 all quantization. So what is the process of quantization? The process of quantization is building the quantum theory that will have the correct tacitin. Suppose you have a tacitin. You discover it by some experiment. You have a tacitin. You have to build the quantum theory that in an appropriate limit would reduce the speculacin. That's the process of quantization. Now, basically every quantization that we have, all the quantization that we discuss in this class, we should discuss quantum mechanics, quantum physics. So you have to say in what limit? You will approach the consequences. Basically all the limits that we have are the features that in the limit each part goes through. They approach the consequences. But you could imagine another kind of quantization, a quantization of the theory in the following way. It's some quantum theory that reduces the classical system I had in the limit in motion. Such a quantization would be harder to guess because you see the way to guess the first kind of quantization is very easy because we have this lovely Feynman idea of all these strings. We just take classical action and use that to construct the quantum theory. But that won't work for these large elements. Why not? Because the classical action is not what appears in the quantum theory. Now, there are some conclusions. Okay? So there is a second of the normal classical elements of quantum systems. There are two. Okay? One, we've exploited a lot. The second one, we've not touched very much. Something, I mean, string theory is very important. But in fact it really means that physics is not touched very much. And that physics is the real word. But you know, one should be very aware of this because a human classical system right when it quantizes may turn out to be in the large empty. Now, there is quite magnificent thing that's happening in the study of quantum theory of lift. It's called the AES theory. You know, the AES psychiatric correspondence is always there. There's a particular theory of gravity. Okay? It's going to be called 2B gravity on AES type. It's a 5. It is, okay? Somebody could have said you're rather interacting with other things. It's a classic theory. But I should tell you, what is its quantization? Possibly there are many answers to this question. But we feel we know for sure that one answer to this question, maybe it's the unique answer, is that the correct quantization of this theory is the large end level of 490 equation. This is a magnificent answer because the unparalleled, you know, every previous attempt at straightforward quantization of gravity has been to take the gravity action, put it in the path integral and do the integral over the metric. If this situation, at least for this particular gravity theory is right, that was wrong because we were trying to quantize gravity in the state of n goes to zero. The right way to quantize, at least one right way to quantize this in this case was the n goes to infinity. And if that's the case, many things have fallen into place. Why is it so hard to find a correct quantum theory of gravity? Because everyone's approaching the problem. It's completely wrong. The right way is n goes to infinity. And that's great, exactly. It's very hard to know if you've got a particular human right. If n goes to infinity then you have to know what the right is. It will be some classical system. That's very hard question to answer a priori. Now in our little foray into the subject, the big subject or potentially great for you, it's a very important question as you can see from the discussion. Sir, so we should get the same answer whether we take h cross going to zero or the right? It could be that there is no quantum system. That in the h cross goes to zero. It gives you this classical answer. But is there a quantum theory that we are trying to get the classical answer of that? I mean, should the final answer be same or what? There are just two different ways of approaching the same. You see, you have to build. All you have in your hands is a classical system. You don't have two different quantum. Until A, C, F, T, in some sense we have zero quantum systems. You don't have a completely well defined quantum system. Whose classical system do you know gives you that? All of the attempts have been to try to build a quantum system. Whose h cross goes to zero? That gives you the equations of gravity. And those attempts have been carried on for eight years. It started with Dira. Dira, like a few weeks after he quantized the equipment and he threw it up his leg and said, that's done. It was a natural thing to do. It didn't work. And starting from there, it has not worked. This approach is not worth thinking. Yes. And cool will be either that's because we've now got the right way of making it work. It will be stupid. All kinds of things are possible. But there is one more possibility. You're just barking up the wrong tree. Now how could we do barking up the wrong tree? We can't say that it is not a quantum theory of gravity. Because we're of the real world. Real world is gravity. Real world is quantum mechanics. And there's this beautiful theorem that quantum theory is a Jedi's code. You know, that it's inconsistent for half of the universe to be quantum mechanics of the other half. Do you know why this is true? Why can't we make this a fun class? Can somebody explain to me why it's not considered consistent? Give me a second example. Why is it inconsistent? Have a classical quantum, you know, why don't it be in the universe so that standard model was quantum mechanical? Why not? Is there something in principle there? Why do we need a quantum theorem? For some reason. How would you argue that that's not possible? Would you believe this question? Unless you can't. Unless you can't fluctuate in the equation. That's probably good. Okay. Forget what experiments are in principle. Why is that important? Let me give you a fine intro right there. This is something that you study in the... All of you have read fine intellectuals and physics, right? Whenever you read fine intellectuals and physics, you know this. This is a beautiful experiment that he talks about. I'll also explain. Okay. In the early chapters of what he begins. As you might be able to see in the film, we do a final analysis of it just to draw out this. The lesson he does is that, he goes down to the screen, and he has an electron gun. A gun shooting away. Okay. And he says, well, now let me do the work. First I'll shut the screen. And I'll see how intensity profile. This is some sort of sensitive screen. Okay. Some photomultiplies. Something intensity. What do you get? You know this. You get this. This thing. I'm not saying that you should manage this thing. Okay. On the other hand, if you shut this screen, you get... Let me draw these better. So let's draw. Now you should keep the both open, and you never had a quantum mechanics. What would you expect for that? If you kept the both open, you'd say, whatever, intensity comes to some of the two. Something will do something like, something, some of these things. And what is the algorithm? The algorithm is where the electron goes through one slit, on the other. Electrons go through one slit. This slit don't care whether that one's open or not. So, well, those that go through the slit, from that angle, those that go through this, from that angle, we saw that we get this. Such perfectly reasonable. Who 100 years ago would have questioned this? This argument. Okay. It's wrong. Thinking person is not really that set. It's those there. Okay. It's uncomfortable. I've got a perfectly reasonable argument. That's nothing but basic element to be wrong. How could it be wrong? So, if you, well, this, this must be inconsistent. If somebody, some experimentalist tells you this is the answer, it's experimented, but it should be wrong. It's a contradiction. But let's chase that down. How does it come to a contradiction? Okay. What he does next, is he puts a little photon on there. Okay. I mentioned some photons here. And this is the measure also of the photons. Now, the photons in the electron goes through the photons of the electron here. Photons go through that electron here. I mentioned the photons I can try to detect. Which electron, which electron? Now, if I can do this experiment, then I have to get this out of the way. Because those that go through here have a problem. Those that go through here, I can die with them. Okay. So, something would be going wrong if I didn't just get this out of the way. Unless, somehow, this guy, non-locally knows about that guy. It's not really. Okay. So, how does he do this? He does this. And he finds the photon. He finds that, if, I mean, this is a photon. He finds that, depending on the wavelength of the photon, he gets, if the wavelength is very large, the interference pattern is not destroyed. If the wavelength is small, the interference pattern is destroyed, but it doesn't happen. Exactly the way that doesn't happen. Let me explain that. Okay. This is just for fun. Suppose the distance between the two slides was very, very long. Suppose the wavelength of the electron was lambda. Okay. And suppose the wavelength of the photon was lambda. Okay. Now, you set this, this thing up. Where do you find the first destructive and what are you doing? If I'm first destructive, according to your knowledge of it, you know, in the experiment, we find the destructive interviews because we know what equation comes up that we could predict that. Okay. And you know, right? Suppose this was a triangle. Uh, theta. And that theta doesn't say, well, this is the difference in path length between the two waves. The difference when that's half of the wavelength, which is destructive to be honest. Theta's small, so we can approximate sine theta, tan theta, blah, blah, blah, by theta. So, theta times d is equal to lambda e by 2. You see this first destructive. Because you've got this four turbulence binding. Okay. The point is that no measurements is entirely unnatural. It affects the electron. But how much does it affect the electron? The main point is that not just the electron, also the photon is quantitative. Because the photon is quantum mechanical. Any particular quantum given wavelength has a minimum energy and a minimum of data. What is the minimum momentum of a photon of wavelength? Lambda. After some true bias, change of path by lambda. Moment of the light, moment of the light. Photon hitting the light from fixed. That's as huge as a small one. So, and that's as huge as the maximum. So, how much will it change its angle? It will change its angle by lambda p by lambda e. Tan inverse of lambda b by lambda e. All are small. Okay. So, delta. The kick angle. This is lambda p divided by lambda e. So, we want to make sure that lambda p is smaller. So, that is delta p. Delta theta is much much less than theta, the typical angle scale of theta p. Otherwise, typically this kicks out. What? Lambda p should be high lambda p. Should be lambda e and lambda p. High lambda p. Love. Thank you. Okay. So, this quantity has to be much less than lambda e by lambda p. Lambda e by lambda p must be much much less than lambda e by lambda e. So, what was this? Sorry. The formula for this structure was what? d sin t raised to lambda. d sin theta. Ah, there is a d. So, lambda e by 2. Yes. Yes, thank you. Yeah. So, theta. Theta. Theta is equal to lambda e by 2. Right. So, it must be much much less than lambda e by 2. Very nice things have to be solved. First thing, nice thing, lambda e. Second nice thing is that you can pre-read this to say lambda p is much much greater. So, you can do this experiment that does not smear out the interior of the planet. Provided, lambda p is much much greater than 2d. And you might say, oh, okay, let's do it with lambda p much much greater than 2d. Then, for a contribution. Oh, we have it. Sorry that you have it. Yes. You say it's a basic factor, a way of motion. You cannot use, cannot build a microscope that has resolution. That is smaller than the weight length of the planet. So, this candle in here, lambda p is much much larger than 2d. Cannot sensitively detect, cannot distinguish between the electron and 2d. Okay. So, quantum mechanics lives very well. It survives. You see, it's on the edge of nonsense. Quantum mechanics. It's poised on the edge. Push it a little and it jumps off. That's how the, you know, it's really took people so long to accept this interpretation of quantum mechanics in a little sense. Okay. And you push it a little and jump off. How do you push it? One way to push it is to have electromagnetism classic, but the electron quantum mechanics. Because why? Because when independent of what lambda p was, you could make the momentum arbitrarily so. That's a feature of classical. It's a classical because it's always in my students probing something arbitrarily small. And then quantum mechanics would make sense. You see, so it's inconsistent. I said this is a colloquial way. You can make it more. In theory of life, it doesn't add too much. This is the basic point. Okay. It's inconsistent. They have a classical system. They have a classical quantum mechanics mixed. So suppose, suppose, suppose gravity was classic. Now you can do the same experiment. Not before darkness. You should tell experimental friends to do that. It's not an easy experiment. Okay. But in principle, it's good. With gravity, it's the same quantum mechanics all the time. Okay. Yeah. So, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so, so so, you see, quantitative mechanics requires everything. So we have gravity. We know it, we know that it should be quantum mechanics. So the master student of quantum theory oğlum. So the fact that things that fail builds one cannot be taken as an indication that one doesn't exist exactly. One must exist. Maybe just think a little. And a new way of thinking, namely in one particular one particular example, they only really successful example, full content is not right, it is basically the ADC of course. And that way gives you complete, gives you a classical limit in this large n, rather than large n. This perhaps is an indication, and this is the correct way to conduct large n, we don't know. Classicalization of what you mean? The classical theory still has a one, there is an extra dimension, there is a dimension to constants. In the classical theory of gravity you say? Yes, in an one time classical theory, there is an h bar, dimension to constant Well, let's see. See, let's take the particular example that we know when, how it's going to happen. So the gauge theory, the four-dimensional gauge theory has no dimension to the consequence, right? Because the gauge coupling in which you can absorb h bar, is dimensionless. So this combination of coupling times h bar, which is all that we are getting, is dimensionless. Now, yes, this dimensionless physics, actually the gravity theory has a dimension, but that can be changed by an arbitrary choice of scale. So basically the way it works is that, there are two-dimensional quantities in the gravity theory, that's the radius of the area space, and there is a straight line on your results. And the appropriate dimensionless ratio of these is related to gauge coupling. That's basically how it works. The field theory would predict every dimensionless number you can build from that. And then there is one arbitrary dimension scale, which is basically a choice of scale. Okay, any other questions? Okay, excellent, sorry. So, I can't remember where this started. You asked a question. I can't remember what the question was. I have answered. You have a question. Sir, was that the story, like, very, very difficult to answer? You asked. Okay, fine. So, but I just want to emphasize that in the last class we've seen this classicalization in the study of the matrix. So how it became classical property? We've got a different classical property, not the naive classic. Okay, and we're going to see more of this kind of work. So, now we're going to continue with this. Now we're going to continue our study of gauge theory. So, for example, gauge theory is in zero plus zero dimensions. And CvH theory is in zero plus one dimensions. Well, in both cases we added matter things. Now, in four dimensions, it's interesting to study gauge theories even just by themselves. The reason that in zero plus zero and zero plus one dimensions, we didn't try to study the pure gauge theory for some point, no matter, is that these pure gauge theories in those two dimensions are empty. Okay, and then we're going to have to explain that statement to you. First, the analysis that I'm doing on the orders of the obedient theory, but it was very easy to generalize for non-requited. So, we're going to have an independent discussion of the structure of Maxwell's equations, the structure of the Hamiltonian equations. Okay, suppose you've got a Lagrangian at the end of the equation. The equations of motion that are following this Lagrangian are d mu. For these linear equations, for these linear equations, we might immediately want to know the answer to the following question. What are the classical solutions of this equation? Well, we need classical solutions. As you know, the solutions are going to be waves. But how many waves are there? What do I mean by how many? In the scalar field, they'll fly, they'll fly. That's it. But computing is that there is one wave. How many waves do we have here? Now, very lightly, if we were in d space, I don't know. You might guess that the answer was d here. That's not the right answer. The right answer is very well. It's d minus 2. Okay, and I'm going to give you two or three different ways of understanding this. Okay, this, by the way, will help us understand why gauge theories in up-to-do equations have basically no very little intrinsic dynamics in them. It's only in 3D or higher dimensions that you have real waves and real local dynamics in the gauge theory. So that's fine. So let's try to analyze this. You see, one reason, of course, why the guess d is wrong is that we could gauge it. So different solutions that are related by gauge transformation are not going to be conducted as different solutions. One way of dealing with this problem is to fix the gauge. And so let's try to understand this. First classically, that's the problem. Okay, but first classically, let's try to understand how to deal with this problem with this issue in different gauges. So one gauge that's often used in any kind of analysis is to set the gauge that we use to make very classical things. I'm setting the gauge because I want to count solutions as distinct or that they're not related by gauge transformation. If the square happens in Maxwell's equations, so this becomes l mu of d mu and d mu goes through with its this a mu. Okay, but the gauge condition up to zero. So this equation becomes the equation l squared. It looks like there are d modes, but that's not true because if you go to momentum space, for instance, this equation becomes k squared and a mu of k is equal to zero. But we also knew that there's no a mu in this equation. So k mu, one condition comes down to number of solutions by one. So that sounds like d minus one means. Okay, sounds like, okay, and I think that's reasonable. We had d guys, we had to get rid of one of them by gauge transformation. So that's one means. That's not the right answer. Why not? From this point of view, from this point of view, from this point of view of this gauge, this way of thinking. The answer is in a small subtlety that's very important. Small subtlety is that this condition does not completely exchange. Look, and the a mu goes to a mu plus l mu lambda. How does this condition change? It changes like s squared of something. So suppose you take a harmonic gauge transformation, a gauge transformation that itself obeys a massless minimally compensated equation. Then such a gauge transformation also continues to obey our gauge distribution. So any two of these solutions that are related to each other by such a gauge transformation, a gauge transformation lambda, okay, such that the s squared lambda is equal to zero, such that the s squared lambda is equal to zero, should be followed with equal solutions, okay? Now, you might think that it's very unlikely that things would come out because this is very particular kind of a solution. And solutions to the equations of motion are very particular kinds of solutions. But these two particular things are the same. Both of them obey del squared and something else, you see. So in particular, suppose you take a mu is equal to k mu. So suppose in particular, suppose you take a mu where lambda obeys del squared lambda is equal to zero. So this mu also obeys the equation. So in the space of d minus 1 solutions in this gauge, there is one gauge for this, okay? So in this space, so what else? In this gauge, how do we parameterize solutions? We parameterize solutions like a mu of k is equal, the solution has k squared. So in k mu, m mu is equal to zero. And you have that a mu goes to a is equal to a mu plus k mu lambda, where every function is supported on k squared. This equivalence that I will give you, one more. This is how you see this in, this is how you see this in the Lorentz Kitchen. Let me see this also in a zero sequence. So if you have mass, if you have mass, you don't have any agent there. So that means the whole thing is totally different. You can consistently impose a condition. You have a massive vector theory where you consistently, the question now depends on how you are making it. You can consistently impose a condition, del dot a is equal to zero. But now it depends on what your theory is. So that's a good question, but we need to specify. So how do you get a result? This was an interesting question. If a of k is not zero, and k squared is not zero, how can the product be zero? What happens? How can product be zero? Because a has support only by k squared. So a of k is zero, unless k squared is zero. Is this clear? Yes. So this is another way of saying that a of x is equal to i of k dot, where k is zero, product k is zero plus k i. So k has support only by k squared. So here we are going to get a problem. Problem? Lumpage. Sorry, a zero equals zero. Doing this last three presents will be useful for context. Because as you saw, our method of getting a Hilbert space under the path of K is getting over. Give a distinguished rule for K squared. So this is a zero equal to zero. For what? Zero. Suppose we put an a zero equal to zero. We've got these equations of motion, we've got d mu. We have a zero equal to zero. Let's work out all the methods. So what is f zero i? f zero i is simply a i dot. That's zero a i. Okay? So we have two equations. Separately first, the equation of three index is zero. That's a value of the index of the three index is zero. Equation of three index is zero. So that is l i of l i e j minus l j is zero. I didn't know about this equation. It's that there's no time limit. What does this equation do? Slice by slice. It allows you to solve for one of the areas in terms of the others. The other equations which were the equations, the equations which were the three index i, those equations are out. They are del minus del zero f zero i plus l j x j i. Now del zero f zero i is minus del zero squared minus because it's crazy, the zero index. f zero i is the lower part of this. Okay, so I have to answer this question. Del j in i is zero index. Sorry. Let me do this question. Yeah. So we wanted a zero index. So del i f zero i f i zero is zero. Yes, I know that. Thank you. I know that. Okay. So this equation is the following equation. It's del i of a i dot. Sorry. When I said I had no time limit, you said that there's no second time limit. There's no second order time limit. So the problem is Hamiltonian system. This is what's called a constraint equation. It's a constraint on positions and momenta. Not an equation because that is how momenta is formed. Thank you. Thank you. Okay. Now let us try to solve these equations. Okay. This tells us that del i f is independent of time. Okay. But now look. Since once again there's this issue of residual gauges. We fix the gauge a zero is equal to zero. Now are there gauge transformations where do not that continue to obey this condition? Answer is yes. Gauge transformations are independent of time. So using a gauge transformation independent of time if we get it one time. So del i a i equals zero. Just like we could say del nu a nu equals zero. Okay. But because by the equation of motion del i a i does not change in time. If we set it to zero at one time it stays zero forever. So now we show the gauge we have actually imposed del i a i equals zero because minus del zero squared a i this term is zero because del i a i is zero. Plus del squared a i is zero. This gauge once again we get two centimeters. Okay. Because there are three a i's and subject to the one constraint k i a i is c. In fact these solutions are exactly what you get by this gauge. If you use the solution gauge is there to set a zero i equals zero. Which you can on selections. Okay. And then you can have fun doing this in many other gauges. I won't bother you. With the similar analysis in many other gauges. But the classical problem of course every correct way of analyzing it can give you the right answer. Okay. Different gauge fixing way at first look different. But the answer is always the same. There are d minus two photonic equations we do. And the reasons d minus two in different gauges seems to have different answers. d minus two always goes to d minus one because you fix one gauge. But then that remaining thing goes back to d minus two either because there is a residue gauge in variance or because of some combination of the residue gauge in variance. And an equation of motion. An equation of motion that's of constraints. Okay. So I just say one. Before I leave the question I can just say one more thing. One quick thing. Okay let's do it again. Okay. Before I leave the question I can say just one quick thing. Okay. I'm not sure. I plan to talk about the two things you see. Two language models on these items. That'll probably be... And we just need to prepare them. Okay that's the answer to this question. Okay. Fine. Yeah. So before... Don't put one quick thing about the structure. One quick additional thing for the structure of Maxwell's equations. Which is very useful to keep in mind. We've already seen this in this particular gauge but this is just not... Or just a little bit. So I'll just say. You see. The equation of motion there d minus two to zero unlogically of two sorts. If you think of them as all you need. There is the free index time which is there mu, mu zero. In this index mu runs over all values. Time and space. It can actually transform into space. Because when this is time, this serves a time. That's zero, zero, zero, zero. So this is going to be written as li fi zero zero zero zero. And without any further analysis you know that this equation cannot have a second derivative in time. Now, equations that do not have a second derivative. You see what is the structure of the classical integration? There was the structure of the classical differential equation. L square phi is equal to zero. The structure is you choose a space like slice. You give phi and it's first time derivative to that. Then that equation will evolve this data. That's what a differential equation does. That was one big thing. Initial... Initial value probably. Initial value is phi and phi dot. Oh my God. This equation takes forever to shoot. Okay? This equation is not finished. It's not telling you how to take a and a dot forward in time. It's doing something else. It's telling you that the data you're allowed to specify on your initial slice must evade this constraint. Because it's some equation that constrain a lot of values of a i's and a i's. Some of the such equations are called constraint equations. So if you slice Maxwell's equations and order a space in time, in a single time condition, the zero component of Maxwell's equations is a constraint equation. We saw this here. We saw the explicit in the a0 and b0 of a2. But it was taken in each room. It was constrainable. The other components of Maxwell's equations as we saw explicitly, all have second derivatives in time. That's what dynamic equations look like. So what is the structure of Maxwell's equations? The structure of Maxwell's equations is that you can't do whatever data you want to maximize. You have to choose the data to constrain that situation. Once you've chosen that data, that data is evolved forward in time when there's a potential time axis. Because what if you take the data that's satisfied to constrain the equation? Evolve the problem in time and find that no longer satisfied to constrain. Again, that would mean Maxwell's equations to be inconsistent. Because we know that we can't have it because it comes from equations. It's a consistent equation. But let's try to understand algebraically how that works. What we need to show is that the constraint equation, that the dynamical equations which evolve, they come forward in time, do not change the fact that the constraint equation is awakened. So what was the constraint equation? Persistence. What was the dynamical? So what we need to check, we need to check what this is. Using only the dynamical equation. So I'm trying to use zero because we're not going to allow us to use this equation. What? Is that the same? What do you do? I use a different equation. If I use a different equation. This is totally different. This is symmetric. Symmetry does nothing. Fine. But what do we know from the dynamical equations? From the dynamical equations, we know that L is zero, F i is zero. Class L, L, J, F i, J. And it's zero. So we can replace this. We take this zero. We replace this with this. And then we have this question. And whether or not we impose a constraint equation, this constraint of object is left in ray and by dynamics. And therefore we impose it with zero. At one time, it will always stay zero. This is the heart of the consistency of mathematical equations. Notice that's what we use here. So the constraint equation, we can solve the constraint equation by saying L i, A i is zero. You know, we can solve it. Put a gauge, C, A i is zero. And the constraint equation is then guaranteed. But that would always be true. Okay? So, in some gauges, in some gauges, the combination of residue and constraint equations that eliminates that secondary effect. Okay? This is how we think the problem is classical. Any questions or comments? The net lesson is that the number of people who are afraid of it in Maxwell's theory, just classically, is n minus two. Okay? Not ds, but be very naive, but d minus one is less than n. Okay? Yes? I think you just have to see in my head how this goes through d minus two which is not logical. Oh, how this goes to d minus two. This was not logical. I mean, we used Yeah. Yeah. That was not meant to be logical. Okay. There'll be a mechanism. Okay? I just wanted this as a separate topic. Okay? That there's d minus two and that there are constraint equations. These are tied, but it's not logical. For instance, a del dot a is equal to zero we never use it. I should be able to just not get better something better to say your question. There's certainly something to say. Other women are not trying to make any mistake. Okay? Good. Sir, so are there several gauge predominant one gauge gives the constraint equation? I would say in this way that every gauge, when you take the analysis through to the end, it's d minus two. And the different things in different gauges that you know, actually I'm using the one loosely because when I say gauge linearly, I should be fixing all gauges. Once you fix all gauge invariance, however you do it, you will see it's d minus two. But I'm using it in colloquially because we often use these other gauges without bothering to carefully fix other gauge invariance. And you just see how badly wrong you can go. If you're not getting it. Okay? Because that was the point I made. Okay, good. Now we're going to go to the quantum theory and we're going to do that in steps. Of course the quantum theory is very easy in the Abelian theory. In all that it's the Abelian theory. But in all the Abelian theory, in theory, you saw in the before I mentioned you've got your million dollars. It's like a foundation to give you million dollars between the Abelian and this theory as a mask. Okay, so it's a very nontrivial theory. Okay? Now we've understood some kinematics and so on. We actually solved the theory. But what we're going to do today if we can manage is to solve the quantum theory in two dimensions. Of course you see that there is a very nontrivial theorem because the number of photonic degrees of freedom was d minus to zero. So over but we're going to put the theory in a circle as you can see. And then there will not be an entire region of the theory. But almost. Okay? Of course, great you can have. We're going to first solve the pure theory and then we're going to add matter to the theory. Fundamental matter in fact we can solve that theory as well. That is quite energy. Fine. Let's take this pure-gauge theory into consideration. Now we're going to do the mechanical and we're going to follow our discussion about the quantum theory. So now in discussion about the quantum theory our Helmholtz base was given. Our Helmholtz base was given by wave function square and diagonal wave function of a object of the condition of the psi area. Equal to psi of gauge transform area. Okay? This is a wave function of a energy. We have two dimensions and we're going to take a two-dimension quantum field theory with time and space memories. As you can see we didn't take this S-1, we have to realize S-1, S-1, S-1, S-1, S-1. Okay? We've got a two-dimension quantum field theory in a circle. Okay? Because two-dimension quantum field theory in a circle and we're going to understand Helmholtz base and the energy is if I solve the shopping problem for the quantum field theory. This is actually surprisingly easy. It's surprisingly easy to that in one you see the effect of a one-dimension and in one-dimension any okay, sorry gauge any gauge field configuration can be turned into a constant by a gauge by a gauge Actually, you might think that any gauge configuration can be turned into zero by a gauge but it will be a little difficult. So first let's do a million theory Suppose you've got A-1 function of your X-1 one-dimension okay? So we've got A-1 which is some function you know our it's a function of X-1 and time of course at any X-1 at any space We were supposed to do these gauge transformations as a function of space Do you remember that discussion we had right about the H-1 space? Okay, we were understanding the H-1 spaces we were supposed to look at A-1 as a function of X-1 and and look at the space of any equivalent configuration of the gauge function A-1 we're going to go into A-1 you know A-1 of X-1 minus A-1 tilde of X-1 to be said to be equal if they're there's a lump My thing that you can always choose a lump that's the best truth for any two ways is just by choosing a lump that's the integral of A-1 minus A-1 okay? So you might think that all gauge configurations are equivalent gauge equivalent in one dimension and in particular equivalent to zero and then we would have basically a wave function and a function to nothing okay? Now this would depending on boundary conditions might be true because it was an hour long but it's not true in X-1 because if we're in X-1 we should only allow those gauge transformations that respect the periodicity of this okay? So we can only relate two A's to each other by a gauge transformation provided the integral of these two A's are the same okay? If the integral of these two A's around the circle was not the same but this this purported gauge transformation would be nearly the gauge transformation because it would not respect the periodicity you are from? It might not It might not Yeah, so then it depends on details boundary conditions okay? But on X-1 it's very clear So now what is the right state? Suppose so not all gauge configurations but all gauge configurations with the same integral they're not the only gauge generating data the only physical data is the integral of A1 because given any integral there's always one configuration that gives you the integral in this constant we can always use gauge transformations these gauge transformations to set A1 equals to constant so what was what was what was what was the rule of quantization the first rule was psi psi is a wave function of of A1 subject to this that psi is function only you know it doesn't care about it depends it's invariant of gauge transformations and therefore we can solve the condition of psi only a function of A1 so now now it works okay? just enough what was the second part of our rule the second part of our rule reverse that it's just the problem that you get from the Yangle's action just setting A0 equals 0 you remember we have this at you remember how we did the quantization we did the quantization for the AIs the A0's treatise is constant background time the integration of A0 give you this constraint okay so the Yangle's action with A0 set B equal to 0 is very simple what is it? it is just A1 equals to 1 by let's say 2 which is great and therefore this problem is extremely simple extremely simple this problem the problem of Yangle's theory of U1 Yangle's theory okay the problem of U1 Yangle's theory reduces to the problem of U1 Yangle's theory reduces to reduces to the quantum mechanics of a non-relativistic body with some effect to value of the mass that generalizes to the non-relativistic and that is one more thing that we have okay so in the non-relativistic case once again gauge transformations can be used to diagonalize A1 and to set B equal to 1 by dimension so psi becomes psi is a function of a matrix commission matrix A 1 and S is simply trace A1 not square and now though it requires a little more work plausibly reduces this problem looks like it is actually right it is the same problem we discussed in the last class namely namely commission quantum mechanics quantum mechanics of commission matrix subject to the gauge constraint that two different A's that related by gauge by a by a constant gauge distribution U A2 was actually the same yes I would so you have a matrix A yes or the matrix A not DL was physical what do you mean yes the matrix is just what you are saying does not respect that no no no yeah yeah yeah yeah no it is not no no what is the physical quantity are the so this matrix A dot DL you see the matrix A dot DL from here to here it is not a gauge invariant until you also place that is exactly what it is certainly so the word what Ranak was saying was that what we should do is today because there is a physical quantity which is the circle of Wilson okay so the path of order exponential of exponential how do you write that the exponential of I in A okay how does this quantity transform the gauge transfer now firstly I have to tell you what this path of order explanation if we were at a billion theory it would just be an order explanation but not a billion theory what this means is the following e to the power I A dx that is e to the power I A dx so as you break up the integral from 0 to let's say 2 pi let's say 2 pi L into little dx regions and you multiply A at this point and dx like this because each of these are matrices they don't commute this is not the same as you know summing something up in the exponent okay why do you have to take an exponent at all you'll see okay the reason we take an exponent is that we want to know how we want to tell something so this quantity how does it transform the gauge transmission so suppose let's take something here so suppose this was this dx to be from x1 to x1 plus delta x1 okay I will do a check that the gauge transformation rule for the A field means that this quantity transforms like ux1 e to the power I A dx the view inverse x1 plus delta x1 this whole object transforms nicely allegations because in between this and that cancels okay so the whole object transforms like this whole path order exponential transforms like u at where we started x start times e to the power path order is pi A dx times u inverse of xn and x start is equal to xn alright we are only going to look at objects that obey you know we are only going to look at gauge transformations that are periodic okay so u inverse of the same let's say this is u over 0 this is to pi n this is to pi n this is 0 so this exponential this integral itself transforms nicely under a gauge transformation but it does transform the only part of the data that the things are better now allowed to depend on are the gauge invariant invariant data the eigenvalues of this object again that's the point very good so one way of saying this is that the eigenvalue data that presents this object has various paths those are there that's the same data as the eigenvalues okay now this way of thinking of it makes clear it makes clear that two different matrices are there okay if you have two different matrices in here such that they are eigenvalues suppose we've got the eigenvalues alpha 1, alpha n yeah and if it's an eigenvalue it's the same as the same okay but suppose I take any amount of these eigenvalues and I put plus 2 pi it also gives me the same thing for this path order exponential because all that I care about is the eigenvalues the eigenvalue is up in the exponent okay so you can see this in various ways but up to gauge transformation you can diagonalize this A1 you can make it constant and matrices okay matrices with eigenvalues that differ by 2 pi are actually equally so what this problem is is the quantum mechanics of now okay so now what was our lesson from yesterday our lesson from yesterday was that when you take matrix quantum mechanics it becomes quantum mechanics of not yesterday basically it becomes quantum mechanics of eigenvalues okay you've got the n different eigenvalues but the whole point of doing all our clever stuff when we start with the eigenvalues became effectively firm now it reveals to that problem except that now the fermions are living in a circle of length 2 pi because we've got this g squared the angle is up here we could absorb that into A effectively these k's in the length of the circle it made the fermions out of standard energy term half twice A1 squared n 12 squared these guys would live on on a circle of length 2 pi g angles okay so this is our final conclusion I haven't been as thorough about it but if you read the papers they take much more care to establish more conclusions but maybe I've been clear about physics okay the n product is this a 2D angles theory on a circle okay a 2D angles theory on a circle and I I I did one thing badly because I did one thing badly one thing badly because it's the ideal values of this object that are periodic object but this object is A times 2 pi where L was the length of the actual physical circle A times 2 pi L at periodicity 2 pi A times 2 pi L at periodicity 2 pi so A at periodicity 1 by L we have to say 1 by L and so the series scale quantum mechanics problem has periodicity G angle no 2 pi yeah it's alpha 1 really good this alpha 1 should be thought of as the ideal values of this object A to ADM then it's correct so the periodicity of the series scale quantum mechanics problem is G angles back okay so you take 2D angles theory we take 2D angles theory on on a circle of a length 2 pi at topic of G angles this problem is identical in fact this problem is the problem of N fermions leaving N identical leaving on a circle of length G angles by angle isn't that very good we have thought of this you have thought this is 2D angles theory actually seems to be funny but actually the physics of this problem this problem is just exactly a reasoning for the N free fermions leaving on a circle L N free ideal leaving on a circle of length 2 pi G by L and therefore radius G by 2 pi this problem of course we know the solution it's a very easy solution 3 angles theory was the first place is this clear ok 2D angles theory was the first place where we saw some some not completely trivial physics as we see it with a purely simple but it was pretty trivial in direct these fermions free fermions have not nothing to do that's why the business is spent of the life of free fermions ok so they know nothing they all know it's very easy ok excellent we saw the general lesson we saw the general lesson the general lesson was that the 2D theory goes solve a problem because it is not locally produced only degrees free from a globe associated with these holonomies around the circle and because of that that produced 2D angles theory from quantum mechanics luckily it turned out to be a solvable quantum mechanics sir if we do it in higher dimensions can we generalize it like 3D, 3D, 3D the problem with 3D is as we discussed the number of photonic degrees of freedom is 1 now we have to do pure physics so we are out of now ok yeah the way you showed it it seems that if some of the eigenvalues are some of the eigenvalues are no but also some squares of eigenvalues because you can take this thing and one twice and some cubes of eigenvalues and some 4 pumps of eigenvalues therefore the eigenvalue is up to computation excellent so now we want to we want to move on to doing less trivial physics with the eigenvalues theories ok and in 2 dimensions that means eigenvalue but now if you take 2 dimensions in the eigenvalues theory and you would indeed the analog of what we did in 0 to 0 dimensions 0 to 1 dimensions so add an eigenvalue frame that problem is already too hard to solve even with a large element ok you know sir there will be problems there is a condition and this one I would give a lot to be resolved they have to be in the theory however Gerard Doft a great theoretical physicist of the 1970s managed to solve a simple one that is date to be in the same and coupled it to fundamental fundamental fundamental matter the large element now there is a basic point about fundamental matter is that we have seen that as we saw I have been seeing in the last 2-3 lectures large and limits are interesting but large and limits for fictor like things but simpler than large and limits for matrix things now the eigenvalues theory always is matrices because gauges are matrices but in 2 dimensions the gauges means almost almost nothing it's almost nothing therefore if you come to fundamental theories it's sort of like for fictor like things ok but in preparation for the analysis the presentation of the next class and then start trying to understand more clearly the large element in the diaphragm finally ok the analysis I do now will apply to different dimensions just to change the analysis it was done by Doft also in a separate manner but Doft in the 1970s looked about 15 15 16 totally amazing you know each of them with great originality great creativity and great technicality which once one after the other we will discuss two of them ok so the general analysis applies to any matrix theory which we scale 5 5 5i's arcane process so we have the analysis so we've got minus n times trace something as 5 7 trace of some function of the matrix because we have 5 whatever this is what I'm going to apply n goes to the matrix 0 0 n goes to the matrix last class the general analysis matrix what we want to do is to understand how perturbations are there because we want to understand counting powers of n let me it will make way little difference but let me just introduce myself suppose we have very mu that's by the whole thing square plus if I'm in diagram I want to be able to efficiently count ok I want to be able to efficiently count the powers of n and to this end I introduce the following notation this 5 field is the the field for matrix 5 i n so this has to be indices the i index the j n so I introduce the following notation for 5 n propagate of this 5 n and broaden out into a double where the top line gets an arrow which we strike the fundamental the bottom line gets an arrow that we strike and come just a bit of notation changing my notation for the propagate of this 5 but this notation is that gauge invariance ensures that these indices can go away yes gauge invariance ensures that these indices can go away what is the 5 cube to the text it's just what you may have to write what may happen we have 5 i j 5 k i because this is 5 i j 5 j k 5 k the final diagrams for these yeah the final diagrams for these for these matrix series look at this the vertex factor looks like this I am going to let me take a sample diagram let me take a diagram like this first I draw an order notation so suppose I take a diagram like this a vacuum bubble diagram and I think it out to double line notation okay how could how could the point is that this way of drawing the diagram makes the n contemplate why let's look at the n contemplate for this time how many propagators like 3 propagators but each propagator comes with a factor of 1 by n propagator is the inverse of the quadratic so we have 1 by n through the power how many vertices do I have on this side 2 we have n because the summation of the index 1 2 3 what's the net power of this diagram let me do the following suppose I had another propagator like this how has this changed it's changed in various ways and I don't know a propagator so it's one of the one additional factor of an order let me so how many different propagators go over there this was just fine so let's draw this diagram which was like this whatever I am doing by doing this I've done this how many propagators totally 1 2 3 4 5 6 7 1 2 3 4 5 6 an additional factor 1 by n q because I've got 3 new propagators how many new vertices do I have 1 2 new vertices okay n squared how many new index lines do I have well 1 new index so this additional power of n cancelled when I complicated this graph in this particular okay what what particular manner was this what it was was just taking this this graph that I drew on the surface and breaking it out into breaking out the so imagine that you're taking this this thing here as a kind of tiling on the surface what I did was to break up the tiling into smaller lines that did not change that did not change the power of n okay now I won't have the time today is how I think we'll stop here but what's the punch line the punch line we will show is that this is generally true that if you take some surface imagine this graph some some you know something you can draw on on the surface and then you just keep changing the tiling of the surface that does not change the power of n however not all graphs describe the same surface imagine if you take this outer line and shock it from in your head and so imagine that the surface includes not just the surface like this but also the other sides that's the correct way of thinking the surface okay then this surface is a sphere and any Feynman diagram which in double drawing notation can be drawn for can be drawn without lines crossing each other on a sphere on a plane if you don't close that on the line I'm through and through so you don't need to at the same power of n however not every diagram can be drawn on a sphere on a plane without crossing each other some diagrams in order to be able to be drawn without crossing each other would need to be drawn on a canvas some diagrams would need to be drawn in a genius beautiful classification of compact two-dimensional they're classified by what's called a genius which is one number it's basically how many holes but you think of it like a pretzel how many holes there are so sphere has no holes the torus is one hole the genus two surface two holes big torus okay and when a beautiful result of top prove doesn't we will see in the next class is that the power of my n for such diagram is my n squared to the power 2 minus n n to the power 2 minus n which is the genus the sphere at genus zero is playing a class that we will look at okay the torus which has genus one will have 8 to the power zero so what we see is something quite beautiful that diagrammatically the one over n limit the large n limit gives us a new parameter which organizes the graphs the final graphs also quantum theory in two graphs according to some sort of each topology has an infinite number of graphs and solving the theory of large n is analogous to summing all the graphs it's not analogous it's the same as summing all the graphs that can be drawn in a sphere this is sometimes called planar graphs the sphere only if you close up the large n the simpler way to say is that can be drawn in a sheet of paper without any two propagators going over it okay so we will prove this in the next class and then we will use this result to try to solve two-dimensional QCD with a flashlight okay thank you very quick questions I have to so your point is yeah yeah yeah but more seriously more seriously you see this is what I I may have said before in the class that you you know one has to be one has to be what is the role what are what can the theoretical physicists hope to you see there are some problems but most problems in the world there are so complicated that likely the theoretical physicist would have to use some crazy computer okay we'll never be able to solve you know not in the history of humanity however that doesn't stop theorists from making progress how do we make progress we take the space of problems and find stops where we can stop and then we do perturbation theory from those points and by finding these spots and doing this perturbation theory we fill out a qualitative picture that helps us qualitatively understand the whole space even though we can theoretically solve ordinary special points but these special points are totally invalid the points of the space of theory is because without them you would have no theoretical after you have this qualitative picture to get numbers in any way it feels like a cheat but it seems like there's nothing better we can do than to go to some computer but without having that qualitative picture that won't work because you will know what to do on the computer you will be able to understand these results ok so this is the realistic view of what a theorist can do ok so QCD in three dimensions with the masses of the quarks as they are is probably the kind of problem that human beings will never solve without a computer ok you just look at the spectrum of results for instance so complicated it just seems very unlikely that those numbers would come just out of the human brain you will be able to get some number so how we can just give up and say well ok surely not that's a cop out for a theorist what we have to do is to find those points we can understand with our mind and only those that we cannot leave so this is what we must do for instance suppose you could solve QCD at large and it had the qualitative features of some quality that would be a huge detail because it would be a new point in this place that helps us fill out ok so some problems of the real world are easy enough to just solve but those extremely rare ok so deforming problems of the real world to solvable problems which do not give up which captures some of the content of the real world ok so next class Monday ok Monday class