Lecture 6 | String Theory and M-Theory

[Music] Stanford University okay so thermionic strings have been a part of the subject since almost the beginning not quite the beginning and as I said they solve two important problems I forgot what they are though so they give you fermions and they get rid of the tachyons they leave you with ten dimensions instead of twenty-six so they don't solve that problem that problem turned out to be less of a problem and more of a feature feature means a good feature then people are expected at a time but we'll come to it and it's a subject known as compactification taking the extra dimensions and doing something with them to make them innocuous or malum Occulus an interesting feature of the theory but we'll come to that not now though what I want to talk about a little bit was not really historical but the scattering of strings the subject really began with studying scattering of particles elementary particle physics was always about scattering of particles not because it's the most interesting phenomena that can happen it's not it's rather dull you send some particles together and a bunch of junk comes out and all sorts of directions but it's about all we can do in the way of experiment and so we try to unravel from the scattering data what was going on inside the collision and inside the collision of course means the properties of particles and so forth so the natural tool of experiment the scattering and the natural thing that a theorist would ask is if I have a theory of particles how do you compute the scattering amplitudes the scattering amplitudes what does the scattering amplitude a scattering amplitude you have some incoming particles that are part of your incoming information I'm going to have time running this way tonight horizontally instead of vertically I don't know why variety particles come in something happens inside a black box and particles go out not necessarily the same number of particles the particles come in and they carry momentum of course they carry other things they carry spin they carry charge they carry labels like for example is it a muon or is it they whatever it happens to be but let's simplify the story and ignore everything except their momentum so particles come in and they carry for momentum for momentum means energy and momentum let's write down what a four-vector of energy momentum is the energy and the three components of momentum of course if we're working in 26 dimensions we have 25 components here allows us to write down four of them that's a four vector relativistic four vector each particle has a momentum and we're going to call it k k mu where mu goes from 1 to 4 or from 0 to 4 excuse from 0 to 3 this is usually called 0 1 2 3 doesn't matter 4 components of momentum now what do we know about the components the four components of momentum of a particle they have something to do with the mass of the particle well of course they are the energy and the momentum what's the relationship between energy and momentum and mass for a relativistic particle anybody remember C is one we will take C equals H bar equals one what's the connection between the components of energy momentum and mass a squared a squared equals P squared plus M Squared or let's write at the following way a squared minus P squared equals M squared and just in order to keep my consuming notation consistent with notations of physicists for many many years I am going to write this as P squared minus y squared equals minus M Squared same formula I've just taken the negative of it now that can also be written in terms of the components of K in terms of the components of K that's what is it it's K naught squared at sorry it's our K vector squared minus K naught squared not standing for the time component e spatial component squared minus the time component squared and this is often just called K squared just call it K squared the left hand side here is the square by definition the definition of the relativistic square of a vector it's called P mu P mu or to simplify it let's just call it K squared it's the space component squared minus the time component squared that's called K squared and for every particle K squared is equal to minus M squared you can't vary K squared of course K squared consists of the energy and the momentum when you say you can't vary K squared it doesn't mean you can't vary the momentum it means when you vary the momentum the energy varies in a certain way and the way that it varies is that a squared minus P squared or K times K naught squared minus k space squared equals minus M squared so that's the first thing about it's not even about collisions it's just about particles you characterize them by their for momentum three components of which are independent the fourth component has to be subject to this constraint now we put in a bunch of momenta let's call this k1 for the first particle that the way I label them I like to keep my notation straight yeah I think I call this one K 1 this 1 K 2 and then out going over here we'll call the incoming momenta K I'm going to call the outgoing momenta Q these are also four momenta or call Q 3 Q 4 but that let's take the very simple case in which 2 particles go to two particles Q 4 these particles come in those particles go out how do you represent momentum conservation momentum and energy conservation momentum and energy conservation are simply that K 1 plus K 2 thought of as four vectors is equal to Q 3 plus Q 4 all components the space components define momentum conservation the time components define energy conservation ok now because of the perversity of physicists where physicists like to do is to redefine the outgoing momentum and think of them as incoming momentum but it's crazy the outgoing momentum are outgoing the incoming momentum are incoming but to do it all we really have to do to make it symmetric with respect to incoming and outgoing momentum just take each qq4 change its sign and call it K minus k3 that means changing the sign of its energy means changing the sign of all of its momentum it's I'd say it's a trick to be able to write this in a symmetric form instead of writing k1 well let's see what it is it's here's what it is K 1 plus K 2 minus Q 4 minus Q 3 equals zero now becomes K 1 plus K 2 plus K 3 plus K 4 equals zero you treat all particles as incoming but you have to remember that the label for the outgoing particles is labeled with minus the actual momentum of the outgoing particle but once you do so momentum conservation is completely symmetric between the four between the four particles a useful trick it's a useful trick that that keeps labeling especially consistent notice that when you change the sign of the momentum to redefine the momentum with a minus sign it does not change the fact that the square of the momentum is equal to minus m squared so we have in this particular process we have four momenta think of the motor-oil incoming although the energies of two of them may be negative the outgoing ones each one of them subject to this constraint the question that a physicist would ask about this collision is what is the amplitude what is the probability the thing the amplitude is a thing that you square the complex number that you square to find the probability for that collision but the collision is a function of a number of variables it's a function or the probability for the collision is a function of the momenta of the incoming particles and a function of the momentum of the outgoing particles so it's a function of the k's let's call that amplitude a it's a thing that you Square and it's a function of all of the case k1 k2 k3 k4 but to some redundant information here first of all the moment that have to be conserved second of all the square of each K has to add up to 2 minus the mass squared so there's really too many variables here there's too many independent variables how many independent variables are they before we impose any constraints each momentum has four variables this is 4 plus 4 is 8 plus 4 is 12 plus 4 more is 16 okay that's a lot of variables for something to depend on fortunately you really don't depend on that many variables let's think about physically now how many variables the scattering amplitude or scattering process depend on well the first thing you can do is whatever the momenta are you can use relativity to go to a frame of reference where the center of masses where the center of mass is at rest in other words with a two momenta the two space components of the momentum are equal and opposite you can always do that if the particles are both moving down the z-axis well you just move fast enough to be halfway in between them if they also happen to be moving in some other direction just move in that direction you can always go to a frame of reference where the particles are equal and opposite the space components in other words the spatial momentum next you can always rotate the system so that the momenta are coming in along the x axis what's left over in the initial state what does it depend on if you know that the momenta are equal and opposite what does the whole thing depend on it only depends it was the initial state depend on the initial state only depends on the magnitude two of the momentum not the magnitude of the square of the momentum I mean sorry not the magnitude of the floor vector the four vector has to have magnitude M square but but magnitude of the space momentum or the energy if you like once you go to the center of mass the only thing left over is the total energy of the collision in the center of mass frame okay so that's one thing EE center of mass now let's suppose that our particles are all of the same kind for simplicity the particles collide what can you say about the outgoing momentum first thing is they have to be equal and opposite Y that's momentum conservation momentum conservation says they have to be equal and opposite what about the energy of the outgoing state has to be the same as the energy of the incoming state what is the only thing that can differ between the incoming and the outgoing particles the angle that's it the angle of scattering theta so the scattering amplitude although it's written in terms of sixteen variables really only depends on two two independent variables those variables should be expressible in terms of relativistically invariant things ah we can think of it in the center of mass but we also be ought to be able to think about it in any frame of reference we should be able to construct two independent invariants relativistic invariants which are enough to completely characterize the scattering so let's talk about the invariance describing a scattering process two particles come in two particles go out let's think of them as all coming in changing the sign of the momentum k what is this k1 k2 k3 k4 right what can you do with a four-vector what can you do with four vectors to construct invariance there's only one thing you can do really you can square them that's about all you can do but you don't have to be talking about k1 squared or k2 squared you could be talking about k1 plus k2 squared right that's a good invariant k1 as a four-vector k1 plus k2 k1 plus k2 squared now remember what that means that means the space components of K k1 plus k2 which I'll label with arrows squared minus the time component which means the energies let's label them k naught K naught means energy k naught 1 and k naught 2 squared you sum the momenta square it you so many energies square it and subtract and that's called k1 plus k2 squared let's see if we can figure out what it is it's an invariant quantity let's see if we can figure out what it is by going to the center-of-mass frame in the center-of-mass frame the momentum of this particle what should we call it well it's k1 it's the space component k1 what about the momentum of this particle it's equal and opposite minus k1 right so the total space component for the momentum in the center-of-mass frame is 0 we don't don't worry about this one what about this one over here that's the energies we didn't talk about it but what about the energies of the two particles in the center-of-mass frame they're equal and opposite they have the same mass the energies are equal the center of mass frame is one in which the particles are perceived as moving with exactly the same them magnitude of the momentum and so the two energies are the same so this just becomes twice K not quantity squared very simple this is the center of mass energy here in the center of mass frame you add the two energies it's the total energy and the square of it is just the square of the center of mass energy so this quantity here which is called minus s it's just called minus s is nothing but the square of the center of mass energy just look at it or s is the center of mass energy Y in the center of mass frame this is zero and this is just the total energy so minus s is just the square of minus the square of the center of mass energy square of the center of mass energy we call s now can you think of any other invariant that you can build how about k3 plus k4 squared we took k1 plus k2 squared how about k3 plus k4 squared it's the same thing why because K 1 plus K 2 is minus k3 plus k4 energy conservation so we don't get anything new there k1 plus k2 squared is the same as K 3 plus K 4 squared it's the energy and this just says that the energy in the initial state is the same as the energy in the final state ok what about K 1 plus K 3 squared that could be something new so let's see what it is let's see if we can figure out what K 1 plus K 3 square it is by working in the center-of-mass frame in the center-of-mass frame all of the particles in and out have the same total energy have the same energy each particle has the same energy particles just get scattered through an angle that's all that happens so they come in and they go out with the same energy but they scatter through an angle okay arm so let's take K 1 plus K 3 squared here it is K 1 plus K 3 squared K 3 squared K 1 K 3 squared how about this what is this in the center-of-mass frame it is K same as K 2 and K 4 but that doesn't help ever I want to know what it is though how big is it all the particles have the same energy incoming and outgoing all they do is scatter through an angle 0 Y is 0 Y not twice K why not place the energy yeah because K 3 and K 4 we flip the sign on them all right so this isn't there and this is just K 1 plus K 3 squared so let's see if we can see what that is k 1 comes in particle 1 comes in and of course also particle to particle 3 goes out here's particle 3 1 goes in 3 goes out but if we label these particles with the k's k 1 and let's call this one now q q 3 let's make it an outgoing particle then really what this is is it's K 1 minus Q 3 squared it's really the difference of the momentum of the incident particle and the final particle it's called the momentum transfer it's a momentum transferred if you were to think of particle 1 particle 3 as the same species of particle then it would just be in the collision it's the momentum transferred from 1 to 3 Bingbing is a momentum transfer and that's the momentum transferred from 1 to 3 that's what this is momentum transfer it can also be expressed in terms of the angle of scattering I will tell you what the formula is the formula the formula for K 1 plus K 3 squared is it's just twice the energy squared minus the mass squared all right we already worked out with what the energy is the energy is the S variable it's twice C squared minus M squared in the center-of-mass frame times 1 minus the cosine of the angle of scattering this is the interesting thing here the angle of scattering is the angle between 1 & 3 all right k1 comes in q3 goes out the particle gets deflected through an angle and it's that deflection angle here it is theta the deflection angle of the particle between the incoming state and the outgoing state and that's what's here why is that interesting well of course that's what's measured in an experiment you put detectors in different places you scatter particles and you find out the probability for them to get deflected to an angle at different values of the energy this variable the k1 plus let's see what did we do we I'm sorry we the definition of s I forgot to erase the s over here the definition of s was k1 plus k2 squared that was the definition of it one plus two coming in and it was the square of the center of mass energy so s equals energy center of mass squared K 1 plus K 3 squared that's called got another label anybody think up a name for it peak T good T it is called T I know you're going to get that and that is some combination of the energy and the momentum transfer it's the center of mass energy squared minus the mass squared big deal we already know what the center of mass energy is times 1 minus cosine of the angle of scattering all right so we have K 1 plus K 3 squared how about K 2 plus K 4 squared is that different well if you go back up to here K 1 plus K 3 is minus K 2 plus K 4 so K 1 plus K 2 squared is the same as K 3 plus K 4 squared it's also T is there another combination we have K 1 plus K 2 we have K 1 plus K 3 K 1 plus K 4 right that's another one all right so one more quantity is K 1 plus K 4 squared what can that be well well if you were going to give it a name what would you call it Oh minus R minus T equals K 1 plus K 3 squared u minus u equals that these are the s T u variables but what's going on here there are only two quantities that the scattering depends on energy and angle of scattering where we seem to have three independent things well the answer is there are not three independent things if you use the momentum conservation and you use the fact that each K squared is M Squared what you'll find is that s plus T plus U is equal to minus 4m squared or something like that there's a constraint among them there are not three independent ones only two these are the two interesting quantities the third one is dependent on the other two if you have to consider the vacuum at all on these collisions will it mean it's the vacuum kind of participating there in the collisions so what the means vacuum vacuums all over the place participates like well what does it mean what do you mean better dissipate oh maybe the vacuum could just slow down a particle going through free space there's Bob well we know that doesn't happen don't know momentum conservation okay well while I'm here yeah it could be I mean it's a possible world the vacuum has friction it's not our world momentum conservation says that particles aren't slowed down by the vacuum have the same energy in the center-of-mass why can't you I mean K 1 and K 2 the center of mass right scattered 5 that way yeah but then the center mass frame is defined as a frame where the moment' are equal and opposite if the momenta are equal and opposite the energies of the particles are the same if their masses are the same now they scatter but they have to go out equal and opposite the momentum conservation reasons yeah so that's our these variables s T and U the other one which is there those are called Mandelstam variables and they're very symmetrically defined Mandelstam Mandelstam Mandelstam Mandelstam is a currently active physicist in Berkeley but this notation comes from the early 60s the very early 60s Mandelstam variables and notice what that part they have they have this beautiful symmetric structure that's what's interesting about them now let's talk about Fineman diagrams from and I'll just tell you what the answer is for certain Fineman diagrams um let's suppose two particles collide this is a Fineman diagram create a third particle of a different mass in here let's call it capital m and then those particles go out like this particles come in I'll find them in diagram so they come in together they coalesce to form the second particle or third particle whatever and then that particle decays and goes out that's a Fineman diagram that Fineman diagram has a value I'm going to tell you what it is it's the product of two coupling constants there's always coupling constants and Fineman diagrams and then there's the propagator of a particle in between and that has a very simple form 1 divided by s minus Capital m squared it has a is it yeah s mine s minus Capital m squared that's the characteristic structure of a scattering amplitude where two particles come together and merge and form another particle and notice it's a function of s the energy process the energy of the process it doesn't depend on T it doesn't depend on the other variable you want one that depends on the other variable you draw a similar diagram except in which instead of one in two merging you have I'll draw it over here one and three merging one two three four it looks like exactly the same diagram except turned on its side but it represents a very different kind of physical process 1 & 2 come in exchange a particle between them and then go out as 3 & 4 what would you guess if you had to guess the amplitude for this processes it's going to have G squared again the G's are the vertices what else exactly 1 over t minus M Squared now you can imagine a third process which would be 1 over u minus M Squared it would look like this R 1 ah let's see that it's a little hard to draw a 1 it looks like this 1 & 4 merge 1 & 4 merge little if they switch the lines like that let's ignore it it's there it's there it's important but it's too hard to draw I don't like drawing cross you know they're not perpendicular it's just a question of which share power particles came together and I hate drawing cross lines on the blackboard so we won't but there is in principle there can be another one this one depends only on the energy doesn't depend on the angle of scattering altogether t contains the angle of scattering s is only a function of the energy that means that when this process happens it to say that it doesn't depend on the angle of scattering means that every angle of scattering is equal equally probable when the particles go in they have equal probability of getting deflected through any angle whatever that seems odd but that is the property of this product and the reason is very simple the particles come in from some direction they form this compound state and then when they decay they forgot in which direction they came in from so they come in they form the composite and then when the composite decays it decay is an arbitrary angle uncorrelated to the initial directions that's this process here this one is different it depends on the energy but also on the angle of scattering this is the one that depends on the angle of scattering and if you work it out it's easy to see that it favors forwards a favor small angle scattering this favors large angles so this one depends on angle and this one depends but notice how similar the two expressions are there's a real symmetry between them in fact if you interchange s and T the scattering doesn't change this is a property of relativistic scattering amplitudes that they have this kind of symmetry they sort of forget which will be incoming in which were the outgoing particles but only if you express it in terms of these kind of invariants good so now we have some basic idea of what it is that a theoretical physicist wants to compute about a model of particles to compare not because he's so interested in it but it's the thing that he can give to the experimentalist and say measure the probabilities for scattering as a function of energy and so forth and here's my prediction okay are you saying that the dependence of s T mu causes this amplitude to what shift around if you I mean you still have three turns but but the value shifts around at the tip vertical depends on the angle of scattering all right so if all that was there was this this would favor a small angle scattering and B not forbid but make it less likely let's see if we can see why oh I think I hear it is the third term is there are bored about it oh well what you say that you didn't want to write it up there or did it you didn't want to write the diagram anymore draw the diagram but if there was a process in which one in four could come together and do the same thing than we would want to put a G squared 1 over T 1 over u minus M Squared yeah they're connected yeah yeah the one over you mind okay so what is U it's interesting U is equal to e center of mass squared minus M squared times 1 but 1 plus cosine theta so both so they're clearly related if you know s you know the energy therefore if you also know T you know the angle of scattering and this is just a function of the energy in the angle of scattering so they're all connected with each other another way to say it is if you add s and young you'll cancel out the angle altogether and you'll get just a function of energy which is clearly dependent only on s right right but they're all but the separate finding diagrams and in general you have to you have to add them all ok but I want to focus on this on this here this does not describe the scattering of mesons for example very well it gives a poor description of the scattering of massan's and the reason is simple the reason is that there are many many different particles that can be produced when two massan's collide whole stacks of them with different mass and different angular momentum we haven't described what the angular momentum would do here where the angular momentum would do would be to change the dependence on the angle of scattering let's not get into it formulas like this are too simple to describe the realistic scattering of mesons it's too easy to excite higher vibrations of these particles here and when two particles collide they could make some ground state that could make some first excited state that could make some next excited state they can make a whole raft of different particles that can go in there and people in the 60s basically beginning sometime around 1965 66 67 trying to concoct mathematical amplitudes with interesting mathematical properties that would represent all of the possible particles which could go in here this was trial by a trial and error it was just making up formulas that try to represent the different particles which should go in here and different particles which could go in here the first attempts tended to beat the ad particle this is called an S channel process why because it involves one of the s minus M Squared particles coming together k1 k2 that defines s this is called the tea channel process k 1 and k 3 coming together so they began by saying let's just add more and more stuff into here s channel and let's add more and more stuff into here tea channel for all the particles that could that could be scattered well that lasted for a certain period of time adding up the various particles which could go in there and then people try to find more comprehensive formulas just by I don't even know what the right word is so kind of curve fitting but a very sophisticated kind of curve fitting and a rather dramatic formula was discovered which contained the physics of all of the particles in the S channel and all of the particles in the T channel just replaced this combination here with but not just one particle but many of them whole towers of them with a formula which I'm going to just this is of historical interest mainly with a formula which I will write down and you can explore its beauty it's great beauty there's very very simple it's called the vinet siano amplitude then let's see amplitude young Italian physicist he was young at the time he's not young anymore pretty still Italian he just I don't know how he made this guess he just just randomly wrote down some things which had some write properties and which really did look like adding up things like this it was a function of s and T everybody know what the gamma function is well the gamma function you don't need to know it but I'm just going to write down the formula for fun the gamma function is a generalization of the factorial function for the factorial function is only defined for integers the gamma function is defined interpolated between integers gamma function for the integers is equal to n minus 1 factorial gamma of n but it's defined for non-integer stool and I won't tell you it's defined by an integral and it continuously interpolates between the integers this is it this is the Veneziano amplitude multiplied by the coupling constant squared if you examine this amplitude you'll find out I'm not going to go into it's mathematics it's mathematics is very simple all you have to know is about gamma functions and I if you if you want to explore it it's it's actually quite a simple construction but I'm going to tell you what it looks like it looks like an amplitude that you would make by summing up large number of particles this is representing some composite particle or some particle it could be in there of different masses in the s channel but it also looks like an O notice that it's it it's it's symmetric it's symmetric with respect to SP exactly like this is the peculiar thing about it is you can represent it as a kind of sum assignment diagrams with all the particles in the s channel diagrams like this but because it's symmetric on the SP you can also represent that you don't add it is also equal to the same kind of thing going this way something odd was afoot about this formula it had all important features that the scattering amplitude should have it could be analyzed as if a whole bunch of particles were produced and then decayed but it could also be analyzed as if a whole bunch of particles were exchanged this was something new this had not been seen before previous to this everybody would have added contributions for SMT and this thing replaced this the question was what is this where does this come from what kind of physics is rise to this what kind of physics can you imagine would give rise to this to a formula like this the answer of course turned out to be string theory the invention and the discovery of string theory was just looking for a physical model which would give this for its answers for its answer for scattering of two mesons or two two particles I'm going to tell you our without a lot of drama what physical model gives rise to the scattering amplitude and I'll show you a little bit I'll show you what the logic that went into it was it wasn't very hard to guess that this was a theory of strings it wasn't too hard just because well it wasn't hard he said exchange particles versus this is you could the s channel D channel exchange or direct production and exchange sometimes called the direct channel in the cross channel the s channel is called the direct channel those are the direct particles come in coalesce and then go out across channel is when the particle jumps across from one side to another I'll show you what the physics of that formula is you begin with two strings I'm not going to the calculation but I'll show you everything that went into it it's a little bit a little bit tedious what you have to know about you have to know a lot about harmonic oscillators and that's all nothing much more than that you start with two strings all right here's a string now remember the string has a coordinate along it which we called Sigma let's draw the Sigma axis over here from Sigma from 0 to PI and here's a string but the string propagates in time let's draw it climb horizontally here's the end of a string at Sigma equals 0 Sigma equals PI these are open strings now the string of course moves around in space-time but is always located between zero and pi that's not spatial position that's just its parameter along the string and the material of the string the particles that make it up are in here each point in the history of the string think of the string is sweeping along here it is if it's sweeping along some space-time sheet it's called the world sheet instead of a world line each point in that world sheet is characterized by a point Sigma and a time which has called tau tau goes this way it's a time and Sigma is a coordinate along the string not a real spatial chord and it's just a label for labeling points along the string panel is this time that we've used previously are the infinite momentum time but that's not what's important what's important is that the idea of a world line becomes a world sheet and instead of being parameterised by a single variable tau it's now parameterized by two variables each point in here has a position in space-time X X mu or just X and we've already worked out what the equations of motion of X are they are the wave equations describing waves moving up and down the string the wave equation let's write it the second X by D tau squared minus the second X by D Sigma squared equals zero that's the wave equation that described the oscillations of the string but we don't even have to think about this we can just imagine that this string is a collection of a large number of particles replace the worldsheet by a bunch of world lines narrowly spaced with springs between them whispers fill the springs between them that's the picture of the evolution of a particle now what we're going to do is begin with two particles we're going to have two particles coming in from the past and we're going to put them right next to each other let's see I think we need another color for the second particle we aren't going to put them right next to each other in space necessarily but just in the parameter space here this goes from zero to PI we need to put another one in going from pi also from from zero to pi here's the other string there's no meaning to the fact that I've drawn them right next to each other I've just drawn them their actual position in space-time might not be adjacent so this just parameterize is this half parameterize is the worldsheet of the left particle this half parameterize is the worldsheet of the right particle and they might be far away from each other that means the x's over here may be very different than the x's over here but with any luck at all with some probability the end of the string might touch the end of that string and when it does they can coalesce that's an assumption that they can coalesce but that's the basic process of string theory that they can coalesce and then form a single string now they really are connected if you like a new spring developed when these touched each other a spring appeared connecting the last the last particle of this string with the first particle that string and the whole thing becomes one string that condition persists for a while it persists for a while until a quantum mechanical event happens and the strings separate again randomly but you can transform around cleverly to make it be at the same place along the string is enough symmetry enough symmetry of the of the equations that you can put this point at the same point the same horizontal level as this point so what do you have that's all right so what is the nature of this process given this it's possible to guess what the answer is for the oh the important quantity in here is the amount of time that the that the strings spends coalesced it's the amount of time that the compound state stays before it breaks up into its final constituents and let's just call that tau now let's call that from this time here tau I'll tell you what we do with it in the end in the end we integrate over it but the what you put in if you're interested in a quantum-mechanical amplitude you want to start with an initial state the initial state let's start with one particle the initial state the initial state can be thought of is the state of a whole bunch of little points namely the points that make up the string x1 through xn let's start with the first string x1 through xn it's just a collection of mass points x1 dot dot dot through X N and the wave function of it is just a function of x1 through xn that's its wave function to begin with in the start what do we know about the wave function does everybody understand why a wave function is a function of the in positions of the particle that's quantum mechanics quantum mechanics says wave functions or state vectors are functions of a position of the constituents what's the probability to find the particles at position X 1 through xn so I star sigh so this is that's what this is this is the wave function of the starting assemblage of particles of course I'm I'm purposefully not taking the continuum limit to show you what goes on the end you'll have to take the continuum limit but let's not do that what do we know about this well the first thing we know about it is that this particle comes in with momentum k1 I'll tell you exactly what that says that says that this wave function contains a factor e to the I k1 times the center of mass position this is the way the wave function e to the ikx is the wave function of a particle with definite momentum what momentum do you use you use the center of mass momentum so what is the center of mass position sorry the center of mass position what's the center of mass position of these points it's the average position X 1 sum of the X is divided by n so you have X 1 plus X 2 dot dot up to X n divided by n that factor and that factor alone tells you that this initial particle here had momentum K 1 now what about the rest of the wave function which depends on the relative coordinates not the sum of them but the distances between the neighboring particles that's some wave function which characterizes the ground state it depends on everything except the sums of the X's it's some wave function let's call it sine naught for ground state and it depends on the X is same exes but it actually doesn't depend on the some of them some of them here the differences between them here differences of X's here they bring X's and so forth here some of them here and this wave function is computable it's just a ground state describing the ground state of all the harmonic oscillators making up the the string then you can really work out that can you know will they love it with enough room on the blackboard I could tell you exactly what this function is as a function of a collection of X's it's not very hard it's a bunch of a bunch of Exponential's it's workable you can do it this is the wave function of the first particle what about the wave function of the second particle it's exactly the same kind of thing except it doesn't depend on these coordinates it depends on the coordinates of the red particles here so let's write it down e to the I k2 now what shall I write shall I write x1 through xn no that's the original particle those those are the original constituents of the first string I want the constituents of the second string so let's write X n plus 1 X n plus 2 all the ways up to X 2 in the second half of the particles is grouped together into the second string also divided by in also x sy not of 1 through n this is X n plus 1 to the end to the end of the chain could that be red it is red or can you read it ok you don't read it now that's the initial state but they say with some probability the two endpoints merge to say that the two endpoints merge simply says that you set x1 sorry X in equal to xn plus 1 you look for that piece of the wave function where the two endpoints are at exactly the same point so you begin with this the next step is to say let the nth particle and the first chain be at exactly the same place as the n plus sorry as the X in being at the same place as xn plus 1 and so we're going to put in here then X n all the others left unchanged that's now not that's the wave function of the state right at the point after the two particles of coalesced when they're coalesced at the point where they coalesce they come together nothing happens to the rest of the chain but the two particles so this is if you like it's the amplitude that the two particles the chain touched this is the amplitude that the two particles the endpoints touched now we have a new starting point which is a function of X 1 through X 2 n it's a state of that many particles and what do we do with it we have to evolve it we have to evolve it using the Hamiltonian the motor Hamiltonian is a Hamiltonian is a thing which updates you from one instant to the next so you take this initial state it's so well defined thing and you propagate it forward in time using the Hamiltonian you take it what's the right rule for updating a state from one from an instant or later instant you multiply the state by something e to the i HT e to the I so you take this wave function then you evolve it you solve the Schrodinger equation for it but that's the same as multiplying it by e to the I the total Hamiltonian times tau but what is the total Hamiltonian the total Hamiltonian is only is just the collection of Springs and mass points a collection of harmonic oscillators is just a collection of harmonic oscillators we know how to do this and that gives us the state of the system after time T what's the last step the last step is to project the final state here it is on to two separate particles again to project it onto a state with two separated particles of momentum k3 and k4 it's very very straightforward if tedious slightly tedious you take the two initial States of the two particles they're well-defined ground states of the particles you insist that the two end points are at the same place that constrains the wavefunction you let it evolve as a single string for a ways and then you let it break up again and that simply means multiplying it again by some final state I'm not going to write it all out it's the same kind of thing and that gives you the transition amplitude to make a long story short you start with the two particles you constrain it so that they're at the same place you evolve it and then you project it on to the final state a very very well-defined thing to do that gives you the amplitude that the strings coalesced for an amount of time tau but how did you choose town those things in black and red there are added together and then you multiply those are multiplied together yeah if you have two separate systems each having its own wave function you multiply to create the wave function of a composite so this is just a wave function of a composite of two particles to begin with you don't set the first particle equal to the last particle here then you look for the amplitude you look for the piece of the wave function where those two particles are at the same place you say aha now that they're at the same place fuse them together and evolve it as if it were a single string a single collection of mass points and Springs for the amount of time tau and then basically you take your scissor and just cut the you cut the spring in the middle and let it evolve after that that calculation is quite doable not even very hard it's just too much for the black board for for one for an evening but I'm going to tell you what the answer is I'm going to write down the answer for you so I'm trying to marry member this is this is the answer it's an integral over D tau then we say that we integrate over the tower I think I said we integrate why do we integrate over tau incidentally why do we add up the amplitudes for all possible times that this composite could exist this is the finding rule of summing over all possibilities the only parameter here is the time that it takes for this for this to break up again and find this rule is sum of all paths which in this case just means integrate over the time that they spent evolving together and that gives you the amplitude for a particle for the two particles in the initial state to become the two particles in the final state the result is an integral it's the integral over the time that they spend together that's it I'll tell you what the integrand is after a certain amount of calculation which is actually not very hard it becomes the integral of e to the Tao that's the time times s plus 1 this s plus 1 is exactly this s it's the S variable or the center of mass square root of center of mass energy of the two particles you have a factor like that you have another factor which is 1 minus e to the Tao to the minus t minus 1 t is the momentum change between the initial particle here and the final particle here remember those momenta are coded in this wave function which I've erased they were coded in the wave function in terms of those Exponential's initial moment sorry the center mass energy and here is the momentum transfer it appears in the formula and then you have to integrate it D tau there happens to be another factor I think of e to the minus tau in the integrand you're right I have the top twice the first term is tell times s plus one yeah no yes yeah that's what you get it multiplies this but I've left that I've left it out here on purpose I left that out here on purpose DCAM now that doesn't look particularly symmetric between tau and between T and s but after it was computed it took about a half hour to realize that you should change variables in this integral change variables from E theta tau from tau to something called Z let e to the power e cuisine okay let eat to the tau equals e and now we like this e to the Tao is Z so this becomes integral of Z ah I'm missing some minus sign no AE to the minus e to the minus tau must be Laura this must be minus here it is e to the minus tau there this thing just becomes Z to the S plus 1 R to the minus s plus 1 e to the minus tau so this is minus minus e to the minus tau Z by definition just a change of variables what about 1 minus e to the minus tau that becomes 1 minus Z to the minus tau minus 1 Oh incidentally this integral goes I think from zero to infinity is that right from zero to infinity yes it does this becomes 1 minus Z to the minus tau minus 1 it was D tau times e to the minus tau hmm it's just easy it's just DZ I mean I got it it's just easy oh my did I get that right I've lost track of whether this is plus or minus here I don't remember but it's just DZ this is the formula and what is the integral go the integral goes from tau equals 0 where Z is equal to 1 - this is - it is - I'm sorry I'm making a mess out of it e to the minus tau is equal to Z - tally goes infinity in other words you have tau equals zero that's when it very suddenly merges and then falls apart instantly the tau equals infinity when the separation is infinite here and what happens when this tile goes to infinity Z goes to 0 so it's an integral from 1 to 0 or 0 to 1 there's probably some sign in here of an integral that looks like this the amazing thing about this integral the whole upshot of it is that this integral is completely symmetric between s and T can you see that how do you see that it's symmetric between s and T right you just substitute for ZZ minus 1 you make a change of variables between Z and Z minus 1 and you see that this integral is completely symmetric so although a completely unsymmetric starting point between energy and momentum transfer somehow it wind up giving a completely symmetric answer between the two of them I lost track of the - I comes from e to the minus tau D tau going to disease yeah I'm lost track of me yeah so you can want to sign that's why you speak with the minus sign somewheres I'm saying e to the minus tau D tau yeah and then you're going to make that just be easy yes so you get a minus L yeah there's got to be a minus sign out here right right then you switch the order of integration and it becomes from zero to one like this that's right that's correct it's symmetric whether or not you switch the order of integration right and that's that's the answer it's symmetric between s and T E and what is it it's a process in which two particles join form a composite which Wiggles around for a while and then breaks up in other words it's the analog of the Fineman diagram in which a composite is formed and then decays composite is formed and then decays but it winds up being completely symmetric between s and T in fact this function is called the Euler beta function it's a function of two variables s and T it's called beta of minus s and minus T it's the Euler beta function it's a famous function of mathematical physics and guess what it's equal to it's equal to the Veneziano amplitude it's exactly equal to the vidit siano amplitude namely this product of gamma functions over another gamma function um how did it get to be that it was symmetric between the two now this is not obvious at all that it's symmetric between the s channel and the P channel completely symmetric between the S channel and the P channel I'm going to come to that next time that has to do with fundamental extremely deep symmetry of string theory called conformal symmetry it's a symmetry which allows you to take these world sheets and deform them in crazy ways as if they were Turkish taffy and stretch them out in different directions and for example turn this picture into a picture which looks much more like two particles coming together in exchanging so probability function of right government or health issues yeah so what's the time it is true that that the beta function evaluated at integers is combinatorial coefficients that the inverse of the combinatorial coefficients the always talking about talking about n factorial M factorial over n plus M factorial which occurs all over the place in combinatorics just as just happens to be the same function no simple no simple uhm no simple connection first of all it's the inverse of it the inverse of it is not a combatant power coefficient there but [Music] just happens it's an integral which defines the same combination of gamma functions there is no simple relationship it's not that something combinatoric went on here at least F to my knowledge and it's certainly not the wave in it C on oh I found it I don't know what magic he pulled up to find it so this was if you like partly historical but part of the important logic of the theory is that number one you can calculate with it it's all is is harmonic oscillators anything can be done with harmonic oscillators it's a bunch of harmonic oscillators you break the process up into pieces and then you integrate over the time in between you can calculate you calculate and you find an integral the integral by at this point by magic has the property that it's symmetric between s and T and somehow looks like not the some but has the features of having processes where particles coalesce that's the direct process I drew here but somehow buried in it is also somehow processes where particles are exchanged and that was the magic of it that was the surprise that a whole new logic of putting processes together to make new processes okay um any questions about this yeah um I know this is only in the beginning we described two strings call us n oscillators and then oscillators they come together to it is those arms remember when you're taking a lemonade goes to infinity to n is the same measure but but still has the property of those are oh yeah so far you would rather the question is when you wrote down the wave equation you had the two points coalescing which is 2 n minus 1 sort of changes from positive it seems inconsistent you lose you lose two not one well edit n plus 1 becomes 1 no actually in the n plus 1 disappear basically if you like I mean I think that's right way to think of how you yeah yeah they eat each other they eat each other like that and you left right so so the new spray the new spring the forms connects these two yeah yeah other way otherwise we'd have big trouble we would lose one for me on right all right so that's a good one way to think about it is that a particle and an antiparticle coalesced and if you thought of these as quarks tens of strings you could think of the last one being a quark which annihilated with the first quirk of the of the other string so yeah you lose two of them but who cares about to win there's an infinite number yeah right oh yeah right but you're like you have to be careful not to lose an odd number fermions that's something that shouldn't happen yeah on this query well we're not going to spend a lot of time on super strings although the things that I will tell you strictly speaking apply to super strings and not to the bosonic string are we going now I'm not going to go I'm into the heavy mathematics of super strings what I think we're going to do next time is I'm going to tell you well I'll tell you a little bit more about string theory and about the properties of world sheets and the symmetries that allow you to stretch this thing like Turkish taffy and make different kinds of finding diagrams out of it but then we're going to move on to another subject called m-theory which is another way of deriving string theory from a totally different vent well I think we have to talk a little bit about compactification about what you do with these extra dimensions and we will maybe next time but then we will move on to a totally different origin of string theory that evolved much much later in time sometime around 1995-96 which started with a completely different picture and in which a great many of the features the more complicated features of string theory are completely transparent and it's called m-theory and we'll we'll start a different starting point which leads to exactly the same the same physics so this process you just described to his face on the clues from two masons joining yes but it's also the way two photons collide in string theory it's the way any two open strings collide in string theory incidentally the analog for closed strings is very similar you have two closed strings and you pick a particle from here and one from here and you require that they be at the same point so that they do this kind of thing and then you evolve it as a single string same rules same kind of rules and that would correspond to the scattering of two closed strings if you scatter two closed strings the intermediate thing that you make will again be a closed does this only apply to strings where the particles are fermions no no no no no no no this this dis calculation was originally done for the boson extreme and and of course it's more complicated when you have to keep track of the fermions also extremely similar extremely similar some slight the differences that are not important right what I should say about this is that the scattering amplitudes we said that there was a photon in a and a graviton in the system all right it looked like there was a spin-2 particle and a spin one particle and you could force them to be massless if you wanted even that much choice there weren't enough components to make a massive particle so you did that well then you take these particles and you collide them and you work out scattering amplitudes scattering amplitudes are very distinctive and characteristic for emission of photons and gravitons they are not just any old scattering amplitudes they have very very definite properties which make them extremely special the emission and the emission and absorption of photons cannot be mistaken for the emission and absorption of scalar particles or other things they said they satisfy some very very important rules those rules originate from the conservation of electric charge conserved electric charges emitting photons there's almost very very little ambiguity in what the emission and absorption of photons look like or what a scattering of photons by charged particles look like or even the scattering of photons by photons and it was at this point calculating these diagrams where it became completely clear that these things which we were calling photons were behaving exactly like photons and the things that we were calling gravitons were behaving exactly like gravitons that they satisfied all the rules for graviton graviton scattering just saying there was a particle that looked like a graviton was a very weak thing when all of the scatterings were constructed and the rigorous tests of whether it was satisfying the rules for scattering of gravitons by massive things but natheless things and so forth fit perfectly exactly then people realized that they really were dealing with something that looked like the scattering of gravitons and photons so the scattering amplitudes played a big role in establishing with you know precision that we were dealing with objects that did behave like photons gravitons and it ok I think we're finished yes where's what oh yeah we haven't talked about that but ok we'll talk about we'll talk about it next time remind me remind me for more please visit us at stanford.edu

Channel: Stanford

Views: 67,661

Rating: 4.9052134 out of 5

Keywords: physics, science, mathematics, 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

Id: qllX-dkGfYI

Length: 84min 24sec (5064 seconds)

Published: Wed Mar 30 2011