Lecture 10 | String Theory and M-Theory

[Music] Stanford University all right we were talking about something called t-duality t-duality was very very important to the history of the mathematical developments of string theory let's go back over it again and discuss it a little more fully and then I want to tell you how it led to the concept of D brains and how D brains have become something of a mathematical tool for studying quantum field theories the kind of quantum field theories but have nothing to do gravity but the kind of quantum field theories that we use in a day-to-day basis to understand hadrian's quantum electrodynamics and even quantum field theories that are interesting to condensed matter physicists we won't get to all of this so obviously we won't get to this today but I'll just try to give you some picture of what it's about all right we we imagine that there was some compact directions of compact means these small ones which we rolled up the simplicity I will imagine that the compactification is toroidal on Torah in other words not necessarily two-dimensional torus a one-dimensional torus is a circle or a line interval with the endpoints being identified a two-dimensional torus is a non erect a parallelogram doesn't have to be straight the parallelogram with opposite sides identified a three-dimensional torus is a cube or not necessarily a cube but a parallelepiped thank you a parallelepiped with opposite faces identified at this point in that point this point and that point and you know in front in the back so in general in any dimension the concept of a torus is a well-defined concept and those are the easiest cases to study those are the easiest cases to study in these cases of course supplement on top of this is the ordinary four dimensions of space-time so this is what's present at every point of space-time ordinary space-time there are other directions that we can move in or that somebody small enough can move in and that's the setup now let's focus on one particular place in space-time and ask what could be there a particle can be there but now let's zero in and zoom in and ask what that particle looks like on scales which are so small that these compact directions become visible so let's start with a simple picture that we had just to just to get started we imagine one large direction and one small direction the small direction now would be called a circle a circle not because it's embedded in two dimensions and looks like a circle but just because its head études tail that when you go around come back and come back to the same place all right so that's our space-time where as time I don't know time is add time into it time is not drawn on the blackboard this is pure space okay in this case two-dimensional but if you like we can generalize at the higher dimensions the compact directions become tore I the non compact directions the big ones just become our three dimensions of space so however many dimensions of space we want now in string theory particles are strings let's take the case of not all string theories are the same they're different from each other but there's a fairly small classification of them the strings theories that I'm interested in right now have only closed strings we're going to come back to what happens when you have open strings in a little while but for the moment only closed strings no end points that's number one so they're like rubber bands they're like rubber bands here they aren't closed and furthermore that what we would call we call oriented oriented means that there's an intrinsic dis difference between going around the string in one direction rather than in the other direction it's mathematically like a rubber band in which on the rubber band we drew a series of arrows to indicate direction around and we'll keep track of that orientation an example of what that orientation would say is when the string splits little splits like this it will split into two strings with the same orientation on the blackboard so in that sense orientation is preserved for these strings they remember a sense of which now of course that doesn't mean that they're in equivalent I mean it may there may be a symmetry and the sense that a string with the opposite orientation may have all the same properties but what it does mean is that you can compare two strings to see whether they have the same orientation or the opposite orientation to bring same with electric charge a positive electric charges behave exactly the same way as well as negative electric charges in in that for example if you replaced in the real world every electron by a positron every proton by an antiproton and with it every Neutron by an anti Neutron chemistry would remain exactly the same or at least to a very very high precision would remain unchanged so in that sense plus charges are identical to minus charges but you can tell the difference between the plotter you can know you can't tell the difference between a plus charge and a minus charge but if you have two charges you can tell whether they are the same or opposite how to see whether they attract or repel you can't tell whether they're both plus or whether they're both - but you can't tell okay so the same is true of oriented strings that they have a sense of orientation if you replaced every string in string theory by a string of the opposite orientation the theory would be the same but if you have two strings you can tell whether they're the same orientation of your episode let's just call them oriented strings then you can wrap them well you can do two things you can have a string which is not wound around become eight some compact direction there it is it is drawn on the surface of the two-dimensional world here it's free to move it can move in this direction or it can move around so it can have momentum and that momentum can be along the direction of the large directions or it can be along the short directions or it can be a combination of both it can when I say it has momentum we can think of that as motion and it can move this way it could move this way or it can even move in a in a helical pattern like that having both components of momentum the components of the momentum in the short directions and the finite compact directions are quantized they are quantized in integer multiples of the inverse radius or the inverse friends the distance around the clothes called a cycle around the clothes cycle do the streets have to go all the way around the cylinder can it be like computer this one isn't going all the ways around what do you call it I thought they went from like zero to two five in close in what space Wow no this is not Sigma this is some X this could be X we could call this direction Y around here let's space that's real space Sigma is just a parameter which changes along the string it's just naming the points along the string you have a rubber band and the rubber band is composed of molecules okay the molecules can be named the first molecule is Harry then George and Fred then Sara and so forth that's Sigma its naming the particles as you go around the string okay and if the string is closed yeah then Sigma comes back to itself yeah the size has to be the same what does the size of the compact Direction have to be the same at every place in the long direction we'll come to that the answer's no no it does not but we'll come to that okay so Sigma varies along here along the string and it goes from 0 to 2 pi but that has nothing whatever to do with whether the string is wrapped around space so yeah think of it I'm sorry I didn't bring a rubber band think of a rubber band we're on the surface of the rubber band we've marked off Sigma equals 0 Sigma equals a small number or twice the small number as we around the rubber band here it is rubber band and we mark off points on the rubber band that's Sigma that Sigma has nothing to do with whether I wrap the rubber band around my wrist or not it's always there it's Sigma it goes around once as the string goes as we follow the string around its own shape Sigma goes from 0 to 2 pi it's a completely separate question of whether the string is wrapped around the extra dimension in the same sense that a rubber band could be wrapped around my wrist you can wrap the rubber band around your wrist if it's an oriented rubber band you can wrap it so that the arrows point this way or so that the arrows point this way so we have two kinds of wrapped strings is a wrapped or while the correct term is wound wound around a compact direction here is one of them it comes around the other side and it points that way okay that's one the other one goes in the opposite direction let's take to wound strings like this to wound strings like this and let me just draw it a little differently let's put do it this way this one goes what can happen to to wound strings which around in opposite direction well remember the basic the basic process of string theory is for Strings to come together and join and split now the rule is that when they join and split orientation is remembered a string like this and a string like this can let's put some more orientation arrows on it let's ask how they can join they can join like that I don't think I need words to describe it I think the picture describes it well enough the lines have to be or the arrows have to be continuous okay what you can't have is a string splitting somehow like that here the line can't be continuous you run into a contradiction as these points come together alright so this is the kind of thing that can happen the opposite namely the time reversal of it can also happen strings which are like this can come together and join and so forth alright that's the basic phenomena of splitting and joining and it's the thing which is governed by a coupling constant so what can happen here what can happen is clear this can happen let's say oh I lost track of the orientation now it's not really wound anymore as from the point of view of topology you can unwind it into this all we've done here is we've taken this we've stretched it around and then joined it on the far side to form the original wound pair well we can undo that and now it's unwound it's just a single unwound string all right now one way to think about it is to keep track of winding number winding number can be defined to be positive four windings which look this way we'll call that winding number one and it was wound the other way we'll call it winding number - I won't try to give you a mathematical definition but winding it on my wrist if if the arrows point this way we'll call that positive winding if they point the other way we'll call it negative winding if I have a single string which is wound like so what's the winding number the winding number is +1 if I have another one forget this one for the moment just throw it over on the side we have another one which is 1 this way that has winding number -1 the two together all together have winding number 0 this has winding number 0 it's not 1 on the round at all so what's preserved is winding number let's take another case let's imagine now two strings which around the same way there's two strings round the same way what can happen to them they have winding number two the sum of the two of them have winding number two what can happen well let's let's let's draw it over here again [Music] these are going this way and they come around here they're both going the same way you can't unwind this you can do something you can have a process of splitting and joining which does this which interconnects them a new way can you see what I've drawn across them we've crossed them but still after they've crossed the arrows are still continuous there's no contradiction in the direction of the arrows this also has winding number two you learn if it's one string with winding number two this is two strings each one with winding number one each one with winding number one but the sum total of the winding numbers is 2 all right so here's again another example winding number is conserved you can't change the winding number although you can change the number of connected components so if you have winding number 100 you could have one string wound around a hundred times you could have two strings each round the same way 50 times you could have a hundred strings each round once and they can all communicate in the sense that you can morph from one configuration to the other but you can't change the winding number so winding number is an absolutely conserved quantity in this kind of string theory never can change let's let's forget why let's come we're going to come back to winding number two moment well maybe we should discuss winding them yeah okay this one over here that can just be thought of as a tiny little particle let's just think of it as a little particle it's a string but it's a particle what's it characterized by it's characterized by components of momentum ties by many things shape and all sorts of other things but in particular let's characterize it by its components of momentum in particular the component of momentum in the in the this direction over here the component of momentum in that direction has to be an integer number that's called in integer number of a quantum of momentum and the quantum of momentum has to do with the circumference let's just call it the radius around here our distance around here I'll call our it's a circumference strictly but I'm just going to call it R because R is the standard term for it the quantum of momentum in this direction and also incidentally the quantum of energy in that direction for matter if these are massless if these are massless particles energy and momentum are proportional to each other for example if they were gravitons if they were graviton moving this way or this way the energy is also them as also quantized in that unit and so the energy of the momentum is the momentum can have either sine it can be plus or minus the momentum the energy is always plus for the same amount n over R it has momentum n over R and it has an energy also n over R that's the these are that's the energy due to momentum and in particular momentum along the small direction here it's quantized so that's a that's unwound particles and the contribution of energy due to momentum oh let's think about now what yeah what is the momentum of a string moving in space basically it's just the velocity of the center of mass it's just the velocity of the center just like a rubber band just like a rubber band its momentum is its mass density which for simplicity we can take to be 1 times it just its mass which we can take to be 1 for a rubber band times its velocity so the velocity the momentum is proportional to the velocity and we can think of this as the X the tau tau is the time variable that describes the motion of the string in the quantum and the simple quantum mechanics of we've described previously and it's got to be in integer multiples now let's think about the wound strings the strings which are wound around what is the typical energy of a wound string well strings have a certain tension let's just work in units in which that tension is one but tension is energy per unit length so for energy per unit length if there's a given energy per unit length then the energy of a wound string is not in / R but it's some other integer I'm going to call the other integer W W stands for winding number times R the energy of a collection of strings which are wound around here are the integer winding number times R whereas the energy of a string which is not wound around the compact Direction is n divided by R so I think we talked about this last time but let's discuss it again a little bit let's suppose that our is very small all right here we have the spectrum of the strings which are not wound the separation between the levels is large large because one over R is large so it looks like this with the separation being of order 1 over R is N equals 0 is N equals 1 and so forth now let's look at the spectrum of the wound strings in other words the energy levels of the wound strings those also come in integer multiples is 0 winding number one winding number 2 winding number -1 winding number minus 2 but now it's W times R so that means if R is large well let's say R is large or I was smart I think I said I was small what did I say I said R is large or small or small okay so that makes the separation large here but it makes it very small here now let's go to the other extreme I know that we talked about this before but I want to reiterate it again let's go to the other extreme this is our much less than some unit of length in string theory the natural unit of length of string theory let's just call it 1 is are much bigger than 1 what do the energy levels look like well 4 are much bigger than 1 1 over R is small and these excitations here are very close together what about these here R is very large so it takes a large energy to wrap around they look like this suppose all you knew all you could measure about these particles namely the things the social I'm talking about the things associated with the compact directions the only thing that you can measure about the compact directions let's suppose was the energy of the particles and you're trying to figure out what the radius of compactification that's known as the radius of compactification what is the radius of compactification well your problem is how do you tell the difference between particles which have energy with respect to the compact direction because they have momentum or because they have winding number you can't obviously just from the energy levels you can't tell so with this particular spectrum there's an ambiguity are you in the theory with a small compactification in which large separations between different momenta small separations between different winding number or are you in a world with a very large radius of compactification and in which the role of winding number and momentum has been interchanged you think well maybe you can do a lot of other kind of experiments maybe you could scatter these particles together see what comes off one of the very very remarkable mathematical facts about string theory is that the inability to tell which kind of world you're in is rigorously correct for everything that you can calculate about these strings collisions between them that that two different theories with very different radii of compactification if you interchange winding with with let's call it with momentum around the compact direction the theories behave in exactly the same way so conclusion arm in certain sense it doesn't make sense to think of the compactification as smaller than a certain size there's a certain size where they cross over and where they interchange and if the size of the compactification is smaller than that it's just entirely equivalent to a larger compactification with or with winding and momentum integer does that mean that the winding number is a mathematical convenience rather than a physical well I don't know what to say it says that it says that that there's a symmetry it says there's a symmetry and that there's physicists use the term duality which means an equivalence between different descriptions of the same thing is there are of interest for these two things of the same in some units it's R equals one and there you have some extra special symmetry that you go that not only can't you tell which kind of theory you're in but you can't tell which are which are which yeah they're the same yes there is a special point now I wrote an equation over here that momentum is equal to the X T tau let me make that a little more precise every point on the string has a motion in particular I'm thinking about the motion around the compact direction but different parts of the string could could in principle be moving different than other parts of the string ah no it could be differentially moving in different directions different there are different speeds and so forth the full momentum that we're talking about when we talk about the momentum is actually the sum of the momenta of all the little bits of string [Music] give me the summit or you can think of as an integral over the entire string of the velocity the X D tau and we checks am I talking about I'm talking about why actually the one that goes around the string dyd cow what about the winding number anybody got a mathematical expression for the winding number imagine now a string which is warmed simplest possibility its wound around here this is the y coordinate the y coordinate is periodic it comes back to itself but the string also has a sigma coordinate the string also has a sigma coordinate and that's embedded in the string if the string is wound in this very simple way there another way of saying it is that Sigma is proportional to Y actually Y divided by the total radius of compactification I guess if Y comes back to itself after going distance R and Sigma comes back to itself over after going around once then the right relationship is that Sigma is y over R it's just another way of saying that as you go around Sigma as you go around the rubber band the rubber band goes around the Y direction well now look at this let's look at the Sigma by dy that's just 1 over R the Sigma by dy let's just take the signal by with the Y the Sigma by dy is proportional to the little spacing between here and here and the total winding number is the sum of a little bit of the Sigma dy as you go around the string in other words the winding number yeah actually this is right R times this and the winding number I believe is what I had an equation where'd it go [Music] Y over R is equal to Sigma so the Sigma by D Y is 1 over R is one of our the total winding number is 1 over R times the integral dy by D Sigma divided by the Sigma does this make sense that the total winding number what's the integral of a derivative what you started usually the integral of a derivative in lots of cases just plain zero if a thing comes back to itself when you go all the ways around but why doesn't go come back to itself when you go all the ways around Y itself changes by 2pi so the integral dy DT Sigma is just a total number of times that the string wraps around that direction so this is kind of curious t-duality this funny duality between winding number and momentum is equivalent to interchanging NNW interchanging R and 1 over R and interchanging the Y D tau with the Y D Sigma funny mathematical construction keep that in mind that t-duality is interchanging winding and momentum interchanging are with 1 over r and interchanging the derivative of y that's a position of the string dy D tau would be y by D Sigma where ya dy by D tau would be y by D Sigma the derivative this is partial derivative it's pretty abstract but this is an exact symmetry of string theory if you make these interchanges nothing changes it saves it so I say once again if the symmetry is dy/dt swap dyd town swap the yv tau for the y by D Sigma now the reason I introduced this will will see why I didn't have to tell you this but I'm telling you now because no is the relationship it's just you interchange them wherever is wherever you saw the expression dy by D tau you replace it by the expression might be Y by D Sigma it's a weird thing to do but in all observable quantities scattering amplitudes energy levels all of the properties of the string theory don't change if you make those changes let's talk about something else yeah the stream don't want to land around the DNI's there warn't anything why we did it get in this nice stable if it can be involved in the other step forward somebody may hit this string over here with some energy hitting it with some energy among other things may stretch it out in fact it may stretch out to look like this you blast it with a lot of energy I've drawn a very neat configuration over here of course but if I blast it with a lot of energy it might be much more complicated than that but still have this kind of configuration now once you've done that and you take into account that the strings split and join then this can separate itself into two strings with opposite winding number once you've created two strings with opposite winding number that I'm no longer connected they can separate from each other they can separate from each other and when they separate from each other well then you've got an isolated string with winding number one and another one that you can just send off the Alpha Centauri and never see it again the answer is that in general you cannot prevent it from happening you may have to in particular if the distance around here is large then it takes a large amount of energy to create these winding strings here it is the well let's say we yeah the energy the energy itself was proportional to our four winding number so with a small amount of energy it's not going to happen you don't have enough energy to stretch the string that far but if you collide two strings with a large amount of energy they splatter all over the place they stretch out and Wilder in the chaotic directions and there's some probability that the that they'll reconnect it will not change the total winding number but it will take a single string or perhaps two strings which are not wild which collide with each other and convert out of them something with two strings of opposite winding number once that happens they can separate and they go off so the answer is generally that that will happen in collisions if you don't have it that way to begin with the same question occurs for electric charges if you start with a neutral world why are there electrons well even if there weren't electrons or their got to be something to start with is nothing there's nothing but maybe you just started with photons you take some photons you collide them together and what comes out is electrons and positrons if you take the positrons and you throw them away you don't throw them away but you push them off to some distant place you left with electrons that's all so you you can't have you can't forbid these things you can't forbid them and eventually you will create them now these winding numbers and these momenta these momentum quantum numbers is quantized momentum or not only in some sense similar to electric charge but they have all of the properties of electric charge in particular two strings of opposite momentum one going around one I do this let's see oh yeah I can do it - screams going around the opposite way with respect to the big dimensions of space so here they are two strings these have the same no these these have the same winding number okay that's easy that's this two strings are the same winding number will repel I'll tell you why in a moment two strings with opposite winding number will attract what is this attraction and repulsion this attraction repulsion actually corresponds to the gravitational attraction and repulsion of the higher-dimensional theory if we live in a world of let's say three dimensions three space dimensions and we add in one more tiny direction then strings which are wound oppositely may be in different places in the big dimensions different places in space we have a string wrapped this way we have a string wrap that way they'll attract each other if they have opposite winding number I'll attract each other in our ordinary three-dimensional sense likewise if they have the the same winding number they will repel each other so they behave like electrical charges what colors where is electrical or a repulsion and attraction come from it comes from the electromagnetic field and of course the electromagnetic fields is deeply connected with photons and so forth where does this attraction come from and the answer is it comes from gravitation but not gravitation in the four dimensional sense but gravitation in the five dimensional sense in the extra dimensional sense in the theory with extra dimensions the Einstein gravity the extra dimensions manifests itself in the phenomenon of attraction between like momenta nope cross without attraction between opposite momenta in the compact direction and repulsion of things with the same moment that's not obviously easy to see I've told you this before it's not it's not something I'm telling you for the first time and in fact you can relate it to Einstein's field equations let's let's not going to relate it to Einstein field equations in any depth I'm just going to discuss the origin of the electromagnetic field or the analog electromagnetic field that's associated with this electrical type behavior of these particles there must be something like an electric field there must be something like a magnetic field if they're behaving so much like electrically charged particles what is it okay so first just ordinary Maxwell equations the electric and magnetic field are describable in terms of a vector potential a four-dimensional vector potential a mooo you build up the magnetic field as the curl of the vector potential and the electric field as related to the time derivative of the of the vector potential it's a four vector it has four components let's go to gravity in five dimensions gravity is described by a metric tensor the metric tensor let's call it G M n why don't I use mu and nu hey I'm saving mu and nu for the four dimensions of ordinary space-time but we're now talking about a five dimensional world so let's use for the five dimensional world M&N but what a lemon in run over they run over the ordinary dimensions of ordinary space-time plus one more direction so we can write out we can write GM n as having various components G mu nu that's the ordinary dimensions and then g mu v v is v extra dimension and then what else g55 those are the components of the gravitational field what about what about g v mu ro g mu v what about g v mu symmetric tensors yeah the same thing okay these are the independent components of the gravitational field in five dimensions well this one here that's just a metric of four-dimensional space-time so just a usual einstein gravitational field nothing new there he only have something that has an index it has a five index here but the five index is this hidden direction we can't see it but it also has a mew index which means that it's a four vector we would mistake this object for a four-vector it has four components it's like a mule let's make that identification because that's that is the correct identification and what about g55 that had that has no components in the usual four dimensional sense so it must be a scalar that scalar is usually called Phi it's called a dilettante or sometimes called a Dilla time but it's almost always called Phi what is g55 it's the component of the metric around the fifth direction it tells you how big the fifth Direction is if g55 is big it's just a metric it's the square of the distance around the fifth Direction is what it is if g55 is big the fifth direction is big if g55 is small the fifth direction is small the metric tensor in five dimensions is a field it can vary from place to place all of these things can vary from place to place this is just a usual gravitational field varying from place to place this becomes an analogue of the electromagnetic field which can vary from place to place and this is a scalar which can vary from place to place but what does that scalar mean that scalar is the size of the fifth Direction that can vary from place to place somebody asked me that before and you can imagine waves in space where the waves of space are not waves of electromagnetism they're not waves of gravity they're waves of varying size for the fifth dimension okay but nevertheless the quiescent ordinary vacuum doesn't have such waves it's fixed okay but in general the size can vary from place to place okay this is an electromagnetic field if this is an electromagnetic field then it's also a gravitational field and it's an analogue electromagnetic field thought of as a graphic a tional field what are the sources of gravitational field there are things like energy and momentum in particular the sources of mixed components of the field like this are components of momentum so the source of this component of the gravitational field is actually the momentum in the fifth direction the flow of momentum in the fifth direction that means that the source of this analog vector potential is going to be the component of momentum going around the fifth direction lesson the momentum quantum number the one where is it the momentum quantum number over here can be thought of as electric charge in an analogue whit's on that and then the Colusa klein theory in which the mew five component of the gravitational field is the electromagnetic field electric charge momentum electric field are component of the metric well good that's easy that was easy [Music] that was easy what about and of course go a little bit further and we can say that that the that the electric that the electric field is associated with the graviton with a graviton itself okay now let's come to the winding number some saris feed and proportional to n or no no n no n is an integer it's related to our it's related to our it's not quantized it's related to the size of that dimension and it can vary from place to place continuously and smoothly called a Dilla Tom and it's a wave field that string theory was right should exist in some form or another but in any case it's part of the mathematics of string theory let's come to the winding number now I told you that the winding number also behaves with this property that opposite winding numbers attract are like winding numbers repel they also behave like electric charges but not the same kind of electric charge as the momentum quantum numbers it's as if there were two kinds of electric fields two kinds of electromagnetism living side by side two kinds of photons two kinds of charges winding number associated with something we could call winding photons and a momentum quantum number associated with momentum photons but momentum photons are just a piece of the gravitational field in other words they're gravitons which are polarized along the Meuse I've directed the field what is the field or the analog of photons which is emitted and absorbed by winding number not by momentum of by winding number then here we know the answer we'd have to go back several lectures for fairly early lectures but let me remind you this is probably something you've forgotten I told you but I don't think I stressed it very hard let's go back to the spectrum of closed strings the closed strings what kind of states are there of closed strings we start with the closed string being totally unexcited and then what do we do at it we excite oscillations on it we excite oscillations with the creation and annihilation operators which create and excite waves on the string they're not creation and annihilation operators particles the creation and annihilation operators for waves moving along the string these are closed strings let me just remind you what kind of operators there are there are creation and annihilation operators I won't think I'll just think about creation operators and I won't put a little plus because we wind up with too many indices a creates a unit of excitation what labels it well first of all the A's are labeled by directions of space directions of space perpendicular to the momentum of the object of the string so they're labeled by directions of space let's call it I I can equal if the string is going down the z-axis it can be X it could be Y or there can be X 1 X 2 or it can be a compact direction Y so the eyes vary over all the directions of space including the ones which are wrapped up that's one thing there now what's the other index that depends on the frequency of the oscillator and that was an integer and it tells you how much energy when you create one of these oscillations creates an energy proportional to n all right so n is related to the frequency I is related to direction in space and one more thing you remember what it was for closed strings would weather the wave is propagating around the string remember the string is oriented there's little arrows whether the wave is going one way around the string or the other way around the string I don't remember there's anybody remember how we label the two let's just call it left moving waves and right moving waves a I in and left and right here are purely symbolic they refer to to the direction on the string not left and right in real space okay now there was a rule I told you the rule I called it level matching it's a rule I'm not going to go into the rule now I think I explained it a little bit what it had to do with but it's a rule and the rule is the amount of energy and left room moving waves must equal the amount of energy and right moving waves that's a fundamental rule of string theory the mathematics goes to hell in a handbasket a few threw violate it and so let's see what kind of states we have we can first of all have the state with no energy in it at all that's a tachyon it's not there in super string theory but in the simple versions of the string theory and it has minus two units of energy we work that out - two units of energy let's attack a bad thing we don't want it what's the first excitation what's the next energy up well the next energy up is to hit it with the lowest frequency oscillator to excite one unit of energy with N equals one so for example we could hit it with a I I can be any one of the directions of space one the lowest oscillator left or a one i right oh but neither of these states are a legitimate option and the reason is because this one has one you of left-moving energy and no units of right-moving energy this one has one unit of right moving energy and no units are left moving energy and so it doesn't satisfy this principle that the amount of energy and left and right after balance okay so these are gone they're not there what's the next thing up the next thing up is to apply two units of the lowest oscillator one left and one right to make sure that you have balanced energy going and left and right and that means a i1 left times a j-1 right I and J are two directions of space okay they are the two directions of space for simplicity let's imagine the directions of space are the usual three directions of space plus one more namely the v direction the thing that I called Y here one more direction of space the one more and we apply this of course to the vacuum these are the things we identified with graviton and other massless particles are with polarizations that have to do with I and J well there is something new when we add this v direction one or both of these indices can now be in the v direction earlier before we discuss the v direction i and j could only be ordinary directions of space and these corresponding the gravitons which have polarizations which are they're like photons they're polarizations and the polarization is characterized by two directions of space but now we have something new we can have AI be the ordinary direction of space and this one being the v direction of space the internal direction the small direction okay now we have something which has a vector in X the fresh direction here we don't even call out a direction in space normally and so it has the same kind of index structure that a photon would have this is like a photon it has a polarization let's say moving down the z-axis this could be polarized either X or Y or however the many many dimensions we have to worry about and this is very similar to a photon now but there are two ways to do it you could have this one being left in this one being right or you can have a i1 right a15 left oh there are two possibilities now and it seems like that means that there are two kinds of particles that behave like photons that are similar to photons the correct thing to do is to add them and subtract them in the sense of quantum mechanics - linear superpositions but whatever there is there's going to be two types of objects which have the index structure of a photon there's going to be two kinds of photon fields the one with the sum that's identified with ordinary graviton moving down the z axis the other one is identified with another field a separate field that also has a structure similar to photon and guess what it is connected with it is the field the electromagnetic like field whose sources are what winding number so one linear combination is associated with momentum and one linear combination is associated with a winding number these two things one of them is associated with gravitons which is which are the field quanta of the gravitational field and the other one is associated with something called B mu nu which is different than the graviton different properties but also gives rise it's mixed component is also like a photon so here's what we can say now what is T duality and this is a field this is a field which exists in string theory called the rom owned the Cobb Ramon field not important what it's called it is another field which appears in same here oh no no no no no wait a minute both of these of our polarized like photons there's one more nobody asked me about what about a 5 5 a 5 a 5 left right that one's the one that behaves like the scalar that's the one associated whose field quanta these are the field quanta of this one over here okay namely the radius of the of the compact Direction here which was which would correspond into the winding number the other corresponding the one with a plus sign corresponds to momentum and that's the one that's associated with graviton the one with the minus sign is associated with with the winding number yeah okay so now we have a more complete idea of about t-duality and involves momentum being interchanged with winding number and involves are being interchange with one over R it involves replacing the X by the Sigma with the X by D tau this is the integral of this is winding number the integral of this is momentum and finally it involves interchanging g mu v with b mu v all right so these theories have moisture though of course not clear that the ordinary world does have such fields we don't know of real particle candidates for all these fields we don't so this is at the moment a mathematical construction and we're exploring a mathematical construction there's lots of ideas about how to use these these constructions but at the moment I think we should regard this as an exploration of the mathematics of the theory there's plenty of room in physics for these objects incidentally but I don't want to get into the phenomenology of them good so there we are that's what t-duality is after five minutes we're going to come back and talk about T duality for open strings and how it leads to a new concept completely new concept called D branes so let's take a break yeah okay so we've figured out what T duality is this very strange interchange of big geometries with small ones and various other things going on I want to concentrate now on the aspect of T duality that has to do with well let's let's erase some blackboard here ah this one now you're going to see the kind of gymnastics that that the kind of deductions that people have made over the years the ste duality is one of them they're implicit in the mathematics there many of them are very surprising we'll talk about d-branes and how one was forced to have these new objects in the theory which are called D branes D stands for dearest lady rish lay had nothing to do with him he was dead for many hundreds of years this should be called pea brains for polchinski but pea brain was already a term in use brain is spelled BR a in II like membrane and one speaks of DN brains these stands for darish lay not for the dimension of space n stands for the dimensionality of the brain a string is a one brain a membrane is a two brain a solid three-dimensional chunk which may be embedded in higher dimensions but solid through is is a three brain and so forth so there's D in brain where do they come from what do they do what's there what's there they're not just made up things they were essential to the consistency of the theory the mathematics absolutely demanded them and the result of knowing about them has been to derive an enormous number of equivalence between different theories and in fact it turned out that those enormous numbers of equivalences turned out to be enormous numbers of equivalences between different kinds of geometric structures equivalences which the mathematicians have no idea existed equivalences between different the calabi-yau manifolds and which the mathematicians were entirely surprised by and they turned out to be able to confirm them but this is some of this is now part of the gymnastics of string theory and the brains have played an enormous leap powerful role also in the applications of it so let's talk about open strings now d-branes have to do with open strings open strings what's the winding number of an open string open strings don't have stable winding numbers if you have an open string on a compact space now this that this picture is sort of shorthand for all of the compact directions with of which there are presumably six in super string theory and all of the open uncompact directions are but let's imagine now that in addition to closed strings there are open strings open strings now have endpoints we can't classify them with winding number it doesn't make sense to classify them with winding number they're not wound so we have to we have to think about what happens to them when we do this process of treaty duality t-duality was a phenomena discovered in closed string theory but surprisingly it also makes sense for open string theory and I'm going to show you what it says there's something rather remarkable something which is confirmed in other ways but this is the simple way to think about it um there's a string an open string and what are the rules for open strings do you remember what the rules are for what goes on at the end of an open string the boundary conditions on the end of an open string no I'm on no I'm on the end of an open string should satisfy the X or dy again this is y this is X any of the coordinates but let's step in particular well let's write them them both the X by D Sigma equals zero dy by D Sigma equals zero and what does that correspond to that corresponds to the idea that the end of the string is a string it's made up of a lot of tiny infinitesimal mass points and supposing the end of the string had some net stretching the X by D Sigma was not equal to 0 that would exert a force on the very last molecule well as you subdivide the molecules into smaller and smaller structures to go to the continuous limit of the string the molecules get lighter and lighter and lighter how much force can you put on a mass on a thing of arbitrarily small mass without having it accelerate with an infinite acceleration the answer is 0 in the limit that means that the string cannot afford to have any net stretching at the end of the string so that that's why that's where these conditions come from the X by D Sigma dy by D Sigma should be equal to 0 and that's the forbid infinite accelerations either in the X direction or the Y direction of course there is another possibility and that's somebody holding the end of the string and applying an enormous force so to keep it from moving but nobody is standing there holding the end of the string then if there's any net tension in the end of the string it will accelerate infinitely top to basically to remove that stretching ok now what about t-duality let's suppose now that this was a small compact direction and we want to replace the theorems very very much smaller extremely small we want to say laden that this is supposed to be equivalent to a theory with a large compact direction here's the large compact collection enormous lis large could be cosmologically large what do I do with these open strings okay well was supposed to replace winding number by momentum but remember that momentum was what was momentum related to it was related to DX by D tau right velocity what was winding number related to the X by D Sigma namely exactly what goes on here I should actually write since I'm mostly interested in the compact directions for the moment let's concentrate on the compact Direction t-duality involves among other things replacing every place you see it be Y by D tau by the expose or dy by D Sigma by dy by D tau so what does it say then about open strings open strings when you replace where when they undergo this process of t-duality what you have to do is you have to change the boundary conditions and the boundary conditions at the end of the string become the X by D tau equals 0 or in this case we're not sorry the t-duality is associated with the compact directions we don't want to do that only the compact directions are being interchanged small with large we're not diddling at all with the uncompact Direction one compact Direction Y goes why the radius in the Y Direction becomes big all right roll was we were supposed to replace the Y by D Sigma but dy by D tau if strings originally moving on the surface had no mam boundary conditions that's dy by D Sigma equals 0 after t-duality they have diversely band rican what does it mean to say that the Y by D tau is equal to zero Oh incidentally this is only at the endpoints this is at the ends of the string it means the endpoints are forbidden from moving so wherever they started they are stuck at where are they stuck well wherever they're stuck there's something holding them there it means that this theory has objects in it which can nail down the ends of strings there are strings there must be if T duality is to make sense there must be objects in the theory which can nail down the ends of strings you start with a theory with no such object and the strings just move freely and after T duality you discover there's some kind of object in the theory which has nailed down the location of the string it's nailed down only the Y component we have not played around with the X's the X's are still free to move the compact direction here is now nailed down so where is it nailed down let's so here's a little space that's nailed down someplace someplace in Y Y goes around this way let's put that place right over here there's something here which is capable of holding the ends of strings nailing them down that object is called a D brane D federa Schley and it wasn't put in by hand instead one asked what if there are open strings in a theory and the theory has t-duality which it must have for reasons that are by now well understood then what happens to the open strings when you interchange the small compact direction for the large compact direction and the answer is that you discover something new you discover some new kind of object which it's holding narrow she's not yeah Oh does it necessary that the y-value be the same for the two implants no I think you're asking whether this can bend I drew it so that it oh yeah yeah yeah yeah I could that's right but the simplest cases where where it's just fixed fixed in some and some position now what position can you do you put it out here here here well the answer is that you better be able to put it anywheres so therefore this has to be a movable object because there was nothing special about one point in space than the other point in space along these axes so there's got to be a movable object which functions as a anchor point for the ends of strings if it's movable then you can be more quite sure that it's also bendable our relativity can't make sense with absolutely rigid objects and so it will also be bendable but let's just not bend it for the moment and strings can attach themselves to it though strings attached to it can exist a string can come or let's let's so the good that's that's a these are called d-branes right this is the first of all supposing we took let's let's take an example supposing we took 10 dimensional string theory which has nine dimensions of space nine dimensions of space and we made one of them compact and then introduced a D brane in this way what would be the dimensionality of the d-brane we'll be 1:9 no eight-eight we've pinned down one direction and left the other nine the other eight directions of space free to move around all right here's a tabletop that tabletop is 2-dimensional if there's an object attached to the tabletop how many directions is free to move in and how many is it constrained in it's free to move in two directions here's a string here's a string with two ends or my finger is at end of a string it's free to move in two directions and constrained in its third Direction all right so if I constrain one dimension if I constrain one dimension it creates for me in 3-dimensional space a two-dimensional surface supposing them now in nine dimensions of space and I constrain the end of a string I constrain one of the dimensions it becomes an eighth dimensional surface that the string can move around now suppose there was no reason for only constraining you can do the same thing with any number of compact directions you can play this t-duality game with any subset of the compact directions you can play that's it's the same game and you can in the same way construct d-branes which pinned down any number of the coordinates at the end of a string can move in all right so let's take three-dimensional space again let's suppose now we've played this game and we've constrained the vertical position by doing this t-duality trick with respect to the vertical direction supposing I also do it now with respect to the front-to-back direction I could strain the motion of the position at the end of a string so it's constrained vertically and this way that means that the end of the string has to move along the the intersection of this plane with that plane so if I constrain two dimensions what kind of brain am i talking about a D in this case D one but if I started with seven there with nine dimensions of space seven so each constraint removes a direction that it's free to move in all right all right for that reason you can have in nine dimensions by playing this game over and over again with the different directions associated with a compact space you can have the eight brains the seven brains the six brains all the ways down to what's the maximum number that you can you can just get rid of the freedom to move in any of the directions of space then what does e then what would you call the brain these 0d0 it's a point or it's a point in space that the string is connected to and it's not free to move along any of the directions that's a d0 brain the zero zero dimension means a point you're going to from the zero brains to the eight brains they all make sense and they're all their existence in some sense is all necessary to the consistency of the theory these zero brains are a new kind of particle an unexpected new kind of power yes no no no not really no no um no no [Music] you couldn't no I'll show you why in a moment I'll show you why in a moment right okay we've played this game with a compact directions but now let's imagine having done this operation let's imagine shrinking and shrinking and shrinking this until it becomes arbitrarily small what happens on this side it gets bigger and bigger and bigger okay so eventually on this side it can be so big that it's completely mistaken for a non compact for a non compact dimension in other words if a dimensional space is big enough for all practical purposes it becomes a non compact what we've demonstrated by this series of arguments is that even in the non compact directions there must be objects which are the anchors of strings they can be oriented along compact directions they can be oriented along non compact directions any number of them and of any dimensionality starting with zero and going up to eight dimensions actually you can go up to nine dimensions but that means they completely fill space and that simply means that we're talking about open string theory in which the open strings are just free to move but they're not intraday we wouldn't call them brains there are things that people called D minus 1 brains but you don't want to know they're not there they're not real concept ok so you start here's a d0 brain it's a point in space and a string in can end on it where's the other end of the string it could be on another d0 brain or it could be stretched out to infinity that's not a good thing because it becomes infinitely massive or it could even come back to the same point the same deep brother the same d-brane that's a d0 brain what's the next one up a d1 brain the one brain something very interesting about it b1 brain it's a line now as you already pointed out it does not have to be a straight line we deduce their existence in this way but once we know that they exist and once we know we're talking about a relativistic theory they have to be bendable if they're bendable they can even bend around on themselves are there structures which are physically physical objects and they have strings connected to them but now let's forget the strings that are connected to them and the strings connector and could be like this let's forget the strings that are connected to them and just look at them this is a one-dimensional object it's called a d1 brain it's a one-dimensional object it can bend what's another name in fact it can not only bend but it can even come back on itself what's another name for one-dimensional objects like this strings but are they the original strings no they are not the original strings they're called d1 brains and they're distinct from the original strings how would you expect they might be the same that's the same all the way around II got points on them to have articles and points that don't I mean you have streets that are attached on placing well I may or may not have strings attached to them but let's go through our moment suppose they don't have strings attached to them they're much heavier in string that makes a lot of sense because well how do they start out they started out as anchors like infinitely heavy things that could that could hold the strings down but they're not really infinitely heavy there they're there their mass depends on some coupling constants and so forth depends on various things they're heavy they're much much heavier than the ordinary strings and so in that sense they're anchors but they are string like the TQ le hold for this one what sir the t-duality holds my friends yeah they also if it holds for the brains too yeah it does hold for the brains ah what's the next one up a D to brain a D to brain is called the membrane membranes I mean the two brains and those are simply sheets like the table top here that string ins can move around on a pairs of strings can move around on them like that these are the two brains next one up the three brains in a three-dimensional world a d3 brain fills all of this filled space alright or filled space into space filling brain people and then the words to space and if we had a d3 brane then that would simply mean that we can have open strings that can just move around they have to be attached to the brain but the brain is everywhere and so it's just open strings that are free to move around everywhere so it's often said that ordinary open string theory is string theory on a space-filling brain but if we're living in a world with higher dimensions then a three brain doesn't fill all of space it's like a two brain in the 3-dimensional space it's a surface and strings can move on it okay that's the idea of d-branes and as I said and as I emphasized they are mathematically important to string theory but they also are the origin of a lot of applications of string theory so let me very quickly tell you what's known you can start with one of these D branes I shouldn't the unconstraint directions that are left over contact directions as well yes yes yes that's correct but that's certainly correct right okay let's imagine a d-brane and I'm going to draw it as a d2 brain there's the d2 brain now you know if you say to yourself this is awfully slick though I really believe that this has to be not from the arguments that I made obviously there's giving you some some sketches of arguments whether string theory is the Right theory of nature is not the issue here the mathematics of this is very very tight by now I mean many many cross-checks many many different things point in the same direction mathematical theorems that have come out of it though the mathematical speculations have been confirmed in mathematics so it's it's remarkable and presumably correct ah all right here's an empty D brain think of it as empty empty means it has no strings ending arm it think of it as empty and think of it as an empty space time or empty space the space now is not the full space but it's just a surface in space but think of it as a space it's empty we can put things into it we can put strings into it like this little open strings and these open strings are free to move around now I'll tell you what the what the basic process of them is the basic thing they can do is their endpoints these are oriented strings so they have arrows associated with them when the end points come together in string theory let's say the end points come together if we have two end points one with an arrow coming into it and the other with an arrow coming out of it it can do the obvious thing namely the end points come together and lift off the surface and form a single string with two end points like that how do we think about that if these strings like this are thought of as particles which are free to move around and in string theory particles or strings these are very much like the original open strings that we started with if we think of them as particles which are constrained to move on the surface or very close to the surface then what we discover is that these particles can split and join two of them can come in join into one one of them can go out join separate into two we're starting to build up something that looks like particle processes we're starting to build up something that looks very much like finding diagrams two of them come in join to form one maybe it hangs around there for a while and then the reverse process happens and it becomes two again we're building up the kind of processes that quantum field theory describes creation and annihilation of particles transmutation of the number of particles and so it mean it's not completely surprising that the mathematics of the interactions between these particles looks very much like quantum field theory in fact at low energies where you don't have enough energy to really excite the vibrations of the strings it is simply quantum field theory where the particles of the quantum field theory of these open strings in fact the open strings behave like photons moving around on the surface then enough photons the sense of the 10 dimensional theory their photons in the sense of a theory which lives and whose particles live only on the surface that's the basic connection between brains and their application and studying quantum field theory our brains like this define quantum field theories and they define quantum field theories in exactly the way that I've just described but yeah let some incidentally these are these open strings connected to the brain here do behave like photons moving around in the brain they behave like photons in the lower dimensional theory but let's imagine now that we have more than one brain there's no reason why you can't have several brains for example parallel to each other here's two brains parallel to each other you can move them closer and closer until they touch in which case they just form a compound brain or you can leave them separated let's leave them separated for a moment let's put three of them in just for fun three parallel brains like that and let's think of a kind of excitations that can move around on them well you can have and the excitations means ordinary strings an ordinary string let's give them names let's give the three brain the three of the the three of these names the names I'm going to give them are red green and what yellow blue sorry blue blue red green and blue this is just the names of the three brains of course we get that four or you could have seven or we could have 15 but I'm particularly interested in having three of them for the moment what kind of strings do we have we can have strings which begin on red and end on red all remember these strings are also oriented keep in mind that they're oriented and so what would you call this string I would call it a red anti red or a red red string this is a red red string and therefore some kind of red red particle we can also have red green particles where green particles and ones let's look like this where one end begins on read the other beer end and the other ends on green so what do we have what are the class of particles we have we have particles that are labeled by two indices two colors for the case of three brains we have three distinct colors and we have particles which are labeled by pairs of colors one associated with the outgoing and the other associated with the incoming does it sound like anything you've seen before gluons gluons and quantum chromodynamics [Music] of course maybe you should keep in mind that this is embedded in a higher dimensional space so to mimic that let's think of lines in higher dimensional space now there's no problem in going from here to here or from here around to here you're worried about passing through here well we have there there will be some rules there will be some rules and the basic rule what is the basic rule is really only one rule but we can use it over and over again I'm a red green string what can it do if it hits a red red string all right let's suppose this is going out here and coming in here let's suppose this is going let's see what I want to have I want to have in here and out here okay what can happen this end can join with this end they can come together and join and form a single string which goes all the ways from here to here in other words the red ant I read the thing coming into red and the thing going out of red one in one out can join and simply make a single string this is very much like a pet like a green red gluon coming together with a red red glue on and forming another red green blue on same rules exactly the same rules as for gluons but the string cannot annihilate or lift itself off the the surface on the other hand if we had this situation here then these can come together and they can lift themselves off the surface that's incidentally why there are eight gluons instead of nine because one linear combination can disappear and it is not stable so the mathematical rules for splitting and joining are exactly the yang-mills quantum chromodynamics rules for gluons okay what about quarks okay so now what about quarks what is a quark in this language a quark in this language only has one color it's either a quark or an anti quark it only has one color it doesn't have three colors I'm sorry it doesn't have two color and anti color color a quark must be a thing which only has one end a quark in this language is a string which ends on one of these brains and goes off to infinity now it doesn't really have to go off to infinity it could go off to some distant brain of a different kind but it doesn't have another end which ends on one of the three that's a quirk and it's either a string coming in or a string going out when a string coming in and a string going out meet each other they can join and just disappear out into the off the brain they can join and disappear from the brain that's annihilation of a quark and an antiquark if there's two quarks they can't annihilate they they're stuck there and that's just that's exactly the same rule as quarks the mathematical structure of the field theory that described these strings moving around is essentially exit with some little extra added ingredients because of supersymmetry it's a super symmetric version of quantum chromodynamics and it is a part of the reason that brains are interesting for exploring quantum chromodynamics I'm not going to show you how they used in detail I'm just showing you what the connection between things is [Music] let's suppose there's only one brain then it's not like quantum chromodynamics what do you think it might be like now we have objects we don't have to name them red red green blue it's just it's just a string it's like well it's like quantum electrodynamics it has only photons it doesn't have this complicated gluon structure these will be photons and what would these things be charged electrons they could be coming in or they could be going out in which case there would be electrons and positrons when two of them come together they can annihilate one last point which is really quite fascinating remember I told you that there are ordinary strings and D strings these strings were these no objects that we discovered these no objects which are a good deal heavier but there are also strings an interesting question is can a D string also end on a brain like this incidentally I'm thinking about three brains because I'm thinking about three I'm thinking about mimicking three-dimensional space the three brains so these are really the three brains interesting question can a D string these string is itself something that ordinary strings can end on but let's forget that can a D string end on a three brain a d3 brane the answer is yes I'm not going to try to prove that for you that's a much more elaborate question but then if ordinary strings the ones we started with make electrically charged particles what do D branes would be strings make that their ends when they intersect the the three brain here can you guess hmm No well they're in some sense the true but they are they got to be something which is similar to what happens when ordinary strings and magnetic monopoles remarkable the magnetic monopoles are much heavier than the electrons because these strings are much heavier magnetic monopoles are expected to be much heavier than electrons they become magnetic monopoles the relation between magnetic monopoles and electrical monopoles in in quantum electrodynamics is mimicked by the relation between these strings ending on these things and ordinary strings on it so I've told you a lot of stuff I don't expect that that you're going to follow every detail but I'm trying to show you how string theorists discovered it took a long time to make I mean that this didn't happen all in one day this happened over a period of twenty years basically or more no more than most twenty years twenty-five almost thirty years that all these pieces were put together by various people a very wide variety of people who saw these connections and today string theory and it's description and time in terms of brains is the primary tool for studying quantum chromodynamics it's very bizarre the whole thing made a full circle it made a full circle from a theory of hadrian's to a theory of quantum gravity to the presence of d-branes there was necessarily there which when you put them together in the 10 dimensional space put together three-dimensional d-branes all of a sudden becomes the theory of of quantum chromodynamics that we that it started out as so that's I don't know what to describe it I describe it as sort of mental gymnastics but I think it's more than that I think it's a process of discovery and unraveling of the of the mathematical structure of this thing it has wide application to quantum chromodynamics other quantum field theories fluid dynamics all kinds of things so far it has not had application to a direct application to understanding the particle spectrum and that's probably because it's just too complicated yeah what determines which of the dimensions that t-duality is applied to you oh you can do it any dimension you please to any compact dimension you please choose one that's convenient but that eventually just winds up saying that these d-branes can lie along any axis and what chooses it history of the universe it's a membrane for magnetic monopoles men magnetic monopoles 3 3 D 3 the universe here from our point of view is a three brain here d3 brane but if from some of the other 10 dimensions which the people who live on this world don't even know about but a string ends over here and happens to be a D string they'll experience that as a magnetic monopole so something's these D strings arranged as well what's that they must be wearing these jeans are also oriented absolutely you know they're also oriented you used two planes for su - yeah and how do you get the feature word it's not the pleasure in other words su - if you have a reflected University physics doesn't work the same way oh you come back charge conjugation violation you're saying I don't really drunk talking about the left-handed and right-handed for particle/anti-particle more interesting yeah there are answers string theory certainly has a potential possibility to describe that but not at this level at this level you need when you have khalaby on manifolds they can have more complicated a symmetries that allow that this toroidal compactification doesn't allow it there's a lot there are many many other objects in the theory which I haven't gotten into this is these are the simplest to describe there are lots of other constructions and other kinds of objects that the theory has to have and some some of them when they intersect other ones break various symmetries and we can probably we can discuss them some other time I'm out of time and out of energy and amount of momentum [Music] [Applause] for more please visit us at stanford.edu

Channel: Stanford

Views: 72,618

Rating: 4.8847737 out of 5

Keywords: Physics, Science, Gravity, Cosmology, Particle, Theoretical, Mathematical, Energy, Path, T-Duality, Frequency, Electrostatic Fields, Momentum, D-Branes, modeling QFT, QCD, EM, Polar, Space, Black Holes, Vacuum, Symmetric Tensor, Neumann boundar

Id: L1_3Xp3rT3c

Length: 107min 49sec (6469 seconds)

Published: Wed Mar 30 2011