the time derivative together.

Okay, and you can straightforwardly generalize to, say, in more and more spatial derivatives.

good so so here so so let's just do the variation so the so we want to zero equal to data s so you want the variation of this s to be zero so now let's just worry s so you have d4x now just remember this not going in density it's just ordinary function because of the locality just ordinary function of phi a and its derivatives.

And we can just do the variation in a straightforward way.

You can just write partial l partial phi a and delta phi a plus partial l partial mu phi a and delta partial mu phi a.

And in this case, so the delta is a variation and the partial mu. So these two operations are independent of each other, so you can exchange them. So you can put actually delta inside.

So this is the same as partial mu, delta phi.

So now you can integrate by part of the second term.

Now you can do it because here it's a little bit awkward because you're moving the partial mu, delta phi a. So again, here the repeated indices, they're all summed. So you should sum them.

So now we can do integration by part here.

So then you get d4x, and you get partial l partial phi a minus partial mu partial l partial partial mu phi a, OK?

And then the whole thing, delta phi a, and then plus boundary terms. terms.

So the boundary terms come from you doing integration by part here.

You're doing integration by part here.

So the boundary term will be proportional to data phi a. Yes?

That's just the arbitrary variation. Yeah, just arbitrary variation.

So we always assume we have the boundary conditions. Okay. And so that the boundary term vanishes.

Okay. So here, we are not going to detail.

So we always choose theta phi a so that boundary terms vanishes.

The term comes from integration by parts.

And the second, just remember, we sheeted indices are summed.

And now, so this has, then we don't have to worry about boundary terms so that boundary term vanishes. Then now we just have this has to be 0.

But this has to be 0 for any variation. So this delta x mu can be any function of x, OK?

Can be any function of x. So the only way this can happen is that this pre -factor must vanish, OK?

So this implies that the delta x partial phi a minus partial mu by a.

So this is the general equation motion for classical field theory. Okay?

Any questions on this?

Okay. So from now on, we will make two restrictions. Okay?

also there are the most, we'll make two restrictions just for simplicity.

Yeah, most of our discussion can be generalized beyond those situations.

Yeah, so the first is that we restrict to fields here which are translationally in warrant, OK? Means that there's no special point.

There's no special means there's no special space time point in this theory, OK?

If you do experiment here in Boston, it's the same as you do experiment in Washington DC, et cetera, OK?

And if your theory is not translation in warrant, and then when you do experiment in Boston, and then will be different from you do it in, say, in New York.

And the second is that we assume it's Lorentz invariant.

So in some condensed matter applications, which you don't have Lorentz symmetry, and then you don't have Lorentz symmetry, but similar techniques we are going to talk about can be applied. right.

And we will also elaborate a little bit more on both of these aspects.

OK, so now let me give you some simple examples of the classical field series, which satisfy these two conditions, examples. So the first one is the Maxwell series.

So for Maxwell series, the dynamical variable So this four vector potential, a mu, so now I'm using the four vector rotation. So that will often suppress the index on x. That means this is a four vector.

And then from the a mu, you can define the field strings.

And e and b then can be obtained from this f mu mu.

And b can be obtained from f mu mu.

And then the action for the E and M can be obtained by sd. So this is just the action for the E and M without charged matter.

So if you do the variation here to get the equation motion, then you get the Maxwell equations without sources.

OK, you get the vacuum Maxwell equations. You get the vacuum Maxwell equations.

And so you see that this action half of the form we mentioned here is corresponding to a local theory, a single integral of space time.

And the second example is Einstein gravity.

It's a lot of classical field theory.

So let me just write down the action.

So this is written as G newton, so G is the metric. Yeah, anyway, if you don't know the Einstein series, OK. So R is a rich scalar.

And if you don't know the Einstein series, OK. And just write it down just to show that this is a local theory involving only a single spacetime integral of some quantities.

And then the simplest quantum field theory called the scalar field series, because I think I erased it, because this is involving the vector field. So this is a vector field. Because at each spacetime point, there's a vector.

Here, the Einstein gravity, the dynamical variable is a tensor, so it's even something more complicated.

So the simplest case is a scalar, which at each spacetime point, you just to have a single value quantity.

And so let's just consider the simplest case, just consider a real value scalar field phi.

So this is just take a real value. You can also consider complex value. But the simple is just a real value.

At each spacetime point, you have a single value. And so that's the dynamical variable.

And so in real life, there are many examples of scalar fields.

So, for example, the Higgs field, which is a celebrated Higgs field, which was discovered a number of years ago, is a real scalar field.

And also pions, okay, the pion particle can be considered as excitations of a scalar field.

Anyway, so, so a simple the simplest family of a scalar field action is the following form.

So here I just write this action down.

But essentially, you can argue this is the most general action you can write down, which are compatible with locality, with locality, and with translation symmetry, and the Lorentz symmetry. So here you see the Lorentz indices are contracted it.

In the Lorentz indices, the derivative are contracted, so that guarantees this theory is Lorentz invariant.

And v is just some function of phi. V here is just some function of phi.

So this is essentially the theory, yeah, the simplest theory you can write down based on localities.

So say we can take v phi, in general, we can take it to be polynomial. Yeah, there's some function of phi.

So when you have a translation symmetry, means that in this theory, if you do experiment, it's the same everywhere.

That implies that v phi, so here there's nothing, there's no parameter here, v phi can must, yeah just, yeah sorry, all parameters in v phi must be constant.

Okay, they cannot depend on spacetime, okay, must be constant.

So if you have some parameter dependent on the spacetime, then of course a different space and a different spacetime point will behave differently, and there will not be transitions in warrants, okay.

So let's try to work out this theory a little bit in more detail.

So let's find this Lagrangian, let's try to find its momentum.

So it's momentum density.

Here, there's only single fields, so there's no index. So pi x would be take derivative of this. So this is L.

So this thing inside the bracket is L. This partial L partial phi dot times derivative of phi.

And so if you look at here, here, at the time derivative, just phi dot square using this four vector notation.

And if you expand it, and then the momentum density, just phi dot x, because partial mu phi partial mu phi is equal to minus phi dot square plus.

And then you can find the Hamiltonian density, pi phi dot minus a l, and then you find that this is given by mean expressed in terms of momentum.

And equation motion, if you apply here, if you apply here, then you find given by partial to the partial square equal to partial v partial phi equal to 0.

And then again, here I'm using a short -hand notation, partial square. It's defined to be partial mu, partial mu.

So this is the same as minus partial g squared plus.

So here, let me also give you some simple examples of the v.

v in the simplest case is you take v just, of course, to be a constant. But the constant does not do anything because of the, yeah. So the simplest case is just v phi equal to linear term.

And from time translation symmetry, this f has to be a constant. It cannot be dependent on spacetime.

If it is dependent on spacetime, and then you will violate the space translation symmetry.

So when you have these, then your equation motion given by partial square equal to f. f is a constant.

So in this case, so essentially, you have some kind of thing, some kind of constant which source this phi. So in this case, we say this system have external force.

we call this f external force.

Because if f is non -zero, then phi cannot be zero. Phi has to be non -zero.

And so in the sense the phi will be always be excited, it will always be generated, non -trivial phi will always be generated if s is non -zero. OK. So here we call it external force.

And so normally, we don't normally would like to consider the five.

Normally, we will consider that normally in the situation of your interest in the situation, there's no external force. OK. The system developed on its own.

So in that case, so if we forget about external force, then the simplest situation would v phi equal to quadratic, some polynomial, phi square. So that's the last simplest function of phi.

And m square has to be a constant, again, from the translation symmetry.

So in this case, then you have equation motion. Partial square of phi equal to m square of phi square. Or m square of phi equal to 0.

So this is a very famous equation. equation.

So this is called the Klein -Gordon equation.

And the next time we will, yeah, you will see very quickly why this is a famous equation, okay?

And yeah, yeah, so this would correspond to the simplest field theory we can consider.

And then you can consider more complicated field series with v5 to be some higher polynomial or more complicated functions. equations.

But keep in mind that this square is special. Because when it's square, this is a linear equation.

But whenever you have something which is not linear or quadratic and you will get a nonlinear equation, and then the story becomes complicated.

Okay. Nonlinear equations are always complicated. And anyway, so this will be the simplest theory we will consider, okay? And, okay, so let's stop here.