Lecture 3: Why Quantum Field Theory（原声文稿）

2026年11月11日 · 阅读需 38 分钟

MIT OpenCourseWare（主讲 Prof. Hong Liu（刘宏））

Transcript of MIT 量子场论·8.323·第03讲：为什么需要量子场论（中文完整文稿）。

Yeah, so last lecture, we discussed loss theorem.

For every symmetry, for every continuous symmetry, this conserved current.

And then we also started talking about relativistic quantum mechanics, how we want to unify special relativity and the quantum mechanics. OK.

So the most immediate idea for that is what's called the right -to -basic quantum mechanics.

And the most immediate generalization of the Schrodinger equation.

So if you have, so at the end of last lecture, we talk about, say, the most immediate generalization of the Schrodinger equation, which, so the, So if you have e equal to say p square divided by 2m, and then you go to non -ratavistic quantum mechanics, the Schrodinger equation, okay?

And now if you have e square equal to p square plus m square for ratavistic particle, and then you get what's called the Klein -Gordon equation.

And again, this psi has the implication of of the wave function. So this describes, and then this.

So if you add the generalization of these, then this means to describe the quantum mechanics of relativistic free particle, say, of mass m, OK, of mass m.

So here, the psi tx is the wave function of a relativistic particle, relativistic particle of mass n.

And we also noticed that this equation actually is the same as the simplest field theory equation.

So we also talked about a simplest scalar field theory, classical.

So here is a simple, a classic, simple scalar field theory.

So this theory you can write action of the form.

So this is the simplest theory you can write down. And then relativistic invariant theory.

And then the equation of motion of the, so this is a, you view this as a classical field.

And again this has the equation of motion have to see exactly the same form as this equation.

So but now here phi, again is a function of tx, now has a completely different physical interpretation, so here is the, this is a classical field, okay, so this is a classical field, okay, so in this case the interpretation of the x in here, I mean here is very different, okay, so not only phi and psi, the physical interpretation are different, the physical interpretation of x also here are different, Here, x is just a label.

It's a label for the location in the space which we define this field.

But here, the x is the eigenvalue of the position operator for this relativistic particle.

And so they have very different physical interpretation.

And so let me just label this equation by 1, and label this by 2, and this by 2 prime.

So we also mentioned that this one has the interpretation of this as the wave function for relativistic, as the equation for relativistic quantum mechanics has a number of difficulties.

So the first is that as you will show in your PSI2, there's no sensible way to define a positive definite probability density.

So if you want to interpret this as a wave equation, then you must have a probability density because in quantum mechanics, probability should be conserved.

And the second difficulty is that there's a negative energy state because of the square.

Because when you take the square roots, then you get the minus sign and then the negative energy the state, which you cannot avoid in quantum mechanics, even though classically you can just throw them away by hand.

And the third thing we mentioned at the end is that for a relativistic wave equation, you can describe fixed number of particles.

So the particle number cannot change.

So this equation describes a single particle, and if you want to describe two particles, then you need to write down a separate equation for a different wave function, but this is for the two particle wave function will be like this, and et cetera.

But this does not really make sense in the relativistic system, because we know that in the relativistic system, e equal to mc squared, in any case, you have enough energy, then you should be able to create particles.

And then that means the number of particles in the given process is not conserved, okay?

So if you want to use your quantum mechanics to describe a process, and then that's, you cannot have a formalism which the number of particles is fixed, which you cannot change.

And so this is actually the most fundamental difficulty. OK, is that you cannot change the number of particles.

And related to this difficulty is this interpretation.

Here, we say, now if you want to do, we say in here, there's a fundamental asymmetry between the t and x.

OK, also, yeah, maybe let me put it as four. There's also fundamental.

An additional difficulty is the fundamental asymmetry between t and x.

So here in the wave equation t is just a parameter which we describe the evolution but x is the eigenvalue of quantum operators, eigenvalues of quantum operators, say a corresponding to say hat by putting a hat we reloaded the corresponding quantum operator.

Yeah, so this is the eigenvalue the oppositional operators.

And this AC means you become even more prolonged, say, if you can see the two particles. You have two x here, but the only one t.

But because of those fundamental difficulties, so if you connect this 1 to 4, so we can we conclude that the relativistic quantum mechanics define in the sense that you write down a wave equation.

And for wave function, it cannot be a fundamental description.

But relativistic quantum mechanics just refers to this kind of wave equation.

Okay, so at most, this can be approximate approximation and most, this is approximate description in situations, say there's no particle creation or allylation.

So in case which your particle number is fixed and then you can use this as approximation, okay, but it cannot be a fundamental description.

For example, later we will talk about the fermionic version of this wave equation.

So this will describe a particle without spin.

So later we will describe the Lannagos equation for electrons for spin half, and then that can indeed be used to describe electron in the hydrogen atom as far as you don't create new electrons, etc.

Anyway, so right of this quantum mechanics can only be described as some kind can see that as approximate description.

But now if you want to unify special relativity and quantum mechanics together, it turns out that the right formulation is just quantum field theory.

So it turns out that the quantum field theory addresses these difficulties.

So it turns out the right way.

So if you want to describe quantum mechanics, say, of relativistic particles of mass m, as we want to do here, it turns out the proper thing to do, which is a little bit unintuitive at first sight, is to start with this field theory, which seemingly have nothing to do with relativistic particle, but to start with this classical field theory, and then quantize it.

It turns out, once you treat this theory as a quantum field theory, and this become a theory of arbitrary number of relativistic particles of mass m.

And so that's the non -intuitive part, and that's the one of the miracle of the field theory is that it automatically gives you a formalism for treating arbitrary number of particles. labels.

And also in field theory, both t and x are parameters, even though x only labels your location.

So both t and x are parameters, and so you can easily make them to be on equal ground to be compatible with special relativity, OK?

Good. So any questions on this?

So we will see that the right framework is quantum field theory, OK?

So finally, as a last motivation for quantum field theory, we quickly describe the last.

So the field theory can also arise as a limit of discrete systems, OK?

And this is the most relevant for kinase matter physics, for example.

So let's just consider, say, some, yeah, let's consider 803 example, OK?

So let's imagine you have this number of particles, a number of the atoms, say, on the chain, OK?

And then they're connected by some springs between them, OK? Okay, so this is the, yeah, consider this to be infinite, okay? And the spacing between them, say, is A.

So atom affects on some matrix point, and the lattice spacing is A, okay?

So, yeah, so we can label the other, say, each particle by their position.

For example, this is x0, this is x1, this is x2, et cetera, okay? And the typical particle is xn.

And the location of nth particle is xn.

And so we can also introduce the deviation between the equilibrium position of each particle. So let's call it eta n.

So now let's consider the dynamics of eta n for this theory.

And so this is just deviation of the nth particle from its equilibrium position. So xn0 is its equilibrium position.

So now if you write down the Lagrangian for this system, what we can easily do is just write t minus v. The t is the kinetic energy.

And v is the potential energy.

So we can just write it as sum over n over all particles.

And then let's assume they have the same mass. Let's write mu eta n dot square. So this is the kinetic term.

And so mu is the mass for each particle.

And then their potential, let's assume at each point that there is also some interaction because each particle are connected by the spring.

And so they are a harmonic force between neighboring particles.

And now let's imagine also there There's a harmonic potential which trapped this particle itself at each location.

So this is a very simple spring and the particle problem which you encounter, say, in 803. Is this problem clear?

I assume most of you have seen this problem before.

And your task in 803 is actually to find the normal modes of this system.

And in 8 .03, you also described that we can, in the a go to 0 limit, say if the lattice spacing is very small, and if you're only interested in the behavior of this system at a very large distance, say the distance is much larger than a.

Then you can essentially choose this system as a continuum.

You don't have to resolve individual particle.

And so we can just, in the equal to a new limit, so it can treat the chain as a continuum of particles, OK?

And so each eta and t, you replace it by eta x t.

So x label is positioned, OK?

So x label is positioned, t describes the dynamics, OK?

So eta is the deviation at location x, and it's dependent on t. So this is the oscillator.

And then sum over n in the Lagrangian, then we can replace it by integral over dx.

And now you just treat this as a one -dimensional continuum.

So it's some integration of dx. But of course, here there's a label.

So for lattice spacing, so the infinitesimal here, the element is A.

So A times the sum over N, you can replace it by dx.

A is the lattice spacing.

And now you can just write this Lagrangian in terms of continuum theory.

Now you can write this Lagrangian continuum theory. And then let's just do it.

So we can write it, yeah, let me just write one more step. So you can write it as sum over a.

We take the a fact out, because the a fact out have to be changing to integration.

And then you have 1 half mu divided by a eta n square minus 1 half lambda a.

So I just slightly rewrite this Lagrangian So that it's easy to take the continuum limit. So we have to take the fact of a out.

But for this term, because this can sense the difference between the two, and we also divide it by a in the downstairs. And then we need to multiply a upstairs, and then there's a in the front.

And now the continuum limit, you can just replace this by an integral.

And now you can just write it as 1 half, of mu tilde eta dot square. So now eta n just replace it by eta x t.

And here, let me call it lambda tilde partial x eta square.

And this term, you can just replace it by the derivative of eta.

And then this is just become 1 half sigma tilde eta square. OK?

And the mu tilde, of course, is mu divided by a.

A lambda tilde is lambda times A and sigma tilde, this is the sigma divided by A.

So the continuum limit is that those quantity has to be fixed. The tilde quantity has to be fixed.

And then we have a continuum Lagrangian.

And then we have a classical field theory.

And this theory is essentially the same as that theory.

So if you take this factor mu tilde out, If we take this factor, mu tilde out, so let me just take this factor of mu tilde out in the front, just up to overall factor.

And here is lambda tilde divided by mu tilde we could, let's call it v square.

And this becomes sigma tilde divided by mu tilde, let's call it m squared.

So the v square is equal to mu tilde is equal to the lambda tilde, right by mu tilde.

And the n squared is equal to sigma tilde, right by mu tilde. OK?

And then this is just essentially identical to that theory when v equal to 1, OK?

So when v equal to 1 become the same, That's just two, equation two.

So we could develop corresponding to relativistic case, speed of light, but in general this can be just a non -rativistic field theory for other values.

So this is, even though this example is very simple, but this is actually a very general way that we can treat many condensed matter systems, which often involve lattice, say because it's solid, you can imagine all the atoms are on the lattice, et cetera.

And if you're only interested in the very macroscopic behavior, then you can treat It's solid as a continuum.

And then now if you're interested in quantum mechanics of such a system, then the quantum field theory then naturally arises.

Good. Any questions on this example?

Sorry? So the limit of the system is the same length as mu? Yeah. So what is that physics next? Is it the strength of?

Yeah, yeah, just corresponding to the case that the, yeah, it just tells you that the relativistic limit is special. Happens at a very special point.

But I guess why is that the relativistic limit, like, to me, lambda is, like, the strength of your spring and then use your mass.

Right. So, you've got to be comparable, how is that?

Yeah, yeah, that's not much you can read from here.

Yeah, yeah, it's just that when you choose some special parameters, then you can have a relativistic limit.

Are there questions? Yes?

Yeah, so these are all?

You can also have tensors or vectors.

So what would you treat?

Oh, you can treat, yeah. For example, you can treat forearms. You can also treat spins.

For example, if you have an icing model, just consider lattice of spins.

And then the average spin and then you can treat as a scalar field and then again you can write down a field theory Yeah, and actually the breakthrough of the phase transition using this metaphysics to understand what phase transition is really about and describe the behavior near the phase transition and precisely coincided with the development of field theory and Yeah, actually increased our understanding of quantum field theory.

Yeah Good. Other questions?

Just to summarize what we have discussed so far, all paths leads to QFT. Okay.

So we have described three paths. It's three paths, but they're pretty general.

First is that the quantum - we offer interesting quantum dynamics of some classical fields.

say, such as, say, electric magnetic field or spacetime metric, if you are interested in gravity, et cetera.

OK, so in this case, we already have the classical field theory.

But we know the world is quantum, and we want to understand what's the quantum version of it.

And the second is that it unifies special relativity class quantum mechanics, OK?

So you need the field theory to unify them.

And the third way is that it's a large -distance description of discrete systems.

So, yeah, just combine all three elements together to cover many, many areas of physics, okay? They cover many, many areas of physics.

Good, so now we can say a little bit about the plan for the whole semester.

Okay, so here is the plan.

So this is just a rephrase of the outline, which so the first thing we do in chapter 2, so here is chapter 1, in chapter 2, we discuss the simplest field theory just squashed with this equation 2, okay, the theory of 2, okay, 2 and the 2 prime, Yeah, the prime is its equation motion.

So yeah, in physics, we always start with a simplest example. We always start with a simplest example.

And so that is the one we will start with.

So what we will see is that this describes, that field theory describes spinness with those three massive particles. particles.

So we will see when we quantize that theory, two, and then we get a theory of free spinless massive particles.

So you say, oh, that's a little bit boring because in this theory it's free.

The particle, by free means they don't intact. The particle they just don't intact.

And then in chapter three, we will add interactions.

We will describe how to treat interactions.

We will introduce interactions and tell you how to treat the interactions between those particles.

Then in chapter 4, we go to the real physics.

So this scalar field is also real. Say, for example, can be used to describe the Higgs boson.

But the Higgs boson maybe is a little bit far from what you normally think about.

So in chapter 4, we will go through something which is much closer.

We'll talk about the theory of electron. So this is called the Dirac theory.

So this theory describes three but spin half particles.

So this is a theory of electron when we neglect these interactions.

OK, so this is the free spin -hard particle.

And then we move on to the Maxwell's theory.

So this is the theory of quantum electric and magnetic field.

So we quantize the Maxwell's theory.

So without source, the vacuum Maxwell's theory. And you find you get free. Again, there's no interaction.

mass is spin -1 particle.

You get a theory of mass is spin -1 particle.

So this is what we call the photon.

Okay, so this is the quantum for an actual magnetic field.

Okay, and then five, sorry, so this should be chopped five miles.

I think I lost my So now go to chapter 6, we combine the 4 and the 5 together, okay, combine electrons, so photon normally will be developed by gamma, combine the theory of electron and photon together and then plus interactions between them and then we get the so -called quantum electron dynamics.

So this is called QED.

So QED is very general, essentially it covers all the quantum phenomenon, yeah, a microscopic phenomenon up to say weak in tachyns and strong in tachyns.

If you go, don't go inside the nucleus and or don't go to a very high energy and the thing that's covers essentially most of the physics and yeah and then we will describe how to yeah and then that will be the the end of this course so do you have any questions on this okay so this is a roadmap yes the chapter lecture loads Yeah. Good. Other questions? Yes?

Yeah, gluon is also massless spin 1, but gluon actually they interact with themselves.

And so gluon is different. So gluon, to describe gluons, you have to wait for quantum field theory 2.

And so the thing about the photon is that the photon don't interact with itself, but the gluons interact with itself. Yeah. Yeah.

So essentially we treat everything except gloves. Yeah.

So now we can just move to chapter two.

Now we are talking about this theory. Okay, so actually I should not erase it.

So now we talk about this theory.

So because this This theory describes three particles. So we call it a free scalar field theory.

So this is the theory we are interested in.

So now we will describe how to contact this theory.

So first, we will quickly go through the quantization of harmonic oscillator, which you should already have done in your p -set. And so we can do it relatively fast.

So the combination of harmonic oscillator in the Heisenberg picture.

So we will see that once we understand this example in the right way, and then quantizing this field theory become trivial.

And the quantizing this field theory become trivial.

OK, so let's start with a harmonic oscillator.

was implicitly I take the mass to be 1 and then take the frequency to be 1, okay?

Yeah, yeah, let me put the frequency here.

Okay, let's take the mass to be 1, okay?

And so, so for this theory, so this is a simple harmonic oscillator which you have seen it maybe for the most of your intellectual life.

And so P will be x x dot is a momentum.

A conjugate momentum is x dot.

And so the Hamiltonian is the p squared divided by 2 dot 1 half omega squared x squared, OK?

And equation of motion is x dot double dot equal to x squared, OK?

So let's first look at this theory, a look at monochrome theory that is a classical theory.

So for classical theory, we know how to solve this equation.

We just need to solve this equation.

So classical solution just given by xt equal to a cosine omega t plus b sine omega t and a and b just some integration constant.

And for convenience, I can also write it is in the complex form, like it is following in, equal to a, exponential minus i omega t plus a star, exponential i omega t. And a is some complex constant.

And again, it's an integration constant. I just re -write the integration constant slightly differently.

OK? So these are just integration constants.

So now, yeah, so this is a complete solution of the problem. So now let's go to quantum.

So when we go to quantum, and then we replace then this classical dynamical variable then become the Heisenberg operator, become the quantum operator, in particular in the Heisenberg picture, and then this operator will depend on time.

And now this equation becomes operator equation.

OK, so now let's maybe I should label my equation. Now this star becomes an operator equation.

Now star is the operator equation for x hat.

So you have exactly the same equation as a classical equation, but now the interpretation is different.

And now the x hat becomes the operator equation.

So now the solution, now let me call this star star, so this still solves that equation, OK?

So this still solves that equation except, so these are just c -lumbers because this is a function of t, these are c -lumbers.

But now x becomes the, so now quantum mechanically, so this becomes, now becomes the quantum solution, OK?

So this still solves that equation. The quantum mechanically, this becomes a hat.

And so this still solves that equation, but these are C numbers.

The left -hand side is the operator.

And it can only be that A hat and B hat are operators. And also, A must be operators. And the star will replace it by dagger.

So now, say, now A, you just go to A hat, and A star goes to A dagger, A hat dagger.

Now these are integration constants for the operator equations.

So now they become constant operators.

So they're just constant quantum operators.

OK, they're just constant operators.

So they are integration constants for your quantum operator equations.

So now the solution, so now this is your quantum solution.

So this is the form we will often use.

You can also use that form, the equivalent. But this is the form we will often use.

You can also, from here, you take the derivative, you can find the P.

Again, this it become an operator equation, you take the deletion x, x, and maybe you find p, et cetera. Okay?

Yeah, so you p hat t, you just take a derivative, okay? And then you can just work it out. It's very easy.

So now, so this equation just, so we already solved the quantum problem, okay?

So, so because we find that the full evolution, full solution into the quantum operator equation, except that we still need to impose the canonical condition.

So this is just equal to i.

So if you plug in the expression for x and the p into here, and then you find that a and a dagger, the commutative is equal to 1.

OK, so this is your familiar creation and relation operator for harmonic oscillator, OK, for harmonic oscillator.

And now we can also use the A to build the Hilbert space.

So because all your operators now, or because x and t, x and p are expressed in terms of a and a dagger, so essentially any operator of this theory can all be expressed in terms of A and A dagger, OK?

And then we can just use A and A dagger, because A and A dagger, essentially, they are fundamental building blocks of your full quantum theory. And then we can also use that to build the Hilbert space.

So the Hilbert space is defined by the lowest state, is evaluated by A.

And then the higher states obtained by acting a dagger on the ground state.

Okay. So this is your full theory. Okay. So this is your full theory.

And so now you can compute anything in this theory just with those knowledge. Okay. Just with those knowledge.

So any questions on this regarding the harmonic oscillator?

So let me just summarize, so this is maybe very familiar, but let's summarize the rule we have been using, okay?

Summarize the steps in context of the harmonic oscillator, and then the same steps can be used to contact the field theory of quantization.

So we make it a general.

So first, so the zero step is that the classical equation motion, the quantum and quantum operate the equation.

Then the first step is to find the most general solution, find the most general solution to classical equation motion. Yeah, just equation motion.

And then, then when you've got quantum, and then you just promote the integration constants in your classical solution in one, in the step one to constant operators, constant quantum operators, okay?

So this gives the, then you have the full time evolution at the quantum level, okay?

Now you know how the quantum operator evolves, and then And you impose canonical condition conditions, okay, so that will tell you the commutators between those integration constant operators, okay, just as we do here.

And then constant operators in two, now you also know the commutation relation among them, and then now it can be used to generate the Hilbert space.

So this step, 1 to 4 are very general, and if you can do it, then you can essentially do it, apply it to any system.

Say 1 degrees, harmonic order is 1 degrees freedom, you can apply it to 2 degrees freedom, 3 degrees freedom, and also to field theory with infinite number of degrees freedom.

Okay, and now we will apply these two field fields.

Yes, for this procedure you cannot, but you can get the finite dimension, yeah, because the finite dimension of Hilbert space don't have the classical analog, so here we start with the classical system and then we quantize it.

And there's the quantum system with the finite dimension of Hilbert space, they essentially intrinsically quantum, and yeah, like spin, spin is an intrinsically quantum thing. Yeah. Yes.

Yeah, yeah, yeah, it's just because they don't have, yeah, yeah, the reason is just they don't have classical counterpart, yeah, yes.

Is it always true that the constant operator is sufficient to generate the entire Hilbert space?

Yeah, because if you think about it this way, that's a very good question, because let's just look at this harmonic oscillator and then try to single -generalize it, because there are integration concepts of the X and the P.

Any operator in your theory can all be expressed in terms of ANA dagger, and then your Hilbert space must be able to generate the hub of space union then because they are the building block of your whole operators yeah yeah the vacuum state here is based on it's come from the from the energy right so so once we solve x and p and then we can write the Hamiltonian in terms of x and p and then you just look for the lowest energy state and then you find the lowest energy state just satisfy this equation yeah and then And then from there, you can find other states.

Yeah, the same thing we are going to use for quantum field theory. OK, good.

OK, good. So now, become a mechanical.

We can just apply this to this theory.

And now let me add here. So here, the canonical momentum density conjugate to phi is called pi before. It's just a time derivative of phi.

Under the Hamiltonian density, you can find it explicitly.

It's pi squared plus 1 half.

And then this is the classical equation motion.

So now let's just solve this classical equation motion.

So this equation can be, so this equation is easy to solve because the translation is symmetry.

You can just do a Fourier transform.

So we can Fourier transform.

So now let's do the Fourier transform.

So 2 prime is solved.

So we can just write phi x equal to exponential minus IET g plus i k dot x, OK?

So then you can see that this just solves the, so this provides a basis of solutions to 2 .0, OK?

It's just a plane wave, OK? Just a plane wave.

So now when you plug this into there, then you just get the dispersion relation.

e squared should be m squared plus k squared.

So we'll denote this as omega k squared.

So omega k is defined to be just the k squared plus m squared.

So e, when you take the square root of e, So you can take a plus minus omega k.

It can be a plus minus omega k.

So we normally call the solution, we normally separate.

So for historical reasons, we normally call define uk x to be exponential minus i omega k t.

So now we have inserted the positive root of E.

So this is normally called the positive energy solution.

Even though this name is a little misleading, we don't define the energy actually, yeah later we will see this is not really the energy of a particle, so this is just a traditional This is just a traditional name.

Conventional name. And then you can define the complex conjugate of k.

Now you have, then corresponding to, you have minus omega k in there.

So we take this complex conjugate.

And so this is called a collective energy solution.

So all together, they form a complete set of solutions. OK, so a complete set of solutions.

So complete basis, a complete set of, yeah, complete basis is formed by UK and the UK star for all k, OK, for all k.

So these are the complete set of solutions to that wave equation.

OK, so that's just the, yeah. Any questions on this?

So this is just like a classical, it's just like a wave, just like a plane wave, which you should also have seen in A -dolls v.

So now we can write down the most general. So this is a basis.

So these are the counterpart of the exponential plus minus l omega t here.

So now we can write down the most general solutions by just putting in the integration constant for the most general classical solutions.

So we can just write phi x equal to integrate of all possible value of k, because this is for all k. So we just sum up all of them.

And so this factor is for just for convenience. It's just a convention. You don't have to put it here, it's just a convention.

And then we have a k u k plus a k star u k star.

So this is just the most general set of solutions with a k and a k star as integration constant.

So this is a full set of integration constant.

So now when you go to quantum level, So now, we can just follow the rule and we find the most general classical solution.

And in the quantum level, we just promote this to be operator.

You just put a hat there and change this to dagger.

So now, this becomes your basis of quantum operators.

So these are the full set of quantum operators.

And this solves your theory, okay, so this solves the operator equation, and this solves the operator equation.

So now the next thing is to impose the canonical commutation relation.

So firstly we have to, now we have to do a little bit thinking, okay.

So far, it's just straightforward. But now we have to do a little bit thinking.

So for finite harmonic oscillator or for quantum system of a single variable, you just have this.

And now we need to come up with a generation of these two fields here.

So, so we need to come up with a generation of that to field theory corresponding to phi x.

So now let me make the time and the spatial coordinates separate.

And this conjugate momentum is phi, it's pi, okay, conjugate momentum density.

So we should do them at the same time.

Remember t is the, it's the same evolution operator, so they have to be evaluated at the same time, or it's equal time, canonical or quantization conditions always at the equal time.

But x is a label of operators, so x, they don't have to be the same. So here can be x, here can be x prime.

So now we have to come up with a generalization of what is this quantity for field theory.

So now we just need to do a little guesswork. You can easily guess it.

But before we do that, do you have any questions on this? Yes?

Yeah, yeah, x is always, so x is always, here is always just the label of the spatial location, right?

Yeah, it's a label for the, yeah, it's your field theory, label, yes.

So in this procedure, I guess you always add?

Yeah, so normally if you have certain order differential equations, you always have some integration constant. Yeah, that's it, yeah. So that carries forward all throughout? Yeah, yeah, yeah.

Are there questions? Yes?

Right, yeah, yeah, so that's a very good question.

So that's just the extension of our usual procedure for the quantum mechanics.

So the usual procedure, even just for harmonic oscillator for single variable system, you have this correspondence between the classical system and the quantum system.

When you quantize the classical system, then the classical equation will become a quantum operation.

Here we just use the same rule, because quantum field theory is just the theory of infinite number of liquid freedoms. We are not changing the rule of quantum mechanics.

And so that's why we just, again, just promote the classical equation into the operating equation.

Other questions? Yes? Yes? What do we understand?

The classical variable? Is there something like that for field theory as well?

Um, yeah, yeah, yeah, there is. Yeah.

So, so classically, you can define the Poisson bracket between the, between the classical field variables, and then, and then quantum chemically just become, uh, uh, uh, quantum commutators.

Yeah, yeah, you can also do that. That's right.

So, so one way to come to this is you first describe, uh, first you need to generalize is your standard Poisson bracket for finite number equals freedom to classical field theory.

And then you can just generalize that to quantum. Yeah, indeed. That's one rule of doing it.

OK, other questions? Good. So we'll just guess the answer. The answer is very easy to guess.

So remember, if you have a single x and a p, that's what you have. OK?

But if you have more than one particles, if you have more than one particles, say just hint, say you have multiple particles, system in quantum mechanics, and then you have xA and PA as your dynamical variable.

So A equal to 1 to, say, n, say the number of particles. OK?

And then your canonical quantization condition just become x, a, p, p, b, t equal to i, delta, a, b.

And different x, a commute, and different p commute.

OK, different p commute, OK?

So now this a and b are essentially just replaced by x and x prime.

So x and x prime are just continuum version of those a and b.

Remember, we can't emphasize the x and the x prime are the labels of your degree of freedom.

So now you can just guess.

So we must have the following scene.

So from here, we must have phi tx, phi tx prime must be 0.

And the pi tx. So pi is the analog of p here.

Yeah, so those are operators.

And then phi tx, which pi tx prime should be something can only be 0 when x not equal to x prime can only be non -zero when x equal to x prime.

It's a gen addition of this.

So now you can guess. So what should this be?

Direct delta? Yeah, this should be just direct delta. But now you can ask the question, why has to be direct delta?

Maybe why should not be, say, the derivatives of direct data?

OK? Say y should not be, say, 100 derivative of Dirac delta.

And that question can be addressed just by from dimensional analysis.

So here we know somehow this must be rated Dirac delta. And now let's decide.

So now you can do a little bit dimensional analysis.

So if you just write down the action.

Yeah, the action I have just erased. Sorry. OK.

So if you look back on the action, I may just outline the idea, because I'm sure you can do dimensional analysis yourself.

So if you look at the action, the action is dimensionless in the lateral unit we are using.

So from there, you can deduce the dimension of phi should be 1 over l, so 1 over the length, OK? OK?

And the fact that the pi, where it's pi, and maybe I'll also erase, is equal to pi dot, means pi should be dimension 1 over L squared, because you take the derivative on time, and then there's a lot of factor of L.

Then that means on the right -hand side here must be something 1 over L to a cube, because there's no other parameters here.

OK, yeah, because here there should be a i, OK?

And if it's a dimension 1 over cube, then it can only be the data function, not a hundredth derivative of a data function.

So this thing should be just a data function, OK?

So this is the convention that there should be a i, and then it should be just the three data functions.

And this, indeed, have the dimension 1 over l cube, OK? Good?

So now you can just plug.

So you have the expression for x for phi.

You take the time derivative of this.

You get the expression for pi. And now you can just plug them into here.

You can just plug them into here.

And then you can find the commutation relation between those a k's.

And so this is a slightly tidier calculation, which is, however, a little bit fun, which of course I will leave you to do.

So, so if you just plug them in and then you can deduce that the following commutation relation between A's.

So, so this is the, I think this is in PSI 2, but I can still change my mind. Yeah, I wanted to put in PSI 2.

So, you find the, the the commutative between a and the commutative between a dagger.

Yeah, so now I will suppress the hat, because at the right hat, I think over and over I will be too tired.

So the commutation relation between a are 0.

And a between a and a dagger, so this give you 2 pi q delta function k.

So this is a 3 delta function k.

So again, this is a straightforward generation.

So if you have multiple harmonic oscillators, so if you have considered the multiple harmonic oscillators before, and then a between the different harmonic oscillators because k prime are just here just corresponding to, essentially you have, yeah, here it just is essentially have infinite number of harmonic oscillators. And each one of them labeled by a k.

So this is just like essentially we find. Let me just write it here.

So from those commutation relations, we conclude this theory when we quantize, after we quantize it, become an infinite number independent harmonic oscillators, decoupled harmonic oscillators, harmonic oscillators labeled by continuous parameter k.

The k is the wave number.

So for each k, there is a.

And a, so between a themselves, it's 0. Between a dagger, it's 0. But a dagger, they're not equal to 0.

And so this is, again, the continuum generation of 1.

OK? Okay, this is a continuum generation of one, because you have a continuous variable, yes.

Yeah, then you cannot say for sure.

No, but you see the quantization condition is in quantum mechanics.

Quantum mechanics, T and X are not on the same footing.

You can require your action to be X and T to be on the same footing.

Once you start to quantize your theory, and then T will have a pronounced row.

The canonical computation relations have to be imposed at equal time.