Lecture 9 | String Theory and M-Theory

Stanford University alright let's um let's start let's try to get to the heart of some of the aspects of string theory which are really very central we haven't talked about them but we set up many of the questions we set up many of the ideas I want to show you first something which has both to do with why string theory has gravity in it and it has to do with why there are such unusual constraints on the allowable kinds of string theories that you can have I'm not going to actually get into 26 or 10 dimensions but I'm going to give you an example which shows just how tightly constrained the idea of string theory is compared to the idea of point particles um point particles can move in curved spaces a very very simple example would just be this is something you can you can look up I suspect in elementary mechanics textbooks ordinary particles moving on a curved surface curved surface could be a sphere doesn't matter some sort of curved surface it could be positively curved negatively curved in fact it doesn't have to be two-dimensional but I'm drawing it as two-dimensional let's say a two-dimensional surface and a particle on it a particle is described by a point on that space a point on a velocity if you want to know the future history of it and it moves on a trajectory moves on a trajectory and it satisfies an equation of motion the equation of motion is some form of Newton's equations are projected onto the surface well how does the point particle moved I'm thinking now about ordinary nonrelativistic particles moving on a curved surface without any forces it the forces of course are there to hold it in the surface but other than the forces needed to hold it in the surface no other forces has a point particle move anybody know goes on a geodesic which is another way of saying it moves as straight as possible it doesn't deviate to the left doesn't deviate to the right it moves along with geodesic that's right if there are forces on it the foreign kind or another ah then it may move in a more complicated trajectory you could imagine that this wouldn't be consistent you could imagine that it might not be consistent how could it fail to be consistent to write down it's just a differential equation just a Newton type differential equation here's a way that it could be in which it of course is not inconsistent it's perfectly consistent as long as the surface is smooth as long as the surface is smooth differentiable all the good things that nice surfaces have but what do you really mean by a motion of a particle satisfying a differential equation as always you mean break up the trajectory into a lot of little pieces replace derivatives by distances and think of a discrete version of it that's first step derivatives are really differences ah and then take the limit take the limit in the appropriate way of more and more little time intervals question how do I know that the limit exists how do I know that the limit exists could it be that as I put more and more points in there that the limit of whatever it is you're calculating the trajectory itself how do I know that the limit exists well these are just described by ordinary differential equations people have been studying differential equations since the mathematician differential was a young man and we know that the solutions of these things are smooth regular and have good limits okay now you can ask a more complicated question you could say wait a minute I'm really interested in quantum mechanics and in quantum mechanics we have a more complicated kind of question instead of just looking for a trajectory we ask questions for example what's the amplitude that if a particle starts here that at a later time we will discover it here we find again there is an action principle but the action principle is not to find the trajectory of least action which gives a differential equation but to use a path integral and the path integral sums over all trajectories including wildly varying ones again a trajectory being defined in terms of breaking it up into little pieces and then taking a limit now how do I know that the limit exists yeah this trajectory is all inside the surface yeah they're all inside the surface yeah everything takes place inside the surface yes good point yeah we're talking about physics in inner surface on a surface whatever you want to call it those features which are completely independent of how the surface that depend on the geometry of the surface but not in anything outside the surface yeah all right so how do I know that this very very complicated path integral structure exists in the sense that it has a limit a limit as you take the a grain structure finer and finer there the answer is much much more complicated but its equivalent to solving the ordinary Schrodinger equation on on the surface and again that's a more complicated kind of equation it's a partial differential equation and partial differential equations are more difficult but partial was a very smart guy and we figured out how partial differential equations work and we know from 19th century physics and so forth that the limits do exist they do make sense string theory is more subtle infinitely more subtle and string theory we're not talking about a point particle we're talking about an extended object now that's the string itself imagine a string which is constrained to move in a surface and this surface does not have to be two-dimensional but as I said I'm going to think about two dimensions here's the string and it's a quantum mechanical string which means it's subject to the vibrations in fact just to just a pick an example let's talk about string theory where the space that the string moves on in other words real space the space that the string moves on is a sphere and it happens to be a sphere with radius R it's a sphere with radius R oh let's do a point particle first in fact let's do a classical point particle first point particle moves on a sphere what is its kinetic energy or its Lagrangian as always it's just one-half MV squared mass times velocity squared which is also the momentum squared divided by twice the mass let's that way P squared divided by twice the mass but now there's a constraint and the constraint is that the particle has to stay on the sphere nevertheless this velocity that what that means is this velocity vector is in the surface of the sphere the momentum vector is also in the surface of the sphere but other than that it's just the standard classical mechanics now if we were doing quantum mechanics then on a sphere spheres the motion of a particle on a sphere moves in a circle of course a great circle great circle of size R the momentum is quantized how do you see that the momentum is quantized as well as various ways to see it but let me just give you a very simple way if the momentum is P and the radius of the sphere is R what is the angular momentum P R P R so the angular momentum L is equal to P times R and magnitude anyway an angular momentum we know is quantized we know that it's quantized in units of h-bar so some quantized angular momentum here I'm not even going to bother right here angular momentum is fundamental because it's quantized so it pays to rewrite the energy in terms of the angular momentum let's write P as L over R and stick it into here then it becomes L squared divided by twice M let's see what is it twice M and now R squared right mr squared you know what them R squared is called moment of inertia the moment of inertia in this case of a point particle over here and the symbol for it I l squared over twice the moment of inertia so what characterizes the motion of this particle is first of all its angular momentum but the moment of inertia the moment of inertia plays a central role in the energy levels in all of the properties of the trajectories but in particular the energy levels when we're doing quantum mechanics and that's an important formula good so the classical mechanics of point particles is described this way and the quantum mechanics of point particles is also described this way L being angular momentum we know how to deal with angular momentum it's quantized okay so all of that is straightforward now we want to talk about a string a string moving on this sphere here it is it's constrained to be in this field sphere the spherical surfaces all there is there is no outside the spherical surface and inside there's only the spherical surface itself however as I said the spherical surface could be higher dimensional but for the moment is two-dimensional and it's going to move around good how do I know that in the limit that I think of the string as a collection of little points how do I know that the properties of this little string have good limits the answer is they don't there's fizzers we know we not that we don't know the answer is that the limits don't exist all right so let me show you why and what happens the first thing is to understand a little more about the size and shapes of strings we want to understand a little more about them so let's calculate how big statistically the string has zero-point oscillations the oscillators in the string are constantly vibrating even in the ground state because they're vibrating they give the string a little size they give the string a little size and I'm interested in trying to figure out how big that string is when the zero-point oscillations are included so how do I do that I go back to a formula that we had on the blackboard maybe the first or second lecture let's do open strings just for simplicity remember this formula the position of a point on the string what is Sigma Sigma is the parameter along the string is equal to a sum over oscillators each oscillator has a frequency associated with it so an in and we have let's call it a plus in the plus a minus in cosine of n Sigma and there's one other thing a square root of n downstairs does everybody remember that probably not but if you go back in your notes yeah they may be factors are two I don't remember in detail whether the factor of two sits in front of it or not but that's not important this is the structure of it now what do I want to calculate I want to calculate in the ground state of the string ground state of the string is the state with no oscillations I want to calculate the mean size of it the average size of it so the average size of it let's calculate the average squared size those it was square X but square X x squared at any point Sigma at an arbitrary point Sigma what does that give us that gives us Sun on in and a some on em we're going to have a plus n plus a minus n times a plus M plus a minus M over square root of n square root of M cosine and Sigma cosine M Sigma I've just written down this expression twice once as a sum over in and once as a sum over m I've done nothing special now what I want to calculate is its average value in the ground state average value in the ground state and quantum mechanics we represent like this o stands for the ground state string and we want to calculate its average value that means we want to calculate the average of this in the simultaneous ground state of all of the oscillations now you might think the answer is 0 right well that's not quite true every oscillator in its ground state quantum mechanical oscillator has a little bit of zero-point energy that little bit of zero-point energy gives it an average fluctuation in position or in this case in in X itself and how do we calculate it let's look at the various terms here we're going to calculate the ground state expectation value of this oh oh the important pieces are these creation and annihilation operators what happens if I take the piece with a plus times a plus can that have an expectation value for R in the R and the ground state well no it can't the reason is to a pluses when they act to the right create a state with two extra units of energy you've raised the oscillator in twice and that has no overlapped with the ground state at all so the term with to a pluses that that can't contribute to this how about the term with to a minuses what does a minus do when it hits the ground state it kills it right it's also true incidentally when a plus acts to the left that kills this so the term with a plus a plus and a minus a minus give nothing what about a plus times a minus this is a plus times a minus right so that's not there but what about the term a minus a plus sounds good let's so look let's look at it a minus times a plus that's still there if it's zero so be it but I don't see any reason why it should be zero okay what does it say it says we create one unit of excitation for the inflator this says we remove one unit of excitation for the enth oscillator can you remove a unit other than the one you put in No so this can only be nonzero if n is the same as M but if n is the same as M what do you get chronica Delta one these are all discrete so they're not continuous variables ah you get one if n equals M and zero otherwise in other words the sum over N and M is only a single sum it's the sum over N of a minus n a plus n over square root of n sorry not school in in square root of n times square root of M that's just in and then cosine 9 squared of n Sigma now cosine squared is always positive and it averages to about 1/2 cosine of n Sigma is a function which looks like this if you square it of course it's above the axis because the square is always positive and averages to about 1/2 that's good enough for our purposes it's never negative you can't get any cancellation and it kind of oscillates around 1/2 so for most purposes it's sufficient just to call this 1/2 and that's about right oh oh now what is this thing this thing is just 1 you create a particle or you create an excitation of the enth oscillator and then you take it out that average value is just 1 so we now have an answer the answer is let these that x-squared now I'll tell you what x squared means in a moment it's equal the 1/2 is not the important thing here just sum on n 1 over n now first of all what is x squared what is X I wrote here XO Sigma but I left out one term there anybody know what the term that I left out is the center of mass motion there's one term which corresponds to 0 frequency which is just the center of mass so really the thing that I've calculated was the average of the position of the string at Point Sigma minus the position of the center of mass think about that for a moment what's it mean you have a screen in this case an open string and the center of mass is may or may not be on the string but different points on the string are separated from the center of mass the average the average separation from the center of mass is a measure of the size of the thing the center of mass is where the center of mass is and if the points of the string are not at the center of mass that means that it's been spread out if this is nonzero it means the string is extended in size and he were calculating it and we find out that x squared is this sum of 1 over N trouble trouble big trouble why is it trouble infinite that's infinite now most of these oscillations are incredibly rapid large n oscillations of very very rapid oscillations maybe we should just say really all those very very rapid oscillations are unphysical cut them off only take the first n the first 25 oscillations into account and see what happens well then we'll only have log of 25 and log of 25 is not a very big number how big is log of 25 log to the base e of 25 not very big it's a small number a moderate number but then we're not doing string theory we're essentially doing a string of 25 particles we're not doing string theory ah we've got to go to the limit N equals 0 now it's an extremely remarkable thing that this causes no problems for the scattering of strings in flat space flatspace it's a real magical very very delicate cancelations that make sure that the string which is essentially infinitely big with rapid fluctuations taking place all over the place doesn't bombard other strings far away even though it's extended out all over the place they're very rapid fluctuations it's an amazing set of cancellations which makes sure that all of this works out but those cancellations don't always work they work in some geometries and not in others so I'm going to show you what goes wrong on the sphere and it doesn't go wrong in flat space first of all the message of all of this is first step terminate this some at some maximum in in my maximum frequency later on we're going to have to repair the damage by letting the maximum frequency get larger now we can ask how does this string behave as a function of N max in flat space well here's the way to think about it as a function of n max the string occupies a certain region of space it's the region of space whose x squared or R whose R squared is equal to the logarithm of n max the higher we make n max the bigger it seems to spread out but very very slowly but nevertheless it seems to spread out well if this is moving in flat space it has a energy which is just 1/2 its mass times P squared it doesn't matter that it's spread out the motion of the center of mass of a billiard ball this big that big moves exactly the same way as the motion of the center of mass of a proton it moves the center of mass well of course it's not the same way as a different mass but in flat space the motion of a particle the center of mass of a particle separates off the fact that it's an extended object you don't have to remember the fact that it's a that it's extended if you're just interested in the center of mass motion but that's not true in curved space in curved space they don't separate off so easily so let's see if we can figure out what's happening here here's the string let's put the string up at the North Pole here the string is up at the North Pole and if I make as my first approximation that there's only one or two oscillating modes well we could start with no oscillating modes if there were no oscillating modes then the string would be a point it wouldn't be spread out at all it'll be a like a just just be a point then the string would just be a point and it would be over here and it would move as an ordinary point particle how does a point particle move it moves with an energy which is the square of the angular momentum over the moment of inertia and what's the moment of inertia it's just proportional to mass times the radius squared times the radius squared of the sphere if we set em to one then it would just be the square of the radius of the sphere so this point here would move around the sphere on great circles and it would move with an energy which would be controlled by the radius of the sphere the radius of the sphere would go into this calculation in terms of the moment of inertia okay now let's take into account that as we add more modes of oscillation the string starts to spread out it starts to get a size and grows that grows with time but grows with more and more oscillations here starts to get bigger it fills up some region which looks about like that incidentally if you ask what does it look like on the average it simply looks like a tangle the more modes that you add the more and the bigger the tangle is its fluctuations on Pappa fluctuations on top of fluctuations and this thing starts they get bigger now think about its motion around the center of mass of it think of the center of mass of it moving on the same great circle the center of mass will still move on the great circle but what about its moment of inertia moment of inertia about an axis is an axis it's moving it's going around that axis the relevant quantity is the moment of inertia about that axis and it's governed by the square of the distance to the axis the moment of inertia about an axis is proportional to a sum it's an integral over the mass distribution each little piece of the mass distribution you take the mass of that little piece and you multiply it by the square of the distance to the to the axis that you're rotating about now notice that this point over here is closer to the axis than this point up here in fact let's go to extremes let's suppose we include so many modes that let's let me draw this better droops down to about over here it's rotating around the sphere this way it's on the sphere it's not inside the sphere it's on the sphere the point on the sphere up here that is a distance R from the from the from the axis what about this point over here how far from the axis is it is it less than or more than R looks less than doesn't it but that's deceptive because we forgot that this here is really a sphere it's really a sphere well it's like moving the point down to here okay with the axis going in this direction it would be the same distance so this this point over here which is out here somewheres is the same distance from the axis as this point okay but how about this point over here it really is closer to the axis than the point up here so what's going to happen when I calculate the when I calculate the moment of inertia the moment is going to be bigger or smaller or exactly the same as it was when I pretend that all of the string was at a single point it's going to be smaller right it's going to be smaller the the mass of the string is the same incidentally that does not change when you add more more modes just oscillates differently but the mass the amount of mass that's there is just the sum of the masses of little point particles and they get up the mass stays the same but the moment of inertia gets smaller meaning of that is that the quantum mechanics behaves you can either say yes I say it too is you could say the string is spread out and like that and therefore closer to the axis or you can just say it's as the string moved on a smaller sphere it's as if all of the quantum mechanics essentially takes behaves as if the string was a point but was a point on a smaller sphere but the more modes that you add the bigger the string gets eventually if you add enough modes it's going to grow so that it's almost as big as the entire sphere in other words well let's let's go to some extreme where it where it fills up a good fraction of the sphere in that case the effective moment of inertia is going to be much smaller than then the moment of inertia would have been if it would have been a point particle well even worse there's no limit or at least as n gets larger and larger the string begins to cover the whole sphere and the effective moment of inertia is actually zero the average position of it is right at the center if the effective moment of inertia is zero well that's a zero in the denominator here that's not a good thing for for the motion of anything so still spread around is a reason to think well it's no it's not zero it's not zero but okay um yeah I know you have to be if it does go to zero but you have to be you have to think about it in a different way what you do is you say the string behaves if you if you only take into account let's say a dozen normal modes then the string behaves as if it were a point on a closest circle but now instead of a dozen normal modes let's take two dozen normal modes then what it does is it starts to behave like a bring on this inner surface here this is not obvious this is what is called renormalization group the normalization group is a way of following the system as you add more and more degrees of freedom what you can follow it honestly is for a small change in n max but once you follow for a small change in in Max and you find out that it's behaving as if it were a lump let's say this big now you add more structure it begins to behave like a like a collection of lumps that also form a string and you just do the whole process over again you say with a dozen normal modes it behaves as if it were a more or less a point on an inner region here now you add more of them and it begins to make itself into a string of these same lumps that has the effect of pushing in and again and eventually it just pushes it to zero so there is no limit there is no limit the limit of a string moving on a curved surface and this is due to the curvature if it weren't due to the curvature you wouldn't have this effect that's due to the curvature of the sphere this happens in any number of dimensions are that as you add more and more modes the behavior does not tend to a finite limit it just tends to behave eventually as the motion of a string on an extremely small infinitesimal sphere which means in the denominator here is zero and that's not a good theory that's what happens so the answer is strings on spheres are not good strings on spheres don't make sense what is the condition for a surface and a surface I'm not I'm using the term in a general sense it could be it doesn't have to be two-dimensional it could be three four ten twenty six dimensions whatever whatever the number of dimensions of space what we mean by a surface is a geometry with a metric the geometry with a metric of romani in geometry oh what are the rules what you really want is a set of rules which approach a limit if you're near that limit then in other words by a limit I mean the limit that as in Max gets bigger and bigger you don't want the answers to change as n max goes from a hundred billion to 100 billion plus one or for that matter for a hundred billion to two hundred billion you want the answer to stabilize that's called a fixed point you want the answer to stabilize and not to change so what you would like is a situation with the geometry that when you do exactly the same thing the geometry does not seem to change in other words we could define an effective geometry for a string we take the geometry and now we sort of average over the properties of an extended object and that defines the motion on a different surface different surface being the effective surface that the spread out string moves on as long as it keeps changing on us as we increase n max there will not be a limit a limit would mean that at some point as you add more and more structure to the string the effective geometry stops changing and tends to some limit that limit will be a good geometry for for string theory to make sense on okay so I'll tell you what one can do this calculation very precisely an arbitrary geometry and you can ask when you add additional modes let's say you add you go up to some n max and you add one more mode how does the effective geometry that the string moves on how does it change well how do you describe a Romani in geometry what's the what's the variables that you describe every money in geometry the metric right the metric tensor that's the description of the geometry G mu nu of X this don't think of it now is the gravitational field although it is the gravitational field it's the metric of space gym you know of X we have a geometry it could be the sphere could be anything else could be a hyperboloid it could be a thing with wiggles in it it might be space-time geometry itself okay a string moving on such a geometry the effective geometry will change a little bit why because the string is spread out a little bit the center of it though will move in a way that responds to a slightly different geometry and so we can ask what's the change in the effective geometry when you add one more mode okay the answer is first of all that it depends on the curvature if the sphere weren't curved this would not have happened this has to do with the fact that the sphere was curved so whatever the change in the geometry is if it were flat it would not change but if it's curved it does change in general so they must be on the right-hand side something which involves the curvature tensor the left-hand side is a tensor with a mu and a new a second-rank tensor symmetric just a matter so the right hand side has to be a tensor it's not gene you knew itself it's some other tensor that has to do with the curvature if it weren't curved nothing would happen so it must be that on the right hand side is something involving curvature curvature is called capital R and the full curvature tensor has four indices it's a thing with four indices are let's say mu nu now let's not call it mu and nu for the moment let's just call it Alpha Beta Gamma Delta you can put all four downstairs upstairs doesn't matter it's often written with one index upstairs and some indices downstairs okay this equation doesn't make sense the indices don't match but is there a thing that you can make out of the curvature tensor which also has only two indices in which is symmetric anybody know the Ricci tensor the Ricci tensor is simply the contraction alpha alpha nu nu that's called arm you knew and it's the Ricci tensor so when a careful calculation let me see is it plus or minus I think it's - - minus the Ricci tensor it corresponds to the sphere getting smaller when you add more structure that's why they as well as a minus there this is the equation that governs how the effective geometry of the string changes as you add more and more fluctuation to the string mathematicians have a name for this equation it's called Ricci flow Ricci flow is a tremendously important part of mathematics and it's an equation that if you start with the geometry and then you change it in accord with this equation what happens is it starts to deform and it flows the geometry flows okay it's called Ricci flow it has very important applications in mathematics I don't know whether it was discovered by string theorists or first discovered by mathematicians I'm not sure but that's what this is it's a fuzzing out of the geometry because the points have gotten fuzzed out it's a kind of diffusion of the geometry which is fuzzing it out the points have gotten fuzzed out by the fluctuations of the strings and the effect of geometry behaves this way is there is there a fixative in the X there so no this is the change when you are the change when you add it's the logarithmic changes the change when you change the cut off by by a factor yeah yeah yeah changing the logarithm of the cut off to be exact but it doesn't matter it's the change as you change the cut off now what we're looking for though is geometries which don't change when you change the cut off we want something stable we want to say we're talking about string theory on this or that space we don't want the description of it to keep wandering away as we add more of these modes so what we really want are geometries which are stable and don't change when we change the when we change things or we change the cuddle off what does that mean that means geometries where the Ricci tensor is zero if the Ricci tensor is zero that's not saying it's well it's not like saying it's flat flat space is does have Ricci tensor 0 of course it has all of its components of curvature equal to zero so flat space is an example of this question is are there other examples of spaces this is called Ricci flat if a space has vanishing Ricci tensor it's called Ricci flat Ricci flatness is essentially the condition that strings can propagate sensibly are on that geometry have good limits as the number of modes goes to infinity all right so there's this a condition on the geometry it's called Ricci flatness it could be the space-time geometry that the string is moving in in fact that's the way you think about it for the moment the space-time geometry has space directions time directions but anybody know what that equation that equation has another meaning yeah it's essentially the Einstein field equations for the gravitational field the Einstein field equations for the geometry it's the vacuum Einstein field equations Einstein in vacuum no matter just gravity and curvature I'll write out the Einstein equations for you and then I'll show you that it's the same as as this equation anybody remember the Einstein field equations something equals R equals T something like that so here here it is a tensor called capital G mu nu it's called the Einstein tensor and the reason it's called capital G is it's got something good gravity but it's also equal the arm you know minus one half G mu nu R okay where our words are R is R alpha alpha or yeah R alpha alpha and that's equal to 10 you know where team you knew is the energy-momentum tensor energy momentum tensor means the energy momentum will cookies the energy momentum of a cup of coffee but not the energy momentum of the gravitational field everything is in here other than gravity okay the vacuum Einstein field equations mean the field equations for the gravitational field when there's nothing on the right-hand side zero that's called Einstein vacuum equations are there any interesting solutions to this equation what are they what are the solutions of this floor child is one but Schwarzschild is a nasty singularity in it so it's that's no well robertson-walker also have singularities but hmm no neither the sitter space requires a cosmological constant just gravitational waves just gravitational waves just that's it gravitational waves it's similar to Maxwell's equations without sources Maxwell's equations without charges though they have any solutions yeah just plain elector just electromagnetic radiation gravitational radiation also exists the solutions of this kind of equation here in four-dimensional space-time in four-dimensional space-time there's one time Direction three space directions the solutions of this include all kinds of things like gravitational waves interacting gravitational waves gravitational waves can do complicated things in Einstein gravity but there's a whole family an infinite parameter family of complicated solutions of these equations right now we're interested in reaching at this doesn't say our mu nu equals 0 but let's take let's take when I put one index upstairs and one index downstairs that's called the trace when I do want to sum over alpha and alpha all right let's um let's raise one index let's write it R mu nu minus one-half G mu nu R alpha alpha equals zero what's G what is this thing here one new where G with one mute downstairs and one upstairs that's a Kronecker Delta right so that's just the kronecker delta do' communal open you know and now set mu equal to nu and some call them both alpha R alpha alpha equals 1/2 - oh sorry is equal to 1/2 what happens to this when you set mu and nu both equal to alpha and some I think you get 4 right 4 for the 4 directions of space so this becomes either plus or minus 1/2 every times 2r alpha alpha it doesn't it does not say let's say to it to world together either plus 2 or minus 2 I can't remember but it's quite inconsistent unless ro 0 right this equation implies that this is called the Ricci scalar or the curvature scalar the curvature scalar must vanish if Einstein's field equations are correct if they're satisfied so this is not there and the Einstein field equations in empty space is just mu nu equals 0 what do we learned we've learned that the only acceptable geometries in which strings make stable sense and in which the geometry doesn't change in change in change in change as you add more degrees of freedom are the solutions of Einsteins field equations you can quantize strings in background geometries which satisfy Einstein's equations and you cannot formulate them correctly in spaces which don't satisfy in Stein's equations that's pretty impressive that out of the consistency condition for Strings to have a well-defined geometry in the limit that the number of degrees of freedom becomes infinite you derive Einsteins field equations for for geometry that's an extremely impressive fact yes in European connection between the 10150 and the idea that that's been cast informally it is very closely connected to conformal invariants but not conformal invariants in space-time conformal invariance of the world sheet is very closely connected with that it is actually the condition for the conformal invariants but they are connected yes but I've showed you the simplest way to think about it that in general because the string spreads out the effective geometry that it sees is different than the geometry you started with and in general doesn't approach a limit if it does approach a limit it's because at the limit the Ricci tensor is zero so Ricci tensor being zero are special okay now we want to come to a subject Oh incidentally I've cheated a little other than cheated I haven't told you the whole story the whole story is that it really does to work except if you're and the right number of dimensions the subtleties of this kind of thing are quite subtle and it really only works that the answers become independent of of n max if the geometry is Ricci flat and in the right number of dimensions ten for super string theory is 26 for so that's the basic principle the answers should have finite limits as n max goes to infinity and that is enough to tell you that the geometries that you move in must be solutions of Einstein field equations there must be gravitational waves and other things now we want to come to the problem of what do we do with ten or twenty six dimensions how do we make sense out of them and the answer is called compactification compactification is simply the process of taking however many dimensions we want to get rid of by getting rid of them we don't really get rid of them but we make them small enough that they become invisible except the very very small things we want to roll them up into little manifolds that are coarse-grained the apparatuses don't see they may affect and they will affect they'll have you know real serious effect on the structure of particles and so forth the structure of elementary particles type of elementary particles that you can have but they won't be visible as directions that that you can move around in alright so let's talk about compactification we have let's take the case of super string theory where there are ten dimensions of space-time which is six too many and one good thing is that the ten dimensions of space-time have only one time and six space and nine space dimensions we wouldn't have we would not know what to do with two time dimensions what um physics with some extra space dimensions that's that's no problem things just get them moving more but what will mean to have more than one time beats me and probably doesn't mean anything so simple string theory is a theory with nine spatial dimensions and one time they kind of ties into what just question earlier was about degrees of freedom versus to mention his face in essence what I was trying to say never mind whether you know there's twelve or only six the idea is that the more moving parts you have the more degrees of freedom you have okay which is not the case for dimensionality of your coordinates and that's because you're looking at commonalities of those degrees in dimensioning your space the more dimensions you have the more ways to thinking old so there it turns out in electronics design you can have multiple clocks so there are sort of multiple times within the design and essentially what singles out a particular signal as a clock signal is that it synchronizes a lot of other signals or ties things together connects to a lot of things which means it really was only one time and a lot of clocks a lot of clocks is not the same as a lot of times where physics would be like if the only question is if you have regions in which they are effectively independent of one another than they would be so the sports you make a theory with two parms I remember how we say two times we mean at every point the space there are two times not a region way out there was one time in a region up there was another time I mean every point in space well I mean I think the amazing thing about our universe is that isn't the geometry space-time geometry everywhere seems to be the same nothing has only one time it only has the same same stand on unity of the lecture okay all right your party that there are people that don't think it's nonsense that's why yeah there are there are people who don't think it's nonsense however it is never there's never become any part of mainstream thinking and more than one time more than one time direction not more than one o'clock okay let's see let's go on yeah so we have these six extra dimensions of space how do we get rid of them and of course yeah you know you know the answer you roll them up the simplest model would be a world in which on the large scale there was one dimension of space-time is there but let's just freeze time one dimension of space that's a long line we can imagine particles on this line particles move up and down the line maybe they bind together maybe they form molecules maybe they do interesting things who knows maybe even they can form life they can signal each other by launching a particle from one group to another group and you can have some possibly interesting physics these people who live on this line who are made up out of these point particles well I always like to say they have a rather boring life each one has a friend to the left and maybe a friend to the right but my joke is that they don't have a social circle because you can't have a circle in one dimension hahaha very very boring life and also they can't you know they can't pass through each other they get they bang into each other very boring but then they look through a high-power microscope and they discover their line is really surface of a cylinder now they are too big to feel the cylinder their great big things which move up and down but that doesn't mean that there aren't smaller things and the smaller things now can move both along the line and perpendicular to the line so there are new degrees of freedom there are new distinctions between particles particles can move this way they can move this way they can even move this way and there's a whole new game this is the most elementary and simple example of compactification another way you could think about it is let's try to think about higher dimensions let's suppose ordinary space was two-dimensional but that when the creatures who lived in this ordinary space magnified it they discovered a tiny third dimension what might that look like okay and a third dimension in the same sense that that the circle here provided a third dimension so here's here's what it might look like there's the world that they think they live on but now they look under a microscope and they discover that there's a direction perpendicular so there's some thickness there okay there's some thickness there I will call the top thing the ceiling and the bottom thing the floor and there's some bulk stuff namely space in between okay now we're going to do something else having edges like this makes trouble for strings the string start to fluctuate and eventually they fluctuate out and get bigger than the than the than the distance between the floor and the strand of the ceiling but we're going to do something the same thing we do when we well what we're going to do is identify the floor with the ceiling we could have done this let's go back to the cylinder the cylinder is really a ribbon in which you identify the flow with the ceiling you say that if you go out here you come back here in other words here's the ribbon and the cylinder is obtained from the ribbon by identifying one edge with the other edge when you go out here you come back in here same thing here the floor and the ceiling are identified so that when you go out the ceiling here you reappear on the floor now this direction there is no edge there's no edge anymore and this perpendicular direction here has the topology of a circle is periodically identified this would be an example of compacta fiying one out of three dimensions can you compactify two out of three dimensions how many big dimensions big ones would be left over if you compactified two out of three one so we'd be back to the world of one dimension but with something new to compact dimensions instead of one let me show you what that would look like a particular way of doing it this is a particular way of doing it take a well let me make it bigger this is kind of a prism an infinitely long prism I would you call this a prison I thought prism I guess no prisons wrong word um rectangular something-or-other rectangular what let me thank you a parallelepiped except that it's infinite in extent okay right now it has edges but what we're going to do with the edges is we're going to identify this edge with this edge and this edge with this edge not just the edges a point over here is identified with a point on the floor a point on the ceiling is identified with a point on the floor a point on the back wall is identified with a point on the front wall his back wall his the front wall so anything that goes out through an edge reappears on the opposite face at the same spot that now has compactified there's no edges left once you make these identifications they really are no edges anymore you draw it you know to draw it on a plane really on a plane you take the cylinder you slit it and you open it up okay so now it looks like it has a floor and a ceiling but then keep in mind that the real geometry is identified and has no edge same here there are no edges anybody who walks out one side reappears at the other and doesn't even notice it so this is compacta fiying two out of three dimensions how about two out of ten or six out of ten yeah you can do the same thing and exactly the same thing and this process is called toroidal compactification okay what does it have to do with a Taurus let's talk about a tour I because tour I are the simplest geometries to use for compactification the edge of this here this face over here is a Taurus the look like a Taurus but more like a donut but topologically it's a donut it's a so let's talk about tour I the simplest Taurus is a circle that's a one-dimensional chorus they set the circle all right you say a circle doesn't look like a Taurus well it don't worry about it we're going to define the thing that does look like a Taurus let's take a Taurus incidentally when we talk about a Taurus we're not talking about a shape embedded in three dimensional two dimensional shape better than three dimensions what we're talking about is a topology we're talking about the topology of the boundary not the interior of the Taurus we're not talking about a solid torus we're not talking about a donut with the dough on the inside we're talking about the mathematical surface of the torus okay and what we're talking about when we see the word torus is the topology of the surface okay there is a torus and let me now do something to it I'm going to cut it over here well cut it over here I'm going to take a scissor and slice it around here now remember we're not talking about the stuff that's inside the torus we're only talking about the surface cut it and open it up what does it look like it looks like a finite piece of a cylinder right but I have to remember with yeah we take a lit up and up all right it's not the same as the torus but it's Topol basically the same as the tourists as long as we remember that this point is identified with this point this point is identified with this point anybody who walks along here and crosses this edge that's over here comes back over here that's over here all right so we've done the same game except we've taken a cylinder now and identified the left cut here with the right cut through the cylinder over here that is now mathematically a torus but we can go another step we can now take our scissor and cut it along here and now we'll open it up and it becomes a rectangle so a torus is topologically the same as a rectangle but they'll have to remember that the rectangle has identified is called a and this is also a II if this point is called B then this is also called B let's follow the trajectory of a particle as it moves on a torus hypothetical particle fake particle moving on on a mathematical torus like this what happens to it let's see we can follow it let's suppose it starts over here moving this way and assume that it moves in a straight line okay no forces on it what happens when it gets to there here reappears at the bottom right and it goes at the same angle because it hasn't been accelerated and it gets to over here gets over here reappears over here now what goes to here where's it reappear back over here right okay depending on the angle and depending on the ratio of the sides of the Taurus it will either if the ratios of the sides of the tourists are rational and some other good things are it will simply repeat itself and eventually find itself on the original line again so that there will be a finite number of bands like this and then repeat itself if the ratios of the sides are in commensurate or if you some increments or an angle then it will just keep going and going and going essentially filling up the entire who are getting arbitrarily close to any point on a torus but this is not the important thing the important thing is that a torus is a way of compact affine space that's exactly what we did over here we took this and identified it with this this and identified it with this so this would be called a to than a compactification of two dimensions keeping the third one uncompact off' ID but the compactification would be a torus tour i are especially simple geometries for compactification ah there are other geometries that you could try and people do I can't draw it quite the same way but you could take a line times a sphere a line times a two-dimensional sphere so that every point on the line a little fly moving up and down that little insect moving up and down that line would be able to move along the line or along a sphere attached to each point on there just as this does is a torus attached to each point here you could do a sphere for reasons that I'll come to spheres are not good things to compare the fire alright so this is the toroidal now can you do the same thing with them higher dimensions let's suppose we had so what do we wanted to get rid of three dimensions to get rid of three dimensions we want to invent the thing called the three torus a three-dimensional torus we want to take three of the dimensions and make them into a torus yes that's not hard you do exactly the same thing you take a cube doesn't have to be a cube a parallelepiped or whatever you call it and now do the same kind of identifications this face over here a is identified with this face over here AE this one over here B with this one B and the backside of a I see this is in the back is identified with C in the front that becomes what is called the three tourists it has the topology of a three tourists again no edges although if you try to draw it you draw it with edges and that would effectively shrink three directions if you made the torus small in any case if you made it physically small it would hide three dimensions at least from your coarse-grain there are measurements so if you're doing string theory in ten dimensions and you compactify three ten space-time dimensions compactify three you would have seven dimensions left over one time and six space that's also not what we want so what do we want to do we want to make us we want to get rid of six dimensions so we make a six torus a six torus I can't draw on the blackboard for you I suppose I could draw a for tourists but there we get a recorder up the blackboard terribly what's in what's to figure that if the for cube called and over here you know what tell ya and then identify faces of that but um or volumes of it you be buying volumes of it but mathematically all you're doing is adding axes and keeping and making identifications a total yacht seems like all these dimensions all these spatial dimensions greater than three come back on themselves yeah is there some periodically identified there periodically identified yeah they come back on themselves is it that's fundamentally I guess well can't go infinitely x y&z key you could have six or ten or nine whatever it is dimensions which don't come back on themselves it just wouldn't be our world you asked why did nature choose to to compactify some dimensions and not others that we haven't got the vaguest idea but we do know that we live in a world of three dimensions string theory or no other theory that I know quantum field theory none of them make sense with more than one time dimension yeah we know we have one real time dimension with there's no more room for four more why beyond the four to assume that the remaining dimensions are spatial what else were they gonna be I'm not well I've got some ideas but my life my point is it just because we can't identify what they are why do we assume they are spatial and curled up as opposed to something that else that we just don't understand because we can understand this it doesn't mean it's the real world it means it's a thing we can study find out its properties see how it behaves and compare it with the real world if you just say maybe it's something and we don't know what it is okay that will stop the dead ah this is easy it's straightforward it's not enormous not only nutty's tour i these tour i are pretty easy they're not enormous ly subtle they're solvable we can work out what particle physics is like we can work out what what the world on in such a setup is like and we can ask how close is it how similar is it to the real world that's the only reason it's a it's a setup that we know and understand it has gravity it has quantum mechanics it has particles it has bosons it has fermions and we can study it it also has black holes so we can study the black holes and find out if they do this or that that's that's the reason what is it laughs it's too symmetric it has supersymmetry and other symmetries that are just two symmetric little right some things that lacks and some things it has too much of when you take the bagel and turn it into a rectangle how much you recover but if you look at the distance of Y we inside it ya don't think about the distances right this is these are when a mathematician speaks about a torus or a physicist for the most part he's talking about a topology not talking about a geometry now in fact the torus when it's presented as a rectangle oh let's let's talk a little bit about how many how many in equivalent tour i there are what is the what are the parameters of a torus alright it's a rectangle so first of all it has an overall area or equivalently it has a length and a high detail within a height okay so it has a width and a height which are two parameters one of those parameters is the size of it you could take it to be the area call out a size parameter the other is the and that's the product of the length times the width times the height there's also the ratio of the width times a high you could call it a what aspect ratio or the shape alright so here is a long thin torus here is a broad fat torus but these are the same these are really the same torus oh that is to say that geometrically identical that geometric lead identical every point on here is equivalent to a point on here this maps to this this maps to this we've just turned the picture on its side and we haven't made a new kind of torus so one thing we can say is there's the ratio of the sides but the ratio of the sides does not go from 0 to infinity it goes from 1 to infinity why 1 to infinity because when it's less than 1 the ratio let's say of the horizontal to the vertical this is bigger than 1 this is less than 1 but they're the same torus so the ratio of the sides can vary from from basically from 1 to infinity well from 0 to 1 either way either way you in fact usually take from 0 to 1 and go from 0 to 1 anything else is there anything else that that you can do to change the torus and the answer is yes here's the game that we haven't discussed let's go back to our piece of cylinder we're going to identify the left edge the left circle over here with the right circle over here but before we do it let's imagine a twist a twist by an angle so that you don't identify this point with this one up here but you identify it with a displaced angular point this one over here is identified with here this one over here identified with that you imagine in your mind before making the identification making a little twist and then making the identification that's something new and what it looks like let's say let's let's do it vertically let's draw it this way like a little twist here before we make the identification so that slides things along the top edge if we didn't do the twist we would have a rectangle if we do the twist then we don't identify this point with this one we identify it with that point this point with that point and we might as well draw it like this might as well draw it as well if that doesn't look very neat that's not quite right yeah all right everything in the bottom gets it identified with a point above it and everything on the Left gets identified but with this offset with this angle so to parameterize tor I there's the aspect ratio of is that there's the size that can be taken to be the overall area then getting rid of the area let's say unit things of unit area there's the aspect ratio and there's the angle these things are called moduli these are the moduli of the torus there are three of them the three moduli of the torus or the overall size the ratio of the height to the width and the angle Bleen the angle of Leaning these have technical names the twist is called a dane twist de hn if they're a mathematician I suppose named Dane the overall volume is called a Kaler modulus and the ratio or aspect ratio is called a complex structure majalaya they have technical meanings as you see them you'll know roughly what they mean flip the ring as well flip it you mean reflect it then you make then you make a coin bottle which scares different topology you mean if you if you take if you take this point over here identify it with a point in the front this point with that point this point with that point and this one with the one in the back that's a Klein bottle that's not that's not that's not a Taurus it's not it's not an orientable manifold yeah yeah yeah yeah you can better that's right that's right you can absorb that kind of thing into the metric on the Taurus you're right there are other ways of the forming the Taurus but those can be absorbed into the metric on the Taurus these Torah are all flat they have flat geometries or they can be given flat geometries they since they can be drawn on the blackboard without stretching them I don't mean the real I don't mean the real doughnut the real doughnut is a truly curved surface it has curvature on the outer boundary it has positive curvature on the inner near the hole here it has negative curvature I'm talking about this this mathematical structure that's been mapped onto the plane in this way it's completely flat triangles on it have a 180 degree of some of their angles even on this one and in particular it's not only flat but it's Ricci flats Ricci flat and it's flat and so string theory on these geometries is well-defined it's a good the good mathematical structure for string theory to exist arm it's by far the easiest way to hide the other dimensions and it's called toroidal compactification it's not good for it's not good for the real world the real world is more complex than that but I thought I would show you what the simplest form of compactification is now I'm going to let's see let's take a five-minute break somebody wanted to make an announcement I can't remember the argued there they were yeah tor i are all a flat tor I are all flat tour I is the plural of Taurus you realize tor I are all flat and any number of dimensions and therefore they're all so Ricci flat and because they're Ricci flat string theory strings moving on tour I are good things they have a sensible well-defined mathematics there are other Ricci flat spaces are where you can take six dimensions that you don't like and get rid of them by compacta fiying them not on a torus or not by replacing them by a tourist but by replacing them by other Ricci flat manifolds in particular there are a class of Ricci flat manifolds which are known as calabi-yau manifolds which have very special properties and where string theory is a good theory these are incredibly complicated they're far beyond the scope of this course but they're also the kind of things that string theorists which have enough complexity and lack of symmetry that they do look more like the real world but they don't really involve any really new concepts that we some mathematical concepts to be sure they're very mathematically difficult objects but no really new principles come out of they just look more like the real world then I'm not going to try to get into them Taurus is good enough for us we can see some very interesting things when we start to explore string theory on a Taurus okay let's let's go let's take is on model for simplicity just the infinite one-dimensional world with an extra dimension that goes around in a loop you see why I call a circle a one-dimensional torus it's just a line element periodically identified alright let's start with that all right we can have a particle which moves around on here and it can have a component of motion horizontally nothing special about that that's just the that's just the momentum along the direction it would have been there even if you hadn't compared it well even if you didn't have the extra dimension you'd have momentum along that direction but you also have momentum along the other direction you can have both of them at the same time a particle moving sort of obliquely so spiraling around this thing would have a component of momentum horizontally and a component of momentum along the circle the components of momentum along along the circle are quantized momenta momenta on periodic spaces are always quantized the easiest way to see it is just to say supposing supposing the circumference of this circle let's just call a circumference 2 pi R that's just that that's just what I'm calling the circumference if I take the momentum along the r direction there's a momentum along sorry along the circumference let's call that P along the circumference PC P along the circumference and I multiply it by R which is the circumference divided by 2 pi I get something which looks like an angular momentum in fact it is a kind of angular momentum it's there's an angle associated with this circle just an angular position along here and P times R that's just like that it is like angular momentum okay that's quantized in integer multiples of Planck's constant so the momentum along the circumference the momenta along the circumference is quantized in units if we set Planck's constant to one then the unit of momentum is 1 over R let's suppose the particle we're talking about is massless if it happens to be a massless particle then its momentum or the magnitude of its momentum is the same as the magnitude of its energy energy for a massless particle is equal to the momentum of course as a factor of the speed of light apart from the factor of the speed of light energy is equal to the magnitude of the momentum this component of the momentum can be positive or negative and it's an integer the implication is that there's a that there is a quantized amount of energy even if the particle is not moving along the horizontal direction if it's moving in the circular direction it has an energy which is n units divided by R all right that energy it's from our point of view why we don't see the circular direction it's just a thing standing still it's a thing standing still but the manifestation of the fact that it's moving is that it has some energy it has some mass mass mass this is the mass now of a particle and it comes in integer multiples of 1 over R the smaller R is the bigger the spacing between mass levels the bigger the spacing between mass levels so by knowing the spacing of course we don't measure any of this but in principle by measuring the spacing between the masses of these different particles how are they different they're different from a macroscopic from our point of view they look different they have different masses from the point of view of a small object you know a small detector which can see inside this thing they're just particles which have different components of momentum in this direction we will call them different particles because they have different masses and the spectrum of them namely the spacing between them tells us how big the internal space is the bigger the internal space the closer the levels let's draw a level diagram energy energy equals zero or mass mass equals zero is right over here then we can have plus 1 unit of rotational momentum around here that would give us a particle of mass 1 over R and so forth what about if n is negative does that mean has negative energy no the the energy is the absolute value 8 I think I erased it the energy is always positive so there's a particle of mass 1 over R here there's a particle of mass 1 over R yeah 2 over our - or 2 over R the spacing between the mass levels is a direct reflection of how big the circle is the smaller the circle the larger the spacing if this circle is very very big then the levels are very very close together so if we discovered excited states of particles which were very very very densely spaced we would say oh it looks like there's an extra-big rather big dimension if we find that the spacing due to this kind of motion is very very sparse then we say that dimension is small right okay so these are the energy levels for the components of mass of air mass of a particle where the particle is simply a particle moving around this direction now you can do something else in a compact space you can have a different kind of particle which from the outside meaning from the big dimensions just looks like a particle but what has an entirely different structure Oh incidentally if all particles are strings if particles really are strings that I might want to draw this as a little string over here but just think of it as an almost tiny little particle let's forget the fact that it spreads out so tiny particle and it moves around the whole center of mass of it moves around and gives it a series of energies in this way but there's something else you can do the other thing you can do with a string and a compact space like this is wind the string around like that here this band around here is once wound around the cylinder of the cup I could take a rubber band and wind around that way that can also move up and down the axis its first of all it's localized it's at a definite place in the big dimensions so it has a location from our point of view it can move it can move up and down it's a particle it's also a particle it's a particle which is wound around the extra dimension what is its mass what does its mass its mass now is not dune - its motion in this direction it's due to the potential energy of stretching a string around that distance how much energy would you expect there to be in a string stretched around here the answer is clearly going to be proportional to the length of the string there's a certain tension and energy per unit length for every string let's set it equal to one we won't worry about units tonight tonight I'll ignore units set I'll set just work in units where the string tension is one if a string is stretched out to a certain distance then the mass that that string has is proportional to the length of the stretched string how about a string which is wrapped around this direction or do you expect its mass to be proportional to the circumference R yeah the radius is covered so this one will have a mass equal to R but proportional to R but you know you can wrap it around several times that I can't do with the the coffee cup but I can draw it I think no no but I want to connect it back together again I know I know I got it yeah it crosses here and connects back here and you see it if I had a rubber band that could die if you can wrap a rubber band twice around here are you pick it twist it and put it on here all right how about this one what's the what's the mass of it how much energy does it have two are twice as much twice as much stretching so we have our we have energy our we have energy 2 R 3 R 4 R if the string has an orientation in other words if you think of the string is having a direction then you can wrap it positively or you can wrap it negatively here's one wrapped positively here's one wrapped negatively so this wrapping or winding number it's called winding number the winding number can be positive imagine you have a rubber band but on the rubber band you draw a little arrows to give it a sense of direction and you're going to wrap it around your wrist you can wrap it around your wrist so the arrows are in this way or so that they run this way so the winding number can be either positive or negative but whatever the winding number is the mass of that string is equal to the absolute as R times the absolute value of the winding number okay so there's another spectrum of particles here and that spectrum of particles has energy levels whose distance is not proportional to one over R but it's proportional to R itself let's suppose R is very small let's suppose that r is very very small then this spacing is large and this spacing is very small it takes very little energy to wrap around a small circle okay but for the particles which are not wrapped but which are not wound around but which are moving around literally moving and have momentum the spacing between levels is large if R is small so we have these two spectra of two different kinds of particles one is called these are these ones here are called Colusa klein particles they were this idea has been around the particles having momentum in an extra compact direction that goes back to beginnings of general relativity Colusa first had the idea 1917 I think and it's grown up a great deal since then but string theory introduced a new idea of wound particles particles wound around and they have very complementary kind of spectra so if you see particles with a large spacing you must also see particles with a small spacing both combinations together would tell you something about the the nature of the compact directions but notice that there's no way to make the spacing arbitrarily large here without having very small spacing a small spacing of course means that it just takes a little bit of energy to create one of these things on the other hand what happens if you take are very big now imagine taking are very big if you take are very big then the spacing of these Colusa klein particles is very small these are very close together now all is very big one over R is very small these become very closely spaced and what happens to these they get very widely spaced why because it just takes a lot of energy to wrap it around a very very big cylinder here so you have these two complementary kinds of particles there's almost a symmetry there's almost a symmetry of the spectrum where you replace R by 1 over R you replace our by one over R and you replace momentum by winding number and what happens you just need to change these two and the spectrum stays the same that seems to be a symmetry of the spectrum of power of string theory this only applies to the extra dimensions there's no way to wrap a string around the X the real x axis it's infinite but as applied to the extra dimensions there seems to be a symmetry where you replace R by 1 over R winding by momentum around that direction and the system comes back to itself that no no no no this is not a vertical versus the horizontal this is the duality between winding number and and momentum and Colusa klein momentum now from what I've told you it's hardly clear that this is an exact equivalence in every possible respect it's just an equivalence of the energy spectrum of some some simple particles in fact it's an exact symmetry of the theory if you take a compact direction and you start shrinking it down smaller and smaller well at some point incidentally the specter will be equal some some radius there's some radius where the spacing between the Colusa klein particles and the winding particles will be the same we could call that R equals 1 in some units at R equals 1 the spectrum are the same otherwise they cross if you try if you start trying to make the radius of compactification smaller and smaller you eventually get to this point where when you cross it you can either think of it as saying that the size is smaller than 1 or you can interchange winding momentum and say it's bigger than one there's no sense in a certain sense there it's not possible to think about string theory where R is smaller than a certain size oh you can think of it if you want to make it small on a certain size but all that happens is it rearranges itself so that it looks like string theory on the larger space does R less than 1 in our greater than one of those different kinds of particles are they all for me answer Oh every fermions bosons they all they all do the same thing so there's 10 or 26 dimensions it's yeah yeah it imparts to every compact direction here I've only indicated one compact direction circle we can apply it the problems with several compact directions but this is a kind of freakish surprising result that's characteristic really of string theory it is not characteristic of point particles moving in closed spaces like this it's the interchange of winding with momentum that hardly seems an obvious thing to do remarkably the entire theory of scattering amplitudes the spectrum of particles the whole works is symmetric under this interchange no no no that both closed strings closed strings closed strings open strings that doesn't make any sense to winder army just unwind themselves no no they're closed both closed strings and the theory I should have said that in a theory of closed strings there's this interchange symmetry open strings are a little more complicated you know and we'll talk about them to be similar to the black hole analogy where dimensions become time life and time becomes an action light nor know what it is is it saying that there's a certain distance where when you try to think of string theory on a space smaller than that it's equivalent to string theory on a space bigger than that trying to shrink the diameter of a compactification - arbitrarily small distances you get frustrated it just rearranges itself so that it looks like the theory our business self energies well they'll say in some sense there's a smaller smaller size scale in string theory but there keeps things from being infinite yes that's true if you have a winding number and what was to say that there is a winding number is that just a theoretical model or there analogy to physical space in the absence of a winding number right any accuracy if you have a compact dimension then it's possible to think of strings which om are you asking maybe you're asking how you make them if you don't have them begin work well I'm asking is if there is no existence of a winding number a winding number or a particle with a winding number article you can make them I'll show you how to make them we do have to start with something so we start with a particle which has no winding number that's a little string that looks like that okay now we take this and we pull it around we stretch it out it's still got no winding number it's not wound around the thing and now we stretch it all the ways around here so that comes back the back side and comes up over here can you see what I've done it still has no winding number the winding number of this piece is canceled out by the winding number of that piece so you take your arm you take the rubber band and you pull it around and then take you know I'm going to take the two pieces here and that's what you're left with but still has no winding number but remember now that in string theory the basic process of interaction is for things like this to become things like this rearrangement so if you sit there for a while this string will rearrange and form that now what does this have it has two particles with opposite winding number okay we still have this net zero winding number one is wound one way one is round the other way but now they're disconnected and they're free to separate throw that one away just eject that out of the system give it some velocity and when it gets off the alpha centauri forget about it and you're left with a with a particle with winding number so whether whether you like them or not you can't forbid them some things you can try to forbid you can say if they're not there to begin with they'll never be there possibly some things you can't forbid because you can imagine that the processes of the theory itself the important processes in the theory can force them on you and these get forced on you of course they always get forced in pairs but you can that that's that's a throwaway half of them and just do some experiments with the Rigby remaining ones and this is with increasing energy I guess then the paluszek line close to straightening gets larger every morning yeah it does it get some energy from not only from momentum but from being stretched yeah it takes energy to do it uh but you know this is the sort of thing which would happen if you collided two particles with no winding number just two Colusa klein particles you collide them hard every possible combination of stuff happens and among the things which would happen is these winding and anti winding things we go flying off it's and the logic is the same as the logic of electric charge supposing the world started with no electric charge does that mean there would be no electrons no any kind of collision that would take place we create electrons and positrons and then the positrons could go flying off to some other place and you'll be left with electrons so the logic is really no different than electric charge you can't not have electric charge because you can always make it in pairs the net charge in the world might be zero for all we know the net winding number might be zero but who cares we can do experiments on you know on a region which contains a winding number so yeah so this has a name incidentally the equivalence between theories which look different usually called a duality this is called t-duality capital-t duality and the T in this case stands for torus the T stands for tourists this is torus duality or T duality and as I say it's a duality which relates compactification Zahn very small geometries to compactification Zahn big geometries and it's a rather startling thing I mean it when it was first discovered it surprised everybody what it means is that geometry defined in terms of how strings the geometry can also often be recovered or partly recovered from the spectrum of modes on a the geometry of a drumhead the shape of a drama there's a famous mathematical question can the shape of a drumhead be predicted from the sound that it makes from the from the spectrum of vibrations the answer is not quite but you can predict a lot of the shape of the drumhead the same question applies to these compactification 'z from the spectrum of particles a spectrum of vibrational energies vibrational and other kinds of energies can you predict the shape and/or the size of the compact directions and the answer is yes to a large extent but there are some ambiguities and some dualities that you can't tell if it's a tiny geometry or if it's a big geometry because these two go into each other that would not be the case for the drum I assure you you would not mistake a small drum for a big drum okay good for more please visit us at stanford.edu

Channel: Stanford

Views: 65,404

Rating: 4.8743458 out of 5

Keywords: physics, science, atoms, energy, string theory, m dimensions, constraints, 26 dimensions, einstein, electrons, protons, neutrons, graphs, closed string theory, particle physics, theoretical, black hole, spin, scattering, sigma, directionality, string, sca

Id: Tsav_xnegTk

Length: 115min 56sec (6956 seconds)

Published: Wed Mar 30 2011