# Lecture 4 | String Theory and M-Theory

Jun 09, 2021Stanford University, okay, we want to move to closed chains, but before we move to closed chains, I have a habit of doing a little bit of math first or a little bit of reminding, and what I want to remind you of tonight is technical. which has to do with the canonical formalism of classical mechanics, this is a reminder and what I am reminding you is the nurses theorem, anyone remembers the nurses theorem, where this theorem is the theorem that tells you about the quantities conserved and their connection with symmetries. is that for every symmetry there is a conserved quantity and in quantum mechanics the conserved quantity becomes what I will call the symmetry comfort generator, but I want to remind you of that before we jump into some questions about

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just so that we'll have, I've got it on the board, okay, let's say we have a Lagrangian, let's see how much you remember, we have a Lagrangian and it depends on a bunch of coordinates and I'll call the coordinates Q, what is the Lagrangian depends on the The Lagrangian depends on the Q and what else does the Q dot the time derivatives, now what is the canonical momentum conjugate with a given Q, but there can be many Qs, so what is the momentum?Let's call it P sub I, that is the canonical impulse related to the Q. partial coordinate of L with respect to the velocity now I'm not going to derive the nurses' theorem for you I'm just going to express it to remind you what it is, suppose you have some symmetry or it didn't have any It doesn't have to be a symmetry, well yes sir, suppose you have some symmetry that involves a transformation on the Q's. It's an infinitesimal symmetry where it just changes things a little bit, so we will write it by saying the change of Q under a particular symmetry operation. we'll just call it variation of Q sub.

I'll just call him. I don't want to introduce another symbol, let's just call it Delta Q sub. It could be some function, it could be the eye function of Q multiplied by some small Epsilon parameter. when you do a small transformation, each Q changes by an amount that could depend on all the other Qs multiplied by a small Epsilon. Which is an example, well, an example is rotation in space, the x and y coordinates of the position of a particle if you rotate a little bit. bit the coordinates, each coordinate changes and changes by an amount proportional to the small angle epsilon would be a small angle here and in that case you will have Delta an example of this, the conserved quantity which in that particular case would be the angular momentum is constructed from the peas and the variations of Q, so I'm just going to write the formula for you, let's call North achar j-- it is also called in quantum mechanics the transformation generator often labeled capital nobody, right?

I didn't think you did it is the impulse multiplied by the change in the eighth coordinate maybe we should do it now let's not do an example this is the formula I want on the board this is the transformation okay time this is the symbol that represents the generating north of the transit of a of the operation in quantum mechanics becomes the operator that creates the small change the action of the operator in a wave function that induces symmetry, then that is the nurse's theorem that quantity is conserved and will be asked why does it arise that there are all kinds of symmetries in

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, but one particular symmetry, let us now forget the mathematics of the northeastern theorem and come to closed strings, closed strings of strings without ends is a closed string, a single one, the blackboard is the X Y axis?X and Y and there is the chord projected in the Should I run it? for closed chains, you'd be an idiot not to use 0 to 2 pi weed. Okay, so some point in the chain we will label Sigma as 0, which point in the chain doesn't really matter, but we choose a point in the chain once and for all and label it Sigma 0 had a small pen and we could mark the screen, we would mark it that Sigma is equal to zero there, then halfway would be Sigma is equal to PI a quarter of the way would be Sigma is equal to PI more than two and three quarters of the way would be Sigma is equal to 3 PI over 2 now we imagine that there is a directionality along the rope the directionality of Sigma increasing there is a feeling of a kind of arrow along the rope that tells us which direction Sigma is increasing and in this way he is running out of ink again now the rope it's like a rubber band and think of it as a kind of rubber band mark we're going to remove those marks before we finish a rubber band marked sign 0 Sigma equals 1 degree 2 degrees 3 degrees 4 degrees always around 360 degrees and that gives us a feeling along the string, but now that feeling along the string, the orientation of the string has nothing to do with the clockwise Nisour spin in the XY plane.

I drew it so that Sigma increased clockwise, but now I take this string, I lift it off the board, I turn it around and I put it back on the board, that's a physical thing I could do with a piece rubber with an elastic band. I put it back in and there it is now Sigma equals zero is over here. I put it back in the same place Sigma equals PI over 2 here Sigma equals PI over here Sigma equals 3 PI over 2 here Sigma increases in this direction now The only point is that I don't want you to get confused when I'm talking about the orientation of the rope, it is not an orientation in space, it is an intrinsic sensation to the rope, there is a directionality along it and that directionality we have to maintain. hint okay, now let's move on to the waves moving along the rope, a wave can move to the right or to the left, but by right and left I now mean in the direction of increasing Sigma or decreasing Sigma.

No, I'm not implying anything about that. spatial orientation I'm just imagining a small movement in the string and a small movement in the string moving in the direction of increasing Sigma. Now I'll call it a right-moving wave if I turn the string around and put it back on the board. The rightward moving wave would be moving in the opposite direction, so there is the question of whether it is moving clockwise or counterclockwise in real space, but if it comes from a slightly smaller Sigma or a larger Sigma, yes, yes, yes. You can think of yes in particular in the original and the origins of string theory if you break the string, the only way to break a string is to produce a quark and an antiquark asks which extreme is the quark and which is the antiquark true, no , that doesn't mean I can't make up string theories where the string is not oriented but then you can't break them, but let's assume for the moment that the oriented string has a sense of increasing on a sense of decreasing, okay, what Can we say first of all?

What are the coordinates that describe the chain? The coordinates are exactly the same as for the open chain, an X of Sigma and a y of Sigma, and if we have more coordinates like what Kaluza and Klein told us, we would add them into and Y of Sigma the XY position the ends of the rope here there are no ends of the rope but there is something you can call the boundary condition is the boundary condition of the relationship between the coordinates at 0 and 2 pi what will x of 0 be equal to x? of 2 pi because it's always the same point, that means if you were to plot Sigma's rotates by 2 pi then X of 2 pi is equal to let's start with a string just flat and with a little movement, it has a little movement, the movement can move to the left or it can move to the right if it moves to the right, that is the D, that is the sense of increasing Sigma, moves to the right, it's a right move, which happens when it gets to the end of the rope, it just reappears at the other end, so it might fall off the end here and you pick it up here and it just continues to circle around and around of the string that would be the image of a wave that moves to the right, so it is a wave that moves to the right and then there are waves that can move to the left, waves that move to the left and waves that move to the right, and both can exist. on the rope, what about the open rope where there are waves moving to the left and waves moving to the right?

Yes, you could imagine a wave moving to the left and a wave moving to the right if you start with a wave moving to the left, but what happens when it reaches the end? in that case it is reflected then in that case you would have left going to the right going to the left going to the right the left enos and the writing this would not be conserved in time are the type of things that are conserved in the time of standing waves , but for a closed string there are running waves, they go to the left and you go to the right, okay, let's talk about writing the oh well, yes, let's take particular types of waves now, but taking the types of shapes.

We have plane waves that move up and down the string, let's say they move to the right or to the left. Now it is convenient to describe the waves not by signs and co-signs but by exponentials. Let's delete this here. Exponentials are simply linear X e to I Sigma. e a In Sigma are exponential, they are just linear combinations of sines and cosines; in the open string case, it is restricted to sines or cosines depending on whether it was dear Ashley or Noah and the boundary conditions for closed strings you may have. both sines and cosines, but you can also have linear combinations of sines and cosines, so you can have exponential waves propagating from left to right, so the type of waves you can imagine are e for In Sigma, which would be, of course, cosine.

Sigma plus I n i cosine n Sigma plus I sine in Sigma we can choose to decompose things into exponentials this way if we want, so let's do that, let's take X from Sigma, what we're going to do is exactly what we did with the open string let's Let's go to the Fourier transform period 2pi is like a general way is that it is a sum of all the integer coefficients, let's call them xn multiplied by e up to In Sigma n can be positive or negative if n is positive, consider it as a wave moving to the right if n is negative, think of it as a wave moving to the left, ok let's say let's just check this, does this really have the property that it returns to itself when Sigma goes from 0 to 2 pi when Sigma is 0 e towards In Sigma is a warning?

What happens when Sigma is 2 pi? instead of n? to set n to 1/2, does it return to itself after one cycle? No, so you need to put integers here and this is the most general form of a periodic function when, therefore, it expands on a line between 0 and 2pi, assuming it's periodic, so, of course, I'll do this to write as a sum of two terms, well, a sum of three terms, the first terms are positive. Alright, this is going to be a sum of N greater than zero, those correspond to waves that move to the right waves that turn to the right now I want to write the waves that move to the left, which I could write in the same way except n is negative or you could write it as a sum of N greater than zero, but let's call it X a waves moving to the left that's beautiful thank you yes you see a signal implores silver right yes yes the time coordinate is generally here e to the plus or minus sign okay so think about these anyway waves moving left and moving waves right, something is missing here N is equal to zero, okay, so let's put it back and what is that, you know what it is hmm that requires zero, but what is its physics?

What is its physical meaning? centergravity sinem center of center of mass center of mass of the string the actual location of the string the center of the string x0 the same for y exactly the The same for and I think I will not write it, but and of Sigma is equal to the same type from sum Y sub n e to In Sigma And from minus n e to minus In Sigma, then you can have This is the general way. What are you doing? You take the Lagrangian for the string, which I haven't written down, but it is exactly. the same Lagrangian that for the open string exactly the same the string is still imagined formed by small points of mass with strings and balls the Lagrangian of the string is written exactly as before the only The difference is that now you integrate from 0 to 2 pi sigma the same that before x times d tau squared palestine minus lagrangian what are the things that depend on time in of the time of these xn, are the degrees of freedom, what do we do with this?

We take this expansion and plug it in here, you can get X point differentiating that gives you X endpoints and you can differentiate Lagrangian, I'm not going to do it, calculate the Lagrangian and what we notice about the Lagrangian, if it's worth what the xn will be, there will be harmonic oscillators exactly as they were before, there will be harmonic oscillators exactly as they were before. Oh, before I do that, yeah, before I do that, let me do something else first, I don't want to. a I don't want to write the harmonic oscillators I want to look at this formula for a minute What is the energy?

This is the Lagrangian. What is the energy? Does anyone know the same thing with a plus sign. This is potential energy. This is kinetic energy. X times tile squared plus X times D Sigma squared same for Y I'm going to write this in a fun way I'm going to write this as X times Tao plus X times D Sigma squared plus minus X times D Sigma squared I think I'm off by a factor of two, probably 1/2. Can you see what's happening? X times D tau squared here, X times D tau squared, we double it, that's it. why I put a 1/2 there is also a DX times D Sigma squared and a DX times D Sigma squared from here and then there is a cross term X times D tau multiplied by X times D Sigma comes with a plus sign here and with a minus sign here and they cancel this is the same as this do you have any idea what the meaning of this decomposition is? well, a wave that moves to the right is a function of a of Sigma plus tau a wave that moves to the left and then I can have it Rob, I can have the opposite, but there is a wave that moves this way is a function of Sigma plus tau a wave moving in the other direction is a wave moving in the opposite direction this is just the stored energy, I think this is waves moving to the left and this is the energy stored in the waves moving to the right waves moving to the right and waves moving to the left are transparent to each other they just cross each other these are linear equations linear equations of motion and this is just writing the energy as the sum of the waves that move to the left and the waves that move to the right on the rope, this will prove to be of some interest to us in a moment.

Now let's get to the harmonic oscillators. Each of these Xs is itself a harmonic oscillator, so now we can stop. There is no need to go over the details of the construction of harmonic oscillators from these things, we can assume that there will be a harmonic oscillator variable for each oscillation mode, as always, whenever there is an oscillation mode, you can excite it. with the creation and annihilation operators, creation and annihilation. Oh, by the way, what is the frequency of a wave that is labeled here? It will be the same as for the open string proportional to n.

The higher ends, the shorter the wavelength modes, the higher the frequency. that will be the same thing, so we have a collection of harmonic oscillators, but now we have double and the reason we have double is because we have waves, each integer or each frequency, now we label two oscillations, one with that frequency moving towards left and one with that frequency moving to the right, well, one of them is labeled N and the other one is labeled minus n. This to me is too much notation, but I can't help it, there were all these things that we have.

To keep track of the little n, here represents the frequency when n is positive, since in the first term it represents, say, waves going in one direction, when it is negative, it represents waves going in the other direction, so Now let's write what the different harmonic oscillators are. First of all, I have create, I'm not going to bother writing the annihilation operators, the annihilation operators just happen, the conjugate, the creation operators can create an excitation that moves, let's say to the right, and then we'll just call to a sub plus n, you can also create a different operator a plus to minus n if this one creates a wave that moves to the right in the tension, then the other one creates the wave that moves to the left in the string, okay , first of all we have that decomposition, what else do we have x y? y I didn't write everything related to Y, but writing Y simply replicates the same thing for X.

What did I call the creation and annihilation operators for the Y oscillators last time? I think I called them B now, so now we have more, we have B more. in and B minus sorry B plus n and B plus of minus n now you could add to this if you wanted to write the form of annihilation operators or with minus signs here. I'm not going to let us get confused with too many pluses and The minus signs a and B represent X and y plus, that's just the creation operator. It's exciting, you can keep in mind that it also exists, but we won't use it anyway and in y, n has to get it right, first of all, n has to do with frequency, frequency is in but More in and - n has to do with waves moving in this direction or in that direction, that is the structure of closed string theory, that is the whole structure, but now we can start to ask what is the spectrum what type of particles what type of excited states we will have in the string what is the analogue of the photon that we discovered in the open string you know what I will do I will just remind you that for the open string we did not have this duplication of minus N and plus n what we had were only operators of creation a and B a plus N and B plus n we begin.

I'll just remind you that we start with the fundamental state of the string which is a string. in its lowest state of oscillation without doing anything, then you excited what you can excite it with. You can excite it to get the lowest energy. You want to excite it with the smallest in Y because frequency is proportional to n and of course frequency and energy are. the same to get the next lowest energy you have to excite it with a plus of 1 and B plus a 1 a plus 1 and B plus 1 this is not an oscillation in some way that is associated what is the straw image here is the z axis here is the x axis and here is the y axis the string comes down the z axis with a large momentum but it is oscillating in the polarizations of photons this is a polarized photon minus I B plus one or oh I being just the complex number, the superposition of two plane polarized waves of polarization without I, it would simply correspond to plane polarization at some angle with I, it corresponds to circular polarization, okay, so we'll have it into account, a plus I B and a minus.

I B created the units of angular momentum plus angular momentum minus angular momentum. Well, where are we now? Let's see what we can do. The first thing we can presumably have is the fundamental state. Now, whatever the ground state is, it has some mass squared, which I'. We'll just call it n zero squared the ground state has some energy M zero squared what excitations can I do right I can excite it with plus one we have to make the excitation using one here we don't want to excite it with two units of energy so we don't want to use a 2 and B 2 ok, what can we do?

It looks like we can make four different states four different states that look like they have two photons or yes, they look like two photons. We could write plus I B and minus IB or minus one and B minus one. Now there are four possibilities to act in the fundamental state a. plus 1 or minus 1 or B plus 1 minus 1 four different states that you can do what they could correspond to what type of structure they have the first question is how much angular momentum they have how much angular momentum they have are they really? Like oh, let's see, this is basically the first thing you can do, they're not hmm, what is that?

Although they do have, they have a unit of angular momentum, they have one of your angular momentum because you could write them as a plus I B. and minus IB one and minus one, so in essence it's just twice what we find here, which It means that there are two types of objects, they both look like photons, right, that doesn't sound so bad, but I'm going to tell you now. Right now this is not the correct image, okay, it is not the correct image, but to see that it is not the correct image, what you have to do is go to the next level number one.

If it were the correct image, then you would have two photons, each behaving. like a photon, there is no zero angular momentum state here, by the way, there is no zero angular momentum around that axis, there is just an s and a B, a plus IB, a minus IB, you can't make the momentum angular is zero from this combination, that means something about these hypothetical photons. means it has no mass, if you have a massive photon then it could necessarily have plus angular momentum around the axis of motion, minus angular momentum around that axis and also zero angular momentum, we discussed last time there is no candidate for that. here no candidate for zero angular momentum is like having two angular momentums one and minus ones and that does not correspond to any possibility other than Escalus particles, so we could say that this theory has so far produced for us a doubling of the spectrum of photons.

Okay, but to see that that's wrong, go to the next level. Let's move on to the next level. What can you do with a power unit? I'm getting a little tired, so I'll tell you what you get, what you get. make is a bunch of garbage, doesn't seem sensible at all, doesn't fall into rotational multiplets, has the wrong number of states to be angular momentum; You know, there is no angular momentum in a single piece, it's just wrong. just the wrong combination of things this is not right this is not right I'm going to tell you now why it's okay it's not okay it's extremely subtle and it's not easy it's not easy well it's very easy but it's a bit complicated not all the things that you can write they correspond to legitimate states of the chain in fact there are restrictions there are rules the rules prohibit certain combinations and I'm going to tell you well you know what I'm going to do I think I will do it I won't explain why tonight, I'll explain why next time unless we have some extra time tonight, but I'm going to tell you what the rule is, let's take a rule and then we'll try to figure out why it's the rule. called level matching and level matching says that the energy moving to the right and the energy moving to the left must be the same now you asked me why the total amount of energy moving to the right has to be the same as the total amount of energy moving to the left, that's step four you'll have to go back to notice the theorem, okay, but let's not do that now, let's just do it as a postulate, suppose that the energy moving towards the right and the energy moving to the left must be equal.

What can we say about these states? Well, let's go to the right, let's see which ones there are. There's a plus one, oh, which has one unit of energy moving to the right, so it doesn't satisfy the level matching rule about negative one. There is one more unit of energy moving to the left. It does not satisfy the level that coincides with B, whether it has one unit of energy moving to the right or one unit of energy moving to the left, so none of these states here that we write satisfy the condition that the energy moving to the left and energy movingto the right of B the same is not good States let's go to the next level let's see what's at the next level at the next level we can do the following things we can take a let's see what we have we have a plus one multiplied by a plus minus one this has 1 unit of motion to the left or one more unit of motion energy to the right and one unit of motion energy to the left oh, that satisfies the level matching condition because it has one unit of motion to the right and one unit of motion to the left. movement to the left, I appreciate this coming out of nowhere, why?

I'm requiring this to be true, but let's get back to it, so this is a good state, okay, how about a plus and how much power does two units have? Alright, next we have a plus, oh sorry, next we have B more than 1 B more than minus 1 same thing perfectly good status levels match the most movement to the left, oh my gosh, energy of movement to the right and we can have a plus 1 B plus minus 1, yes, B plus minus 1, a positive, you know, a movement to the right. a little bit of energy and a left moving a little bit of energy, this is also a good state and how about a plus minus 1 B plus 1 Oh is also good, as long as 1 and minus 1 match, these all have 2 units of energy because they have two oscillators of the first frequency now, is there anything else with the same energy? a 2 so let's write here a 2 plus oh, this is a 2 plus a minus 2 plus Oh, how about B plus all of these states have two units. of energy, but do they satisfy the corresponding level?

Noo they have two units of energy moving to the right or two units of energy moving to the left, so these are illegitimate states, cross them out, what's left, what's left, are these, okay, now the I'm going to rewrite a little differently. I'm going to rewrite them in terms of a plus IB a plus IB or the operators that do circular polarization we could write exactly the same states in a different basis in the form a plus IB 1 1 plus plus multiplied by a 1 plus I B plus 1 what do you think someone does? I have an idea what one more IB did for the circular polarization of the photon, which meant angular momentum plus 1.

What's up with this? Whatever you did for the photon, you did it twice. It rotates 2 2 units of angular momentum around the axis. No. no, you mean here no, sorry, thanks, that's it, yeah, you're right, otherwise it wouldn't be level, man, no, no, no, sorry, that's right and I don't have it right. Plot 1-1, this gives you more of a unit. of energy this gives you minus one no this gives you a thing that moves to the right this gives you a thing that moves to the left the correct time the left time matches in level that is the correct way to think about it what plus you can have the same thing except with a minus B times a minus B times minus IB, how about that?

What would that do if this is like two right-handed photons adding the spins to this is like two left-handed photons adding the angular momentum minus two around the z axis, okay, so first of all we have angular momentum, we have angular momentum, there were four independent states here, so we are missing something for linearly independent combinations, what else can we do? We can do, let's see, we can do, let's just write a 1 plus I B 1. I'm not going to put the plus and forget about the plus to minus 1 minus I B minus 1 this is a plus I B times a minus IB left right moving to the left how many units of angular momentum generates this is like superimposing a right circularly polarized photon with a left circularly polarized photon two zero photons so this has zero angular momentum this has angular momentum let's call it N equals 2 this has angular momentum M equals - 2 this has M is equal to 0 and the other possibility is to change plus and minus one to make this a one minus IB 1 multiplied by minus one plus IB minus one those four possibilities are linear combinations of these how much angular momentum here 0 again, for what we have two states with angular momentum 0 and 2 states or one state with angular momentum 2 and one state with angular momentum minus 2 now let's remember what we're doing: we have a particle shot down the z axis, whatever it is, it seems to get from a state with angular momentum to the state with angular momentum minus 2 and then maybe some pieces with angular momentum 0 what's missing how can there be something on the face? seems wrong if spin 2 there then there must be a spin 2 pi then this must somehow represent a spin 2 particle how many states does a spin-2 particles have 5 and when they come in M is equal to 1 0 minus 1 minus 2 we have one candidate for spin zero we have two candidates for spin zero but we don't have any candidate for spin one and minus one no, no, no, this is what we said, could it be a spin-2 particle?

Well, could it be a spin-2 particle? Forget, we call it a graviton, let's just call it it could be a spin-2 particle, a spin. -2 particle has five states if it were moving down the axis there you will declare it or the angular momentum around that axis would come in a multiplet of five possible states very well, we did not find the correct five possible states we found four states but not of the right type, not even close, we are missing the M equals one and minus one, those would be part of the spin two doublet, right, what do we conclude from that or everything is a disaster and doesn't work or what is the spin particle? 2 what massless massless because massless particles come only in states of maximum and minimum angular momentum.

That is a general fact about a graviton like a photon that would only have a left and right polarization. hand polarized, the only difference with a photon is that the right hand polarized graviton has two units of angular momentum and the left hand which has minus two units, gravitons do not come with one angular momentum or zero angular momentum, so What we have here is the only possible interpretation, if this makes sense, is that there is a graviton here, but we are left with two states with zero angular momentum. What could be the only thing that could be two particles that have zero angular momentum?

That's two particles with zero angular momentum one of them a linear combination I think it's this Also, this is called dilettante it's a scalar particle it's there in the spectrum of string theory it has the same mass as the graviton, i.e. mass zero and the other particle is called action They are both familiar particles, very, very familiar particles to phenomenologists and they have one important common characteristic: they have never been discovered. It's a fact about them. In the most formal mathematical structure of the theory, they begin as massless particles. The interesting question is: can they? you can get rid of these somehow without getting rid of this and the answer is yes, these are not necessarily there, this is necessarily there, so what we find then for the closed string is something new, we find massless spin-2 particles , the only thing we found.

I haven't explained why this level matches. Why do we require this funny rule that the amount of energy flowing to the right must be the same as the amount of energy flowing to the left? Maybe we'll go over it. It is something that is not complicated. It is subtle, well, left and right does not mean in space, it means along the rope to the right and the condition that the energy moving to the left and the energy moving to the right are the same. I'll tell you what we're going to drink, let's take a break for a few moments. minutes I will tell you what the condition is.

I'm not sure if I'll refer it or not, but I'll tell you what the condition is that means we'll direct it hmm, okay, we're onto something. technical details of the string spectrum and so on, and the only missing piece was this level match. I'm going to tell you what level matching is, now what it means first and then I'll derive it for you, if you want. recognize the steps in the derivation or not I don't know at least one formulation uses the northeastern theorem maybe I don't even have to use the earth theorem but we'll see, the question is whether the point Sigma is equal to zero really physically special point or no matter what point you call Sigma equals zero, there is another way to pose the question of whether the state of a string should be invariant by changing what you call the origin of the Sigma coordinate.

Now it's not obvious what the point is. The answer is: it could be that there is a special point on the rope that is marked with a little piece of ink that you know as and that point may be special. It could also be that there is nothing special about it. any point and that the theory has to be symmetric or invariant when changing the parameter Sigma, that's what it boils down to and if the states of a chain are invariant with respect to changing that parameter, that is a fundamental question, it is clearly a fundamental question that I have.

I haven't said exactly what it means yet, but I'll say it now. We could start by thinking the string is a discrete collection of points and then have it instead of having X of Sigma and Y of Sigma, we would have X of I. Let's call it X sub I and Y sub body What does X sub I and Y sub I sub I and Y sub I are just the positions of point I units along the chain? It could run from 1 an to n I now our next, what is the quantum wave function of a straw of a string?

Well, if we think of the string as just a collection of point particles, at least temporarily, then the wave function of a string, the quantum state, the quantum state vector of a system, what would it be? be a function of the Why didn't I start with x2 there and cycle and let's say sigh of x2 dot dot dot dot what about the quantum wave function of a string that is completely symmetric with respect to Rearrange the X's. I don't want to rearrange them in the sense of putting x3 between x1 and x2. That's too violent. You need a point to open the rope, but simply cycle them with a rope that does not have a preferred point. must have the property that the wave function when expressed in terms of 4 and so on, which are also equal sine of X 3 dot dot dot X n i a i plus 1, that's one possible What we could demand from a string theory is not just one possible thing we could demand, we get into really big trouble if we don't do that, the whole work just falls apart, okay?

What does that have to do with the level not matching anything? obvious but let's leave this here everything is fine suppose that now I go from the discrete chain to the continuous chain what is the corresponding here instead of saying that the symmetry is or the operation of interest is X I go to X I plus 1 module n going around in what is correct if the index I is replaced by the continuous variable Sigma correct, then it says that the wave function which is a function of X of Sigma is a function of a function what is a function of a function called? a functional so the wave function is a function of , Sigma plus Epsilon now Sigma.

Also epsilon of course means something, you are at point 2 pi and you want to change it, don't you know? It means a small rotation of the Sigma circle. Well, let's see if we can figure it out. Actually, this is just. some simple law of course we have to change all these . so this is actually an operation that changes all the Xs, changes all the Xs by moving them to the neighboring points. Well, let's see if we can write this another way. Let me rewrite it by writing Sigma's X sigh minus. s and of X of Sigma so it is equal to zero, it is a small change in the wave function if I make this small change where each equal to zero, okay, let's work What we have done here is taken as a function of a variable, it is actually a function of a continuous set of things, but let's treat it as a function, how do we calculate what is happening here?

Well, we write that this is the small change in sigh when you change X at the Sigma point multiplied by the change in Sigma plus Epsilon to come out of partial Sigma of X with respect to Sigma multiplied by Epsilon, so this is the change in s and when you change X with respect to Sigma multiplied by Epsilon now that Sigma Am I talking about Sigma, the origin,Sigma Pi, a sigma or whatever? What does it mean here? What should I do? This is a separate equation for each Sigma. No, you should add them all. You are saying the change. inside when you change something at one point plus the change inside when you change it at the next point plus the change inside when you change it at the next point and so on, this should really be integral D Sigma, the sum of all the changes inside when you change a little bit each X should add up to zero that's what it says we can get rid of the epsilon now what is the back to quantum mechanics quantum mechanics again we have P and Q, but how is a P related to the corresponding Q in quantum mechanics minus I h-bar times D times DQ?

Moments are the action of a moment on a wave function to differentiate it. with respect to the corresponding coordinate, so whenever you see that a wave function has been differentiated with respect to a coordinate, let's forget the h-bar. Whenever you see a wave function that has been differentiated with respect to a coordinate, you can rewrite it as the action of the corresponding coordinate. impulse in sigh well, here we have the differentiated wave function with the coordinate at the Sigma point, well, what should I write that? Since it's B Sigma, what does that mean? P Sigma. Let's think about what P Sigma means.

We have a rope and it has a lot. of points rope here each small point of mass has its own momentum P of Sigma and what I find is that the condition for the wave function to be invariant under fist parameterizations of the Sigma axis, in other words, simply shifting the Sigma axis is the condition that a certain integral P of Sigma multiplied by XD Sigma is equal to zero, but P of Sigma is nothing, but where is it? There is nothing more than the velocity X point of Sigma or XD tau, the integral of a sigma of xt tau multiplied by remarkable that such a condition exists and it is quite remarkable that what it says is that there is no preferred point on the Sigma axis, but now look and compare it with this, by the way, I wrote energy here, these should be integrated on the left, remember what it was this, remember these, the energy moving to the left and the energy moving to the right, the energy moving to the left and the energy moving to the right, okay, the sum of the energy moving to the left and the energy moving to the right is just the energy.

What about the difference between energy moving to the left and energy moving to the right? What remains when you take the difference instead of the sum. Let's take the difference between energy moving to the left and energy moving to the right. What's left? This is okay, I don't even have half of it here, but if you were to take the difference on the left for whatever reason, let's take the difference of the energy moving to the left and the energy moving to the right, then the squares tau X times D would be to cancel the DX by D The squares of Sigma would cancel, but the cross terms would add up and we would have exactly this, so the condition that the wave function does not change when the Sigma axis is shifted ends up being nothing more than the statement that the movement to the left The energy and the correct energy in motion should be the same, it is not so curious, that is the condition if you have a string or if you have a string theory that says that there are no preferred points or that the Sigma axis has no preferred point. point and that the wave function or the state vector of the string does not change when you just arbitrarily come and change the parameterization of Sigma, change the Sigma axis, then that becomes the condition that the energy moving to the left and the energy moving to the right are equal. was the level matching condition, the energy moving to the left is equal to the energy moving to the right, so the content of discarding all these states that do not match the energy circulating in this way and the energy that circulates that way is the content of saying that we are talking about a string theory in which the string does not have a preferred point along its direction Sigma does not have a special point that is called Sigma is equal to 0.

I consider that this is a very beautiful fact in many ways and it discards a very large portion of the spectrum that what you can deduce from the strings happens to pull only the Porsche, which you cannot unite in angular momentum, multiplets of sensitive particles that it pulls, for example , those, well, throw away large portions of the other spectrum, leave the photon and the proton. wisely the photon because photons are open strings and it doesn't care about this, leave the graviton and leave the dilettante and leave the action like the massless strings and then whatever else was there so you can find out for yourself what it will do. be on the next level, the next level, and see if you can make any sense of them anyway.

I think that's what I wanted to do today. It's a lot, but you'll see there are some things you can't understand, but I had to be guided to them. I think someone asked me an interesting question. Well, there were two interesting questions. One of them was simply a mix-up, but the mix-up was obviously because I had gone fast. I just want to remind you again and again what I mean by movement to the left and to the right I have used ter or clockwise and counterclockwise I reverse the burials I want to distinguish them one of they had to do with N and minus N and that had to do with waves moving along the plus Sigma axis or the minus Sigma axis.

I had nothing to do with orientation and space. The other way I used it was talking about the polarization states of photons and talking about circular polarization. I said right circular polarization. Left circular pole that was actually in genuine space x and y space, so we shouldn't confuse those two, even though I use the same terminology for them. The other question I was asked, we've been going over some of the technical details of string theory, what is it, what, what type? of constructions enter it ask me what I don't know whether to call it a philosophical question or not it is not a philosophical bazaar of Asafa by definition bad questions this was not a bad question it was whether one way or another there is any either There is no evidence or reason to believe that, in some sense, strings are the most fundamental things.

Good ropes are made of other things. I think it's a question of, in the march of reductionism, whether there is any sense in which strings are the most fundamental things. the world and there is nothing smaller than them or they are not made of anything and I gave an answer that it was the best hedge or the last thumb waffle the last waffle and we have learned that that is not a good question so let me say a little on this I will return to the topic of mana poles for a moment remember the monopole why because I want to ask the question which is more fundamental the electron or the monopole is a more fundamental monopole or is the electron more fundamental and I want to raise this question of " okay", so let's assume that there really are monopoles in quantum electrodynamics.

It is easy to formulate quantum electrodynamics so that there are monopoles in it and ask which is more fundamental. Now let me remind you what I told you. before the electric charge is multiplied, the monopole charge has to be equal to PI so that the Dirac chain, which is the solenoid that is connected to the monopole, is invisible, this is the condition that if you have a monopole and it is connected to a long rope is the only mathematical way to make a monopole that the charged particles rotating around the rope do not detect phase changes e multiplied by q is equal to 2pi that means that if the electric charge is very small now first nothing if the electric charge is very small, then we can do quantum electrodynamics the way we've all learned to do it, by finding diagrams etc.

Feynman diagrams are not very effective whether the electric charge is large or not, yes, because each Fineman diagram contains a lot. of vertices each vertex each vertex has an e squared and the probability if E is large then it means that the Fineman Dai have the values of the Fineman diagrams become larger and larger as the size of the diagrams becomes larger. larger and not They do not converge, they cannot be added, they do not converge to anything, so finding diagrams is explicitly a tool for studying theories with small charges, it simply will not work, it will be useless for theories with large charges, on the other hand, here We have a theory. which has a small electric charge, suppose the electric charge is small, but if the electric charge is small, the magnetic charge is very large.

If we tried to exchange electric charge and magnetic charge, we might think that electric and magnetic fields are this kind of Same thing, not the same thing, but they are interchangeable. Maxwell's equations are the equations for electric and magnetic fields. They are completely symmetrical to each other. Some negative signs, but those are the ones you can handle. Electrical and magnetic, completely parallel. with respect to each other, then, assuming that the theory has magnetic charges, how do we know which of the two types of charge, electric or magnetic, is more fundamental? Then you could say, "Okay, let's go back and try to work with magnetic monopoles as fundamental charges." The findings generate additional sighs by exchanging electrical charges and magnetic charges.

You could do it, it's perfectly doable, but you will find that if you try to make the Fineman diagrams in terms of the magnetic monopoles, because the magnetic charge is large, they would not converge correctly. It is useful to think of electrical charges as fundamental objects. Now another thing is that the magnetic charges are large, which suggests that the mass of a monopole will be large. Why, because they have associated electric and magnetic field energy, the field energy of a magnetic charge. will be much larger than the energy field of an electric charge and will therefore be heavier because they interact strongly, meaning that a magnetic charge will be very effective at emitting a photon.

An electrical charge will emit a photon approximately once. One hundred and thirty-seven percent of the time the magnetic charge will emit a photon 137 times squared stronger, so this magnetic charge will be surrounded by an incredibly dense sea of photons, but the photons will interact very strongly with pairs of charges. Magnetics form pairs of magnetic charges and this will turn the magnetic monopole into something very, very complicated with all kinds of internal structure and, in fact, will spread it over a larger volume, make it heavier. It is complex and will make it useless as a starting point for finding diagrams.

Does that mean that the magnetic field and magnetic charges are somehow less fundamental? Well, I guess it's a matter of taste, but this is what I can tell you. you could start gradually changing the parameters of the theory, increasing the electric charge by just one number in the equations, you could imagine increasing it slowly, in fact you can actually imagine slowly increasing the magnitude of the electric charge and at some point they will equal out by The tail. beyond that tail it will become smaller than what happens the magnetic charge becomes more magnetic charges begin to play the role that electric charges originally had electric charges become complicated heavy things magnetic charges become simple light things about The question of whether something is really one must ask whether it is composite or fundamental to know in the following way: one must ask whether it is useful to think of it one way or another and whether it is useful or not may depend on the values of the parameters in the theory on which it may depend. values that may depend on the environment there may even be situations where there are control knobs that you can turn that end up converting magnetic charges into something more fundamental into electrical charges.

This can really happen. You can imagine this happening, so there is no invariant. The definitive answer to the question of which is more fundamental: magnetic charge or electrical charge. It's a question of which one is useful. I remember this question coming up unsuccessfully at a resolution conference once in Texas. Oh, it must have been 20 or 25 years ago. I don't remember and he was giving a

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on the Higgs boson and the question was whether the Higgs boson is fundamental or is it composite and he was describing a theory in which the Higgs boson is composite and Eugene Wigner, the famous Eugene Wigner. who raises his hand and said vos means composite and I said that means uh, you know, things made of small pieces and so on, the arbor vos means composite andexplain the rollover again, there is enough, it doesn't mean compound, what compound means is that it can't be divided, we can't decompose, it can't be decomposed, I think it can't be decomposed, sorry, fundamental means it can't be ke and compound it means it can break down, you can split it, it can fall apart, so I asked them well, Eugene, do you think that means that the hydrogen atom is fundamental because you have a ground state?A hydrogen atom is essential because it cannot disintegrate. No, no, no, no, and we started talking about the proton. can't decay it's fundamental and the neutron isn't fundamental because it can't decay, the whole argument got really crazy, no one was making any progress in explaining what and everyone from his brother had some opinion on what composite and fundamental means and finally They pulled him out with hooves. who was always the most sensible person that these things stood up and simply said that something is fundamental when it is useful to think of it as fundamental and everyone shut up because they knew he was absolutely right that is now our fundamental strings there is a set of parameters in the theory where the strings are made of little things called d-branes, you change the parameters and you will find that the d-branes are made of strings, so it is a question of usefulness, there are ranges of parameters of the theory where it is useful to think on strings as the most fundamental objects, you can change the parameters and find parameter ranges where there are things called d-branes that had previously been thought of as big, fat composite objects similar to minor poles. they transform into tiny things and the strings themselves explode into large, thick compounds.

This is a lesson that I think physics has been teaching for some time now, the kind of continuous march of reduction of small things made of smaller things made of smaller things and some The ultimate sense of what is the most fundamental thing that does not seems to be the way things are going. The way things go is that things transform into each other when you change the parameters in the theory and what was fundamental can become composite. What was composed. It may be fundamental what the last lesson to be learned is, no one knows, but it is a pattern that has been emerging for the last 20 years, you know, both in quantum field theory that electrons are here and in the theory of ropes, so hopefully we can see something.

I think we are closed in open strings. I think what we are going to do is jump to a new topic which is the area of strengths, of course, but seen from a totally different angle than the angle that is called in theory. Okay, now I'm going to tell you what M-theory is and then how, in theory, what we don't have becomes string theory. We haven't talked about why 26 dimensions, why 10 dimensions, we can talk about that, but we can't talk about everything, so I thought I'd tell you what M theory is and how it relates to string theory.

Well, I think we're done with tonight. For more information, visit us at stanford.edu.

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