OK. So welcome. So this is the first series of lectures on string theory. These lectures will mark at least two new next year. But depending on various things, including your interest in the course, we might continue the subsequent next time. I guess it might be a year and a half long. The plan for the first part of the course by the first part I need to main is to go through a treatment of the dual district. So the main textbook, the textbook that will be most useful in this course, is a textbook in string theory. Well, Tritsky has written a couple of really nice textbooks in string theory. We will follow Tritsky's book. Mainly chapters, we might have to go through large parts. And this will be the most useful thing in the lectures. Other useful references are the book made use of what's in it. And another book that I found useful by the string theory was the book by Loose and Tyson. Through the course, we'd also follow other references that I'll tell you about as we get through. OK. So as far as books, though, these will be the main now. We'll make references for our course. Good. Any organizational questions or comments before we start? So the plan is to have the classes on Tuesdays and Wednesdays. Say that, 2.30. OK. This Friday is enough class, or? This Friday, we couldn't take all of that. How about having it this Friday and then from next week, then Tuesdays? So we have the classes. OK. So what are we going to do? Well, so let me give you a five-minute spiel on why we're studying string theory. As all of you know, quantum theories of three of the four basic forces of nature have been successfully developed, successfully tested and accelerated experiments, and worked very well. These quantum theories are the quantum theory we like to learn. The dynamics and the gauge theories that govern the weekends from forces. Yes, you do, and yes, you do both theories. OK. These theories are based on a classical assumption, but we've understood how the quantum mechanics from these theories, but quantum physics. OK. Now, well before we got understanding of, for instance, the classical, the grand deal of the weekend's strong force, we had good controls over the classical action for gravity. The Einstein action was written down in the early part of the last century, many years before strong forces of the mid-classes were even begun to be used. And we have very good understanding of the classical physics of gravitation. In some limits, Newtonian physics, in more extreme limits, it's given by Einstein's theory of gravity, these are very well understood theories, and we understand the main theory of a very nice ways. OK, well, however, as all of you know, the way that electrodynamics, first Maxwell wrote down his classical theory of electrodynamics, that was written in nicer ways at that time, and then various people started to delock and then find and treat that over and over again. So the people understood how to convert that into quantum theory, but quantum physics was a country. So whatever the, it's natural, a natural cause of revenge would have been that somebody would have, somebody smart would have taken the Einstein Lagrangian and context it. OK, that has never successfully yet been that program of taking the Einstein Lagrangian and trying to make a quantum mechanical theory based on it, just by following the usual procedure to converse the quantumization, that some of you know it, others of you would study it, but quantum theory has never successfully achieved it. OK, there are various technical reasons for this lack, for this achievement, for this lack of achievement, which we won't try to go into at the moment. OK, but though many people have tried it, it has never since, this program has never successfully won it. OK, but the first one possibility is that we haven't been smart enough. The right young brilliant guy will come in and explain how to make a quantum theory that has the Einstein Lagrangian as its classical limit. And so then we already have been, at least we have a quantum theory of gravity. However, this, the program of trying quantum gravity has proved directly, has proved so hard and has left some little insight of all the other stuff. Right? OK, yeah, it might have really not led very much. So you might think that there is another possibility here and that other possibilities are spotless. There is no such thing that it's just misguided effort to try to make a quantum theory of gravity by quantizing the gravity Lagrangian. OK, a consistent quantum theory and an analog would be something like this. You know, in the study of the weak, in the study of the weak interactions, in the study of the weak interactions, there was a phenomenological stage in the study of that theory in which people realized that many experimental results could be, could be, could be obtained using four-fold interactions. OK, now if you, if you, if you try to write down a Lagrangian, so some, some action which has four-fold interactions, the detailed contraction structure doesn't matter, slightly four. And you try to make a quantum theory based on this Lagrangian, to run at various levels. Then you, some of them are analogous to the problems you run up to the instructor, trying to study gravity, trying to work nice gravity. And the, the way we understood how to make a quantum theory of this, this interaction was not by correctly quantizing the Lagrangian, but by realizing that there were more degrees of freedom than just the fermions. There were, in addition to the W bosons of the weak interactions, integrating out the W bosons in some approximate regime gives you the theory of the four-fold interactions. But the theory that we know how to make sense of, the theory that we're so well-tested in accelerated experiments, is a theory with fermions interacting with W bosons. This is not part of the basic interaction theory. Just phenomenologically effective at, interactively integrate out some things. So in that case, the lesson was that we should try to make quantum theory based on this action. We should instead look at a larger system, which you can understand and make consistent. And when you take the appropriate limit of that larger system, it recovers the physics of this, of this phenomenological motion, okay? So that's, there's a second possibility that at least most consistent quantum theories of gravity. Perhaps there's some, about, okay, we'll get to that later in this course, but at least most consistent quantum theories of gravity should be far off as larger systems, systems with degrees of freedom, that go beyond the ground on it.  Reduces to the physics of gravity. So this is the second possibility, integrate. Brilliant theory is, and a very, well, is a formalism that implements possibility number two. Okay? It is a consistent, as far as we can tell, consistent quantum theory. There's many, many more degrees of freedom than just the ground on it, okay? But there are no energies. Reduces to a theory of gravitons that are interacting with some small amount of power. If, within the string theory, it seems clear, or there are all of the statements that I'm making, are statements that, at various points in the history of the development of the field have been challenged. Some of these challenges have come very recently, two, three months ago, so. Everything I'm saying is based on current understanding, which may turn out to be wrong. Next bit from Aldersena. But, but, but, but, but, but, as far as we understand, if we look at string theory and we take the low energy effect of action, the action of gravitons interacting with the other stuff, there's no sense in which you can make a consistent quantum theory just out of those fields. You need all the fields of string theory to make it all work. Yes, the effect of dynamics, at low enough, at least, is that a threat. Is the program that string theory implements, okay? And it's a program where you're going to study in this course. Okay, now, one or two comments, one or two more, it's just placement comments before we start the data stuff. The first thing I want to emphasize is that string theory, as you will see as we go through, is in some sense badly made. Firstly, the string part of that name is not entirely accurate. Secondly, the theory part of the name is not entirely accurate. Strings happen to be just some of the excitations within this theory that we're going to build, okay? But also, the statement that it's a theory is a little inaccurate in my opinion, especially from the point of view of the experiment, this string theory is more formalism than a theory, okay? As we will see, string theory, we build it out of the theory, but we've done it out of many, many different vacuum states. The physics of each of these vacuum states will be inequivalent. In, you know, learn if you observe us, sitting in one of these different vacuum states will see completely different physics. This is just where you are in the space of vacuum art from the point of view of a learning experimenter is like parameters. So what string theory will turn out to do is to build a whole class of different effects, low-energy theorists. And in some sense, it can be thought of in my opinion. Yeah, in fact, we more accurately thought of as a formalism rather than theory, okay? So it's a formalism within a wage, where it will prove possible to build consistent quantum theories of gravity, okay? And we're going to study this formalism in this course. The second thing I want to say on, immediately, is that, as I've already alluded to it, it will turn out that the string formalism would build thousands, you know, an infinite number of consistent quantum theories of gravity. Okay, so in that sense, string theories of formalism, it's like quantum field theory. You can't ask, what is the predict, does quantum field theory reproduce the hydrogen atom? That statement makes no sense, it's quantum field theories of formalism. A particular quantum field theory does reproduce hydrogen, the physics of hydrogen. And in the same sense, string theory, at least not in my kind of understanding of the subject, and in my opinion, it's a formalism. And the question, which is of deep interest to all of us, but does string theory explain the real world, is as imposed as a question of does quantum field theory reproduce the spectrum of the hydrogen atom? You can't string theory by itself not reproduce the real world, it's a particular implementation of string theory, a particular theory within the string formalism that may or may not be the theory of the real world. Now, all of us who study string theory, both of that is the case. But we have no expert, not at all, no experimental evidence. No, no terribly pressing reason yet to believe. No overwhelming reason yet to believe, but that is true. So string theory is a formalism, but has its successes so far, have been that it has been a consistent formalism within which it has proved possible to make consistent quantum theory of gravity, a whole zoo of quantum theory of gravity, and in my opinion, it's the only successful in the dense way by human beings to make successful quantum theory of gravity. But one way by quantum theory of gravity, what I mean is clearly that it's consistent of quantum mechanics, and that low energies is effectively described by gravity. I don't mean something that implements quantization of the lines of energy, okay? So all the successes, and we see many of them as we study this course, all the many interesting things that are going to study string theory, all the successes have been formal successes. They have been successful at relating one formalism, like gauge theory to another formalism, like gravity, and understanding one of the things about these very formalism. But string theories are no complete success yet in implementing the theory of the real world. So rich vacuum string theory, if any, is the theory of the real world. It's happening on completely unknown, okay? And there's no overwhelming experimental reason to believe any of these is the theory of the real world, okay? So string theory has been successful as formalism for quantum gravity. It's also developed as phenomenology. And formalism within the way of phenomenology can be placed. It's the only, in my opinion, the only formalism that we know of within which you can try to make the theory of the real world. It's the only formalism that has both quantum mechanics and gravity, okay? But it has had no complete success yet in reproducing the real world. So you should not think that all of the calculations between our calculations about the theory of the real world yet, we don't know that actually, okay? So these are the general placement remarks that I wanted to make about string theory. Any questions or comments before we continue to do today's theories? Please. In terms of background, would you recommend coming over to this course? Okay, so the things that would be essential are an understanding of quantum mechanics, a very good understanding of quantum mechanics. That is what would be essential. Okay, things that would be useful is exposure to general theory of relativity and exposure to quantum physics, okay? Now that these would be totally essential, they would be useful. Good. Other questions or comments? And that's it. Okay, so let's start. So in the next 20 lectures or so, what we're going to try to do is to undertake one that, okay, one more, one more, situational model of stuff, okay? So as you will see, the basic idea of string theory is that the various particles that make up the real world, whereas this is an in-roading to string theory, come by quantizing the motion of a little string. Just like quantical theory can be thought of in the rising of quantizing the motion of particles. Okay? So the next several lectures, at least 20 or so, will occupy us with the process of consistent quantizations of strings moving around in space time, okay? But before we get to that, before we get to that, let's back up a little bit and ask how we can quantize the motion of a bosonic relativistic particle moving in flat space, okay? So with a situation that's R, let's say, D minus one comma one, by which I mean we're in flat space with D minus one spatial dimensions at one time. And I want to understand how to quantize the motion of a relativistic particle moving around in this back. Now, so in order to do that, I want you to write down an action. The S is equal to something, okay? But even before I come up with the action, I'm going to talk about the variables. What are the variables? Now all of you know how to think about this, because this is the kind of question you ask the point of view that takes up, is that? The variables of the problem are the position of the particle. It's a bosonic scalar particle with no internal elements. So the position of the particle, let's call it Xi of D, is the standard set of variables in what type of it that you use in trying to make the mechanics. Using these variables, you break matters to Lorentz invariant, because you choose variables to be space and T to be a parameter. Lorentz invariant mixes space and type of element. So this doesn't look like the kind of thing you want to do if you want to preserve manifest Lorentz-istic invariance, okay? So unmotivated as it may sound in the beginning, unmotivated as it may sound in the beginning, okay? Let us start by trying to make a formalism in which the variables are not Xi of V, but X mu, where mu runs over all spatial as well as the type of element, as a function of tau.  There is nothing else in the problem, the space and time. So tau is a fictional parameter. And as soon as we set up the problem in this way, we must make sure that we set it up in a way that is independent of what tau is, okay? So any action we write down for these variables here, any action we write down for these variables has to have the property that it is invariant under the forward transformation. Tau is equal to f of tau prime. Which I mean, the action function, written as a function of X of tau prime, you take the same functional form, there's no tau as if it's some parameter that tells you where, you know, parameterizes the path of particles, some arbitrary parameter that's x star from zero and goes to one in some given process. There's nothing physical or invariant about it. So we should be able to change parameterization and will without changing physics. It's a scam. That's the idea of the idea, in principle. How do we write down an action? That's the idea of the idea, in principle. So let me try to write down one action that has this property. So once again, tau is an arbitrary parameter that parameterizes the words lines. We want to write down an action for x mu of tau prime. Now how do we write down an action that does not depend on what tau is? A simple way to do that is to make our action a geometric one. To understand this, let me give you the example that will turn out to be what we want. Let's choose the geometric quantity of a length. So suppose I have some part parameterized by tau, let me write down an expression for the length of that part. So the length of the part, let's say for the moment that we're working in Euclidean space, we get a minus sign, right, in cosmic space. So if we're working in Euclidean space, well, we know that if there are two points and the difference between the coordinates of these two points is dx mu, then the length between these two points is simply dx mu, dx mu squared. The metric is one, the idea. I tried to write tau, so what we could do is put a d tau here, I put a d tau there, but we don't really want the tau in the wrong. That's just fiction, that's fake. So we multiply the divided by dels, we put a d tau, we include the integration sign. So this object is an expression for the length of the part, length of the part traversed by this particle, and it's clearly independent of rescaling of redefinitions of the variable tau, because we're writing about it, is this clear? And what you guys can satisfy yourself is clearly responsible that if I write if tau is equal to f of tau prime, you make this change of variable, realize this action as a function of tau prime, you get the same expression as function of tau prime. It's obvious, it's obvious we've done it, but I want you to get this. Again, it expresses the dimension quantity, the parameter which is a fake. I mean, now let's move to Euclidean space, let's move to Vincas space. Okay, write down the action that will actually, you have been doing this. This very transformation does not single-legit go beyond that. Yes, yes, you would have to, that's right. You would have, it's not a multi-valid, that's right. So, so Suresh's comment is a good one. The transformation will have to have the property that tau is equal to f of tau prime is a one-to-one function. It doesn't go back, that's true. You see, basically the point is that suppose you take, suppose you take this following path, then you choose a parameterization that does this. Then that changes the length of the path. You see, along this parameterization, you're not computing length of the path. You're computing length of the line that you're traversing. It's just larger than the length of the path. See, one, one honest variable that really parameterizes the path. And then all variable changes that will be allowed will be honest values or changes of that honest variable, namely one-to-one, very much so. Good point, thank you. Okay, good. You'll see that that's needed because the square root here will bring out an absolute value, if you try to do this X space. You'll have an absolute value here that will have to cancel with a number there. So that absolute value, that's all. But we'll do it in this case. All right, good. So now let's move to Minkowski space. So, we want to do the same thing from Minkowski space. It's a really interesting part that I find like a particle's move, okay? So, in our metric dimensions, let me immediately set up a metric dimension. A metric dimension will be minus plus that. A metric dimension, we need to include minus sign inside the square root in order to be taken the square root of a particle, okay? So now metric dimension S is equal to minus of the X mu by tau, the X mu by tau. And now we can put any constant we want behind this action. That doesn't change anything. It will turn out to be useful to have a negative constant as we look at this moment, okay? So this is the action that we're going to start with. The action for the motion of a random mistake particle. Random mistake, stay there. Okay, good. Is, and of course, that's a big topic. Any questions, comments, or any discussion about what this action is. So, now, see. Before we go on to the quantization of this action, let me first convince you of something very familiar. Okay? So, one thing we said was that the parameterization of the part should be faithful. That is, any given part of the parameter should also have a single point of the part. Okay, now, if the particle always undergoes time like trajectories, I'm going to do, so if the particle always undergoes time like trajectories, then there's one obvious parameterization of the part, namely the time, okay? So that's why we can use any parameterization of the part. Let's choose for the better to use time. Okay, so what do we get? The di zero by d tau is one. But, when we contract that to the metric of minus one, which cancels this, right? So, as equal to minus m integral of one minus e i squared, where e i, several of you, identified this and have studied already the motion of relativistic particles, and now this is the famous action for the motion of a relativistic particle of mass m. But just to help you have it, let's check that at least it reproduces the correct non-relativistic limit. In order to do that, of course, all of this course, speed of light is one, h cross is one. Okay? We'll reinstate it with great difficulty if we need it. So, we as real humans speak for that. Okay, so let's see if we get the right non-relativistic limit. Let's expand out the square root. So, let's expand out the square root to get minus m plus m to the square root two plus higher orders. Minus m is a constant. It doesn't affect physics. mk squared by two is the correct non-relativistic non-relativistic Lagrangian. Okay, so we started with the Lagrangian that was guaranteed to have the right non-relativistic limit and in fact, well, is the famous Lagrangian for the motion of a relativistic particle. Okay, good. So, the starting point makes physical sense. Even though it looks really formal, but I think that's interesting. Questions, comments, and others. I shouldn't say that because this course has been videoed, I'm not allowed to stray beyond this line or this line. If somebody catches me doing that, please push me back. But, now that we've agreed that we've got a good Lagrangian, we're gonna try to coordinate it. Okay, how do we do that? Well, that's just to implement the standard procedures of canonical coordinate. So, what are our standard procedures of canonical quantization? The first thing we do is to write down this action in canonical. The first thing we do is to compute the canonical moment, the canonical moment, that can be, which is del l by del x. Okay, so, what's that for the Lagrangian? Well, what do we get? So, there's a mu i square root of minus x mu dot x mu dot. So, dot is good in order to get a fixed time. Now, time, okay, the two minus signs of x. Now that we have the canonical momenta, you might think that in order to make one to begin, it's all we have to do is to replace canonical momenta by del x operators, okay, and coordinates by the x, the coordinate operator, and with root. However, things are not quite so simple. We know that it should be so simple because if that were true, that the answer to quantum mechanics would be an inverse space of functions of d variables. Whereas in the non-dimensional limit, we know the right answer is the inverse space of functions of d minus 1 variables. We've got the wrong number of variables in the problem. So, things should be so simple. We should see that somehow. How do we see that? We see that from the following way. You see, you can get the moment in here and then construct the following object, construct free mu t. Okay, so what do we get? What do we get? We get m squared x mu dot x mu dot divided by square root of minus x dot mu x dot mu. So, this thing is minus m squared into one. Sorry? Sorry, it was not that... You can denominate it with this. Oh, thank you. Lovely. Lovely. So, we just get minus minus m squared. And therefore, we say that we get mu free mu plus m squared. Does it work? What do we find? You see, from the formal point of view, p mu's were canonical momentum of our action. They labeled points in phase space. Points in phase space give you all the loud initial conditions. Gastic mechanics.  The momenta has to allow you to assess by surface. People study these canonical quantization with such relations. In the standard implementation of canonical quantization, x mu's and p mu's are completely independent variables. They're no relations between them. We find the system in which that's not true. The momenta are constrained to a basis in which we implement the quantization of such a system. Now, this deep general theory developed by Dirac to study the quantization of canonical systems, there's a little book he's written on lecture notes and quantum mechanics, which is a real pleasure to read. If you're interested, I strongly recommend that book to you. We're not going to go into some deep theory, we're just going to follow up. How do we do the implementation of this system? Well, actually, something else we should do before. We've noticed that we've gotten still strange. Ignore that and proceed for a moment. Compute the Hamiltonian in the system. We want to compute the Hamiltonian in the system. Hamiltonian in the system is p mu h mu dot minus m. Can anyone quickly do the algebra and see what this gives us? Good. Can you see why somebody gave me a reason for that to be zero? Very good. I haven't done it in general, it's shifts in tau. But tau is a fake variable. So shifts in tau should do nothing. At a more mathematically technical level, you see that Lagrangian itself was an homogeneous function of degree one and derivatives. Therefore, this quantity is guaranteed to be the Lagrangian. For any homogeneous function, you add the derivative over there, you build times variable of that function, you get degree times the function. So this was guaranteed to be zero. So, you see, we've run the system with two odd properties. First, the Hamiltonian in Spanish. Secondly, the system has some strengths. These two properties of course relate. If you were a master of Dirac's techniques of quantization, you would understand that this prescription for the part made with this system would be the following in-jewel prescription, which I'm going to show you in a minute, which says, go ahead and quantize your theory, ignore the constraint. But implement the constraint as an operator equation that physical states have to satisfy. This is a so-called first-class constraint. I'm not even going to explain what that means. This is the general procedure for quantizing systems. You don't have to know the general theory. If some form will come, as always, what's it doing when you have a classical theory and you try to make one mechanical theory? It's to make any quantum mechanical theory that is consistent and has the right classical limit. So if you find a procedure that works, it's as good as anyone else's procedure, no matter how fancy it is. So that's what we're going to do. We're trying to find a procedure that works. So I'm clear on this. So firstly, suppose we just did the quantization. So what's the answer to the Hilbert's face? The answer to the Hilbert's face is functions of x mu. With the norm we will discuss in a moment. The answer to the Hilbert's face is function of x mu. You might say, well, hold on. If ordinary quantum mechanics, we have functions of our variables and also time. In this case, it shouldn't be the function of x mu and tau. The answer is no, because the Hamiltonian is zero. The Hamiltonian is what changes the state as you move in time. The state doesn't change. It's not a function of x. So the Hilbert's face, it normalizes the functions of the given norm between the discussive point. Now, not all weight functions of this form are physical. Only those that satisfy this constraint. What does it mean for a weight function to satisfy this constraint? Well, you know, if mu is equal to minus i del mu, I've almost wrote it in each cross there, which would be a garden of air. So it's minus i del mu. So let's put that in. So the constraint that we have is psi of x mu, actually done by minus i squared to minus 1, del mu, del mu plus x squared. I'm sorry, that's too close here. What sense is this making? In order to see what sense this is making, let's try to go back and use much more low-brow approach to the quantization of this system. Let's go back into the grand deal. s was equal to minus m squared with a 1 minus pi squared. x i dot squared, where we chose to take it out implementation of what that was. How do we do the quantization of this system? Well, if we do the quantization of this system, we can have the variables that is x i. So what is pi? Pi is equal to m x i dot divided by squared with a 1 minus x i dot squared. And the Hamiltonian, the Hamiltonian is equal to where we compute it, but you know what the answer is. Pi squared is perfect. Yeah, this is something along the middle. Now if we were to work out the quantum mechanics of this system, what we would have is the base, folks, this is not good. So what we would have is the function of the function that's i, but that doesn't look like zero. So it's also a function of time. This wave function is a Schrodinger equation. What is the Schrodinger equation? Pi d by dt plus i d by dt. On psi is equal to h of psi, h of this. So that's minus del i squared, plus m squared on psi. Now let's take this equation and squared act a little bit i d by dt of that. That's how it is. Then we square the operator. What we get?  On psi is equal to the same, which is minus of the same. So you see, these two methods of quantization have given us the same answer. They've given us the same answer, but it wins. You see, the constraint, the paper working with constraint in our first wave quantization was simply the Schrodinger equation that described time evolution in more physical, more crude, but more. So we've got the same answer working in both ways. Now this has given you a feel for what these constraints do. The problem was, you see, what was the problem? The problem was that we started with too many variables. We predicted that our variables of the system were the time of the particle as well as the position. That was not true.  So this has given us what replaced the Schrodinger equation for the more physical wave quantizing. And there was no Schrodinger equation in the less physical wave quantizing, because I haven't done it for this year. So we've got two ways of quantizing the theory that give us the same result. But give us almost the same result so far, because I've not discussed the scale of the problem. Now this is not in me. When we do it properly and stringically, if we do it very properly, I'm just going to make a couple of throwaway comments about the scale of the problem, just such that the wave function that we naturally get using... So now that we've seen this, let's divide our board into two. That's physical. So s is equal to minus m, one minus v i squared. And then s is equal to minus m, square root of minus x. We've got a wave function and doing the quantization here. Let's give it a different name. Let's say the wave function... I'm sorry, I call it psi here, but I'll change it. Natural norm for scale of products in this system. Well, we know we've all studied quantum mechanics. There's no bigot. If we work in momentum space, the norm is just, you know, up to some two pi's between what it says about. It's a pi of pi star of p times psi of p, v minus 1 of p. This is equal to pi psi. And it's just the usual thing we do, the usual measure of quantum mechanics. Some of the normal wave functions in this way of quantum mechanics. You see, in that way of quantum mechanics, in everything, the natural thing you would think about, you say everything was a function of d then. So you might be tempted to write, well, d, d, d. Oh, now I've got the psi. Let's call this pi prime. Sorry, sorry. Phi is our wave function to get to this point. I think psi is equal to this. You might think that the natural thing could do is to say psi star p, psi of p. Except that makes no sense, because we know that this psi is the size of the strength, and that it only has norm zero support. On those momenta, the wave equation p squared plus x squared equals zero. So while you can say this, this would just be zero. So let's enhance the contributions of that surface. And let's do the Lorentz-Ingerian-Tamana. What? That's a nice aspect, really. It's clearly Lorentz-Ingerian-Tamana. It's a nice norm, but I'm going to leave it as an exercise for you to check. These two norms are only the same. Are the same, only if these two wave functions are not quite equal. In fact, the conversion that works, well, can somebody do it in real time? Well, the conversion that works is that psi of p is equal to square root of, and I'm going to leave this as an exercise. We list these exercises, by the way. So the first exercise was explicitly demonstrating bearings of the reaction. The second exercise was to check that this variable transformation equates these two norms. These are the same systems. The same system in the wave function is not quite the same, same up to a literary definition. This was a by-the-way aside which we understand very well when we study string theory. That's what I wanted to say about these two ways of quantizing the system. Any questions or comments? Now, before we turn to the quantization of string theory, I want to talk about one or two slightly...  It's very simple system. We know everything there is to know about it. But I want to approach the quantization of the system from many different ways, points of view, so that when you see the same kind of... when we do more complicated things by quantizing the string, you will have many options in your hand and use the one that is most convenient. Okay? It's not that I expect to learn any new physics about a free particle. We know everything there is to know about it. But anyway, we're going to kill this problem to death. So let's go through it now. So, to do is to take the action, the fancy action, and make it even more fancy.  Now, when you think of something as simple as a free particle, you can deal with ugly and it's okay.  Okay? So, question. Can you take this action and rewrite it in a way exactly equivalent that makes it look pretty? Answer, yes, and let me do it. Okay, so I'm going to have to look up the signs. Now, if you consider the O8 s is equal to m by 2 mu dot x, mu dot 5 e, you can ask me what is e, and I say, well, it's a new variable. Instead of dealing with the system with d variables, we're now dealing with the system with d plus 1 variables. Okay, we've decided we have.  So now this looks like an odd system. This looks like an odd system. So, before we let us check that this system has something to do with that system. Okay, in order to do that, what we need to do is to notice that it's very simple to solve for the equations of motion of e. Basically, because the the action is no data, it is in time with respect to e. Okay, so the equation of motion of slowly, slowly let's easily solve. So let's write down the equation of motion of e. So the equation of motion for e is minus 1, that comes with differentiating this guy, minus 1 over e squared that comes with differentiating this guy in x mu dot x mu dot is equal to 0. Okay? So that tells you that x mu dot x mu dot is equal to minus minus of this equal to e squared or e which by dimension we choose to be a positive number, is minus x mu dot. You have to keep your wits about you to the minus side. Otherwise you might think it was 0. Okay, so now let's substitute this. Okay, so we substitute this back into the action, we have x mu dot x mu dot and then we got this x mu dot x mu dot we will write that as minus of minus x mu dot x mu dot and then there was divided by square root, so this is this and then we got another minus minus square root of x mu dot x mu dot minus so the action this is to simply equal to m minus m, square root of minus x mu dot. So it's the same action. It's the same action once you solve for e using this equation of motion as previously. For both of you if you are uncomfortable about working with actions once you put in equations of motion something you in certain situations shouldn't be out of contact with what you're doing. I encourage you to check that the equations of motion or that you get from these two actions are the same. Okay, e is simply to determine what x is once you've got that into the equations of motion of x you get the same equations of motion. That's exactly the systems I just had in here. The reason we've done this is by introducing a d plus 1 variable in the system of action d minus 1 variable. It's that the action looks more simple like this. Okay. The equation of motion for e is to demand that the moment are the same for both the actions. So the moment I hear is x dot by e basically and if you wanted to be x dot by root of x x dot then wouldn't I give the value of e? Sorry, I didn't understand. So you write down this action in d plus 1 variables demand the momentum that we get from the action for the same as the momentum from the other action. Yes. It's a consistency check that I have to work. But then d is the value of e determined by dynamics. Which we check it. So the way I do work what you're saying is that now that we've done it we can plug it to the equation for all the momentum and we'll find the same. Everything will be the same. Once you've done it. So why are we doing it? Actually I'm not putting it over I mean if you have increased one parameter in the space of function others and you're setting that fixing that once you're fixing that parameter you're reducing the number of parameters by one. Well you should in general be uncomfortable about plugging solutions into action because you get equations motion from an action by performing all possible variations of the action. Okay but if you see that equation as a constraint so you're again reducing all the parameters by one so you're going back to that. So there are some situations in which it's correct but there are other situations in which it's not correct. There's an analogous question you can ask. It's not the same question but it's analogous and all of us are familiar with that. It's a very reminder of that. The analogous question is when can you plug in a gauge condition into an action? Suppose you decide to work with a gauge a0 equal to 0 into an action and you plug that into the action that you get from that action by varying with respect to A i all correct equations of motion but you miss an equation of motion again you miss an equation of motion because there was a variation of A with respect to A0 which would give you the answer to what you missed. So that's the kind of thing you have to worry about. In this situation I completely agree with this. It's correct. Fine. But now let's move on. So now we're going to make a quantum theory course into this action. We follow the usual canonical procedure. We follow the usual canonical procedure and what does that give us? Well, the first thing we see is that there's no time derivative there's no x dot tau derivative of e. So p e is 0. What about the canonical data? Okay. So p mu is what? Well, we get m x mu dot. Okay, now what? Now the next thing we see is that if we compute okay, so this is already constraint that a momentum on the edge of a particular variable is identity 0 is telling you it can't be at all a phase pressure but it's on some kind of a phase pressure. You have at least one constraint in our system. Okay? And I think that you might have thought that's it. There are no other obvious constraints in the system. However, there is one more constraint in the system which is more subtle than the initial constraint. How do we see that? In order to see that, that's going to be the Hamiltonian. The Hamiltonian in our system is what? Well, the Hamiltonian in our system makes mu dot p mu minus the random. So that's going to be e times p mu p mu by m square by there's an e outside there as well by graph. There's a square of both that's constant just reverse the side. And this part is exactly like the standard we're going to do in a table place like me. Remember that we had a constraint in our system that p equals e to 0. This is a constraint on phase space but now we have Hamiltonian that is not zero in the system. So suppose we start with an object with a constant constraint time variation reactant with the operator that generates time variation it could leave this constraint space. That would be bad because we should demand that our particle always sits in the constraint map. So let's check whether it does or not. How do we check that? The way to check that is to check what the force on bracket of p e constraint is done. That's very simple because the only thing that we can do is to check what the force on bracket is proportional to p mu p mu plus m squared. So it's not zero unless p mu p mu plus m squared. The second constraint we have to impose on our system is dynamic. Now we can check that that's it.  The system is closed. We impose these two constraints. p e is equal to zero. p mu p mu plus m squared is zero. Nothing else. Now let's make our quantum mechanics to the system. States are functions of x mu e and f. We impose the constraint lambda minus zero. To state that off. Once we impose the constraint what's the velocity constraint p is equal to zero that's d by d e minus i d by d e of psi of x mu e of the momentum. e is equal to e. The solution to this is we function so much so it affects me. It was a second constraint. Now it uses this problem exactly to the quantum mechanics. How this works? We start with the system with just one variable. But for each additional unphysical variable that we imposed we had an additional constraint. And then we back down to the right answer when we do things carefully. Questions or comments about our third way of quantization? Just to show you what varieties of quantization are possible we will do a fourth way but I promise you that's it. This fourth way of doing this quantization is a fourth way of quantization. Is there a general rule you can expect with these three variables? Just put d by d r of psi r x mu r equal to zero. That's probably what you seem to be doing here. For example, the first way you have psi r as a function of x mu comma tau. I mean t, for example. The constraint equation was d by d t of psi equal to zero at the end of the day. Right. That's right but as this is a Hamiltonian that's provided. And the constraint then generally not true dynamics in the system. I know that you want to make very general rules. No. But please go. No, no, no. Okay, fine. That's different from... So you just demand, for example, that you stay on the space of constraint when you do a time evolution and that gives you back the first constraint equation. That gives you back the first constraint. You see, the constraint that p e and c put a zero was not consistent under time evolution unless it was also true at least what the sense was. So there's no consistent way of doing the quantization with constraint quantization unless it was both constraints. So dividing the equation of motion from both actions would be the same at every instant of time, essentially. Isn't that the statement that p equal to zero always? Because p equal to zero is a necessary condition for both actions to be the same, I think. p, p, p equal to zero, it cannot be the equations of motion. Oh, I see. So it's not... You see, there's nothing you're putting by hand. Okay. We're doing quantization. We're following the rules. We find that constraint. And we want to inverse the condition that the constraint is... In that event, you know, there's nothing arbitrary here. Each one of those, there's the answer here. We always do things that are sensibly like that and then answer them. Let's do the last one. This last one is a problem that is root to quantization. In the previous way, we didn't have to worry about the fact that the Hamiltonian was identically zero was an obvious constraint. Well, it was not a constraint. We just told you that time evolution is true. But now, h is not zero I didn't take that in. It's not zero identically. However, you're only dealing with states on which the Hamiltonian matches. So time evolution for physical states is true. Okay? I'm pushing this line further. Mentor of quantization uses the same action that we started with. Okay?   I'm just following my notes. So, reaction distance minus plus m x mu dot x mu dot by e minus e. Okay, double. Whatever. And now, I want you to notice suppose you make the sum. The substitution x of dot whole is equal to x prime of f of equal to, well, prime makes something very important. e to the dot of a to the dot df by d equals and I realize the action with this value will change. Okay? At number three. Look, these are two minutes. Okay? I want you to check that this action can be written by two mu by e to the dot minus e to the dot to read the definition of what this dot is. This is really df. Yes. So, isn't that dg by df dot? What? Isn't that d, for the e-transformation? This is df by, df by d. Okay? What I'm saying is that this action here is an initial invariance. That if you change the coordinate, okay? Go on. Why is that df d now again? Well, this is a rule, a transformation rule under which the action is invariant. We see why you see, as we will see in more detail very soon, e is playing the role of a metric on this one-dimensional rule. Okay? More precisely, it's playing the role of a square root of a metric. And this factor is representing the transformation property of the square root of a metric under a coordinate. That's just a variable in the initial one-dimensional rule. So, if I were to do the sports where I'm quantizing myself, I wouldn't have put that df by e to the dot. I would have just done the same kind of thing for x and e to the dot. And I wouldn't have got the right answer. All you have to do is to observe that there is any variance in the action. If you didn't make the variable change, you would have got a different action. That doesn't give you anything, right? What does that give you something when you do find any variance? Well, you know, the way to do it is to think about the solution to the equation of motion. You remember what's going to be minus x mu dot x mu dot. Okay? So now if you make this variable change to an x and if it changes in use as an e, you get this. Okay, so... It was the only consistent. We ought to rewrite the other direction as e to the dot if they were to be changed. Right. Good. So, it's the same thing. So, e times d tau must be invariant. Same. Okay, so how about you see it? We've discovered that this action has only parametrization in variance. Under a certain, you know, change of our variables. Now, there's only parametrization in variance as we talked about right at the beginning. Okay? It's just a change in the labeling of your path. It's a change in the labeling of your path. And therefore there's a gain symmetry. So, for those of you who are not familiar with the term gauge symmetry, what do I mean? What I mean by this is if a gauge symmetry is the word used to describe really iniquivant descriptions of the same physical state. Okay? What's the physical state under consideration? Physical state is the physical solution as the path of the particle in space time. You make two different parametrizations of the same path. You have different functions x mu of tau, but it's the same solution. So, if you deal with bad variance or redundant variables, it often happens that two different functions describe the same physical thing. Any situation in which that happens is referred to in physics as a situation in enjoying a gauge symmetry. The reparametrization in variance and the reparametrization in variance is a gauge symmetry must. Okay? Now, one way of dealing with systems of gauge symmetry is to fix a gauge. It's to say that well, there are many different functions that describe the same physical state. So, let me see in the space of gauge organs you know, choose some condition that cuts each gauge open exactly once. Choose one function per state. So, put a condition on the functions that I'm looking at such that this condition chooses exactly one fits out one function per equivalence orbit. I'm thinking too much about whether it's varying and whether it's possible because this is not our main job. Here we have a little object particle. Well, let's try to fix gauge. You see, because an e changes with f this way, you might think that it's possible and we won't even be able to get that to me, okay? Let's try to set e is equal to 1. It was possible, right? Why was it done as a solution? Somebody remember? Square root k, so that was possible. So, let's set e to 1. Okay? What a gauge choice? Once we've done that action, let's say, so s is equal to m by 2 x is down to mu dot x is down to mu by 1 minus 1. So, it's like a lot of reactions, basically, it's like non-realistic mechanics. So, if you might think, well, let me do my quantum mechanics by making this choice of gauge. What is this gauge? I'm doing my quantumization. What we get is that it's equal to m x mu dot and that h is equal to p mu p mu plus m squared by 2. This is the first time that we've done something.  We've done something fishy and again, and then ended up with a fishy answer. We've ended up with a fishy answer because this one has a function of d variables and now we don't use lot 0. So, the wave functions are actually aboundly functions of d plus 1 variables. Oh, this was bad. That's wrong, but there's something that's even worse. Let's write this out in detail. This is equal to minus 0 squared plus p i squared by 2. It's the unbound difference. It was one of the original operators. Square of an original operator, that's negative. So, we've ended up with a crazy system that has no ground state. Okay? Somebody tell me what we've done wrong and how we've fixed it. We substitute the gauge into the action. This is based on the discussion we had about substituting a 0 into 0 into the action. How do we fix it up? Exactly. Why have we missed when we substitute the gauge into the action? Well, we miss the equation of motion with respect to varying, with respect to the thing that we substitute. What's this action? With the equation of motion and in the equation of motion, then it's equal to 1. Then it will be okay. But what was the equation of motion for me? The equation of motion for me was x mu dot, x mu dot minus divided by 1 because it is 1, minus 1 equals 0. That's n squared. That's the ninth thing. The ninth agent of the action. But we remember to look at the equation of motion from the degree of freedom that we've eliminated. Then we get the right answer. Because now we're back to our second question. However, I can't remember which is the way of quantizing. This problem was so simple that we could carry out any way to want to the quantizing in each of five minutes. We assume, in fact, that now we insert the problem and different ways of quantizing will have different degrees of complexity. So you want to choose the one that is best suited to your problem at hand and then you have to do it right. To summarize, we've had four different ways of quantizing the relativistic part. The first way was to write out the relativistic action. Write out where d value was the x mu's. There was Hamiltonian was 0, so it was not a function of tau. There was one constraint that gave us the effect of the true equation of quantization. The second way was the physical way of writing this Lorentz non-imperial interaction where we saw we got the same answer. The third way was introducing this auxiliary variable e and doing everything honestly and we saw we got the same answer. The fourth way was introducing the auxiliary variable e. Fix and gauge. If we did things wrong, we got the wrong answer but we fixed gauge, did the quantization and then imposed the constraint and now we've got four different ways of working each which give us the same physics. We keep all these in mind when turning to the problem we really want to study, namely how to quantize the motion of relativistic action. Questions or comments? Comments? This is a really good time to ask questions and comments because we're moving on to the next stage now. We're going to move on to the quantization of strings. Please discuss it before we use it. Anything you want to discuss about now? Please. If there's no question about this stuff can I ask something more some of the logical things? If you're saying that string theory is a theory, it's not really a theory since it's sort of your experiment describing the real world and what kind of experiment are you looking at? Well, you see what string theory is in my opinion, this is a big statement but in my opinion string theory when we understand completely it's better to be the correct framework for all consistent thought. So it's a framework within which consistent calculations take place. So with this framework my this will lie in theory in real world but within this framework will lie every calculation you might want. So if you want to perform the calculation of checking to see whether the angle theory it's a calculation that even troops really do in this framework. If you want to find the calculation of computing the viscosity of the fluid that makes up the angles for Yang-Mills theory it's a calculation you can do within this framework. You see string theory has a galactic nature about it. String theory techniques have been used in mathematics to significantly impact entire branches of mathematics. For instance in the Greek in Rene Vismas once estimated that if you look at top algebraic geometry and ask which of the papers that are written there today what fraction of the papers written there today could not have been written had string theory not existed the answer would be about 30%. That's a big impact to the field. People who are trying to study the experiments at rake today heavy ion collations not a model of what's going on from string theory. People who are this is much smaller but to be of such level even with condensed matter theories and trying to understand the dynamics about quantum phase transitions is using calculations from string theory to help them. So at the very least that's the string theory. It's a calculation of framework of great versatility that can be brought to bear many different physical systems. That is at the level at which it is expected experiment today. Will it do more for that we all know? As all of you know this big experiment with the LAC is out. We're going to get into this whenever it does. Let's say in three years we know something about what's going on. If it turns out that supersymmetry is a future in the real world then even though supersymmetry is not a prediction it's not a low energy supersymmetry it's not an extreme prediction or string theory. The study of string theories the idea of supersymmetry first supersymmetry action was the one that went over the chain of string. How these ideas are going to impact experiment is hard to say at the moment. The most direct way in which you might is if you find the right implementation of the real world with the string theory. Even without that the study of string theories already had significant impact on studying physics in various different fields of mathematics. My opinion is that you should keep having more because it's most powerful in my opinion the computational technique that human beings have hit a lot. This is my technique other people could say this is my technique. Other questions on it? I did not understand. What? Right. No, I don't. If I understand you right and the various gauge conditions can determine the measure of the path. We did not reach such subtle issues yet. That's like a small addition to what's going on. This is much more basic. My answer is I don't think that's the same issue. That's like choosing a particular value of E is basically cropping up from choosing a particular form for F. Yes, you have to choose 16 or E times. In fact you choose F so that we have E times one. If that that says E times two more. Now E is basically for the if this was our main job we would do a very thorough analysis when such a choice is possible as I call the redundancies when we move the worksheet which is our main study of interest we would do a very thorough analysis but for now we just don't want to get into this for the next two lectures. I'm not going to worry too much about when it's possible. There are more subtle issues we deal with in the system. The configuration of this system works. I mean you know the answer, you get the balance you must have had, right? One mistake is a case in that fact. OK. Other questions or comments otherwise we turn to the study of the stream. Good. So let's turn to our study of stream. Something I shouldn't say, of course is that the equation that we get can come to me from zero. Of course the famous equation for the quantum mechanics of an energetic state of particles is the Klein-Guyer equation. That's the famous thing, of course. It's the correct way to study the state of things. We get this just by quantizing nothing. It's the particle problem. Fine. Now what we're going to try to do now is to quantify the motion of what's called a bosonic stream. What is the word bosonic? Well it means basically it means that the result of this quantization will give us particles at space-time that are only bosonic. Actually a more technical level that means that on the world sheet you'll see what that means, we'll only deal with bosons. We'll come back. We'll come back to improving this. We want to measure this real world for a lot of fun. It's very interesting. But we're going to use round one. There are no technical complications to try to quantize a bosonic stream. Okay. So, how do we go? How do we go? So what's a stream? A stream is an object whose world whose history in space-time is too damaged. It's an object with a loop but also moves in pattern. You can think of the world of all the emotions in space-time. And what you want to do is just write down an action for the motion of the stream. So an action that depends on the world volume, cylindrical world volume. So, first let's hit upon the parameters. Let's use sigma as one of the parameters. Sigma of course, as always, we will never say much about the parameters except for this one statement. Sigma will be chosen to be to lie between 0 and 1. Every field will be periodic in sigma. What we're doing by saying this is enforcing the topology of our of our services. Enforcing the way of standing signals. So, that signal is one of our variables. Everything is periodic. Sigma 0, 0, 0, 5. Second variable is tau. We say nothing is tau of minus infinity to infinity. Nothing is periodic at tau. You ought to do a little close strength now. At the moment, yes. Okay. So, good. Of course. Because you could trade. You could use one of the coordinates in space in some sense. Some. You can use time for tau and some other spatial some other functional spatial coordinates for sigma. So, actually, the real number of variables is d minus 2. We're not just going to find out the forward quantization procedure. Do it right and get the answer. Understand that we are actually having these are variables. Next question was the action. When we study the motion of particles in order to choose the action, whether we chose as the action the geometric quantity. That is a good thing to do because it ensures that our action was refactored. It's purely geometrical, so to say, that kind of thing. Let's try to do the same thing for a strip. You know, some geometric quantity depends on the two-dimensional worksheet. Well, what quantity do we have in our hand? You could write that meant me. But let's choose the simplest one, the analog of the length of the line. The analog, that's the area of the strip. Let's write down an expression to the area of the strip at the surface. Once again, let's do it first in Euclidean space then we pick it up to write minus times to go to the industrial space. Okay? So, how do we get that area? See, as far as we've got some worksheet and we vary sigma a little bit we move in this direction of the worksheet. We vary tau a little bit, we move in this direction of the worksheet. Okay? This spectrum, what was it? Well, it was deltas in one. That's zx0 by zx0. This industrial vector was deltas tau and zx0 by deltas. What is the area that we suspect? Well, this is an elementary question. It's an elementary question which you could ask high school students and he would think and say, well, it's parallelogram. It's parallelogram generated by two vectors. So, the area is a little bit times zx0. a is the length of the first vector b is the length of the second vector times zx0. Okay? But, you know, we want to address this in slightly more right as it is going very well. So, it will be useful for us because this is square root of a squared b squared, you know, 1 minus cos squared. Why is that useful? This question can now be written as dot root. This is a dot a. This is b dot b. And this is a dot b squared. So, the area is square root of a dot a into b dot b minus a dot b square. The quantity under the square root is simply the determinant of a dot a b dot b a dot b. So, let's remember what a and b were. a and b were these quantities. Okay? Dot means contraction of these matrices. So, what we concluded is that the area is square root of the determinant del x mu del alpha x mu del delta x mu So, you understand what this matrix is. It's a matrix whose indices are alpha and beta. So, two cross two matrix and we've written this down. It was a nice call, actually. Fine. Now, in we were, as we discussed, we would be interested in strings and moving in this nice time like fashion. Okay? Therefore, one of the two over a and b would be a timeline. So, for instance, if we chose things that were of armor, this quantity would be negative. So, clearly, if we want to find a solution, we want the minus sign there. And so, finally, we would choose the action to be T, no, with what's sign, with what's sign minus things.