String Theory and M-Theory | Lecture 5
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Susskind, Leonard
(October 18, 2010) Professor Leonard Susskind delivers a lecture concerning plonck variables and how they relate to string theory in the context of modern physics. String theory (with its close relative, M-theory) is the basis for the most ambitious theories of the physical world. It has profoundly influenced our understanding of gravity, cosmology, and particle physics. In this course we will develop the basic theoretical and mathematical ideas, including the string-theoretic origin of gravity, the theory of extra dimensions of space, the connection between strings and black holes, the "landscape" of string theory, and the holographic principle.
 Stanford University. We haven't discussed units at all. And let's discuss units a little bit. Hadron physics, we've discussed the idea that a Hadron is a string and that it can be excited. That it can be excited by setting it into rotation or setting it into vibration, by exciting the harmonic oscillators that make up the stringy character of the proton or whatever it happens to be. There's a certain energy scale or a certain amount of energy that each excitation will give you. In particular, the energy jump from the ground state to the first excited state. How much is it? What does it depend on? Well, in a string theory, there's really only one parameter. We talked about it a little bit. We discussed the idea that if you stretch a string, let me just go back a little ways, if you stretch a string from one point to another, then it behaves pretty much like a spring. It does behave like a spring. It has an energy if this is a non-relativistic string, just an ordinary non-relativistic string, a rubber band, or an idealized rubber band. Its energy, its potential energy, let's call it E, what does Hooke's law give for the energy? Some k, which is some spring constant, times the separate times the square of the distance between the end points. Missing something, factor of two. Yeah, that's the potential energy that's in a string. And we've identified energy, this non-relativistic energy, we've identified it not with the mass of the string, but with the square of the mass of the string. Why? Because we're thinking about this in a frame of reference where the string is moving very, very fast, and when non-relativistic physics actually works for the motion of the string perpendicular to the direction of motion. So let's keep that in mind. We're looking at a string that's going down the z-axis like this. It's been stretched to a distance x, or let's call it L. Distance L, this becomes L squared. And we've identified the energy with the square of the mass, L squared. There is some spring constant. That spring constant, I'll just call k, it says that the mass of the string, which in the rest frame, in the frame of rest at which the string is at rest, that mass is the usual energy. That's going to be the square root of that spring constant times L, perhaps divided by the square root of 2. I'm not interested in the square roots of 2 now. So the mass of a stretched string is proportional to its length times some parameter, which I've called the square root of k. The square root of k has another name. It's called the tension. It's the tension in the string, the energy per unit length. Take a mass, and we have a string at rest. We're not looking at it in this frame in which it's moving fast. Its energy is its mass. We stretch it, we stretch it out, and the energy of it grows proportional to the length. It's like surface tension, except now it's linear tension instead of surface tension. Energy proportional to length, the coefficient is called tension, the tension in the string. And so there's a tension, which is just the square root of k. We don't even need to call it square root of k. We can just call it tension. Now, how much energy do we get for each oscillation? For each oscillation, oh, yeah, what are the units of a square root of k? Let's work in our favorite units in which the speed of light and plonks constant are equal to 1. In those units, energy has units of 1 over length. So if energy has units of 1 over length, what's the units of kappa here, of square root of kappa, the string tension? Yeah, it's energy squared, or 1 over length squared, energy squared, if you like. So this object over here, let's call it t, t for tension, that tension in the string is the thing which sets the fundamental scale. And it sets the scale for everything. It sets the scale and has units of energy squared. And I'll tell you what it does. It tells you that each excited state, each time you excite the string by one unit, it adds to its mass squared, essentially that tension. So the thing which has units, the thing which determines the units of the theory, are this string tension. The bigger the string tension, the bigger the jump in mass when you excite something. The same thing is true of an ordinary spring, incidentally. If you have an ordinary spring, let's take the mass of the spring to be 1, the spring constant, call it k. The frequency is just, well, it's the square root of k, k over m. The frequency is proportional to this tension. And of course, the energy that you bump up the spring by every time you excite it is also proportional to omega. So the energy jump, the energy jump between the ground state, the first excited state, next excited state, and so forth, is controlled, the unit is controlled by this string tension. OK, you could ask, what is the string, and string tension is force per unit, or sorry, is energy per unit length. You know another name for energy per unit length? Energy per unit length. Force. Force. Force times length is energy. Force times work, force times distance is work. Right. So all this tension is, if you pull this apart, that's how much force is pulling you back. That's how much force is pulling you back. It's the force within the string. Another way to say it is, if you took one of these strings at the surface of the earth, and you anchored it at some place and suspended away from it, it would be the weight that the string could support. A heavier weight would just shrink, or would fall down. A smaller weight would be pulled up. OK. That's the character of these strings. If you had one in the laboratory, the weight that it can support is independent of its length, incidentally. You see that from this formula. You see that from a formula that mass or energy, same thing, energy is equal to spring or tension times length. Force is energy per unit length. Energy per unit length is force. And so the force that this string can support is independent of its length. That's the character of these strings. OK. So then you could ask, how much weight at the surface of the earth, how much weight could a hadron, a meson, if you could somehow anchor one of the quarks in a meson to some support over here, the ceiling, how much weight could a meson support? The answer is about a truck. About a, I can't remember if it's an 18-wheel truck or a 16-wheel truck, or maybe it's just a half-ton panel truck. I don't remember, but it's that order of magnitude. That's what a meson could support. So they're pretty strong. They're microscopic little things, but they're pretty darn strong. And the stronger they are, the higher the frequency of the oscillations, and so the larger the gap between the lowest energy state and the highest energy state, how do I actually know this? Did anybody ever support a truck with a hadron? Of course not. What we actually know is the amount of energy that it takes to excite string. And from that, we deduce what the tension is. And from the tension, I can then tell you what you can support. These are not the strings that string theorists imagine are associated with really fundamental particles like gravitons, photons, and so forth. Those strings are much, much stronger. Very much stronger. The tension in them is vastly larger. If, again, you took the Earth and you supported a weight, namely the weight of the whole galaxy. Now, of course, this is nonsense. It's not the galaxy that's going to move. It's the Earth that's going to move. But if you could concentrate a mass of the galaxy in some small volume and make the Earth heavy enough, then you say, if I make the Earth heavier, I increase its gravitational field. Well, not necessarily so. We can make the Earth much bigger, but keep its gravitational field the same. If you could somehow keep the acceleration due to gravity the same, then the weight that you could support would be about the weight of the galaxy. So the strings that we're talking about in fundamental string theory are vastly higher in tension, much higher in tension. The meaning of that is that the energy in an excitation to excite the string is much, much larger. These are much stiffer strings. They have much, much larger spring constants. And therefore, much larger frequencies of oscillation, if larger frequencies of oscillation, then larger energies in exciting them. How much energy? Well, the thinking goes that it's probably somewhere up near the Planck energy. If we're talking about gravitons, it's the gravitational scale. Does everybody know what the Planck energy is? I know you've heard of it. Do you know actually what it is? First of all in numerically and second of all, what did actually where it comes from? Times c squared. That's a fairly big energy. It's enough to, that's a car bomb. Boom. How much was it that I didn't hear? How much did we didn't hear? 10 to the minus 5 grams. 10 to the minus 5 grams. And then you take that and multiply it by the square of the speed of light. But when I asked you if you know what the Planck energy is, or the Planck length, or the Planck whatever it is, I was asking a more theoretical question. I was asking, do you know what it is in terms of the other constants of nature? Maybe we should go through that. I mean, this is worth doing. Also, how long are these strings? OK, that's the other question. That's the other question. The stiffer they are, the smaller they are. Now, of course, you can stretch them out to any length. There's no limit to what you can stretch them to. But if you ask how much in their ground state, remember, things in their ground state oscillate. How much do they oscillate in their ground state? And what's the mean fluctuation in the size of these strings, the vibrations of them, then the larger the spring constant, the smaller they are. They're much stiffer, much harder to pull apart. Pulling this thing apart by one centimeter will cause a lot of it. The smaller, the bigger the spring constant is. So how big do people think they are? Oh, some ways of order of the Planck length. Planck mass, Planck length. How fast is one vibration? The Planck time. OK, so let's talk about these Planck units. There's some reason to believe it's a little bit smaller than that. Maybe a factor of 10, maybe a factor of 100 smaller, but that's a fine point. So let's talk about what the Planck length is. I didn't prepare this, so we're going to have to work it out in real time. What is the Planck length? What are Planck units, first of all? What are any units? In physics, we need to have three units, mass, length, and time. And our usual choice of units is the usual units that we use in the laboratory are not determined by any fundamental physics. The meter is a convenient unit for measuring rope. That's where it came from, measuring rope. You measure a rope like that, a cloth, or whatever it is. So no doubt it originated from the length of a human arm. Probably. A foot, of course, is also a similar kind of unit. And I assume it came from somebody's foot, the king's foot. So what's the real physics then? Or what's the real science of what a meter is? The real science is how many atoms does it take to make a length of an arm, a useful arm? There's nothing whatever to do with any fundamental physics. It has to do with biology. How long it takes to make an arm? Not a very deep unit. That's mass. Mass, same thing, a kilogram is a kind of weight that you can manipulate in the laboratory. And a second is about the time that you could measure with a pendulum. That's it. That's where they came from. And they have no deeper meaning than that. We would like, for really fundamental reasons in physics, to choose units which have some very fundamental meaning. Now, for example, and if we do, then many of our equations will be much simpler. For example, if the size of a proton comes into various equations in nuclear physics, and the size of a proton is 10 to the minus 13 centimeters, if we work in centimeters, or 10 to the minus 15th meters, there's going to be 10 to the minus 15th all over. And it's not just 10 to the minus 15th. It's 1.73498. And physics is going to be very messy. And the other hand, there's nothing to prevent us from saying, let's use the unit in which the radius of the proton is exactly 1. Then nuclear physics will turn out to look a little bit simpler. And it will. Nuclear physics will turn out to be a little bit simpler. But atomic physics won't look a hell of a lot simpler. The physics of sub-nuclear particles, sub-hidronic particles will not look a lot simpler. There's nothing universal about the proton. The proton is just some particle, that's all. Why use the proton instead of some other particle? No good reason. So there's nothing really universal about the proton. And the rules of physics will not in any way be especially simple if we use the proton. Time, same thing. So we would like three units which have some deep fundamental significance. It's equivalent to saying we would like to choose three constants of nature and set them equal to 1. It's completely equivalent to choosing three units to choose three constants of nature and set them equal to 1. Constants of nature now mean dimensional constants. The size of the proton is 1, but that's not a good one. You want to use things which are very universal, three constants. There are three constants in nature which are truly universal. And I'll tell you what they are. The first constant is the speed of light. Why do I say it's universal? Because there's a rule. And the rule is nothing can move faster than the speed of light, nothing. Everything move is every velocity of every material object is bounded by the speed of light. The use of the word every there tells you there's something fundamental about it. That there's something, whether it's protons or electrons or photons or anything else, they all are bounded by the speed of light. So there's something universal about the speed of light. Yeah? Well, I think that the constancy of the speed of light is constant. The constancy of it? Well, yes. The fact that it's the same in every reference frame and the fact that it's the same for all particles. Now, when I say it's the same for all particles, you say, wait a minute, wait a minute. Protons don't move with the speed of light. The limiting velocity for a proton is the speed of light. So in that sense, it's very universal. So that's the first thing. We want to set c equal to 1. Sounds like a good thing to do. And we do that all the time, of course. The next thing, which is very universal, is Planck's constant. Let me give you an example. Planck's constant is not just some any old constant like the radius of a proton, but it's connected with the uncertainty principle. For every object in nature, the uncertainty principle between the momentum and its position is greater than or equal to h bar. There may be some four or some two in there, I don't remember. But it doesn't matter what object you're talking about. You could be talking about an electron. You could be talking about a proton. You could be talking about a bowling ball. They are all constrained by the same uncertainty principle. And so in that sense, h bar, and it doesn't much matter whether you use h bar or h, the one without the 2 pi in it. That's not the point here. The point is that Planck's constant applies to everything in the world. It's universal. So the next thing you might want to use, not might, but we will, is to set h bar equal to 1. So what's the last really universal, is there another truly universal constant? The gravitational constant. Remember, according to Newton, every object in the universe, no matter what it's made out of, no matter what it is composed of, every object in the universe has a gravitational force, which is the product of the masses divided by the distance squared times the universal constant g. So again, something which applies to everything, are they units, are they units in which c, h bar, and g are equal to 1? Now that's the same as asking the question, can you combine c, h bar, and g into a combination that has units of mass? Can you combine them into a combination that has units of length? And can you combine them into a combination that has units of time? And the answer is yes. Let's see, let's do length. We want to do length. Tell you what, even better. Let's do length squared. Let's see if we can find a combination of g, h bar, and c, which has units of length squared. OK, so here's what we do. Just some simple dimensional analysis. Everybody here knows how to do dimensional analysis? OK, we'll do some simple dimensional analysis. We want to find a combination g to some power, let's call it p, h bar to some other power q, and c to some other power p, q, r, which has units of length, I'm going to choose length squared. Let's say it has units of length squared. Now, this doesn't mean it's equal to length squared. It means it has the same units. Sometimes people put a bracket around a thing like that to indicate its units. The units of g to the p, h to the q, c to the r, should be length squared. I'm choosing length squared to avoid a square root in the final formula, that's all. There is another reason. It is widely believed that the fundamental unit is a unit of area, not a unit of length. But we'll come to that some other time. But let's see if we can find p, q, and r. OK, first question is, what, now we have to figure out what the conventional units of g, h bar, and c are. The units of c, that's easy. What's the units of c? That's length over time, right? What about the units of Planck's constant? Momentum distance. L squared m over t. L squared m over t. Yeah. L squared is mass. Momentum times distance, right? Delta x, delta p. Delta x, delta p, so that's a length. And then a momentum is a mass times a velocity. And a velocity is a length over time. So that's length squared over time. Length squared mass over time, that's Planck's constant. And the last one, the one that I can never really remember, these two I remember, the one I can never remember is g. But how do we figure out what the units of g are? Acceleration equals mass times g times. Yeah, we use one of the equations that g appears in. All right, so here's an equation that, let's see. Which was your equation? Acceleration equals. All right, acceleration equals. Acceleration equals. Yeah, we'll get that in a minute, but that's equal to. Mass g over r squared. Mass g over r squared. So that's length squared. Acceleration is what? Length per time squared. And again, equal means, doesn't mean equal. So it's L cubed over t squared divided by m, it looks like. Right? OK, let's now go through the exercise. By the way, that's like Kepler's law, where the mountain. You're right. You're right. You're right. It is like Kepler's law. OK, so now we have on the left side, length squared. Now let's go g to the power of p. What's g again? So that's L cubed over t squared times m in the denominator. All right, so that's L to the 3p, t to the 2p, m to the p. Right? Do I get that right? That's g to the p. Now what about h bar to the q? h bar to the q is length squared, length to the 2q, m to the 2q. No, m to the q. And then t to the q? Do I get that right? OK. And then the last one is speed of light to the rth power, which is length to the r over time to the r. Now all we have to do is find three constants so that everything on the right-hand side cancels out except two powers of length. OK, so the first thing you can see is mass only appears in two places, mass to the q and mass to the p. So what does that say? It says p better be equal to q. So let's just set p equal to q. We now know that. And we can cancel out the masses. That's done. Now what else do we have? We have to get rid of all the times it looks like, huh? All right, so how many times do we have? We have 2p, 2p plus q is the same as p. So that's 3p plus r equals 0. 3p plus r is equal to 0. So that tells us that r is equal to minus 3p. Did I do that right? Let's just check. We have 2p, 3p plus r equals 0. r equals minus 3p. OK, I hope I'm not doing this right. Minus 3p. Did I make a mistake? Yeah, yeah, yeah, but q is p. We've already figured out the q is p. So if I'm not mistaken, I think we can get rid of all the t's, p is equal to q. So let's see what we have. We have p is equal to q. That looks like 5p. Boy, why not recognize that? 5p. L to the 2q. That's taken care of. And now r is what? r is equal to minus 3p. Yeah, and you're done. Oh. 3p minus 2, 5p is? Yeah, you guys, 5p minus 3p and you're done. 2p. We have 2p, right? We get 2p. All together 2p, 2p. And that now tells us what p is. p is equal to 1. OK? So let's see what we have now. p is equal to 1. It tells us that the plunk area, this is the plunk area. The plunk area L squared is g h bar looks like over c cubed, right? C cubed. C cubed. This is usually expressed by saying the plunk length is the square of this. OK. The square root of it. Thank you. Let's put in some numbers. My problem is I can never remember what plunk's constant is. Anybody know plunk's constant? 10 to the? 10 to the minus 34. Good. 10 to the minus 34? Yeah. OK. Yeah. g is 6 times 10 to the minus 11, which is about 10 to the minus 10. Order of magnitude, 10 to the minus 10. What is h bar? 10 to the minus 34? Yeah. 10 to the minus 34. 10 to the minus 34. In usual MKS units, 10 to the minus 34. And what is c cubed? C is 3 times 10 to the 8. 3 times 10 to the 17. Oh my god. How many? Times 10 to the minus, how big is it? 24. 10 to the minus 24? I think it's more like 10 to the minus 25, huh? Yeah. 10 to the minus 25. So how many square meters is this? Pretty small. Pretty small. Yeah. Some tiny, tiny number. OK. The plunk length is very small. If we had done it right, we would have gotten 10 to the minus 35 meters. So 10 to the minus, it should be 10 to the minus 70th square centimeters. I'm not sure I got everything right. About 70. All right. So that's small. That's smaller than anything. What's that? Good. OK. Now the plunk time. That's the plunk length is the square root of this. What about the plunk time? Think of the plunk length as being the size of a little thingy. There's a thing the size of the plunk length. What do you think the plunk time is? It's the time for a light ray to cross that thing. What else could it be? So it's the time for a light ray to cross that little distance. So does that mean we have to divide it by the speed of light? Yeah, we have to divide it by the speed of light. So the plunk length, L-plunk, is 10 to the minus 35 meters. The plunk time is 10 to the minus 43 or 42, something like that, 42 seconds? 10 to the minus 44, 43 seconds, something like that. Now what about the plunk mass? We haven't figured out what the plunk mass is, but we could do exactly the same calculation. Work out what the plunk mass is. From H bar. And the plunk mass would be about 10 to the minus 8 kilograms. So this one's big. That's a big mass. 10 to the minus 8 kilograms is a observable thing. It's a little dust grain. It's a little dust grain. These are impossibly small, small times and small distances. OK, this is a plunk unit. Things measured in these plunk units are measured in some very, very fundamental units. OK, I'll tell you some things. The universe has a radius. The known observable universe, the horizon size, is about 10 to the 60th plunk length. 10 to the 60th plunk length. The age of the universe is also about 10 to the 60th plunk times. And good, that's a gallon of, sorry, a tank of gasoline is about an energy content. A tank of gasoline is about the plunk mass. So that means you could drive across the country with 10 plunk masses of energy. Right, that's what it means. These are the units that string theory is in. The size of a vibrating string, the fluctuations in it due to quantum uncertainty, order of magnitude, the plunk length. The frequencies of oscillation or the time, not the frequency, but the period of oscillation of order 10 to the minus 43 centimeters. Sorry, sorry, 10 to the minus 43 seconds. Right, and the amount of mass that would be involved in exciting a fundamental particle such as a graviton or an electron. Let's take the electron. If the electron is a string, you could ask, how much energy does it take to excite it? The units are expected to be somewhere in this range, 10 to the minus 8 kilograms. Now, that's a huge number. How many GEVs, giga electron volts is that? One other useful number to remember is that one plunk mass is about 10 to the 19th GEVs, which means 10 to the 19th proton masses. One plunk mass is about 10 to the 19th proton masses. 10 to the 23rd proton mass is a little gram of water or something. 10 to the 19th, which is 4 orders of magnitude smaller, is a tiny, tiny little droplet, but visible, quite visible. Well, if you have good eyes. So exciting an electron by a particle physics collision. You say, well, we don't need to do a particle physics collision. We'll just explode a tank of gasoline in the vicinity of an electron, and it will start the electron vibrating. Yeah, it's just rather hard to get the energy to be concentrated in such a small distance. So these are the reasons that that. Is it above its ground state? Yeah, above its ground state. At its ground state? No. Well, remember it's mass squares which come in integer multiples. All right, so it's mass squares which come in integer multiples. So if you want to know what the unit is, square the plunk mass, and that's the amount that it takes to bump you up. But to go from the ground state, if the ground state is almost massless, then it's about one plunk unit of energy. That's why you can't excise the electron. Right. Right. It's just much, too much energy to pump into such a small volume. And for this reason, that string theory is way, way direct detection. How would you detect that a thing is a string? The same way you detected a proton as a string, you hit it, you set it into vibration, and you discover that there's a bunch of excited states where the mass squared is proportional to the angular momentum. That's the kind of thing you'd like to do with an electron. That's the kind of thing that you'd like to do, or be right with you, Michael. That's the kind of thing you'd like to do with a photon. But the increase in mass here is just way beyond what can be done. In order to have the structure of the string that we've been describing, wouldn't the string as a whole have to be somewhat bigger than this? Right. Mm-hmm. Yeah. Question? Question? This is just a bunch of arithmetic over here. What makes us think that when we get a number out, 10 to the minus 35th, that it has anything to do with the length of a string? It's a guess. It's a guess. It's a guess based on a number of things. No, no, it's not just, no, no. No, it's not just numerology. It could be wrong. It could be a thousand times less. Nobody thinks that the string length scale could be smaller than the Planck length. That looks quite impossible. But that the string length scale could be larger, and the energy scale could be lower. That is possible. How much lower? Well, I think from the things that we know about particle physics, we can't say with any precision at the current time really at all. But from the precision with which particle physics standard model and so forth seem to work, I think there's plenty of evidence that the energy scale is very, very high. Whether it's Planck scale or a hundredth of the Planck scale or a thousandth of the Planck scale, we don't know. There is another scale in physics. There is another scale in physics which keeps coming up over and over again. We've talked about it. It's the unification scale. If you remember it all from the last quarter, we talked about the scale at which the various coupling constants seem to come together. The scale at which the electro-weak forces and the QCD forces seem to merge into a common structure experimentally. And there, there are experiments. They're not experiments to get to that energy, but experiments to extrapolate. The extrapolation of what we know about physics seems to say that ordinary quantum field theory probably holds to something like a distance scale of roughly a thousand times larger than the Planck scale. So we do have evidence that ordinary quantum field theory without any stringy structures and so forth, or equivalently that the electron is fundamental to scales perhaps a thousand times smaller, sorry, a thousand times larger than the Planck scale. But that's still pretty small. Then you wouldn't be able to suspend the galaxy. Maybe you could only suspend, oh, I don't know, some globular cluster somewhere. It seems that a Planck-weft can be thought of as the fundamental unit of small stream of time. But certainly a Planck mass is not a fundamental unit of mass. No. No. Planck mass is believed to be the mass of the lightest possible black hole. One Planck area, one unit of entropy. And how long would it take to evaporate? One Planck time. Here today, gone. Today. You showed the coupling coefficients unifying. I believe one time you put down a fourth one for gravity, so that comes down. Is that especially? What's the coupling unit of gravity? No, that's okay. I'll tell you what the various pieces of evidence are. I'm not going to explain them in detail. This is just to remind you. If you plot the coupling constants of which coupling constants, the ones that we measure in the laboratory, the electric, let's just call it the electric coupling constant, E squared really, the weak interaction, sorry, the electoral weak, I'm getting, there are three of them. There's u1, su2, and su3. There are three of them. This is energy this way. And energy is the same as inverse length. Inverse length. So as energy gets large, wavelengths get short, energy, high energy means small distance. Okay. We measure these coupling constants at some energy scales. Remember, coupling constants are things which change with energy or change with distance scale. We measure them somewhere where in laboratories, which means 50 GeV, 100 GeV at most, we measure them here, we get three numbers. We also measure slightly indirectly the derivatives of these things. In fact, we can compute the derivatives of them. We can compute the derivatives of them purely theoretically, but with good confidence because we know the theory here very well. What we find is that if you extrapolated these, they would all cross at about a common point. That point, and that point is way, way out here at an energy which is not the Planck scale, but it's about 100 or maybe a thousand times, somewhere between 100 and 1000 times smaller than the Planck scale, I think maybe more like 100. So this could be M Planck roughly divided by 100, somewhere in between. And the Planck scale would be over here. Or this is a logarithmic scale. On a logarithmic scale, it looks like this. So on a logarithmic scale, a given gap can mean a very, very big change in energy. And it looks like this. Now, of course, we don't know with certainty that there's not all kinds of things going on at higher energy here which would muck this up badly. We don't. But what we do know is that if there is nothing in here to muck things up badly, and there's some reason to believe that, that these coupling constants would just come together at the scale. The meaning of that is that quantum field theory, conventional quantum field theory, seems to be working to about here. There's other evidence for this scale coming from neutrino masses and a few other places. But it's far from tight. It will become much tighter if LHC is lucky and makes the right discoveries. It will become much tighter and will have a much better knowledge about this extrapolation. So this is one of the things we're going to learn from LHC, is how reliable this extrapolation is. Planck mass being out here. And this in itself doesn't tell us where the string length scale is. What it tells us is that the stringy character of particles does not become important until energy is above this. So somewhere is between here and here. If string theory is all right, somewhere is in here at the scale of Schmitts. Remind me again what the three coupling constants are. There's what is it? QED, QCD, and gravity? No. QED, QCD, and the weak interactions. A weak interaction. At one point you did put down, I think you put down something for gravity. Yeah. Yeah. If you were to plot the gravitational coupling constant behaves differently, has a different character. But let's see. It starts very, very small. And it goes up something like this. And it would cross. It doesn't cross at the same point. It crosses somewhere as nearby. But basically the place where this one gets to be about order one, that's the Planck scale. So it's really exactly the same statement to say that the gravitational constant, the running coupling constant, crosses at roughly the same place as to say this unification scale is not too far from the Planck scale. Is that the Planck capital G? Yes. So does capital G change as you start to measure the higher and higher interactions? No. Capital G does not change. But capital G is not exactly the right measure of things. It's capital G. All right. Let's do, I'll answer your question. Let's look at the force law between two objects. In electromagnetism it's E squared divided by R squared. And that's the force. And that's the electromagnetic coupling constant here. In gravitation it's G M squared over R squared. So the thing which is analogous to E squared is G times M squared. But what M should you use there? Well, the M that you use there is associated with the particular scale of physics that you're studying. All right. So you can see then in that sense this quantity here increases like M squared. Yeah. So this is the relevant thing here. It's a dimensionless thing. OK. That's units. Where on that scale if you could build accelerators big enough is that they start producing nothing but little black holes. Real black holes? Yeah. So the small ones, you know, if you build galaxies. Oh, it's all the way out there. The clock mass. Yeah. There's two ways to produce black holes. One way or another you've got to have a lot of pressure to squeeze things into a small volume. One way is kinetic energy. Small objects with lots of kinetic energy blast them together and you have a chance of making a black hole. The other way is not kinetic energy but gravitation. Gravitational potential energy pulling things together. That takes a star. So to make a black hole using gravity you need something as heavy as a star. To make a black hole by the collision of particles you need energies of order of the plunk energy. Now if you think about that for a minute, imagine an accelerator of a, I don't know, is it fun to, let's explore the following question. How big, what kind of accelerator would it take to do experiments of the plunk energy? To do interesting experiments of the plunk energy? Well, basically the energy of an accelerator, of a linear accelerator, scales linearly with the size of the accelerator. So you get a certain number of GEV per unit length. Slack is a good example. It takes approximately two miles to accelerate a thing to a hundred GEV. So two miles for a hundred GEV. Two miles, one hundred GEV. The plunk energy is about ten to the seventeenth times bigger than this. It's ten to the nineteenth GEV, so it's ten to the seventeenth times larger. So with just the parameters of the Slack accelerator, you can multiply this by roughly ten to the seventeenth. So it would take ten to the seventeenth miles. How big is ten to the seventeenth miles? Oh man, ten to the seven, ten to the, I think ten to the thirteenth miles is about a light year. Ten to the thirteenth kilometers or something like that, if I remember. Ten to the thirteenth kilometers, ten to the seventeenth, ten to the fourth light years. Galaxy size, galaxy size to accelerate something up the plunk mass. We're not going to do it, at least not tomorrow. But suppose you could build such an accelerator. Next question, what kind of luminosity do you have to have? An accelerator has to have a luminosity. The chances of, what we want to do now is we want to collide particles head on to make a black hole. But remember, the radius of that black hole that we want to make is itself the plunk mass, the plunk radius. So that means we've got to aim these particles to collide within a plunk distance. Well, accelerators don't do that. That's not the way accelerators work. The way accelerators work is you just get a lot of particles going to the left, a lot of particles going to the right. And you get enough of them so that some will collide at the distance scales that you're interested in. In other words, you need big luminosity. What kind of luminosity would you need? Oh, I don't know, let's say, what's a slack luminosity? 10 to the 13th particles per second or something, anybody know? I don't know. Well, 10 to the 13th particles, but that's just, but you know, the distance scales there are pretty big. How much better are you going to have to do? I don't know, let's say 10 to the 20th particles. 10 to the 20th particles colliding, each with a plunk energy. 10 to the 20th tanks of gasoline per second. Now, yeah, 10 to the 17th barrels of gas per second to fuel it as big as the galaxy, we ain't going to do it. That's what we would need to explore small black holes and something pretty close to it to explore string theory directly, a really direct test. The question is, are there indirect tests? All right, I think the indirect tests will pretty much be very theoretical. And unless somebody constructs a very, very convincing string theory that gives rise to the standard model exactly in some very computable way, the hope of direct tests, I think, is very remote. So that's what we're facing, that's what we're up against. Okay, we didn't get where I really had intended to go. In fact, we didn't even touch on what I intended to touch on. Especially these extra dimensions that you had talked about, aren't they in the range of the tanks in this third minus 25? Or which dimensions? The extra dimensions of the... Yes, it is believed, again, without really terrific evidence. But again, it's the same kind of evidence. It's the same, where is our plot of coupling constants. If there were extra dimensions, and the extra dimensions had any sizable distance, if they were at all sizable, then ordinary quantum field theory would break down when particles begin exploring the extra dimensions. The conventional use of quantum field theory, as we use it, would break down at points where the energy became large enough to explore these extra tiny dimensions. If it's true, and if we continue to get evidence that quantum field theory makes sense as it is out to these distances here, then we'll know that the extra dimensions must lie somewhere between the Planck length and maybe a thousand times bigger than the Planck length. So exploring extra dimensions will also be something that is extremely indirect and extremely hard. Why couldn't it be significantly... or is it magnitude less than the plot point? What's the argument? Or is it magnitude... oh, that's another question. That's another question. The theory really doesn't work at all under those circumstances. Let's put it this way. If you tried to make them smaller than the Planck length, you would wind up making their mass larger than the Planck mass. Making a small thing with big mass means you make a black hole. So if you're trying to think of strings which were much smaller than the Planck distance, and their mass would accordingly be larger than the Planck mass, you would be talking about black holes. You wouldn't be talking about strings. That's the problem. They would collapse. The mass and their small distance would cause gravitation basically to turn them into black holes. So that's the reason that string theorists do not ever consider the possibility that the string scale is smaller than the Planck mass. And I think it's a very good reason. Did you say that the mass here is derived from the choice for L and T? From your what? Well, you gave us why we have an L and why we have a T. Yeah, yeah, but you do exactly the same thing. You say, let's make a mass squared out of these parameters here. And I forget exactly what it is. We could figure it out. But the others were a physical choice. Sorry? The other length of time were a physical choice. What do you mean they're physical? They're all physical? Well, yeah, but I mean the time, for example, is the time to cross L and A. And that's a physical quantity. Well, each one is a physical quantity, too. And what do you say about mass? You just put it down. Does it follow from our choices from L and T? Yes, it does. Well, no, it doesn't follow by itself from L and T. But if you asked, ah, great, let's see. Once you have L and T, you can get mass from either H, R, or G. So that is the way to think about it. Take two particles whose distance apart is the plonk length. Now, because their distance is so small, the uncertainty principle says that they have a lot of energy. The uncertainty principle says if things are very well localized, then they have a very large momentum. The fluctuations in the momentum are very large. So take two particles whose distance is known to be within the plonk distance. And that says something about their momentum. Their average momentum is extremely large, and from that you can compute what their kinetic energy of these two particles are. That kinetic energy will be the plonk mass. So two particles, if you try to take two particles and localize them to a distance comparable to the plonk distance, the answer will be that the energy that you have to put in to do that will be the plonk mass. So it's just uncertainty principle. We can work it out, but, you know, okay. Let's take a rest. I'm going to tell you next about some mathematics. I'm going to tell you about the mathematics of conformal transformations. Why? Because that's the basic mathematical tool of string theory. Well, we have our choice. I can either tell you why there are 24 or 26 dimensions in string theory, or I can start to tell you about conformal transformations. We'll get to both of them at some point, but... We haven't clapped the thing together yet. Yeah, as long as it's not on Thanksgiving. Thanksgiving is always Thursday, so I think the answer is yes. Okay, we have 45 minutes, and I will try in 45 minutes to give you the easy version of why there have to be 26 dimensions in bosonic string theory. The easy version is not easy, and it doesn't really get to the root of it. The hard version is very hard. But that doesn't mean that we can't talk about it qualitatively, but I thought I would do what I call the easy version, which is not easy, and not satisfying, but nevertheless is correct. When people see this, they walk away and say, gee, that's sort of magic, but in the bad sense. But it is correct, and it is sort of the way that the need for extra dimensions was discovered. It has to do, if you remember, in order to get the photon to come out right, we had to do something odd. We had to take the mass squared of the ground state of the oscillating string. Remember, let's take the open string, take in the case of the open string. The mass squared had to be minus one unit. That was so that when we applied a creation operator to create the photon polarization, it came out massless. Minus one, one unit on top of that, gives us something massless. We haven't talked yet about how we get rid of this particle of negative mass squared. I assure you that can be gotten rid of. But let's not worry about it. Let's agree. The mass squared of the ground state is minus one. Why is it minus one? How do we get minus one? Well, harmonic oscillators have zero point energy. So the ground state energy of this vibrating string should be nothing but the sum total of all of the zero point energies of the vibrating string. Zero point energies are all positive. And in fact, they depend on the frequency. Anybody remember what the zero point energy of a harmonic oscillator is? A half h bar omega. Let's forget h bar, set it equal to one. One half omega, and do you remember the frequency of the nth harmonic oscillator? It was just n. Each one of the oscillating modes of the string, each one of the oscillating modes, the nth oscillating mode has frequency n. So the ground state energy of the nth oscillator is just n over two, in some units, is n over two. Now this doesn't sound good. We have a lot of oscillators, one for each integer. We have to add up all that energy. It's not quite easy to imagine how we can add all this up and get minus one. So there's something strange going on. But actually we don't have to get minus one. Well, we have to get to something a little bit different. Let's go back to the energy of a very, very fast moving system. The energy of a very, very fast moving system has, first of all, it has the overall momentum. Moving down the z-axis, it has the overall momentum. And then on top of the overall, and that's a conserved quantity. And we really don't care about it very much, since it never changes during the processes that take place in particle physics. And then for a particle which is not moving, whose center of mass is not moving in the plane, what's the rest of it? The mass squared divided, I think, by twice the momentum. And that's why we began to identify the energy, the non-relativistic energy, with m squared rather than m. This was the energy, and to make precise this idea of very rapid systems which become non-relativistic, you have to let p become extremely large. Basically you have to take the limit of an infinitely large momentum. In the limit of infinitely large momentum, this is the form that the energy takes. You say, well, I'm going to take the form of infinitely large energy. This thing just goes to zero. But what you really do is you take e minus p, you subtract the momentum from the energy. That's all right. This is a perfectly conserved quantity. It never changes. If it never changes, you don't care very much about it. And then multiply the whole thing by, let's say, 2p, and that gives you m squared. This 2p here, the reason you multiply by 2p, is that's got to do with time dilation. The faster a thing moves down the z-axis, the slower its internal motions go. The slower the internal motions, that means the smaller the internal energies. For example, if you have an atom, what's the ionization energy of a hydrogen atom? 13.5 electron volts or something. Now you take that atom and you speed it up and you send it down. In other words, the energy difference between the ground state of the atom, let's not take the ionization energy, the energy difference between the ground state of a hydrogen atom and the first excited state is what? Oh, 3 or 4 volts, something like that. 3 or 4 electron volts. Now what happens as you boost the atom? What happens to the overall energy of it? It increases, of course. Right? That's this. But what happens to the energy difference between the first excited state and the ground state and the first excited state? Does it get bigger or smaller? Smaller. And it gets smaller with this one over P here. And in fact, the energy of that atom or the differences of energy between the atoms are proportional to the square of the mass of the atom divided by 2P. So that's what this formula is. Well, we don't care about this piece. That just has to do with its overall motion down the z-axis. Just subtract it off. It doesn't have to do with the internal motions. And then the P in the denominator here, that's just associated with the slowing down of the internal motions. So let's just multiply through by it. And this is the way we really think of energy. This quantity is the energy and it's proportional to the mass squared. Okay, the point is now that the actual energy of the ground state has an infinite piece, a piece we, or let's put this piece back. Let's put, let's not subtract this. Let's add this back on the side. Plus, what is it? Plus P squared, plus 2P squared. Let me just get this right. Let's see. Let's take P times E is M squared plus 2P squared. Okay? Notice that first of all is the interesting thing which is the mass squared, which we're interested in. But there's also this infinite piece here, this infinite piece. So it's not really quite true that the energy has to add up to minus one. It would be okay if it added up to something like minus one plus an infinite constant that was similar to P squared. So I'm going to show you how that works. I'm going to show you how that works, how you can add up one plus two plus three plus four plus five. This is all the internal energy and get something which looks like a constant, namely minus one, which is what we want to get, plus an infinite piece that can be absorbed into something that's already there. This is a trick of physics that takes place all the time. It happens all the time in all kinds of contexts where you get some infinite answer and you don't know what to do with the infinite answer, but you realize that the thing which is becoming infinite already has a constant piece in it that you just add the extra infinity to the constant piece. An example is the energy of vacuum, the energy of just a mass of a particle or a self-energy, all sorts of things like that, where you can hide infinite constants into things which are already there. I'm not saying this is a satisfactory state of affairs. It's just something that we do all the time. The energy in this room, it has zero point oscillations from all of the photons and everything else, not from the photons that are here, but from the oscillations of the electromagnetic field. How big are those oscillations? The energy? Infinite. What do we do with it? We just subtract it off by saying we could add a constant to the energy of the room and it wouldn't make any difference, so just get rid of it. The same thing here. We're going to get a constant piece which is just infinite but constant. As long as it's constant, it never affects anything. Plus a term which looks like this with an additional mass squared. Let me show you the mathematics and we can then discuss another term, the physics. The mathematics is tricky but easy. The mathematics is just a question of adding up 1 plus 2 plus 3 plus 4. It's just a sum here. 1 plus 2 plus 3 plus 4. There's an overall factor of a half outside that comes from the half h bar omega, but the basic calculation is to add up all of the integers. Obviously infinite. Let's see if we can extract out of that infinite piece an infinity plus something finite. This is always the way quantum field theory and relativistic quantum mechanics works. 1 plus 2 plus 3 plus 4, this is not something we're going to easily add up, is it? Let's do something to it and then take a limit. Let's do something to it so that it makes sense and then take a limit. Let's multiply each one of these. Let's introduce a small constant, epsilon. E to the minus epsilon is a number less than 1. But when epsilon is very, very small, this number is close to 1. Let's add to it plus 2 E to the minus 2 epsilon. This is just a trick for making the sum converge. Each term in it, we're going to put in something which gets smaller and smaller, but when epsilon goes to 0, it'll just give us back the original sum. So epsilon is going to go to 0. In fact, you know what? Well, yeah, all right, let's leave it this way. 1 plus 2 epsilon, what's the next one? 3 E to the minus 3 epsilon plus 4 E to the minus 4 epsilon. But now, if epsilon is not 0, then the sequence of terms E to the minus epsilon, E to the minus 2 epsilon, E to the minus 3 epsilon do get smaller and smaller. Now, if epsilon is small, it takes a long, long time for them to shrink. So if we were just to plot the E to the minus epsilon, E to the minus 2 epsilon, as a function of the integers, it would fall very slowly if epsilon is a small number. But eventually, the E to the minus N epsilon will win, and it will make each term successively smaller and smaller, and you'll be able to add them up. So what you do is you add these up, and then after adding them up, you let epsilon go to 0. It's a mathematical trick. Let's see what we get. It takes a bunch of steps, and I'll show you, but the steps are elementary. Okay, this can be written as the sum from N equals 1 to infinity of N, E to the minus N epsilon. E to the minus 1 epsilon times 1, E to the minus 2 epsilon times 2, and so forth and so on. That's the sum. Now, the trick is to get rid of this N. First trick is to get rid of this N, and the way to get rid of this N is just to take the sum without the N in it and differentiate with respect to epsilon. What happens if we differentiate with respect to epsilon? What does a derivative with respect to epsilon give? It brings down minus N. So this is equal to minus, we really want to put minus here, this is equal to the original sum we were interested in, N, E to the minus N epsilon. That's the first step. Next step is to observe that we can add the series up. This whole series is a geometric series. It starts out, let's just see what it is. It has an E to the minus epsilon plus an E to the minus 2 epsilon plus an E to the minus 3 epsilon, which also can be written in another form. It's E to the minus epsilon times 1 plus E to the minus epsilon plus E to the minus 2 epsilon and so forth. I pulled out a factor of E to the minus epsilon. Do you know what this is? Exactly. This is a geometric series, 1 plus a number plus the square of the number plus the cube of the number. And so we can add all of this up. And what do we get? We get E to the minus epsilon divided by 1 minus E to the minus epsilon. That's the thing that we want to differentiate with respect to epsilon. That's this. I put a minus sign in front of it. This is something you can do at homework, but I'm going to show you roughly what happens. But in the end, we're interested in small epsilon. Let's expand it for small epsilon. Let's first expand it for small epsilon and then do the operation and then epsilon go to zero. We're interested in small epsilon. So what is E to the minus epsilon when expanded in terms of epsilon? 1 minus epsilon plus epsilon squared over 2, we're only going to need things to power epsilon squared. We won't need things beyond that. And of course, there are more, but we won't need them. Now what about the denominator? What's in the denominator? We have 1 minus, okay, so let's write out E to the minus epsilon minus 1. I think it's plus epsilon, minus epsilon squared. Let's go to the next one. I think we're going to need the next one, as you'll see. Epsilon squared over 2, right? What's the next one? Let me just check the sign. 1 minus 1 plus epsilon minus epsilon squared over 2 in the next one. Epsilon cubed over what? Epsilon cubed over 6, right? Now we have trouble. The trouble is the ones cancel. The denominator is proportional to epsilon. Let's factor out one power of epsilon. Let's see. This is the object whose derivative we want to take. Let's factor out a factor of epsilon in the denominator. Since everything in the denominator has epsilon, let's factor it out, 1 over epsilon, and then this becomes 1 minus epsilon plus epsilon squared. Did everybody follow that? Okay. Plus higher terms, but as I said, the higher terms are going to disappear when we go to small epsilon. They won't be important. Was it a minus sign? No, it was a minus sign. 1 minus the 1d1 sub-skeleton. Well, I thought I got it right. All right, should we do it over? 1 minus, put in brackets, 1 minus epsilon plus epsilon squared over 2 minus epsilon cubed over 6. All right, everybody happy with that? Now the ones are going to cancel. And the minus times the minus will make plus minus plus. They have no idea if that's what I had written down before or not. Then we pull out a factor of epsilon, and that turns this into 1 minus epsilon over 2 plus epsilon squared over 6. All right, now the trick is we're expanding everything in powers of epsilon. But here we have a thing in the denominator. I don't want this in the denominator. I want in the numerator. So how do you deal with a thing in the denominator like this? You use the formula that 1 minus a small number, let's just call it s, is equal to 1 plus s plus s squared plus s cubed and so forth. Here's your s. Oh, did I leave out something? No, I think I got it right, didn't I? I think I did everything right. I think I did everything right. So what we do is we take what's in the denominator here, and we expand it to get all the epsilon dependence in the numerator. Let's see what we get. I think we're going to get 1 plus epsilon over 2 minus epsilon squared over 6. But then there's a term coming from squaring the small quantity here. If I keep things to order epsilon squared, all I'm going to get is, let's see, I think plus epsilon squared over 4, I think. But you can expand it out yourself, something like this. Plus higher order in epsilon. No, I think it was epsilon squared. It was just epsilon squared, not epsilon cubed. No, the thing which was epsilon cubed in the denominator here became epsilon squared when I pulled out the epsilon here. And I was purposefully keeping everything to power's order epsilon squared. I know that the next order is not important. OK, so let's see if we can combine this together. I think the chances that I'll get this right are negligible. But all right, so we get minus d by d epsilon, 1 over epsilon times, OK, 1 times 1. Let's put everything that you get from 1. Plus epsilon over 2 minus epsilon squared over 6. Oh, can we combine these two together? What's epsilon squared over 4 minus epsilon squared over 6? Epsilon squared over 12. Right? OK. How did you do that so fast? Who did it? You know? What? Took me about 30 minutes to get that epsilon squared over 12. OK. I know how to do it. 1 plus epsilon minus epsilon, sorry, 1 plus epsilon squared over 12, right? That's this, this, this, and this. Then we get minus epsilon minus epsilon times 1. And then minus epsilon squared over 2. But then from here we get something of order epsilon cubed. And I'm not interested in keeping anything of order epsilon cubed. That's too high an order. I don't care about it. We'll see why in a moment. And then we have plus epsilon squared over 2. And nothing past that because the next one would be epsilon fourth. I think this is everything to order epsilon squared. Should it be plus? No? I think it should be plus, right? Yeah. Right. Looks right. 1 times epsilon squared over 12. OK. Now, epsilon over 2 minus epsilon, that's minus epsilon over 2. I'll get rid of this and these two cancel. So there's our formula right there. As you'll see, anything of higher order is not going to be interesting to us. OK. Look at, let's first focus on the one that's easiest. The easiest term is this epsilon over 2. It gets multiplied by 1 over epsilon. That means it's just 1 half. And what happens when you differentiate 1 half with respect to epsilon is 0. So this is, we don't have to write it. It's not there. OK. Now we have a term which is 1 over epsilon. That's bad news. This is something infinite at epsilon equals 0. And remember, we're going to go to epsilon equals 0. All right. What's the derivative of 1 over epsilon? Minus 1 over epsilon squared, right? Minus 1 over epsilon squared. Oh, and there's another minus sign here. So that's plus 1 over epsilon squared, I think. Yes. And then 1 over epsilon times epsilon, one of the epsilon's cancel, and then you differentiate with respect to epsilon. What do you get? 1 over 1 minus 1, 12. This is a famous formula which is usually stated half jokingly that 1 plus 2 plus 3 plus 4 plus 5 is equal to minus a 12. We tend to forget about this term over here. And the reason is that this term combines, well, this term is an infinite constant which really doesn't, I would have to convince you that where it, what happens to it, it gets absorbed into the p squared, the p squared term in the energy, the infinite term in the energy. And a careful analysis, a really careful analysis of it, allows you to absorb this into an additive constant in the energy that doesn't do anything. For the moment, you'll have to accept that. The real answer, the really right answer, the really right answer is that in good string theories, properly defined string theories, this isn't there altogether. But I'm not going to tell you why now. I'm not going to tell you why now. I'm just going to tell you this infinite constant is not important. I know the famous story of Dirac and Pauli, that Dirac calculated the vacuum energy and found out that it was infinite. And they said, well, since it's infinite, I don't care about it, we'll throw it away and Pauli turned around to him and said, just because it's infinite doesn't mean it's zero. Right. In this particular case, Dirac wins. Dirac wins, but this is something we will have to come back to and explain why this one over epsilon squared. Remember, epsilon is going to go to zero. This is going to be an infinite constant. And in this particular case, the infinite constant really is not important. And I'm not going to pursue it right now. I'm going to tell you the answer is minus a 12. Now, this is not yet a particularly good answer. This is not a particularly good answer yet. Remember, what we wanted to get was altogether minus 1. We wanted to get this minus 1, which when we excited by one unit of energy will give us the massless particle. In fact, the minus a 12 is not quite right. And the reason is because of this factor of a half from the 1.5 H bar omega. So it's actually 1 over 24. 1 over 24. That minus 1 over 24 is the energy of the ground state, including the zero point oscillations, but throwing away a certain infinite term, and I have to explain it way later. But, okay. How do we get, oh, have I left anything out? I left out one thing. Remember, for each mode, there is not one oscillator but two oscillators. The a's and the b's. Remember the a's and the b's? The x oscillators and the y oscillators. So actually, it's twice this. Twice this, one for the x oscillators, one for the y oscillators. It's still not good. It's only 2 over minus 24. We really want to get minus 1. How do you get minus 1? You say there weren't two dimensions of oscillation. There were 24 of them. 24 directions of oscillation plus the direction going down the z-axis, plus time make 26. That's where the 26 is. Now, this is a crazy story. At the time, nobody believed it that this is a little bit too silly to be true, but in fact, the mathematics did fit together. Mathematics. And this is the simplest way. This, which is fairly complicated already, is the simplest way to see. And it is historically the way it came about that requiring, let's say there were d dimensions of space perpendicular to the direction of motion, and requiring this to come out to be minus 1, just so that when you excite it, you can get the photon, that was the original argument about d having to be 24, or the total space time dimension having to be 26. Now, as I said, the arguments, this was hardly a convincing argument at the time. It's virtue is that it's easy to present, relatively easy to present. What is the history? What was the history of this introduction? How did this come about? Yeah, well, I was just wondering, 1990s or... Oh, God, no. 1969. Yeah. No, 69. No, no. No, this goes back to 1969, I would say. Maybe early 70, but I think it's still 69. So it's for a long time? Yes. Well, by this point, people were just exploring the mathematics of string theory. And they had... So you did have a regi trajectory? Well, experimentally, you had the regi trajectories. But at some point, when the mathematical structure was put in place, people began to explore it for its own sake. It was realized rather quickly that there was some kind of funny thing going on that there was a spin one particle, one unit up, one unit up from it, but it didn't have all the states that it needed. It had two polarizations and not three polarizations. And it was realized rather quickly that this could only be the photon or something like a photon, but that then left the question, what was this zero point energy that had to be minus one? And it was realized fairly quickly that that required 24 dimensions of oscillation. Now, as I said, this was by no means a convincing argument. Other much, much, much more convincing arguments came about, but they were highly mathematical, really highly mathematical. And I'll tell you about them in words, but not now. We need one more mathematical concept. Is there a similar trick to get eight? Yes, eight. Yes, yes, there is. It's a similar trick to get eight. But before I tell you that, let me go back to the closed string. Remember the closed string, you'll have to excite twice. You'll have to get spin two. You'll have to hit it with two oscillators. That meant that the ground state had to have minus two units of energy. But now we have a problem. d over 24, if d is 24, is only minus one. What's the answer? 48 is minus 48. 48 is minus one. Why 48? Because you get a minus two, one-twelfth, right? You get the minus two. Yeah, but wait a minute. I know, but you only have 24 directions. We can't change the number of directions in space when you go from closed strings. No, no, no, the same theory, the same theory has both closed and open strings. You can't change that. Right, you can't do that. No, we'll agree. The theory has 24 dimensions. Now what happens to the closed string? Secular parallel. Secular parallel, alright. Well, something like that. Remember, you have twice as many oscillators for the closed strings. You've got the ones that go around to the left, and you've got the ones that go around to the right. Everything gets doubled once again. You have not only an oscillation for every direction of space, but you have one for right-going modes and one for left-going modes. So in fact, it's true that you get twice as many, which means that the ground state energy is minus 2, which is just exactly the right thing so that two units of oscillation will bring you up to massless again. That began to smell better. It began to smell better, but still far from a convincing argument. The real thing has to do with something called conformal invariance. So we're going to take up conformal invariance next time. What conformal invariance means? If you want to prepare yourself, learn a little bit about complex function theory. Very little, not much. We're going to do a very little bit. Learn what the Cauchy-Riemann equations are. If not, I will explain it. And we will come to why string theory is the theory of Turkish taffy. You know what Turkish taffy? You pull it this way, you pull it that way, you stretch it out that way, you stick it together, and somehow it all stays together. It's conformal transformations. Anyway, I think we're finished for tonight. Cauchy-Riemann equations. Cauchy, Cauchy. Just learn a little bit about them.
 Stanford University.  We haven't discussed units at all.  And let's discuss units a little bit.  Hadron physics, we've discussed the idea that a  Hadron is a string and that it can be excited.  That it can be excited by setting it into rotation or  setting it into vibration, by exciting the harmonic  oscillators that make up the stringy character of the  proton or whatever it happens to be.  There's a certain energy scale or a certain amount of  energy that each excitation will give you.  In particular, the energy jump from the ground state to  the first excited state.  How much is it?  What does it depend on?  Well, in a string theory, there's really only one  parameter.  We talked about it a little bit.  We discussed the idea that if you stretch a string, let me  just go back a little ways, if you stretch a string from  one point to another, then it behaves pretty much like a  spring.  It does behave like a spring.  It has an energy if this is a non-relativistic string, just  an ordinary non-relativistic string, a rubber band, or  an idealized rubber band.  Its energy, its potential energy, let's call it E, what  does Hooke's law give for the energy?  Some k, which is some spring constant, times the  separate times the square of the distance between the end  points.  Missing something, factor of two.  Yeah, that's the potential energy that's in a string.  And we've identified energy, this non-relativistic energy,  we've identified it not with the mass of the string, but  with the square of the mass of the string.  Why?  Because we're thinking about this in a frame of reference  where the string is moving very, very fast, and when  non-relativistic physics actually works for the motion of  the string perpendicular to the direction of motion.  So let's keep that in mind.  We're looking at a string that's going down the z-axis  like this.  It's been stretched to a distance x, or let's call it L.  Distance L, this becomes L squared.  And we've identified the energy with the square of the  mass, L squared.  There is some spring constant.  That spring constant, I'll just call k, it says that the  mass of the string, which in the rest frame, in the  frame of rest at which the string is at rest, that mass  is the usual energy.  That's going to be the square root of that spring constant  times L, perhaps divided by the square root of 2.  I'm not interested in the square roots of 2 now.  So the mass of a stretched string is proportional to its  length times some parameter, which I've called the  square root of k.  The square root of k has another name.  It's called the tension.  It's the tension in the string, the energy per unit length.  Take a mass, and we have a string at rest.  We're not looking at it in this frame in which it's  moving fast.  Its energy is its mass.  We stretch it, we stretch it out, and the energy of it  grows proportional to the length.  It's like surface tension, except now it's linear tension  instead of surface tension.  Energy proportional to length, the coefficient is called  tension, the tension in the string.  And so there's a tension, which is just the square root of  k.  We don't even need to call it square root of k.  We can just call it tension.  Now, how much energy do we get for each oscillation?  For each oscillation, oh, yeah, what are the units of a  square root of k?  Let's work in our favorite units in which the speed of  light and plonks constant are equal to 1.  In those units, energy has units of 1 over length.  So if energy has units of 1 over length, what's the units  of kappa here, of square root of kappa, the string tension?  Yeah, it's energy squared, or 1 over length squared, energy  squared, if you like.  So this object over here, let's call it t, t for tension,  that tension in the string is the thing which sets the  fundamental scale.  And it sets the scale for everything.  It sets the scale and has units of energy squared.  And I'll tell you what it does.  It tells you that each excited state, each time you excite  the string by one unit, it adds to its mass squared,  essentially that tension.  So the thing which has units, the thing which determines the  units of the theory, are this string tension.  The bigger the string tension, the bigger the jump in mass  when you excite something.  The same thing is true of an ordinary spring, incidentally.  If you have an ordinary spring, let's take the mass of the  spring to be 1, the spring constant, call it k.  The frequency is just, well, it's the square root of k, k  over m.  The frequency is proportional to this tension.  And of course, the energy that you bump up the spring by  every time you excite it is also proportional to omega.  So the energy jump, the energy jump between the ground state,  the first excited state, next excited state, and so forth,  is controlled, the unit is controlled by this string  tension.  OK, you could ask, what is the string, and string  tension is force per unit, or sorry, is energy per unit  length.  You know another name for energy per unit length?  Energy per unit length.  Force.  Force.  Force times length is energy.  Force times work, force times distance is work.  Right.  So all this tension is, if you pull this apart, that's how  much force is pulling you back.  That's how much force is pulling you back.  It's the force within the string.  Another way to say it is, if you took one of these strings  at the surface of the earth, and you anchored it at some  place and suspended away from it, it would be the weight that  the string could support.  A heavier weight would just shrink,  or would fall down.  A smaller weight would be pulled up.  OK.  That's the character of these strings.  If you had one in the laboratory, the weight that it  can support is independent of its length, incidentally.  You see that from this formula.  You see that from a formula that mass or energy, same thing,  energy is equal to spring or tension times length.  Force is energy per unit length.  Energy per unit length is force.  And so the force that this string can support is  independent of its length.  That's the character of these strings.  OK.  So then you could ask, how much weight at the surface of the  earth, how much weight could a hadron, a meson, if you could  somehow anchor one of the quarks in a meson to some  support over here, the ceiling, how much weight could a  meson support?  The answer is about a truck.  About a, I can't remember if it's an 18-wheel truck or a  16-wheel truck, or maybe it's just a half-ton panel truck.  I don't remember, but it's that order of magnitude.  That's what a meson could support.  So they're pretty strong.  They're microscopic little things, but they're pretty  darn strong.  And the stronger they are, the higher the frequency of the  oscillations, and so the larger the gap between the  lowest energy state and the highest energy state, how do  I actually know this?  Did anybody ever support a truck with a hadron?  Of course not.  What we actually know is the amount of energy that it  takes to excite string.  And from that, we deduce what the tension is.  And from the tension, I can then tell you what you can  support.  These are not the strings that string theorists imagine are  associated with really fundamental particles like  gravitons, photons, and so forth.  Those strings are much, much stronger.  Very much stronger.  The tension in them is vastly larger.  If, again, you took the Earth and you supported a weight,  namely the weight of the whole galaxy.  Now, of course, this is nonsense.  It's not the galaxy that's going to move.  It's the Earth that's going to move.  But if you could concentrate a mass of the galaxy in some  small volume and make the Earth heavy enough, then you  say, if I make the Earth heavier, I increase its  gravitational field.  Well, not necessarily so.  We can make the Earth much bigger, but keep its  gravitational field the same.  If you could somehow keep the acceleration due to gravity  the same, then the weight that you could support would be  about the weight of the galaxy.  So the strings that we're talking about in fundamental  string theory are vastly higher in tension, much  higher in tension.  The meaning of that is that the energy in an excitation to  excite the string is much, much larger.  These are much stiffer strings.  They have much, much larger spring constants.  And therefore, much larger frequencies of oscillation,  if larger frequencies of oscillation, then larger  energies in exciting them.  How much energy?  Well, the thinking goes that it's probably somewhere up  near the Planck energy.  If we're talking about gravitons, it's the  gravitational scale.  Does everybody know what the Planck energy is?  I know you've heard of it.  Do you know actually what it is?  First of all in numerically and second of all, what did  actually where it comes from?  Times c squared.  That's a fairly big energy.  It's enough to, that's a car bomb.  Boom.  How much was it that I didn't hear?  How much did we didn't hear?  10 to the minus 5 grams.  10 to the minus 5 grams.  And then you take that and multiply it by the square  of the speed of light.  But when I asked you if you know what the Planck energy is,  or the Planck length, or the Planck whatever it is, I was  asking a more theoretical question.  I was asking, do you know what it is in terms of the other  constants of nature?  Maybe we should go through that.  I mean, this is worth doing.  Also, how long are these strings?  OK, that's the other question.  That's the other question.  The stiffer they are, the smaller they are.  Now, of course, you can stretch them out to any length.  There's no limit to what you can stretch them to.  But if you ask how much in their ground state, remember,  things in their ground state oscillate.  How much do they oscillate in their ground state?  And what's the mean fluctuation in the size of these  strings, the vibrations of them, then the larger the spring  constant, the smaller they are.  They're much stiffer, much harder to pull apart.  Pulling this thing apart by one centimeter will cause a lot  of it.  The smaller, the bigger the spring constant is.  So how big do people think they are?  Oh, some ways of order of the Planck length.  Planck mass, Planck length.  How fast is one vibration?  The Planck time.  OK, so let's talk about these Planck units.  There's some reason to believe it's a little bit  smaller than that.  Maybe a factor of 10, maybe a factor of 100 smaller, but  that's a fine point.  So let's talk about what the Planck length is.  I didn't prepare this, so we're going to have to work it  out in real time.  What is the Planck length?  What are Planck units, first of all?  What are any units?  In physics, we need to have three units, mass, length,  and time.  And our usual choice of units is the usual units that we use  in the laboratory are not determined by any  fundamental physics.  The meter is a convenient unit for measuring rope.  That's where it came from, measuring rope.  You measure a rope like that, a cloth, or whatever it is.  So no doubt it originated from the length of a human arm.  Probably.  A foot, of course, is also a similar kind of unit.  And I assume it came from somebody's foot, the king's  foot.  So what's the real physics then?  Or what's the real science of what a meter is?  The real science is how many atoms does it take to make a  length of an arm, a useful arm?  There's nothing whatever to do with any fundamental physics.  It has to do with biology.  How long it takes to make an arm?  Not a very deep unit.  That's mass.  Mass, same thing, a kilogram is a kind of weight that you can  manipulate in the laboratory.  And a second is about the time that you could measure with  a pendulum.  That's it.  That's where they came from.  And they have no deeper meaning than that.  We would like, for really fundamental reasons in physics,  to choose units which have some very fundamental meaning.  Now, for example, and if we do, then many of our equations  will be much simpler.  For example, if the size of a proton comes into various  equations in nuclear physics, and the size of a proton is  10 to the minus 13 centimeters, if we work in centimeters, or  10 to the minus 15th meters, there's going to be 10 to the  minus 15th all over.  And it's not just 10 to the minus 15th.  It's 1.73498.  And physics is going to be very messy.  And the other hand, there's nothing to prevent us from  saying, let's use the unit in which the radius of the proton  is exactly 1.  Then nuclear physics will turn out to look a little bit  simpler.  And it will.  Nuclear physics will turn out to be a little bit simpler.  But atomic physics won't look a hell of a lot simpler.  The physics of sub-nuclear particles, sub-hidronic  particles will not look a lot simpler.  There's nothing universal about the proton.  The proton is just some particle, that's all.  Why use the proton instead of some other particle?  No good reason.  So there's nothing really universal about the proton.  And the rules of physics will not in any way be especially  simple if we use the proton.  Time, same thing.  So we would like three units which have some deep  fundamental significance.  It's equivalent to saying we would like to choose three  constants of nature and set them equal to 1.  It's completely equivalent to choosing three units to choose  three constants of nature and set them equal to 1.  Constants of nature now mean dimensional constants.  The size of the proton is 1, but that's not a good one.  You want to use things which are very universal, three  constants.  There are three constants in nature which are truly universal.  And I'll tell you what they are.  The first constant is the speed of light.  Why do I say it's universal?  Because there's a rule.  And the rule is nothing can move faster than the speed of  light, nothing.  Everything move is every velocity of every material  object is bounded by the speed of light.  The use of the word every there tells you there's  something fundamental about it.  That there's something, whether it's protons or  electrons or photons or anything else, they all are  bounded by the speed of light.  So there's something universal about the speed of light.  Yeah?  Well, I think that the constancy of the speed of  light is constant.  The constancy of it?  Well, yes.  The fact that it's the same in every reference frame and the  fact that it's the same for all particles.  Now, when I say it's the same for all particles, you say,  wait a minute, wait a minute.  Protons don't move with the speed of light.  The limiting velocity for a proton is the speed of light.  So in that sense, it's very universal.  So that's the first thing.  We want to set c equal to 1.  Sounds like a good thing to do.  And we do that all the time, of course.  The next thing, which is very universal, is  Planck's constant.  Let me give you an example.  Planck's constant is not just some any old constant like  the radius of a proton, but it's connected with the  uncertainty principle.  For every object in nature, the uncertainty principle  between the momentum and its position is greater than or  equal to h bar.  There may be some four or some two in there, I don't remember.  But it doesn't matter what object you're talking about.  You could be talking about an electron.  You could be talking about a proton.  You could be talking about a bowling ball.  They are all constrained by the same uncertainty principle.  And so in that sense, h bar, and it doesn't much matter  whether you use h bar or h, the one without the 2 pi in it.  That's not the point here.  The point is that Planck's constant applies to everything  in the world.  It's universal.  So the next thing you might want to use, not might, but we  will, is to set h bar equal to 1.  So what's the last really universal, is there another  truly universal constant?  The gravitational constant.  Remember, according to Newton, every object in the  universe, no matter what it's made out of, no matter what  it is composed of, every object in the universe has a  gravitational force, which is the product of the masses  divided by the distance squared times the  universal constant g.  So again, something which applies to everything, are  they units, are they units in which c, h bar, and g are  equal to 1?  Now that's the same as asking the question, can you  combine c, h bar, and g into a combination that has units of  mass?  Can you combine them into a combination that has  units of length?  And can you combine them into a combination that has units  of time?  And the answer is yes.  Let's see, let's do length.  We want to do length.  Tell you what, even better.  Let's do length squared.  Let's see if we can find a combination of g, h bar, and c,  which has units of length squared.  OK, so here's what we do.  Just some simple dimensional analysis.  Everybody here knows how to do dimensional analysis?  OK, we'll do some simple dimensional analysis.  We want to find a combination g to some power, let's call it  p, h bar to some other power q, and c to some other power p,  q, r, which has units of length, I'm going to choose  length squared.  Let's say it has units of length squared.  Now, this doesn't mean it's equal to length squared.  It means it has the same units.  Sometimes people put a bracket around a thing like that to  indicate its units.  The units of g to the p, h to the q, c to the r, should be  length squared.  I'm choosing length squared to avoid a square root in the  final formula, that's all.  There is another reason.  It is widely believed that the fundamental unit is a unit of  area, not a unit of length.  But we'll come to that some other time.  But let's see if we can find p, q, and r.  OK, first question is, what, now we have to figure out what  the conventional units of g, h bar, and c are.  The units of c, that's easy.  What's the units of c?  That's length over time, right?  What about the units of Planck's constant?  Momentum distance.  L squared m over t.  L squared m over t.  Yeah.  L squared is mass.  Momentum times distance, right?  Delta x, delta p.  Delta x, delta p, so that's a length.  And then a momentum is a mass times a velocity.  And a velocity is a length over time.  So that's length squared over time.  Length squared mass over time, that's Planck's constant.  And the last one, the one that I can never really remember,  these two I remember, the one I can never remember is g.  But how do we figure out what the units of g are?  Acceleration equals mass times g times.  Yeah, we use one of the equations that g appears in.  All right, so here's an equation that, let's see.  Which was your equation?  Acceleration equals.  All right, acceleration equals.  Acceleration equals.  Yeah, we'll get that in a minute, but that's equal to.  Mass g over r squared.  Mass g over r squared.  So that's length squared.  Acceleration is what?  Length per time squared.  And again, equal means, doesn't mean equal.  So it's L cubed over t squared divided by m, it looks like.  Right?  OK, let's now go through the exercise.  By the way, that's like Kepler's law, where the mountain.  You're right.  You're right.  You're right.  It is like Kepler's law.  OK, so now we have on the left side, length squared.  Now let's go g to the power of p.  What's g again?  So that's L cubed over t squared times m in the denominator.  All right, so that's L to the 3p, t to the 2p, m to the p.  Right?  Do I get that right?  That's g to the p.  Now what about h bar to the q?  h bar to the q is length squared, length to the 2q, m  to the 2q.  No, m to the q.  And then t to the q?  Do I get that right?  OK.  And then the last one is speed of light to the rth power,  which is length to the r over time to the r.  Now all we have to do is find three constants so that  everything on the right-hand side cancels out except two  powers of length.  OK, so the first thing you can see is mass only appears in  two places, mass to the q and mass to the p.  So what does that say?  It says p better be equal to q.  So let's just set p equal to q.  We now know that.  And we can cancel out the masses.  That's done.  Now what else do we have?  We have to get rid of all the times it looks like, huh?  All right, so how many times do we have?  We have 2p, 2p plus q is the same as p.  So that's 3p plus r equals 0.  3p plus r is equal to 0.  So that tells us that r is equal to minus 3p.  Did I do that right?  Let's just check.  We have 2p, 3p plus r equals 0.  r equals minus 3p.  OK, I hope I'm not doing this right.  Minus 3p.  Did I make a mistake?  Yeah, yeah, yeah, but q is p.  We've already figured out the q is p.  So if I'm not mistaken, I think we can get rid of all the  t's, p is equal to q.  So let's see what we have.  We have p is equal to q.  That looks like 5p.  Boy, why not recognize that?  5p.  L to the 2q.  That's taken care of.  And now r is what?  r is equal to minus 3p.  Yeah, and you're done.  Oh.  3p minus 2, 5p is?  Yeah, you guys, 5p minus 3p and you're done.  2p.  We have 2p, right?  We get 2p.  All together 2p, 2p.  And that now tells us what p is.  p is equal to 1.  OK?  So let's see what we have now.  p is equal to 1.  It tells us that the plunk area, this is the plunk area.  The plunk area L squared is g h bar looks like over c cubed,  right?  C cubed.  C cubed.  This is usually expressed by saying the plunk length  is the square of this.  OK.  The square root of it.  Thank you.  Let's put in some numbers.  My problem is I can never remember what plunk's constant is.  Anybody know plunk's constant?  10 to the?  10 to the minus 34.  Good.  10 to the minus 34?  Yeah.  OK.  Yeah.  g is 6 times 10 to the minus 11, which is about 10  to the minus 10.  Order of magnitude, 10 to the minus 10.  What is h bar?  10 to the minus 34?  Yeah.  10 to the minus 34.  10 to the minus 34.  In usual MKS units, 10 to the minus 34.  And what is c cubed?  C is 3 times 10 to the 8.  3 times 10 to the 17.  Oh my god.  How many?  Times 10 to the minus, how big is it?  24.  10 to the minus 24?  I think it's more like 10 to the minus 25, huh?  Yeah.  10 to the minus 25.  So how many square meters is this?  Pretty small.  Pretty small.  Yeah.  Some tiny, tiny number.  OK.  The plunk length is very small.  If we had done it right, we would have gotten 10 to the  minus 35 meters.  So 10 to the minus, it should be 10 to the minus 70th square  centimeters.  I'm not sure I got everything right.  About 70.  All right.  So that's small.  That's smaller than anything.  What's that?  Good.  OK.  Now the plunk time.  That's the plunk length is the square root of this.  What about the plunk time?  Think of the plunk length as being the size of a little  thingy.  There's a thing the size of the plunk length.  What do you think the plunk time is?  It's the time for a light ray to cross that thing.  What else could it be?  So it's the time for a light ray to cross that little  distance.  So does that mean we have to divide it by the speed of light?  Yeah, we have to divide it by the speed of light.  So the plunk length, L-plunk, is 10 to the minus 35  meters.  The plunk time is 10 to the minus 43 or 42, something like  that, 42 seconds?  10 to the minus 44, 43 seconds, something like that.  Now what about the plunk mass?  We haven't figured out what the plunk mass is, but we could  do exactly the same calculation.  Work out what the plunk mass is.  From H bar.  And the plunk mass would be about 10 to the minus 8  kilograms.  So this one's big.  That's a big mass.  10 to the minus 8 kilograms is a observable thing.  It's a little dust grain.  It's a little dust grain.  These are impossibly small, small times and small distances.  OK, this is a plunk unit.  Things measured in these plunk units are measured in some  very, very fundamental units.  OK, I'll tell you some things.  The universe has a radius.  The known observable universe, the horizon size, is about  10 to the 60th plunk length.  10 to the 60th plunk length.  The age of the universe is also about 10 to the 60th plunk  times.  And good, that's a gallon of, sorry, a tank of gasoline is  about an energy content.  A tank of gasoline is about the plunk mass.  So that means you could drive across the country with 10  plunk masses of energy.  Right, that's what it means.  These are the units that string theory is in.  The size of a vibrating string, the fluctuations in it  due to quantum uncertainty, order of magnitude, the plunk  length.  The frequencies of oscillation or the time, not the  frequency, but the period of oscillation of order 10 to the  minus 43 centimeters.  Sorry, sorry, 10 to the minus 43 seconds.  Right, and the amount of mass that would be involved in  exciting a fundamental particle such as a  graviton or an electron.  Let's take the electron.  If the electron is a string, you could ask, how much energy  does it take to excite it?  The units are expected to be somewhere in this range, 10  to the minus 8 kilograms.  Now, that's a huge number.  How many GEVs, giga electron volts is that?  One other useful number to remember is that one plunk  mass is about 10 to the 19th GEVs, which means  10 to the 19th proton masses.  One plunk mass is about 10 to the 19th proton masses.  10 to the 23rd proton mass is a little gram of water or  something.  10 to the 19th, which is 4 orders of magnitude smaller, is  a tiny, tiny little droplet, but visible, quite visible.  Well, if you have good eyes.  So exciting an electron by a particle physics collision.  You say, well, we don't need to do a particle physics  collision.  We'll just explode a tank of gasoline in the vicinity of  an electron, and it will start the electron vibrating.  Yeah, it's just rather hard to get the energy to be  concentrated in such a small distance.  So these are the reasons that that.  Is it above its ground state?  Yeah, above its ground state.  At its ground state?  No.  Well, remember it's mass squares which come in  integer multiples.  All right, so it's mass squares which come in integer  multiples.  So if you want to know what the unit is, square the plunk  mass, and that's the amount that it takes to bump you up.  But to go from the ground state, if the ground state is  almost massless, then it's about one plunk unit of energy.  That's why you can't excise the electron.  Right.  Right.  It's just much, too much energy to pump  into such a small volume.  And for this reason, that string theory is way, way  direct detection.  How would you detect that a thing is a string?  The same way you detected a proton as a string, you hit  it, you set it into vibration, and you discover that there's  a bunch of excited states where the mass squared is  proportional to the angular momentum.  That's the kind of thing you'd like to do with an electron.  That's the kind of thing that you'd like to do, or be right  with you, Michael.  That's the kind of thing you'd like to do with a photon.  But the increase in mass here is just way beyond what can be  done.  In order to have the structure of the string that we've been  describing, wouldn't the string as a whole have to be  somewhat bigger than this?  Right.  Mm-hmm.  Yeah.  Question?  Question?  This is just a bunch of arithmetic over here.  What makes us think that when we get a number out, 10 to the  minus 35th, that it has anything to do with the length of a  string?  It's a guess.  It's a guess.  It's a guess based on a number of things.  No, no, it's not just, no, no.  No, it's not just numerology.  It could be wrong.  It could be a thousand times less.  Nobody thinks that the string length scale could be smaller  than the Planck length.  That looks quite impossible.  But that the string length scale could be larger, and the energy  scale could be lower.  That is possible.  How much lower?  Well, I think from the things that we know about particle  physics, we can't say with any precision at the current time  really at all.  But from the precision with which particle physics standard  model and so forth seem to work, I think there's plenty of  evidence that the energy scale is very, very high.  Whether it's Planck scale or a hundredth of the Planck scale  or a thousandth of the Planck scale, we don't know.  There is another scale in physics.  There is another scale in physics which keeps coming up over  and over again.  We've talked about it.  It's the unification scale.  If you remember it all from the last quarter, we talked about  the scale at which the various coupling constants seem to come  together.  The scale at which the electro-weak forces and the QCD forces  seem to merge into a common structure experimentally.  And there, there are experiments.  They're not experiments to get to that energy, but experiments  to extrapolate.  The extrapolation of what we know about physics seems to say that  ordinary quantum field theory probably holds to something like  a distance scale of roughly a thousand times larger than the  Planck scale.  So we do have evidence that ordinary quantum field theory  without any stringy structures and so forth, or equivalently  that the electron is fundamental to scales perhaps a thousand  times smaller, sorry, a thousand times larger than the Planck  scale.  But that's still pretty small.  Then you wouldn't be able to suspend the galaxy.  Maybe you could only suspend, oh, I don't know, some globular  cluster somewhere.  It seems that a Planck-weft can be thought of as the fundamental  unit of small stream of time.  But certainly a Planck mass is not a fundamental unit of mass.  No.  No.  Planck mass is believed to be the mass of the lightest possible  black hole.  One Planck area, one unit of entropy.  And how long would it take to evaporate?  One Planck time.  Here today, gone.  Today.  You showed the coupling coefficients unifying.  I believe one time you put down a fourth one for gravity, so that  comes down.  Is that especially?  What's the coupling unit of gravity?  No, that's okay.  I'll tell you what the various pieces of evidence are.  I'm not going to explain them in detail.  This is just to remind you.  If you plot the coupling constants of which coupling constants, the  ones that we measure in the laboratory, the electric, let's  just call it the electric coupling constant, E squared really, the  weak interaction, sorry, the electoral weak, I'm getting, there are  three of them.  There's u1, su2, and su3.  There are three of them.  This is energy this way.  And energy is the same as inverse length.  Inverse length.  So as energy gets large, wavelengths get short, energy, high  energy means small distance.  Okay.  We measure these coupling constants at some energy scales.  Remember, coupling constants are things which change with energy  or change with distance scale.  We measure them somewhere where in laboratories, which means 50  GeV, 100 GeV at most, we measure them here, we get three  numbers.  We also measure slightly indirectly the derivatives of these  things.  In fact, we can compute the derivatives of them.  We can compute the derivatives of them purely theoretically, but  with good confidence because we know the theory here very well.  What we find is that if you extrapolated these, they would all  cross at about a common point.  That point, and that point is way, way out here at an energy which  is not the Planck scale, but it's about 100 or maybe a thousand  times, somewhere between 100 and 1000 times smaller than the  Planck scale, I think maybe more like 100.  So this could be M Planck roughly divided by 100, somewhere in  between.  And the Planck scale would be over here.  Or this is a logarithmic scale.  On a logarithmic scale, it looks like this.  So on a logarithmic scale, a given gap can mean a very, very  big change in energy.  And it looks like this.  Now, of course, we don't know with certainty that there's not  all kinds of things going on at higher energy here which would  muck this up badly.  We don't.  But what we do know is that if there is nothing in here to  muck things up badly, and there's some reason to believe that,  that these coupling constants would just come together at the  scale.  The meaning of that is that quantum field theory,  conventional quantum field theory, seems to be working to  about here.  There's other evidence for this scale coming from neutrino masses  and a few other places.  But it's far from tight.  It will become much tighter if LHC is lucky and makes the right  discoveries.  It will become much tighter and will have a much better knowledge  about this extrapolation.  So this is one of the things we're going to learn from LHC,  is how reliable this extrapolation is.  Planck mass being out here.  And this in itself doesn't tell us where the string length  scale is.  What it tells us is that the stringy character of particles  does not become important until energy is above this.  So somewhere is between here and here.  If string theory is all right, somewhere is in here at the  scale of Schmitts.  Remind me again what the three coupling constants are.  There's what is it?  QED, QCD, and gravity?  No.  QED, QCD, and the weak interactions.  A weak interaction.  At one point you did put down, I think you put down something  for gravity.  Yeah.  Yeah.  If you were to plot the gravitational coupling constant  behaves differently, has a different character.  But let's see.  It starts very, very small.  And it goes up something like this.  And it would cross.  It doesn't cross at the same point.  It crosses somewhere as nearby.  But basically the place where this one gets to be about order  one, that's the Planck scale.  So it's really exactly the same statement to say that the  gravitational constant, the running coupling constant,  crosses at roughly the same place as to say this unification  scale is not too far from the Planck scale.  Is that the Planck capital G?  Yes.  So does capital G change as you start to measure the higher  and higher interactions?  No.  Capital G does not change.  But capital G is not exactly the right measure of things.  It's capital G. All right.  Let's do, I'll answer your question.  Let's look at the force law between two objects.  In electromagnetism it's E squared divided by R squared.  And that's the force.  And that's the electromagnetic coupling constant here.  In gravitation it's G M squared over R squared.  So the thing which is analogous to E squared is G times M  squared.  But what M should you use there?  Well, the M that you use there is associated with the  particular scale of physics that you're studying.  All right.  So you can see then in that sense this quantity here increases  like M squared.  Yeah.  So this is the relevant thing here.  It's a dimensionless thing.  OK.  That's units.  Where on that scale if you could build accelerators big enough  is that they start producing nothing but little black holes.  Real black holes?  Yeah.  So the small ones, you know, if you build galaxies.  Oh, it's all the way out there.  The clock mass.  Yeah.  There's two ways to produce black holes.  One way or another you've got to have a lot of pressure to squeeze  things into a small volume.  One way is kinetic energy.  Small objects with lots of kinetic energy blast them together  and you have a chance of making a black hole.  The other way is not kinetic energy but gravitation.  Gravitational potential energy pulling things together.  That takes a star.  So to make a black hole using gravity you need something as heavy as a star.  To make a black hole by the collision of particles you need energies  of order of the plunk energy.  Now if you think about that for a minute, imagine an accelerator  of a, I don't know, is it fun to, let's explore the following question.  How big, what kind of accelerator would it take to do experiments  of the plunk energy?  To do interesting experiments of the plunk energy?  Well, basically the energy of an accelerator, of a linear accelerator,  scales linearly with the size of the accelerator.  So you get a certain number of GEV per unit length.  Slack is a good example.  It takes approximately two miles to accelerate a thing to a hundred GEV.  So two miles for a hundred GEV.  Two miles, one hundred GEV.  The plunk energy is about ten to the seventeenth times bigger than this.  It's ten to the nineteenth GEV, so it's ten to the seventeenth times larger.  So with just the parameters of the Slack accelerator,  you can multiply this by roughly ten to the seventeenth.  So it would take ten to the seventeenth miles.  How big is ten to the seventeenth miles?  Oh man, ten to the seven, ten to the,  I think ten to the thirteenth miles is about a light year.  Ten to the thirteenth kilometers or something like that, if I remember.  Ten to the thirteenth kilometers, ten to the seventeenth, ten to the fourth light years.  Galaxy size, galaxy size to accelerate something up the plunk mass.  We're not going to do it, at least not tomorrow.  But suppose you could build such an accelerator.  Next question, what kind of luminosity do you have to have?  An accelerator has to have a luminosity.  The chances of, what we want to do now is we want to collide particles head on to make a black hole.  But remember, the radius of that black hole that we want to make is itself the plunk mass, the plunk radius.  So that means we've got to aim these particles to collide within a plunk distance.  Well, accelerators don't do that.  That's not the way accelerators work.  The way accelerators work is you just get a lot of particles going to the left, a lot of particles going to the right.  And you get enough of them so that some will collide at the distance scales that you're interested in.  In other words, you need big luminosity.  What kind of luminosity would you need?  Oh, I don't know, let's say, what's a slack luminosity?  10 to the 13th particles per second or something, anybody know?  I don't know.  Well, 10 to the 13th particles, but that's just, but you know, the distance scales there are pretty big.  How much better are you going to have to do?  I don't know, let's say 10 to the 20th particles.  10 to the 20th particles colliding, each with a plunk energy.  10 to the 20th tanks of gasoline per second.  Now, yeah, 10 to the 17th barrels of gas per second to fuel it as big as the galaxy, we ain't going to do it.  That's what we would need to explore small black holes and something pretty close to it to explore string theory directly, a really direct test.  The question is, are there indirect tests?  All right, I think the indirect tests will pretty much be very theoretical.  And unless somebody constructs a very, very convincing string theory that gives rise to the standard model exactly in some very computable way,  the hope of direct tests, I think, is very remote.  So that's what we're facing, that's what we're up against.  Okay, we didn't get where I really had intended to go.  In fact, we didn't even touch on what I intended to touch on.  Especially these extra dimensions that you had talked about, aren't they in the range of the tanks in this third minus 25?  Or which dimensions?  The extra dimensions of the...  Yes, it is believed, again, without really terrific evidence.  But again, it's the same kind of evidence.  It's the same, where is our plot of coupling constants.  If there were extra dimensions, and the extra dimensions had any sizable distance, if they were at all sizable,  then ordinary quantum field theory would break down when particles begin exploring the extra dimensions.  The conventional use of quantum field theory, as we use it, would break down at points where the energy became large enough to explore these extra tiny dimensions.  If it's true, and if we continue to get evidence that quantum field theory makes sense as it is out to these distances here,  then we'll know that the extra dimensions must lie somewhere between the Planck length and maybe a thousand times bigger than the Planck length.  So exploring extra dimensions will also be something that is extremely indirect and extremely hard.  Why couldn't it be significantly... or is it magnitude less than the plot point?  What's the argument?  Or is it magnitude... oh, that's another question.  That's another question.  The theory really doesn't work at all under those circumstances.  Let's put it this way.  If you tried to make them smaller than the Planck length, you would wind up making their mass larger than the Planck mass.  Making a small thing with big mass means you make a black hole.  So if you're trying to think of strings which were much smaller than the Planck distance, and their mass would accordingly be larger than the Planck mass,  you would be talking about black holes.  You wouldn't be talking about strings.  That's the problem.  They would collapse.  The mass and their small distance would cause gravitation basically to turn them into black holes.  So that's the reason that string theorists do not ever consider the possibility that the string scale is smaller than the Planck mass.  And I think it's a very good reason.  Did you say that the mass here is derived from the choice for L and T?  From your what?  Well, you gave us why we have an L and why we have a T.  Yeah, yeah, but you do exactly the same thing.  You say, let's make a mass squared out of these parameters here.  And I forget exactly what it is.  We could figure it out.  But the others were a physical choice.  Sorry?  The other length of time were a physical choice.  What do you mean they're physical?  They're all physical?  Well, yeah, but I mean the time, for example, is the time to cross L and A.  And that's a physical quantity.  Well, each one is a physical quantity, too.  And what do you say about mass?  You just put it down.  Does it follow from our choices from L and T?  Yes, it does.  Well, no, it doesn't follow by itself from L and T.  But if you asked, ah, great, let's see.  Once you have L and T, you can get mass from either H, R, or G.  So that is the way to think about it.  Take two particles whose distance apart is the plonk length.  Now, because their distance is so small, the uncertainty principle says that they have a lot of energy.  The uncertainty principle says if things are very well localized, then they have a very large momentum.  The fluctuations in the momentum are very large.  So take two particles whose distance is known to be within the plonk distance.  And that says something about their momentum.  Their average momentum is extremely large, and from that you can compute what their kinetic energy of these two particles are.  That kinetic energy will be the plonk mass.  So two particles, if you try to take two particles and localize them to a distance comparable to the plonk distance,  the answer will be that the energy that you have to put in to do that will be the plonk mass.  So it's just uncertainty principle.  We can work it out, but, you know, okay.  Let's take a rest.  I'm going to tell you next about some mathematics.  I'm going to tell you about the mathematics of conformal transformations.  Why?  Because that's the basic mathematical tool of string theory.  Well, we have our choice.  I can either tell you why there are 24 or 26 dimensions in string theory,  or I can start to tell you about conformal transformations.  We'll get to both of them at some point, but...  We haven't clapped the thing together yet.  Yeah, as long as it's not on Thanksgiving.  Thanksgiving is always Thursday, so I think the answer is yes.  Okay, we have 45 minutes, and I will try in 45 minutes to give you the easy version of why there have to be 26 dimensions in bosonic string theory.  The easy version is not easy, and it doesn't really get to the root of it.  The hard version is very hard.  But that doesn't mean that we can't talk about it qualitatively,  but I thought I would do what I call the easy version, which is not easy, and not satisfying, but nevertheless is correct.  When people see this, they walk away and say,  gee, that's sort of magic, but in the bad sense.  But it is correct, and it is sort of the way that the need for extra dimensions was discovered.  It has to do, if you remember, in order to get the photon to come out right,  we had to do something odd. We had to take the mass squared of the ground state of the oscillating string.  Remember, let's take the open string, take in the case of the open string.  The mass squared had to be minus one unit.  That was so that when we applied a creation operator to create the photon polarization, it came out massless.  Minus one, one unit on top of that, gives us something massless.  We haven't talked yet about how we get rid of this particle of negative mass squared.  I assure you that can be gotten rid of.  But let's not worry about it. Let's agree.  The mass squared of the ground state is minus one.  Why is it minus one? How do we get minus one?  Well, harmonic oscillators have zero point energy.  So the ground state energy of this vibrating string should be nothing but the sum total of all of the zero point energies of the vibrating string.  Zero point energies are all positive.  And in fact, they depend on the frequency.  Anybody remember what the zero point energy of a harmonic oscillator is?  A half h bar omega. Let's forget h bar, set it equal to one.  One half omega, and do you remember the frequency of the nth harmonic oscillator?  It was just n.  Each one of the oscillating modes of the string, each one of the oscillating modes, the nth oscillating mode has frequency n.  So the ground state energy of the nth oscillator is just n over two, in some units, is n over two.  Now this doesn't sound good.  We have a lot of oscillators, one for each integer.  We have to add up all that energy.  It's not quite easy to imagine how we can add all this up and get minus one.  So there's something strange going on.  But actually we don't have to get minus one.  Well, we have to get to something a little bit different.  Let's go back to the energy of a very, very fast moving system.  The energy of a very, very fast moving system has, first of all, it has the overall momentum.  Moving down the z-axis, it has the overall momentum.  And then on top of the overall, and that's a conserved quantity.  And we really don't care about it very much, since it never changes during the processes that take place in particle physics.  And then for a particle which is not moving, whose center of mass is not moving in the plane, what's the rest of it?  The mass squared divided, I think, by twice the momentum.  And that's why we began to identify the energy, the non-relativistic energy, with m squared rather than m.  This was the energy, and to make precise this idea of very rapid systems which become non-relativistic,  you have to let p become extremely large.  Basically you have to take the limit of an infinitely large momentum.  In the limit of infinitely large momentum, this is the form that the energy takes.  You say, well, I'm going to take the form of infinitely large energy.  This thing just goes to zero.  But what you really do is you take e minus p, you subtract the momentum from the energy.  That's all right.  This is a perfectly conserved quantity.  It never changes.  If it never changes, you don't care very much about it.  And then multiply the whole thing by, let's say, 2p, and that gives you m squared.  This 2p here, the reason you multiply by 2p, is that's got to do with time dilation.  The faster a thing moves down the z-axis, the slower its internal motions go.  The slower the internal motions, that means the smaller the internal energies.  For example, if you have an atom, what's the ionization energy of a hydrogen atom?  13.5 electron volts or something.  Now you take that atom and you speed it up and you send it down.  In other words, the energy difference between the ground state of the atom,  let's not take the ionization energy, the energy difference between the ground state of a hydrogen atom  and the first excited state is what?  Oh, 3 or 4 volts, something like that.  3 or 4 electron volts.  Now what happens as you boost the atom?  What happens to the overall energy of it?  It increases, of course.  Right?  That's this.  But what happens to the energy difference between the first excited state  and the ground state and the first excited state?  Does it get bigger or smaller?  Smaller.  And it gets smaller with this one over P here.  And in fact, the energy of that atom or the differences of energy between the atoms  are proportional to the square of the mass of the atom divided by 2P.  So that's what this formula is.  Well, we don't care about this piece.  That just has to do with its overall motion down the z-axis.  Just subtract it off.  It doesn't have to do with the internal motions.  And then the P in the denominator here, that's just associated with the slowing down  of the internal motions.  So let's just multiply through by it.  And this is the way we really think of energy.  This quantity is the energy and it's proportional to the mass squared.  Okay, the point is now that the actual energy of the ground state has an infinite piece,  a piece we, or let's put this piece back.  Let's put, let's not subtract this.  Let's add this back on the side.  Plus, what is it?  Plus P squared, plus 2P squared.  Let me just get this right.  Let's see.  Let's take P times E is M squared plus 2P squared.  Okay?  Notice that first of all is the interesting thing which is the mass squared,  which we're interested in.  But there's also this infinite piece here, this infinite piece.  So it's not really quite true that the energy has to add up to minus one.  It would be okay if it added up to something like minus one plus an infinite constant  that was similar to P squared.  So I'm going to show you how that works.  I'm going to show you how that works, how you can add up one plus two plus three plus four plus five.  This is all the internal energy and get something which looks like a constant,  namely minus one, which is what we want to get, plus an infinite piece  that can be absorbed into something that's already there.  This is a trick of physics that takes place all the time.  It happens all the time in all kinds of contexts where you get some infinite answer  and you don't know what to do with the infinite answer,  but you realize that the thing which is becoming infinite already has a constant piece in it  that you just add the extra infinity to the constant piece.  An example is the energy of vacuum, the energy of just a mass of a particle or a self-energy,  all sorts of things like that, where you can hide infinite constants into things which are already there.  I'm not saying this is a satisfactory state of affairs.  It's just something that we do all the time.  The energy in this room, it has zero point oscillations from all of the photons and everything else,  not from the photons that are here, but from the oscillations of the electromagnetic field.  How big are those oscillations? The energy? Infinite.  What do we do with it?  We just subtract it off by saying we could add a constant to the energy of the room  and it wouldn't make any difference, so just get rid of it.  The same thing here. We're going to get a constant piece which is just infinite but constant.  As long as it's constant, it never affects anything.  Plus a term which looks like this with an additional mass squared.  Let me show you the mathematics and we can then discuss another term, the physics.  The mathematics is tricky but easy.  The mathematics is just a question of adding up 1 plus 2 plus 3 plus 4.  It's just a sum here.  1 plus 2 plus 3 plus 4.  There's an overall factor of a half outside that comes from the half h bar omega,  but the basic calculation is to add up all of the integers.  Obviously infinite.  Let's see if we can extract out of that infinite piece an infinity plus something finite.  This is always the way quantum field theory and relativistic quantum mechanics works.  1 plus 2 plus 3 plus 4, this is not something we're going to easily add up, is it?  Let's do something to it and then take a limit.  Let's do something to it so that it makes sense and then take a limit.  Let's multiply each one of these.  Let's introduce a small constant, epsilon.  E to the minus epsilon is a number less than 1.  But when epsilon is very, very small, this number is close to 1.  Let's add to it plus 2 E to the minus 2 epsilon.  This is just a trick for making the sum converge.  Each term in it, we're going to put in something which gets smaller and smaller,  but when epsilon goes to 0, it'll just give us back the original sum.  So epsilon is going to go to 0.  In fact, you know what?  Well, yeah, all right, let's leave it this way.  1 plus 2 epsilon, what's the next one?  3 E to the minus 3 epsilon plus 4 E to the minus 4 epsilon.  But now, if epsilon is not 0, then the sequence of terms E to the minus epsilon,  E to the minus 2 epsilon, E to the minus 3 epsilon do get smaller and smaller.  Now, if epsilon is small, it takes a long, long time for them to shrink.  So if we were just to plot the E to the minus epsilon, E to the minus 2 epsilon,  as a function of the integers, it would fall very slowly if epsilon is a small number.  But eventually, the E to the minus N epsilon will win,  and it will make each term successively smaller and smaller,  and you'll be able to add them up.  So what you do is you add these up,  and then after adding them up, you let epsilon go to 0.  It's a mathematical trick.  Let's see what we get.  It takes a bunch of steps, and I'll show you, but the steps are elementary.  Okay, this can be written as the sum from N equals 1 to infinity of N,  E to the minus N epsilon.  E to the minus 1 epsilon times 1,  E to the minus 2 epsilon times 2, and so forth and so on.  That's the sum.  Now, the trick is to get rid of this N.  First trick is to get rid of this N,  and the way to get rid of this N is just to take the sum without the N in it  and differentiate with respect to epsilon.  What happens if we differentiate with respect to epsilon?  What does a derivative with respect to epsilon give?  It brings down minus N.  So this is equal to minus,  we really want to put minus here,  this is equal to the original sum we were interested in,  N, E to the minus N epsilon.  That's the first step.  Next step is to observe that we can add the series up.  This whole series is a geometric series.  It starts out, let's just see what it is.  It has an E to the minus epsilon plus an E to the minus 2 epsilon  plus an E to the minus 3 epsilon,  which also can be written in another form.  It's E to the minus epsilon times 1 plus E to the minus epsilon  plus E to the minus 2 epsilon and so forth.  I pulled out a factor of E to the minus epsilon.  Do you know what this is?  Exactly.  This is a geometric series, 1 plus a number plus the square of the number  plus the cube of the number.  And so we can add all of this up.  And what do we get?  We get E to the minus epsilon divided by 1 minus E to the minus epsilon.  That's the thing that we want to differentiate with respect to epsilon.  That's this.  I put a minus sign in front of it.  This is something you can do at homework, but I'm going to show you  roughly what happens.  But in the end, we're interested in small epsilon.  Let's expand it for small epsilon.  Let's first expand it for small epsilon and then do the operation  and then epsilon go to zero.  We're interested in small epsilon.  So what is E to the minus epsilon when expanded in terms of epsilon?  1 minus epsilon plus epsilon squared over 2,  we're only going to need things to power epsilon squared.  We won't need things beyond that.  And of course, there are more, but we won't need them.  Now what about the denominator?  What's in the denominator?  We have 1 minus, okay, so let's write out E to the minus epsilon  minus 1.  I think it's plus epsilon, minus epsilon squared.  Let's go to the next one.  I think we're going to need the next one, as you'll see.  Epsilon squared over 2, right?  What's the next one?  Let me just check the sign.  1 minus 1 plus epsilon minus epsilon squared over 2 in the next one.  Epsilon cubed over what?  Epsilon cubed over 6, right?  Now we have trouble.  The trouble is the ones cancel.  The denominator is proportional to epsilon.  Let's factor out one power of epsilon.  Let's see.  This is the object whose derivative we want to take.  Let's factor out a factor of epsilon in the denominator.  Since everything in the denominator has epsilon,  let's factor it out, 1 over epsilon, and then this becomes 1  minus epsilon plus epsilon squared.  Did everybody follow that?  Okay.  Plus higher terms, but as I said, the higher terms are going to disappear  when we go to small epsilon.  They won't be important.  Was it a minus sign?  No, it was a minus sign.  1 minus the 1d1 sub-skeleton.  Well, I thought I got it right.  All right, should we do it over?  1 minus, put in brackets,  1 minus epsilon plus epsilon squared over 2  minus epsilon cubed over 6.  All right, everybody happy with that?  Now the ones are going to cancel.  And the minus times the minus will make plus minus plus.  They have no idea if that's what I had written down before or not.  Then we pull out a factor of epsilon,  and that turns this into 1 minus epsilon over 2  plus epsilon squared over 6.  All right, now the trick is we're expanding everything in powers of epsilon.  But here we have a thing in the denominator.  I don't want this in the denominator.  I want in the numerator.  So how do you deal with a thing in the denominator like this?  You use the formula that 1 minus a small number,  let's just call it s, is equal to 1 plus s plus s squared plus s cubed and so forth.  Here's your s.  Oh, did I leave out something?  No, I think I got it right, didn't I?  I think I did everything right.  I think I did everything right.  So what we do is we take what's in the denominator here,  and we expand it to get all the epsilon dependence in the numerator.  Let's see what we get.  I think we're going to get 1 plus epsilon over 2 minus epsilon squared over 6.  But then there's a term coming from squaring the small quantity here.  If I keep things to order epsilon squared, all I'm going to get is,  let's see, I think plus epsilon squared over 4, I think.  But you can expand it out yourself, something like this.  Plus higher order in epsilon.  No, I think it was epsilon squared.  It was just epsilon squared, not epsilon cubed.  No, the thing which was epsilon cubed in the denominator here  became epsilon squared when I pulled out the epsilon here.  And I was purposefully keeping everything to power's order epsilon squared.  I know that the next order is not important.  OK, so let's see if we can combine this together.  I think the chances that I'll get this right are negligible.  But all right, so we get minus d by d epsilon, 1 over epsilon times, OK,  1 times 1.  Let's put everything that you get from 1.  Plus epsilon over 2 minus epsilon squared over 6.  Oh, can we combine these two together?  What's epsilon squared over 4 minus epsilon squared over 6?  Epsilon squared over 12.  Right?  OK.  How did you do that so fast?  Who did it?  You know?  What?  Took me about 30 minutes to get that epsilon squared over 12.  OK.  I know how to do it.  1 plus epsilon minus epsilon, sorry,  1 plus epsilon squared over 12, right?  That's this, this, this, and this.  Then we get minus epsilon minus epsilon times 1.  And then minus epsilon squared over 2.  But then from here we get something of order epsilon cubed.  And I'm not interested in keeping anything of order epsilon cubed.  That's too high an order.  I don't care about it.  We'll see why in a moment.  And then we have plus epsilon squared over 2.  And nothing past that because the next one would be epsilon fourth.  I think this is everything to order epsilon squared.  Should it be plus?  No?  I think it should be plus, right?  Yeah.  Right.  Looks right.  1 times epsilon squared over 12.  OK.  Now, epsilon over 2 minus epsilon, that's minus epsilon over 2.  I'll get rid of this and these two cancel.  So there's our formula right there.  As you'll see, anything of higher order is not going to be interesting to us.  OK.  Look at, let's first focus on the one that's easiest.  The easiest term is this epsilon over 2.  It gets multiplied by 1 over epsilon.  That means it's just 1 half.  And what happens when you differentiate 1 half with respect to epsilon is 0.  So this is, we don't have to write it.  It's not there.  OK.  Now we have a term which is 1 over epsilon.  That's bad news.  This is something infinite at epsilon equals 0.  And remember, we're going to go to epsilon equals 0.  All right.  What's the derivative of 1 over epsilon?  Minus 1 over epsilon squared, right?  Minus 1 over epsilon squared.  Oh, and there's another minus sign here.  So that's plus 1 over epsilon squared, I think.  Yes.  And then 1 over epsilon times epsilon, one of the epsilon's cancel,  and then you differentiate with respect to epsilon.  What do you get?  1 over 1 minus 1, 12.  This is a famous formula which is usually stated half jokingly  that 1 plus 2 plus 3 plus 4 plus 5 is equal to minus a 12.  We tend to forget about this term over here.  And the reason is that this term combines, well,  this term is an infinite constant which really doesn't,  I would have to convince you that where it,  what happens to it, it gets absorbed into the p squared,  the p squared term in the energy, the infinite term in the energy.  And a careful analysis, a really careful analysis of it,  allows you to absorb this into an additive constant in the energy  that doesn't do anything.  For the moment, you'll have to accept that.  The real answer, the really right answer,  the really right answer is that in good string theories,  properly defined string theories, this isn't there altogether.  But I'm not going to tell you why now.  I'm not going to tell you why now.  I'm just going to tell you this infinite constant is not important.  I know the famous story of Dirac and Pauli,  that Dirac calculated the vacuum energy and found out that it was infinite.  And they said, well, since it's infinite, I don't care about it,  we'll throw it away and Pauli turned around to him and said,  just because it's infinite doesn't mean it's zero.  Right.  In this particular case, Dirac wins.  Dirac wins, but this is something we will have to come back to  and explain why this one over epsilon squared.  Remember, epsilon is going to go to zero.  This is going to be an infinite constant.  And in this particular case, the infinite constant really is not important.  And I'm not going to pursue it right now.  I'm going to tell you the answer is minus a 12.  Now, this is not yet a particularly good answer.  This is not a particularly good answer yet.  Remember, what we wanted to get was altogether minus 1.  We wanted to get this minus 1, which when we excited by one unit of energy  will give us the massless particle.  In fact, the minus a 12 is not quite right.  And the reason is because of this factor of a half from the 1.5 H bar omega.  So it's actually 1 over 24.  1 over 24.  That minus 1 over 24 is the energy of the ground state,  including the zero point oscillations,  but throwing away a certain infinite term,  and I have to explain it way later.  But, okay.  How do we get, oh, have I left anything out?  I left out one thing.  Remember, for each mode, there is not one oscillator but two oscillators.  The a's and the b's.  Remember the a's and the b's?  The x oscillators and the y oscillators.  So actually, it's twice this.  Twice this, one for the x oscillators, one for the y oscillators.  It's still not good.  It's only 2 over minus 24.  We really want to get minus 1.  How do you get minus 1?  You say there weren't two dimensions of oscillation.  There were 24 of them.  24 directions of oscillation plus the direction going down the z-axis,  plus time make 26.  That's where the 26 is.  Now, this is a crazy story.  At the time, nobody believed it that this is a little bit too silly to be true,  but in fact, the mathematics did fit together.  Mathematics.  And this is the simplest way.  This, which is fairly complicated already, is the simplest way to see.  And it is historically the way it came about that requiring,  let's say there were d dimensions of space perpendicular to the direction of motion,  and requiring this to come out to be minus 1,  just so that when you excite it, you can get the photon,  that was the original argument about d having to be 24,  or the total space time dimension having to be 26.  Now, as I said, the arguments, this was hardly a convincing argument at the time.  It's virtue is that it's easy to present, relatively easy to present.  What is the history?  What was the history of this introduction?  How did this come about?  Yeah, well, I was just wondering, 1990s or...  Oh, God, no.  1969.  Yeah.  No, 69.  No, no.  No, this goes back to 1969, I would say.  Maybe early 70, but I think it's still 69.  So it's for a long time?  Yes.  Well, by this point, people were just exploring the mathematics of string theory.  And they had...  So you did have a regi trajectory?  Well, experimentally, you had the regi trajectories.  But at some point, when the mathematical structure was put in place,  people began to explore it for its own sake.  It was realized rather quickly that there was some kind of funny thing going on  that there was a spin one particle, one unit up, one unit up from it,  but it didn't have all the states that it needed.  It had two polarizations and not three polarizations.  And it was realized rather quickly that this could only be the photon or something like a photon,  but that then left the question,  what was this zero point energy that had to be minus one?  And it was realized fairly quickly that that required 24 dimensions of oscillation.  Now, as I said, this was by no means a convincing argument.  Other much, much, much more convincing arguments came about,  but they were highly mathematical, really highly mathematical.  And I'll tell you about them in words, but not now.  We need one more mathematical concept.  Is there a similar trick to get eight?  Yes, eight.  Yes, yes, there is.  It's a similar trick to get eight.  But before I tell you that, let me go back to the closed string.  Remember the closed string, you'll have to excite twice.  You'll have to get spin two.  You'll have to hit it with two oscillators.  That meant that the ground state had to have minus two units of energy.  But now we have a problem.  d over 24, if d is 24, is only minus one.  What's the answer?  48 is minus 48.  48 is minus one.  Why 48?  Because you get a minus two, one-twelfth, right?  You get the minus two.  Yeah, but wait a minute.  I know, but you only have 24 directions.  We can't change the number of directions in space when you go from closed strings.  No, no, no, the same theory, the same theory has both closed and open strings.  You can't change that.  Right, you can't do that.  No, we'll agree.  The theory has 24 dimensions.  Now what happens to the closed string?  Secular parallel.  Secular parallel, alright.  Well, something like that.  Remember, you have twice as many oscillators for the closed strings.  You've got the ones that go around to the left, and you've got the ones that go around to the right.  Everything gets doubled once again.  You have not only an oscillation for every direction of space,  but you have one for right-going modes and one for left-going modes.  So in fact, it's true that you get twice as many,  which means that the ground state energy is minus 2,  which is just exactly the right thing so that two units of oscillation will bring you up to massless again.  That began to smell better.  It began to smell better, but still far from a convincing argument.  The real thing has to do with something called conformal invariance.  So we're going to take up conformal invariance next time.  What conformal invariance means?  If you want to prepare yourself, learn a little bit about complex function theory.  Very little, not much.  We're going to do a very little bit.  Learn what the Cauchy-Riemann equations are.  If not, I will explain it.  And we will come to why string theory is the theory of Turkish taffy.  You know what Turkish taffy?  You pull it this way, you pull it that way, you stretch it out that way,  you stick it together, and somehow it all stays together.  It's conformal transformations.  Anyway, I think we're finished for tonight.  Cauchy-Riemann equations.  Cauchy, Cauchy.  Just learn a little bit about them.
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