Lecture 2: Symmetries and Conservation Laws（原声文稿）

Let us start, first I forgot to mention last time, I will always use the metric convention for Minkowski space time to be given by this one.

So the time component have minus one and then the spatial component is positive one.

And yeah, so also let me just first remind you what we did in last lecture.

So, in last lecture, we talked about principle of locality, which is a powerful principle.

And so, this naturally leads to the concept of fields.

And so, I will generally denote fields, say, using this kind of notation.

Yeah, in the, when we talk about fields abstractly, we will use this kind of notation.

and so phi just an a just label for different fields and also so the fields depend on spatial coordinate and also depend on time.

So the physically you can think of this spatial dependence as the label for the fields for the degrees of freedom.

So at each spatial point this is the freedom and the time describe its evolution, okay?

So essentially this is like, so classical field is like classical mechanics, because freedom, but now it's the infinite number of degrees of freedom, okay?

So the classical field theory is like classical mechanics, but now it's infinite number of degrees of freedom, because now you have some finite number of degrees of freedom for for each point in spacetime, okay.

And the principle locality also implies that the action for such kind of dynamical variables, so the fields are all dynamical variables, and also implies that the action has a local form.

So that the action have the form that is always a single spacetime integral, and some function which is called Argonian density, t, which is a function of the fields, say phi at some point, at the point x, which, and then also its derivatives.

And so because of locality, then the action has a very simple form.

And then I think classical mechanics, you can introduce canonical momentum, conjugate to your dynamical variable.

So we have momentum density, they conjugate to phi A at each point.

So this is the phi dot, which is the time derivative of phi.

And then you can also introduce the Hamiltonian density, which is the pi A by dot A.

So A is summed and the minus L. And the Hamiltonian is obtained by integrating this over space.

Yeah, and also the equation motion.

When you do the variation of the action, you find the equation motion is given by the following general form, the equation motion is given by this general form.

And when we study field theory, as we study, say, any other subject, we always start with simplicity examples, okay?

And the simplicity example would be, say, a single scalar field, if you can see the single scalar field, which you have low index, we just have five, not dependent on spacetime point x.

And then the simplicity action for such a scalar field, you can just write down based on a general principle, say, Yeah, so you can have this form.

So we discussed last time the simplest action for scalar field have the following form, partial mu phi, partial mu phi, and the plus m square phi square. And the sign here is determined because of the metric condition.

So this is the simplest scalar field theory we can write down.

So here I have written here a quadratic function for phi, which we wrote as a general function last time.

And here the kinetic term comes from the Lorentz symmetry.

So this is the simplest derivative term you can write down with respect to Lorentz symmetry.

You can have more complicated terms, you can have this term, for example, squared. Okay, but that would be more complicated, so this is the simplest one.

And so this would be the simplest field theory we will study first, okay?

And yeah, so in this case, for this example, the momentum density just equal to pi dot.

It's time derivative of pi, and then the equation motion, It's given by a linear equation phi, it's given by that.

So this is a very simple theory.

It's very simple equation motion, but data we will see. I actually will teach us a lot about general field theory.

So it's a very quick summary of some of the main points of last lecture. Do you have any questions?

So phi here is a real field. Yeah.

Yeah, so phi just take a real value.

So now we talk about the second topic related to our discussion of classical field theory. Symmetries.

So when we have a theory, say with such an action, we say the theory has a symmetry.

So the definition of the symmetry is the following.

So symmetry is some transformation of each field phi a into say some function of phi.

So this new phi a can say depend on all possible fields.

And so some transformation of your dynamical variables.

So here, prime just means a different function does not mean derivatives.

Which leaves the action in worries, okay?

So whenever happens, and we say this is a symmetry. symmetry.

So for example, in this example, let me give you a label, label to this equation and we call this equation star.

So in the example of star, say for example the symmetry includes say translation.

So if you imagine we take a coordinate transformation, we do a shift in coordinate by a constant shift.

Let's imagine we do a constant shift of the coordinates, and a mu is a constant, so a mu is a constant vector, and now if we assume that the phi transforms as a following, so when you change the the coordinates essentially just change the label on the field.

But the value of the field should not change.

So that means that the phi evaluated at the original point should be the same.

So that means the nu transformed phi evaluated at the nu point, nu x prime point, should be the same as the value at the original point.

Okay, so this is the, yeah, so suppose the phi transformed like this, so the, so scalar field should transform as this under the general, so this is the example of a space -time transformation, okay, and the general space -time transformation as this equation, okay. So the, so you change the label but the value of phi should not change.

So the new phi evaluated at the new location should be the same as the value of phi at the original location.

And so you can easily check yourself.

Okay, so you can easily check yourself. I will not do it here.

So this exercise for yourself, that actually this action is it's invariant under this transformation with this change of x.

Any questions on this?

So second transformation, which this is invariant, is the Lorentz symmetry.

So Lorentz transformation is another kind of spacetime transformation on the spacetime coordinates.

If you take x mu to x mu prime, you're going to lambda mu nu x nu, so lambda nu nu is the constant Lorentz transformation matrix, okay, the constant Lorentz transformation matrix, and again under such a coordinate transformation phi will be transformed this way, phi should transform this way, and now you can check yourself, okay, okay, and because of this contraction of partial mu, partial mu, and this is actually a symmetry.

Okay, again, this is left as an exercise for yourself, and in fact, I think in your P side problem, you will do something similar.

Any questions on this?

So examples like this kind of transformation for translation and the following transformation. So this is what we call the continuous symmetries.

So, so continuous symmetries, okay, so, so these are the transformations, yes, I should also give a label to this equation, so let me call it just star, star, star.

So, so, uh, uh, continuous symmetries are transformations like star, star, star involving continuous parameters.

And so both 1 and 2 are clearly continuous symmetries.

So you can easily see from here. So here the transformation is A, and A is four numbers, okay, it's a vector, it's a four vector.

And a can be arbitrarily changed. So it's a continuous parameter, okay? You can take it to be zero. You can take it to be point one in all directions, etc. So it's something can be continuous change.

And similarly, the load -range transformation contains continuous parameters.

So do you remember how many continuous parameters does a load -range transformation contain in four dimensions yes no yes yeah six it's because you have three rotations because you have three direction three spatial directions you have three rotations and then you also have three boosts yeah so together six so here there's six continuous parameters and here there's four.

Okay, so here is this four. So both are continuous symmetries, okay?

And in contrast, this Lagrangian, this action has a lot of symmetry, which is not continuous. Can you say what that symmetry is?

Good. Priority is a good, it's a good answer, yeah.

But But there's still something slightly simpler.

So notice that here it's all quadratic in phi. And what do you observe?

So here there's a symmetry, a lot of symmetry.

It's phi goes to minus phi, because it's quadratic in phi.

And also, of course, there's a symmetry called the parity, which in the case x goes to minus x.

So the spatial direction x goes to minus x, and then phi again transform as this okay and you can also check this is also symmetry so in both of these cases they don't involve continuous parameters and so these are called discrete symmetries okay so there's no continuous parameters and yeah in this case both of them are called the z2 in the z2 means that if you do the transformation once they go back to the identity okay so here if you do to the twice, yeah, if you do it twice and go back to it, then if you do it twice and go back to it, then I think this is the same.

So here is Z2 transmission.

Also, symmetries can be also separated into, say symmetries can be separated into continuous symmetry, or discrete symmetries depend on whether you have continuous parameters or not.

But symmetries can also be separated into other different, And there's a lot of classification of the symmetry.

It's called the gauge symmetry and global symmetries.

So they are global symmetries corresponding to the transformation parameters are spacetime independent.

So both of these are examples of global symmetries because they are just constant. They don't depend on spacetime. And the lambda don't depend on spacetime.

And you can also have local symmetries in which case the transformation parameters are space -time -dependent.

So an example of the local transformation which you may remember is the so -called gauge transformation in E and M.

So e and m, you can, yeah, anyway, so later we will see example, we will go back to e and m again and you will see examples of this local signature.

Yes? It changes the action but it doesn't change the equations of motion. Did that, so it's similar? Yeah, yeah, as far as it goes with the action.

Yeah, that's a very good question.

So, by definition, if it leaves the action environment, we say it's a symmetry. And in general, actually almost always, if it leaves the action environment, actually also leaves the equation motion environment.

Yeah, but it's not guaranteed, yeah. Yes, purely from a mathematical point of view, it's not guaranteed. But for all physical examples, it's almost always the same. Other questions?

And now there's a very simple, but a very deep connection between the symmetry and the conservation laws.

So that you may have already landed in... Actually I forgot when I landed myself.

So some of you may have landed in high school, some of you may have landed in Adelaide, etc.

Anyway, so there's a connection between the symmetry and the conservation laws.

And so any conservation laws can be understood as a consequence of symmetry.

So in classical mechanics, you may remember that the time translation, this leads to energy conservation.

And the spatial translation, symmetry, then leads to the momentum conservation.

And if it's a rotational symmetry, then it's to the angular momentum conservation, okay? So this should be something you already know, say from your classical mechanics.

But in classical field theory, this can be generalized, okay? So here, there's a Lawson theorem.

So first discovered by I. Let me loss the theorem.

You said any continuous global symmetry leads to conservation laws.

OK, leads to a conservation law.

And so no matter what kind of symmetry, in addition to those things you are familiar with, no matter what kind of symmetry.

But any time you have a continuous global symmetry in your system, then you have a conservation law.

Okay, so now let me give a proof of this node, it's also a theorem, okay, so before I prove it, do you have any questions regarding its statement?

It's like the definition. So you said I could continue a symmetry. You wanted to continue a seal on a parameter.

But if you wanted to write out an equation, star, star, star in a way that specializes to continue a symmetry, there are ways to do that.

I'll show, like, just, like, how to check something. Oh, yeah, yeah. You just check whether there's a continuous parameter.

You just check whether there's a parameter or not.

Does this answer your question?

Yeah, normally you can just see whether there's a parameter.

If you write down the transformation explicitly, you will be able to just see. If we go down here, there's no transformation parameter.

You can just see under there, for the Lorentz transformation and translation, you can just see the explicit.

So, important aspect of a continuous symmetry is that because you have a continuous parameter, So, you can continuously relate it to identity. So, the trivial transformation, you just don't transform at all, right?

So, in this case, the Mu is equal to zero. And so in this case, the trivial transformation, just the lambda Mu is the identity matrix.

And then you can imagine a slightly, a pending on a slight infinitesimal rotation or infinitesimal boost. and then that's corresponding to a so -called infinitesimal transformation.

So here you can also just translate a mu a little bit, so very close to the identity.

And so essentially for any global continuous symmetry, it has an infinitesimal form.

It's just when the transformation parameter is very close to the identity, very close to the zero.

And so for example, for general transformation like this, if the transformation is close to the identity, then we can always write it in the following form.

Phi a goes to phi a prime, which is equal to phi a, and with some parameters, small parameter epsilon, and some function f a, and then say phi b, and the derivative of phi b, et cetera.

okay so this epsilon is the infinitesimal transformation parameter okay so if you have a continuous symmetry there's always say an infinitesimal transformation which are close to the identity so so this epsilon corresponding to this case would be for a mu is very small or for the Lorentz transformation case it's corresponding to rotation angle is very small or the boost is very small yeah so the epsilon can be any of those parameters and then this f it can be some arbitrary function okay this f can be up some arbitrary functions of your fields good so so in this case yeah so in this case the data phi okay so the yeah so the symmetry transformation so we have used the rotation data hat phi a so this symmetry because then given by the epsilon f a okay so so so data hat here implies a symmetry transform infinitesimal symmetry transformation okay so now let's consider during a variation of s under this data hat okay since the by definition of the symmetry because of the this is a symmetry that means that So the delta hat s should be equal to zero, okay? Because the variation of the action should be invariant under the symmetry. And so now let's look at what's the consequence of it.

So now we can just worry this L.

So this means that the variation of the L, the Lagrangian density, must be a total derivative.

Okay, since all, every transformation is proportional to epsilon, yeah, so epsilon is a small parameter.

So when we do the transformation, we only need to keep track linear order in the, in epsilon.

So, so the transformation of the a l, so, so must also be proportional to epsilon.

Okay, so, so, so since this must be invariant, that means the variation of the a l must be a total derivative.

two. So k mu would be just some, so for some k mu, okay, for some k mu. So there must be some k mu.

And this k mu can also be zero. In that case, the Lagrangian density is invariant. Yes?

Is delta hat the variation under the transformation? Delta hat is a spec. Delta hat just delodes such a transformation.

Just delodes such transformation. This is not the general... this is not the general variation. This is very specific. This is a symmetric transformation.

And so, under the symmetric transformation, your action should be invariant by definition, because this is a symmetry.

And that in turn, because the S is the integration of L, then that means that the transformation of L, as long as density, must be a total derivative.

And here, we only keep track of everything linear, all in in epsilon, and so this must be proportional to epsilon times the total derivative, okay?

And so we must have this structure, yeah, for some k -ball, okay?

Which can be zero, which can be zero, okay?

Good, so now let's look at what's the implications of this equation, okay? So this equation is the constraint imposed by the symmetry, okay? Okay, now let's look at what are the implications of these constraints.

So now we can just do the variation of L.

Now we can just do the variation of L.

And maybe I will keep that equation when I will do this board first.

So if something is symmetry of your action for your system, would you like directly vary the thing, like parameter, and then show that your action doesn't change, or would you do that, like, if it was one, and then show that?

So we want to use the fact that such a transformation is a symmetry to derive what is the constraint on the Lagrangian, okay, what is the constraint on the theory.

So we assume that somehow the theory have a symmetry and then we want to see what is the constraint this put on your theory.

Yeah, that's right, we're proving in the law of theorem.

Good, other questions?

So that equation is the constraint imposed by the symmetry. And we want to see what this equation tells us.

And for this purpose, we just do the variation.

So the delta hat L, because L is just a function of phi.

And then we have partial l partial phi a that a hat by a and Then we have l partial partial mu by a and Then partial mu head by a okay So we just did the variation and now Okay, and the so yeah and now we are going to use the equation motion okay so so from the equation motion yeah from the equation motion and partial L partial phi a this is just equal to partial mu partial L partial partial mu phi a okay so we just plug in this into here so when you plug in this into here now you notice is this become a total derivative.

So you have partial L partial mu by A and delta hat by A.

So after using equation motion, this variation of L have the following form, which is the total derivative.

And this should be equal to the right -hand side. It should be equal to the right -hand side.

And then here we find, so this should be equal to epsilon partial mu k mu.

Okay, so we combine both sides together, plug in that this is equal to epsilon times FA, and then we conclude.

Okay, then we conclude partial mu, partial L, partial partial mu pi A.

F A minus k mu, we think of it as mu, okay?

So if we call this j mu, and then we have a conservation equation, partial mu, j mu, we think of it as mu.

Okay, so we have a conservation law.

So remember, a conservation law is just to have a vector which satisfies this equation, and then, yeah.

So more explicitly, yeah. Yeah, so the j mu is just this combination, okay? J mu is just this combination. So any questions on this?

This is quite a strange condition because it's that the derivative with respect to space time is zero, but you can have a conservation of the derivative with respect to time in here for that state. Yeah, yeah, yeah, I will, yeah.

Yeah, this is the stronger version than your normally. Yeah, so this is the space time version of the conservation law from your normally classical mechanics.

So this is a field theory version of that. Yeah, yeah, I will elaborate that equation a little bit. Yes?

So when it transforms, the effect is?

So here, yeah, yeah. So here is a general formulation.

So this transformation can be general. it doesn't have to be, say, a spacetime transformation we said earlier, this is just some abstract symmetry, just some arbitrary transformations.

Is there a change in the field of this function and not the space change? Yeah, because spacetime variable is a dummy variable, in the action you can always get rid of that.

The x change, x is just a dummy variable in your action. because you integrate over x, yeah.

Yeah, that can include the, normally, yeah, you can in principle have as many derivatives as you want, but normally if we have action which only contains first derivatives and your symmetry transformation only involves first derivatives.

That's not discussing the case.

Yeah, in principle you can have it. I just, yeah, I'm just writing this for simplicity.

Yeah, maybe you can ask after the class. Yeah, it's not clear. Other questions?

Okay, good. So let me elaborate a little bit on this equation.

So this equation you may have seen in the, in EMM, so, which you often call the continuity equation.

If you write this equation separately, has the following form partial 0 j0 plus partial i or the divergence of the spatial component.

So j mu, if we write it in terms of the explicitly, then you have this form.

So this is so -called the continuity equation, just the time variation of the density.

The 0 component, they consider some kind of density, it's the same as the divergence of the current, okay?

And in particular, you can define a charge, which is the total spatial integration of j0.

So, if you integrate both sides over the total volume, and then this just becomes partial 0, q.

two, and this term become a total derivative, and then you can convert it into a boundary term using the Gauss law, and then normally the current advantage is at infinity, and then you just have the charge combination, just have charge combination, okay, so this is like the field, so normally in classical mechanics you just have this equation, okay, And this is the field theory version of your conservation.

that how it represents? Sorry? Is there a way to tell me what G represents?

J represents? Yeah, very represent. Yeah, yeah, yeah. If I give you a specific, yeah, you will do it in your PSAC.

So if I give you a specific Lagrangian, if you start a specific transformation, and then you will be able to find all those quantities, you will be able to find the k explicitly, and you will be able to find all those quantities explicitly then you can find the j.

It's because if we have here, so if we have here, if we have data s equal to zero and then you can just put the data in and then this quantity then is integrated over that. If this is zero then this has to be a total derivative.

Yes. It's rather global symmetry than this part of the equation.

Right, yeah, you're good, this is a very good question.

So yeah, so I will not have time to go into here. Let me just very quickly mention. So this is called the first nozzle theorem.

And when epsilon is spacetime independent.

And so when epsilon is not spacetime independent, and the story is a little bit more complicated, so the epsilon will be inside, the epsilon will be inside this total derivative, you cannot take it out, etc.

And then the story changes a little bit, and then there's something called the second law of the theorem, and actually in that case, when you have local symmetries when epsilon depend on spacetime, instead of finding conservation laws, you find that your equation motion, not all, Or you find some part of the equation of motion are redundant.

Yeah, and if you, it's not difficult, but actually in principle, I can put it in the p -side problem. If you really want to see it. Yes?

That's about the transformation of the equation of the identity. What happens to the transformation of the equation?

Good, good, good. So, this is the QBT, QT beauty of physics, which you may already have seen in quantum mechanics.

So, whenever we see a symmetry, no matter how complicated that symmetry is, say some continuous symmetry, it's enough just to understand that symmetry near the identity, near to be the infinitesimal transformation.

It's because any finite transformation you can build up from just adding up the infinitesimal transformations.

So, once you know the infinitesimal transformation, actually, essentially, up to some global topological structure, essentially, determine the full finite transformation. Yeah.

No, then there's always a symmetry, right? Because each step is a symmetry, and so you add them up, you add them up, still a symmetry.

Or if you do different, yeah, there's still a symmetry.

Yeah, because by definition, if something transforms its invariant, it will do a lot of transformations still invariant.

Good, other questions? Yes? For this grid symmetry, we don't have this, if you don't want to do something similar. Sorry? For this grid symmetry.

For this grid symmetry. I have to do this, and I will do something similar.

Yeah, in general, for this grid symmetry, if you don't have such a conservation law, you don't have such a conservation law.

You can still, sometimes you can still define some discrete quantum number, which is conserved, make it a parity, which in the parity case, but you don't have a current. Okay, so here, Jmu is called a conserved current, but you won't have a current.

Good, good, so this concludes our discussion very quick discussion of the important features of classical field theory.

And with this preparation, and now we can start thinking about going to quantum mechanics, talking about the quantum field.

And so here is a good place to pause a little bit from what we already said about the classical field theory to anticipate a little bit about the quantum field theory.

So here I will restate our goal for quantum field theory, which I said a little bit in the last lecture.

So now we have seen the classical fields and our goal is to understand the quantum version of this story.

So classical field theory is the classical mechanics with infinite number of degrees of freedom and now we want to quantize this infinite number of degrees of freedom and now this becomes quantum fields.

And now we want to understand the quantum dynamics of such a field, such kind of five fields, okay.

And for example, the example is the E and M, we have an electric magnetic field, okay, and you know the Maxwell equations, you can solve the Maxwell equations, so this defines the classical field theory, and then we will tell you how to actually quantify such a system to understand the electric field and the magnetic field quantum mechanical.

Good, and that actually would be our goal, okay, so we will start the simplest field theory, just the scalar field theory, and then we go to the Dirac theory, which describes the electron, and then eventually we will go to the theory, which describes the electron -magnetic field, so this is called QED, okay, quantum -magnetic dynamics, so that will be the end point of this course.

Okay, so now let's say a little bit how we go to this classical field theory to quantum field theory. Okay, so some just some general remarks.

So before doing that, let's think a little bit how we go to from classical mechanics with a small number degrees, even how you go to quantum mechanics.

Okay, so let's recall, so in classical mechanics, so let's just consider the simplest case, you just have a single degree of freedom, say xt, okay, just one dimensional particle whose motion just describes the xt.

And then, of course, the goal of classical mechanics is just to solve the equation of motion over xt.

So if I write down the equation for xt and solve it, then I'm done. I'm done. You solve the classical mechanics.

So when you go to quantum mechanics, what do you do? So this xt then become an operator.

So become an operator. So the classical dynamical variable now become a quantum operator. And the equation motion of xt, if you remember, then do you remember what it becomes?

Yeah, it exactly becomes the Heisenberg equation.

So it becomes a Heisenberg equation for xt, for this operator xt now. Okay?

Yeah. So in quantum mechanics, there are two, normally two ways to describe it. So first is the Schrodinger picture.

In this case you have a wave function which is a function of x and t and then you have an operator which is x.

And then of course you also have conjugate momentum which is defined to be say x dot.

And so the wave function is a function of the eigenvalues of, so x here should be understood So that's eigenvalues of this operator x hat, okay?

It's the eigenvalue of x hat.

So in the shorting of picture, you solve the evolution for psi, you solve the equation for psi, and once you find psi, then you can calculate any quantities you want, okay?

You can calculate any expectation values, any amplitude, et cetera. But the second way of approach it is the Heisenberg picture.

In this case, the dynamical quantity is your operator, now you're assuming the operator depends on time. So in the Schrodinger picture, your x is just some constant operator, it does not depend on time.

But in the Heisenberg picture, now your operator does not become time dependent.

And the equation motion, and you solve the Heisenberg equation, okay, and then the state is invariant, okay? State does not evolve with time, okay?

So you solve the Heisenberg equation for those operators, and once you solve the Heisenberg equation, and then you can evaluate, again, your expectation value in any states you are interested in, et cetera.

So any questions on this? This is a very quick review of what you did to go from classical mechanics to quantum mechanics, okay?

Okay, so now in field theory, in field theory we do a similar thing, so similarly in field theory we have classical fields, okay, we have classical fields, and then when you go to quantum, this becomes quantum operators, okay, so this is now quantum operators.

Remember, in field theory, this x is always just labels of the space point, okay? It's not a dynamical variable, okay?

The notation is a little bit, yeah, here in classical mechanics, this is a dynamical variable, but in field theory, this is just labels, okay?

So they're not dynamical, a dynamical variable phi itself. So phi become an operator, okay?

Okay, and then now the classical equation motion for phi will be derived and then become the Heisenberg equation, okay, for phi hat.

So again, here you can do two picture, you can do the Schrodinger picture, you can do a Heisenberg picture.

So for QFT, the Schrodinger picture, again you look at the wave functions of your dynamical variables, the eigenvalue of the dynamical variables.

So what's the generation of this psi xt?

So remember here psi is a function of eigenvalue of your dynamical variables, OK?

So now if we push this analog further, when you go to quantum field theory in the Schrodinger picture, now the wave function should be a function of phi x and t.

So these phi x are eigenvalues of phi hat x.

So in the Schrodinger picture, phi hat does not evolve this time.

So you have the wave function, which is the function that will become a functional, because this itself is a function of space.

And you solve the Schrodinger equation for this, and then you have the operator, and then you have the operator, and then you have operator phi x. x.

Okay, so that's what you do in the Schrodinger picture, in the Schrodinger picture.

And in the Heisenberg picture, you forget about the wave function, okay? You look at the evolution of operators.

So the Heisenberg for the field theory, the Heisenberg picture for the field is that you look at the evolution of x and t.

So now this just obeys the Heisenberg equations, which is just the quantum motion of the classical equation for phi.

So you look at the evolution of this, and then the state does not change this time. OK? The state does not change this time.

Good. Any questions on this?

So in the shorting picture, so remember, Remember the x is a label for phi, but time is evolution, but in the shorting of pictures, the operator don't evolve. And so there's no time here. So we don't have time here.

And we just have the, just the analog of the x here in the shorting of picture for the classical, yeah, for quantum mechanics.

And then the time dependence is in your wave function.

So the wave function is a function of possible values, eigenvalues of this operator phi law.

and then the Heisenberg picture again, then you just focus on the operator equations, and once you solve the operator equations, and then you can calculate the expectation values in any state you want.

Okay, so now you can already maybe see a difference a little bit. So, if you do the Schrodinger picture, you have to deal with this beast, okay?

Which is the wave functional of all possible values of some function in space, okay?

And if you have multiple fields, then this is a hugely complicated thing. And you need to write down the Schrodinger equation for it, et cetera, okay?

But here, but in the Heisenberg picture, where we just solve the analog of the classical equation motion, which we have already written down, and you just now interpret it as a quantum operator equation. So which one you think is simpler?

So that's why in quantum field theory, we almost always use Heisenberg picture.

We always use Heisenberg picture, and we don't even rarely think about the wave function.

Even though sometimes this can be useful in some problems, But for the most of the time, this is much easier. So that's what we are going to do.

And so in quantum field theory, we will just use the Heisenberg picture almost all the time.

From now on, I will not talk about the Schrodinger picture.

Good. So it's very important. You remind yourself about quantum mechanics in the Heisenberg picture.

And because most of your quantum mechanical classes before, maybe it's all in the shorting of picture, solving the wave, solving the shorting of equations, et cetera, but now you have to change the perspective to think everything in terms of the Heisenberg picture.

And then that will make you, then in quantum field theory, much easier. Okay?

Good. Any questions on this?

So in the hiding bird picture, if we're not concerned about the wave function and all, then how do we determine, for example, the probability or the expectation value of where x is at some point in the momentum? Good, good, good. That's a very good question.

So this is actually also related to the kind of questions we want to solve in quantum field theory.

So in quantum field theory, we often work with the vacuum states.

So for example, here, yeah, so when you consider the QED, most of the time, in the quantum electric and magnetic field, they are in the vacuum state.

And so we just consider the psi in the vacuum state.

And so we don't have to consider, yeah, yeah, that's it. So normally in quantum field, they prefer the states we are interested in.

That's a lot of reason that the Heisenberg picture is convenient. You don't have to consider the general state in general. Yes?

What is the physical meaning of the idea? Oh, this is just some state you are interested in.

So you have a Hilbert space, but now you just don't, When you study the evolution, you only evolve operate, you don't evolve states. So that's the difference between the Schrodinger and the Haydnberg picture.

But for example, in quantum mechanics, you get the sufficient value of some operators with the factor of state. And we get like resonance in the resultant stuff.

So what about here? Yeah, it's the same thing. Yeah, same thing. Yeah, at this level, there's no difference between quantum fields here and the quantum mechanics.

Yeah, just think about quantum mechanics in terms of Heisenberg picture. Translate everything you learned about the Schrodinger evolution, et cetera, in terms of Heisenberg picture. Yes? Hm?

Yeah, that's right. That's right. Yes? Can I just say we're interested in the vacuum state and what sorts of measurements?

Yeah, so it's again, so yeah, it's a very good question.

What I should have said is that the state we are interested in are states which are close to the vacuum states. So you excite the vacuum a little bit.

And yeah, it's not, yeah, of course in the vacuum you don't have anything. And we are since close to the vacuum states.

Yeah, and later when we discuss things you will see good good.

Okay, so before going to quantum field theory Let's also make some remarks Have a short discussion on right to visit quantum mechanics okay so So naively if you have special relativity quantum mechanics So, if you want to generalize, so most of the quantum mechanics you learned is the non relativistic quantum mechanics, okay.

But now if you want to incorporate special, we want to combine the special relativity with quantum mechanics. And then what should you get? What do you think you should get?

I expect, yeah, just whatever come to your mind, that's the right answer.

That's the correct answer, but I was hoping some of you will say it's relativistic quantum mechanics.

So, laivably, when you combine these two, in particular, if you read some old quantum mechanics books, they do discuss relativistic quantum mechanics, so laivably that's what you get.

when you combine these two, and you say, oh, we just get relativistic quantum mechanics. But that's actually not correct statement.

Actually strictly speaking, the reason now you don't actually learn much about the relativistic quantum mechanics because strictly speaking, relativistic quantum mechanics does not exist.

And whenever if you want to combine quantum mechanics with special relativity, actually you get quantum field theory. So the quantum field theory is actually forced on us. If you want to unify special relativity and the quantum mechanics, okay?

And even if you don't want to talk about fields, okay?

And even if you don't want to talk about fields, if you just want to talk about particles, still if you want to unify these two, then you actually automatically base two quantum field theory.

So now let me just explain why this is the case, so that you have some better appreciation of the quantum field stream.

So let's try to apply what you did for long -righted vistic quantum mechanics, try to generalize it to derive some righted vistic quantum mechanics, suppose you are the people in the 1926, the quantum mechanics just was discovered. And you say, oh, people have understood the non -right -wistic quantum mechanics. Now let's generalize to special relativity.

OK, so in non -right -wistic quantum mechanics, how do we derive the Schrodinger equation?

The way we do this is we start with your dispersion relation for a non -right -wistic particle.

then we say let's take e to i partial p, okay, p.

So say if you can have a vector become, yeah, so let me forget about h bar. This p then becomes spatial derivatives. And then this equation is become the, a Schrodinger equation for the free particles.

Okay, so this is become the Schrodinger equation for, for free long -range visiting particles. Then you can add the potentials, etc.

OK, so that's how you derive your Schrodinger equation for long relativistic quantum mechanics.

So but if you are someone, say, in the early days of quantum mechanics, say, now let's try to generalize it to special relativity, and then it's easy to do, then in the relativistic case, then you have, you just thought it was relativistic at this point in relation. equation.

You say, let's do the same thing.

Let's do the same thing. This becomes i partial t, and this p becomes this one, and then we can just write this equation.

Now let's combine this derivative together or put all this on the same side, and then what you find is partial mu, partial mu.

That's what you get. Does this equation look familiar?

This This is the simplest scalar field theory equation motion we have written down. But here, the interpretation is very different.

So earlier, we wrote down this equation. We say this is Klein -Gordon equation.

This is the Klein -Gordon equation. So this is the equation motion for the simplest free field scalar field theory we wrote down before. But here, the interpretation is very different.

So remember, the psi is not the field.

In quantum mechanics, the wave function for single particle sitting, so psi is the amplitude for particle at spatial location x at time t, okay, so this is, and it's square, then it's its probability, okay, and this does not describe a field, okay, this is the wave function of a single particle, okay, even though they have the same equation as the field theory we wrote down earlier and so the so that's what a Klein -Gordon did okay so the Klein -Gordon they try to generalize non -rativistic quantum mechanics, rativistic quantum mechanics then they wrote down this equation they say oh now we are immortal okay because we wrote down the first equation for for for for rativistic quantum mechanics and So soon realize, actually this equation, if you want to interpret it as an equation for the wave function, there are actually a lot of problems.

There are various problems, yeah, by saying a lot of it, maybe it's a little bit of exaggeration. There are various problems.

Yeah, so let me call this equation, I think I have used up my stars, I think to have a four star is a little bit too much, let's just call this equation one for this section.

So some immediate difficulties of interpreting one as a wave equation, okay, as a wave equation for relativistic particle, let me just save time, okay, so this is the wave fraction for non -rativistic particle, if you want to generalize the quantum mechanics, the relativistic quantum mechanics, then you will interpret this as a wave equation for single relativistic particle, okay.

So first, so first is that, so if Remember, so in this equation, okay, if you go back to your early days of this long -right -based quantum mechanics, so you remember from this equation you can derive an equation for the conserved probability, okay, a conserved probability, and that equation tells you that psi squared should have the interpretation of the probability, OK?

So but for that equation, you can show there's no quantity as can be used as probability density, OK? So probability density, by definition, should satisfy two conditions. So the first condition is that it's non -lactive.

And the second condition is that it should be conserved, okay?

The probability should be conserved, otherwise you violate the, yeah. So, you can show that this equation actually does not allow.

This equation allows such a quantity, but this equation does not allow, okay? And I will not go through that myself.

I think that will be in your PSI 2, okay? That will be in your PSI 2, you will show it yourself.

And the second difficulty is that if you square that, if you find the energy, so the energy if you take the square root, then the energy in principle can be, say, plus minus, you have two solutions, okay, E equal to plus minus this quantity.

So in contrast to that equation, there's only positive energy, okay, a long range of it just have positive energy. So classically, you can just say, let's throw away the second branch.

We can just throw away the second branch by hand.

Classically, but quantum mechanically, so classically, just ignore the second branch, the negative branch.

But quantum mechanically, this is not possible. Because you have the equation there, then you automatically have the negative solutions.

Okay, so quantum mechanically and then you have dispersion relation like this So that dispersion relation have the following form is the e as a function of p Okay, so you have positive branch then you have a negative branch and Then you remember in quantum mechanics you can have the then you have energy levels and you have energy levels and And a particle here, they have a higher energy than particle here.

Okay, and so here the energy level is higher than energy level here.

And there's always a non -zero probability quantum mechanically for a particle to go from some higher energy level to the lower energy level.

Okay, just like in the hydrogen atom, if you excite it, and they always go to the lower energy level. So in quantum mechanically such thing cannot be avoided.

it, okay? So you cannot just throw this branch away, okay?

And so this will lead to instability because the energy can be infinitely negative, okay? It can be infinitely negative.

And then all your particles, we all go to infinitely negative energies and then your system will be a big problem, OK?

So these are the two, the most prominent problems.

And then people tried many different ways, tried to avoid them, et cetera, including some very ingenious solutions, et cetera, so we were not going to them.

But let me just mention this.

That's actually, even if you can address those problems, still the relativistic quantum mechanics will not make sense for a very fundamental reason.

So these are the more superficial reasons why that equation does not quite work.

But there's actually more fundamental reason why relativistic quantum mechanics, even as a concept, does not make sense.

Okay, so by definition, if you want to interpret this kind of thing as a wave function, okay, so what's the interpretation of the wave function which we already said?

So this describes the amplitude for a single particle at some point, okay, at some point x at time t. But now if you have two particles, what do you do?

So, you introduce the location for particle one and particle two and the t, okay?

So, this describes two particles. And if you want to describe three particles, then you have to introduce more x. Okay, remember that's what you did in non -righted basic quantum mechanics.

So, in non -righted basic quantum mechanics, this makes sense, okay?

You can just, because there's no mixing between the different branches, say single particle, two particles, because the particle cannot be created and destroyed in a non -rativistic system.

But in a rativistic system, you can always create particles.

Okay, if you have enough energy, you can create new particles, pair, create electrons, okay, happens in accelerator all the time, or electrons, they can annihilate into photons, okay?

Okay, so the particles are not conserved, okay, not conserved. So this kind of wave function concept don't even make sense.

So if you want to describe, so that means whenever you have special relativity and quantum mechanics together, you must have a framework which can describe arbitrary number of particles at the same time, okay? Because particle numbers can change all the time, okay?

because of annihilation and the creation effect. And that cannot be achieved by this kind of concept, that wave function.

It cannot be achieved by wave function. It turns out, miraculously, that can be achieved by field theory.

It turns out that field theory, once you quantize it, automatically give you a framework to describe arbitrary number of particles in the unified manner.

single particle, two particles, arbitrary number of particles, and so this is one of the magic of field theory. So that's why the field theory plays such important role in particle physics.

It's because the excitations of fields automatically provide the mechanism to describe arbitrary number of particles. Okay, so we will stop here today.