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https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/8.06-spring-2018.zip
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PROFESSOR: OK, time for us to do one example, a non-trivial example, which is the ionization of hydrogen. It's a fun example, and let's see how it goes. So ionization of hydrogen. Ionization of hydrogen. Very good. So we're going to think of a hydrogen atom on its ground state sitting there, and then you shine an electromagnetic field, and if the electromagnetic field is sufficiently strong, the electron is going to be kicked out by interacting, typically with the electric field of the wave. So the wave comes in and interacts. So we usually think of this in terms of photons. So we'll have a hydrogen atom in its ground state. And then you have a harmonic e field, and the electron could get ejected. So typically, you have a photon in this incident. We think of this in terms of electromagnetism, although our treatment of electromagnetism was not going to include photons in our description. But we physically think of a photon that this incident on this-- here's a proton. Here is the electron going in circles, and the photon comes in, a photon gamma, and it kicks the electron out. So the photon has energy e of the photon is h bar omega. So when we think of putting a light beam, and we're going to send many photos, we have to think of each photon, what it's doing-- whether the energy is very big, whether the energy is very small, and how does it affect our approximation. So half of the story of doing such a calculation is understanding when it could be valid, because we're going to assume a series of things in doing the calculation. And the validity still won't turn out to be for a rather wide range of values. But we have to think about it. So a couple of quantities you'll want to consider is that the energy of the electron, energy of the electron that is ejected is h bar squared k of the electron that is ejected, k squared over 2m of the electron. And it's going to be equal to the energy of the photon minus the ground state energy of the electron in the hydrogen atom. So the hydrogen atom here is energy equal 0. Ground state is below 0. So if you supply some energy, the first part of the energy has to be to get it to energy 0, and then to supply kinetic energy. So the kinetic energy is the energy of the photon minus what this called the Rydberg, or Ry. The Rydberg is 13.6 eV with a plus sign. And that's the magnitude of the depth of the well. The Rydberg is e squared over 2a0 or 2 times the Rydberg is e squared over a0. That's what was calculated, and this is actually equal to h bar c the constant alpha times h bar c over a0. remember, the constant alpha was e squared over h bar c. So those are some quantities. OK. So we need the electron to be able to go out. The energy of the photon must be bigger than a Rydberg. OK, so conditions for our approximation. One. We're going to be using our harmonic variation. We said in our harmonic variation, Hamiltonian delta h was 2h prime cosine of omega t, and we want this h prime to be simple enough. We want to think of this photon that is coming into the atom as a plane wave, something that doesn't have big spatial dependence in the atom. It hits the whole atom as with a uniform electric field. The electric field is changing in time. It's going up and down, but it's the same everywhere in the atom. So for that, we need that-- if you have a wave, it has a wavelength, if you have an atom that is this big, you would have the different parts of the atom are experiencing different values of the electric field at the same time. On the other hand, if the wavelength is very big, the atom is experiencing the same spatially independent electric field at every instant of time. It's just varying up and down, but everywhere in the atom is all the same. So what we want for this is that lambda of the photon be much greater than a0. So that-- 1. So this means that the photon has to have sufficiently long wave, and you cannot be too energetic. If you're too energetic, the photon wave length is too little. By the time it becomes smaller than a0, your approximation is not going to be good enough. You're going to have to include the spatial dependence of the wave everywhere. It's going to make it much harder. So we want to see what that means, and in the interest of time, I will tell you with a little bit of manipulation, this shows that h omega over a Rydberg must be much smaller than 4 pi over alpha, which is about 1,772. So that that's a condition for the energy. The energy of the photon cannot exceed that much. And let's write it here. So that's a good exercise for you to do it. You can see it also in the notes just manipulating the quantities. And this actually says h omega is much less than 23 keV. OK, 23 keV is roughly the energy of a photon whose wavelength is a0. That's a nice thing to know. OK. While this photon cannot be too energetic, it has to be somewhat energetic as well, because it has to kick out the electron. So at least, must be more energy than a Rydberg. But if it just has a Rydberg energy, it's just basically going to take the electron out to 0 energy, and then you're going to have a problem. People in the hydrogen atom compute the bound state spectrum, and they computer the continuous spectrum in the hydrogen atom, in which you calculate the plane waves of the hydrogen atom, how they look. They're not all that simple, because they're affected by the hydrogen atom. They're very interesting complicated solutions for approximate plane waves in the presence of the hydrogen atom. And we don't want to get into that. That's complicated. We want to consider cases where the electron, once it escapes the proton, it's basically a plane wave. So that requires that the energy of the photon is not just a little bit bigger than a Rydberg, but it's much bigger a Rydberg. And saw the electron, the free electron does not feel the Coulomb field of the proton. And it's really a plane wave. So here, it's a point where you decide, and let's be conventional and say that 1 is much less than 10. That's what we mean by much less 1/10. So with this approximation, h omega must be much bigger than 13.6. So it should be bigger than 140 eV. That's 10 times that. And it should be smaller, much smaller and 23 keV, so that's 2.3 keV. So this is a range, and we can expect our answer to make sense. If you want to do better, you have to work harder. You can do better. People have done this calculation better and better. But you have to work much harder. I want to emphasize one more thing that is maybe I can leave you this an exercise. So whenever you have a photon in this range, you can calculate the k of the electrode, and you can calculate how does the k of the electron behave. And you find that k of the electron times a0 is in between 13 and 3 for these numbers. When the energy of the photon is between those values, you can calculate the momentum of the electron, k of the electron. I sometimes put an e to remind us of the electron. But I'll erase it. I think it's not necessary. And that's the range. OK. So we're preparing the grounds. You see, this is our typical additive. We're given a problem, a complicated problem, and we take our time to get started. We just think when will it be valid, what can we do. And don't rush too much. That's not the attitude in an exam, but when you're thinking about the problem in general, yes, it is the best attitude. So let's describe what the electric field is going to do. That's the place where we connect now to an electric field that is going to produce the ionization. So remember the perturbation of the Hamiltonian, now, it's going to be the coupling of the system to an electric field. And this system is our electrons. So it's minus the electron times the potential, electric potential, scalar potential. Now, needless to say, actually, the electron that is going to be kicked out it's going to be non-relativistic. That's also kind of obvious here. You see h omega is 2.3 keV. You subtract 13.6 eV. Doesn't make any difference. So that's the energy of the emitted electron roughly, and for that energy, that's much smaller than 511 keV, which is the rest mass of an electron. So that electron is going to be non-relativistic, which is important, too, because we're not trying to do Dirac equation now. So here is our potential, and then we'll write the electric field. Let's see. The electric field. We will align it to the z-axis to begin with. 2e0 cosine omega t times z hat. These are conventions. You see, we align it along the z-axis, and we say it has a harmonic dependence. That's the frequency. That's the frequency of the photons that we've been talking about, and the intensity, again, for these convention preference will put 2e0 times cosine omega t. So some people say ep, which is the peak e field is 2e0. That's our convention. Well, when you have an electric field like that, the electric field is minus the gradient of phi, and therefore, we can take phi to be equal to minus the electric field as a function of time, times z. So if you take the gradient minus the gradient, you get the electric field. And therefore, we have to plug it all in here. So this is plus e, e of t, z, and this is e. e of t has been given 2e0 cosine omega t. And z, we can write this r cosine theta. In the usual description, we have the z-axis here, r, theta, and the electric field is going in the z direction. So z's are cosine theta. So I think I have all my constants there. So let's put it 2 e, e0, r cos theta cos omega t, and this is our perturbation that we said it's 2h prime cosine omega t. That's harmonic perturbations were defined that way. So we read the value of h prime as this one. e e0 r cosine omega t. No, r cosine theta. OK. We have our h prime. So we have the conditions of validity. We have our h prime. Two more things so that we can really get started. What is our initial state? The initial state is the wave function of the electron, which is 1 over square root of pi a0 cubed e to the minus r over a0. That's our initial state. What is our final state, our momentum eigenstates of the electron? So you could call them psi, or u sub k of the electron. And it would be 1 over l to the 3/2, e to the ik of the electron times x. These are our initial and final states. Remember, the plane waves had to be normalized in a box. So the box is back. u is the wave function of the plane wave electron. And we could use a plane wave, because the electron is energetic enough. And it has the box thing. This is perfectly well normalized. If you square it, the exponential vanishes, because it's a pure phase, and you get 1 over l cubed. The box has volume l cubed. It's perfectly normalized. This is all ready now for our computation.
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https://ocw.mit.edu/courses/8-851-effective-field-theory-spring-2013/8.851-spring-2013.zip
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. So, so far we've recently been talking about examples in SET2, and we're going to continue to do so today. So the example that we did last time was the plan photon form factor. That did not have any soft degrees of freedom. It just had colinear and higher degrees of freedom. So it was a particularly simple example of something we could think of in SET2. We'll start with a slightly more complicated example, this decay, B to D pi, where we have both colinear and soft degrees of freedom. This was an example that we mentioned at the very beginning of our discussion of SET, and now we're going to see how factorization looks for it. And then we'll talk about something called the rapidity renormalization group, which has to do with situations in SET2 where the separation of degrees of freedom is a little more complicated than in the previous examples. And we'll see that there can be a new type of divergence that shows up. And that new type of divergence leads to a new type of renormalization group. So B to D pie. So there's going to be, in some sense, three hard scales of this problem. The mass of the B quark and the mass of the charm quark are going to be taken to be much greater than lambda QCD, so we'll have an HQET type description of the B quark and the charm quark. And also the energy of the pion, which is in some sense, related to the difference of the B quark and the charm quark mass, will also take that to be much greater than the lambda QCD. So just by kinematics this thing is proportional to MB minus MC roughly. You could say, it's the difference of the squares of the hadron masses. OK. So let's first-- we know how to treat this decay if we were integrating out the W. This is a weak decay, so B is changing to charm. Integrate of the W boson, run down to the scale MB, which is the larger scale here, that's the electroweak Hamiltonian. So that's what we'll call the QCD operators, which are the relevant description at the scale of order MB. Some pre-factor. And I'll write the operators in the following way-- a slightly different basis than we used previously, or a long time ago when we were talking about this particular case. So just a different color basis, singlet and octet. OK, so that's our description, where p left is projecting us onto, the left-handed components. So what we want to do is we want to factorize the amplitude. This is an exclusive process where we make a transition between specific states. So we'd like to separate scales in the D pi, and then we have O0 or O8 in this matrix element. So we have two matrix elements, one with O0 and one with O8. And so, what could it possibly look like? Well, we already talked about the degrees of freedom here. The D is going to be soft and the B is going to be soft. So this is soft, this is soft. This is going to be colinear. And so, if it's going to factorize, and the soft degrees of freedom are not going to talk to the colinear, the kind of thing that you would expect to show at leading order in the lambda expansion is that you have the following kind of process. Let me reclaim this space. So here's a heavy quark. Here's one of these operators. Here's the valence quarks in the pion. Another heavy quark-- this was B charm, U and D. And there's an anti-quark, and we have to address this by gluons. And if it's going to factorize, then the way that we should dress it by gluons is as follows. We would have soft gluons here, and they could interact, if you like, between things in the B and the D, because the B and the D are both soft. We can also have back and polarization diagrams. And that is going to factorize from things in the pion which are going to be colinear. So we have our colinear gluons and colinear quarks inside here, and maybe there's some Wilson lines too. So we would expect some kind of picture like that. And that's actually going to be what we do find. But exactly what happens at this vertex, what kind of convolutions there are, that we have to work out. All right. So what factorization in this context means is that there's no gluons that are directly exchanged between the B to D part and the pion part. So that it effectively factorizes into a matrix element that's like a B to D transition and a vacuum to pion transition. So you can even guess what kind of objects this would depend on. If you have something like this green thing, that's a B to D form factor. So we expect a B to D form factor. And for the pion, if you have something like this, well, we already talked about something like this when we were talking about gamma star, gamma to pi zero. So for the pion, we expect the pi zero-- the light cone distribution, which is the sort of leading order operator for the pion. So we'd expect a 5pi of x. And we'll see that we do indeed find that. So the B and D have P squared of order lambda QCD squared for their constituents. The pion is colinear, and its constituents have P squared of order lambda QCD2, but they're boosted. We can again use SET-- this is SET2, but we can use SET1 as an intermediate step, just like we did-- we talked about last time. So let's do that again. So match-- step one was to match QCD onto the SCET1. So there was some hard scale. And the harder scale in this case could be any one of these three. So collectively I just denote them by Q. And so what's going to happen in that matching is we take these operators O0 and O8, and we have to match onto SET operators. So let me call the SET operators Q0. I just there's-- because of the fact that there's a heavy quark in the way that that works, there's two possible spin structures here. But we'll see that actually only one of them has the quantum numbers in the end. So we have heavy quark fields, charm, and bottom, that we can imagine that type of operator. And then the colinear part, we can dress it with the Wilson line, as always. And let me put the flavor upstairs. And let me put in the most general Wilson coefficient that I can think of for this process. This could also depend on V dot V prime. I didn't denote that. But in general, it will. And it could depend on any of the scales Q. So it can depend on the large momentum of the colinear fields, as always. And there's one combination that's not restricted by momentum conservation. So there's one combination that's not related to these scales, and that's the, in my notation, the P bar plus operator. And then there's likewise, there's another thing with the TA. So same thing, TA, everything the same, TA. And then this has got a different coefficient like that. And so, what are the Dirac structures just to be explicit? The heavy one-- so the light one is going to be M bar slash over 4 in my notation, the 1 minus gamma 5. And for the heavy one, you could have either 1 or gamma 5. And that's because you originally have a left handed-- originally here you have a left handed guy between the charm and the B. But remember that a mass term connects left and right. So after you integrate out the mass of these quarks, you don't know whether-- you don't know the chirality anymore in this. So that's why you can have both the possibilities 1 and gamma 5. So if you put any other Dirac structure-- so here you could use chirality. And so, you know that these guys should be left handed still, and that's why that this should be this structure. You know it should be an M bar slash, because any other structure that you would stick in here between them would give you something that's power suppressed. Because you know that N slash, when CN 0, and also CN bar gamma [? per mu ?] with any kind of P left or whatever, is also 0. So you'd have to have something more complicated. All right, so there's those operators. And when you do this matching, it is non-trivial in the sense that these two operators-- it's not diagonal. It's not like O0 goes to Q0, and O8 goes to Q8. As soon as you start adding loop corrections these two mix, and then they give you some contributions to these coefficients C0 and C8. OK? So what you mean by octet operator in the electroweak Hamiltonian is different than what you mean by octet operator in the SET1 factorized result. But that's just a complication that you deal with when you are doing the matching. This guy can be proportional to the Wilson coefficient C0F and C08, and that's not really a big deal. Any questions so far? STUDENT: [INAUDIBLE] PROFESSOR: Yeah STUDENT: So is the point of matching to get one of them as being [INAUDIBLE] to get the softs? PROFESSOR: Yeah. STUDENT: So you're going to distribute them-- PROFESSOR: That's right. I'm going to do that right now. STUDENT: OK, so that's-- PROFESSOR: Yeah. So then step two, field redefinition in the SCET1. And let me not put superscript zeros, let me just make it as a replacement. So then we get RQ0 again. And OK. For this guy, it's exactly the same, because in the Q01 comma 5, all the Y's cancel. In the octet guy, it's not quite that way, because there is-- because we do have-- in that case, we do have the Wilson and lines getting trapped by the TA. So let me write out that case. So we have-- so that's what we would get after the field redefinition for two operators. OK. So the next thing to do would be to instead of calling them Y's call them S's. So, but there's one more thing I can do too. So these are, remember, these are soft fields and these are colinear fields. So this isn't factorized, because we have contractions between these Y's in the fields over here. Gluons can attach to heavy quarks. So in order to factorize we want to move those Y's from there over to here. And we can do that. So here's how that works. This is a formula that I could have told you earlier. So if you have a Y that lies around a TA, that's just actually the adjoint Wilson line. So this is a formula that relates fundamental Wilson lines and an adjoint Wilson line. So in the adjoint Wilson line, you'd build it out of matrices. If you like, they're like this. So the matrix indices would be B and C, and instead of having fundamental indices for the TA alpha beta, you have an FABC, and this is the kind of thing that you would exponentiate. But other than that it's the same thing. And there's just a color identity relating them. So because of this identity, you can also write down another identity, which is Y dagger TAY. This guy is-- so Y dagger TAY is just the other YAB-- this guy's an orthogonal matrix, TB. So if you reverse the indices then you get the opposite way. And also, this guy is-- remember this is a matrix just in the AB space. So if I use this formula in here, that allows me to take these Wilson lines here and move them over here. Right? Because I can take them, write them as a Y, and then the Y is just something that doesn't care about-- it just moves over because it's contracted with that index A. I guess I've got some problems with capitals and small letters. Let's make them all capital. So then not going to move it over here. And then I can convert it back to a Y dagger Y if I want to. OK. So we can take this guy and this thing and write it over there as HV Y TAY dagger HV prime. I was careful about that. This was the prime. And then all the soft gluons-- all the ultra soft ones are over here in this matrix element and all the colinear fields are over there. So if you like, you could say, we get this, and then we get our colinear matrix element that has TA, but it has now no ultra softs. And so we have a product of colinear and ultra soft things tied together by one index A. OK, so that's just a little color rearrangement. It's useful, because now they're really factorized. And now when you take matrix elements, the matrix elements will factorize. Oh, sorry, before we take matrix elements, let's switch to SCET 2. So this is, again, an example where it's trivial. Because what we have is, we have one type of operator that we're considering, this weak transition, and we don't have a time limited product of any type of two operators in SET2. We just have a single operator that has both types of fields, and then we have the Lagrangians So this is, again, simple. So we have one mixed operator plus L0 colinear and L0 soft. And those things are already decoupled, and so this is simple. And so, we simply replace Y's by S, renaming it soft instead of ultra soft. Really nothing is changing. And these colinears, we just put them down onto SET2 colinears from SET1 colinears. OK, so to make it look like I've done something, I'll write it out again. But there's really nothing happening except that now the fields are in SET2, with the correct SET2 scaling. So there's no Wilson coefficient that's generated by this stuff-- there's no additional Wilson coefficient because of that fact. So these are SCET2 now. And similarly for the octet. So here we would really have the-- OK. So now we can take matrix elements. STUDENT: [INAUDIBLE] PROFESSOR: Yeah? STUDENT: What do the coefficients look like when they're not just 1? PROFESSOR: So they would be functions of say, plus times minus momenta. So we could have-- if it wasn't 1, what would happen is effectively-- so yeah, we talked a little bit about this last time, but let me remind you. If it wasn't 1, that would happen in a situation where you had something like this. Some off shell field, O, say like this. So this is a T product of two things rather than just one thing that mix off the colinear. And this field here was off shell in a way that basically, this field here is an off shell field that would be a product of the plus and minus momentum of these guys. And so, you could get something that's effectively living at this hard colinear scale below the hard scale in the problem, from sort of T products of soft and colinear operators. This is getting a little sketchy, but-- since here we only have one operator, that couldn't happen, because you just start attaching softs. And colinear is been-- it's already factored. So there's no way that you could sort of get this intermediate off shell guy. STUDENT: [INAUDIBLE] PROFESSOR: So this guy would be colinear, right? And the way to think about this is like this. This is just a diagram that exists. But then you go over to the SET2, and then you have your change of colinear to the right colinear. But this guy doesn't change. He's still hard colinear. STUDENT: OK, so it's a matching when the material actually has a different scale than matching-- PROFESSOR: Yeah. This scale would show actually up. And so if it doesn't show up-- so this would be kind of a situation, an going from step one. And actually, we could have done some examples where that happens. But I'm choosing to do this rapidity renormalization group instead. Basically this happens if you look at matrix elements, and you could look at matrix elements where you have some subleading interactions. And there are examples in exclusive decays where you could have this happen. One example is if you look at B0 to D0, pi 0, just having all neutral charges, then actually this will happen. It'll be-- and it'll be more complicated than what I'm telling you. But you can derive a factorization theorem for this. It's power suppressed relative to the one we're talking about, because the one we're talking about always has a charge pion, and it turns out that that happens at leading order, whereas the neutral pion process with the neutral B and neutral D is something that's power suppressed. Hope I'm remembering that right. Yeah. I am. All right. So number four, this is an aside. Number four, take matrix elements, and here we find actually that one of the matrix elements is just 0, the one with the octet. So let me write the nonzero ones first. I wasn't too careful about the two different-- about this. So there's some guy that's just giving a convolution between the Wilson coefficient and 5 pi. And then this guy, which is some normalization factor times a form factor. These things are all mu dependent in general. And this thing here is the Isgur-Wise function, which is the HQT form factor. And W0 is kind of the kinematic variable that that form factor can depend on, which is V dot V prime, the labels on the fields, and that encodes the momentum transfer. Which here is just related to the kinematics. So W0 is some function of MB and MC, which I'm not going to bother writing down. STUDENT: [INAUDIBLE] PROFESSOR: Just some constant. I mean, it could have some kinematic factors that makes up the dimension, like some-- depends on my gamma. Yeah, it's just some number. STUDENT: [INAUDIBLE] PROFESSOR: OK so these are the singlet operators. So in the singlet case, we have initial state and final state. In all cases we have initial state and final states, which are color singlets. And these operators are color singlets. In the case of the octet, we also have color singlet states. And we factorize such that we do have a matrix element for example. So let me just write one of them, and no. And this is 0, because there's nothing that could carry the index A in this matrix element. So the octet matrix element's 0. It's important that we factorized it for that to be true, right? If we had D pi, then we'd have a color singlet operator here, B. So we wouldn't have been able to make this statement in the original operator in the electroweak Hamiltonian. This would just not be true. But once we factored it and put all the ultra soft fields here, that everything that's going to be contracted together, then we can make this statement. We couldn't even really make this statement when the Y's were on the other side. We had to move them over here to ensure that this statement is completely true. OK. So color octet operator is color singlet states. OK, so then you just put things together and we can multiply these two things to get the final result. So if we write it as a matching from the electroweak Hamiltonian, there are some normalization factors. So grouping together these factors of F pi, E pi, and M prime, which I'm not worrying so much about, there's a Isgur-Wise function, and then there's a single convolutional between the hard coefficient, which is kind of like our example of the photon pi on form factor. And then the slight [INAUDIBLE] distribution. So we see an example where it showed up in a totally different type of process from the one we were considering previously, thereby showing us kind of the universality of that function. And then there would be some power corrections to this whole thing that we're neglecting, that go like lambda QCD over those hard scales. OK, so this is the Isgur-Wise function. And actually I did write down what the W would be. So this W would be that, some-- you can write it in terms of the meson masses like that, so it's the Isgur-Wise function at max recoil. And this function is measured, for example, in a semi-leptonic transition. So you can imagine that the pion distribution function, or properties of it, were measured in the photon pion transition. Is Isgur-Wise function is measured in the B to DL new transition. And then you can make predictions for this B to D pi. OK, so that gives you an example of how you would use these factorization theorems. So this applies to basically-- this type of factorization that we just talked about, it applies to a lot of different things with charged pi minuses. Or you can make the pi minus a rho minus, and that wouldn't really change anything. So you could have B0. If you wanted to look at charges, you could have B0 to D plus pi minus. Or you could have B minus to D0 pi minus. And there's a third one, which is the one we were talking about over here, B0 to D0 pi 0. So there's three different ways-- three different B to D pi transitions depending on the charges. And what we've derived applies to charge pions or charge rhos. The neutral ones end up being power suppressed. And you can see that kind from our discussion there was kind of never a-- there was-- it just writing down the leading order operators, well, maybe you have to work a little harder. But effectively, the leading order operators don't make this transition with the two charges being the same such that you could get a pi 0, so you have to do something more in order to get that case. All right. So questions? STUDENT: Can you [INAUDIBLE]? PROFESSOR: You can if you want. Oh, MB over MC. So MB and MC, we're treating both of them as avoiding the same in what we've done. So if we wanted to some logs of MB over MC, we'd have to do something a little different than what we did. You'd have to first integrate [? O to MV, ?] treat the charm quark as a light quark. You could do that. STUDENT: Would it make the SCET [INAUDIBLE]---- PROFESSOR: It turns out that-- STUDENT: --analysis different? PROFESSOR: --yeah. So those are single logs, actually, they're not double logs. And it's related to the fact that you have massive particles. When you have massive particles you don't get the extra singularity. And people in HQT worried about something-- logs of MB over MC for a while-- and then after a flight of doing enough calculations they realized it was totally irrelevant, and you should just not bother. You should just calculate the alpha S corrections, treating MB and MC as comparable, and summing the logs-- if you sort of think of leading log as being more important than the order FS calculation, that misleads you. Sometimes the sign is even wrong. And so there's sort of a general experience that something logs of MB over MC and HQT is not even-- STUDENT: Just because they're single logs? PROFESSOR: Just because they're single logs. I mean, that's one thing that makes it different than, say, the double logs that you would resum in this process. Actually, these double logs are also single logs. So you could decide whether or not to resum of them. But either-- whether or not you do resummation, this is still useful, because you could make a prediction for this decay rate, and it works really well. All right. You can actually also make predictions for these decay rates. You can predict actually the relations between the D0 and D star 0 using the factorization-- subleading factorization theorem. OK, so let's move on to our second topic, which will take the rest of today. And that's rapidity divergences. OK. So when we were talking about SET1, and we were talking about loop calculations, we saw that there was a subtlety where when we were doing our colinear loops, that could double count the ultra soft loop, if you remember. So kind of schematically, I could say, that this true CN was sort of a CN naive minus a CN 0 bin. So you could do a calculation ignoring that, but then you have to be careful, and there's a subtraction. And that subtraction avoids the double counting with the ultra soft. So if you think about there being ultra soft amplitudes and colinear amplitudes, this avoids a double counting. Now, we never talked about whether there's something analogous to that in SET2, and we just did a lot of SET2 examples without ever even saying those words. So why it's actually-- for what we've talked about so far-- OK to ignore this issue. But in general, it's not OK. So if we go back to our picture of the degrees of freedom in SET2, have this hyperbola, and you could have softs, you could have some colinears. And then the example that we just did, it's like these were kind of the relevant modes. And in general, you might have some guy down here as well. So these are the degrees of freedom in the SET. And effectively what's happened is, if you want to think about double counting, you're sliding down this hyperbola so this hyperbola is kind of at a constant invariant mass, say, lambda QCD squared, or it could be lambda QCD. And unlike the case in SET1, where this guy lived in a different hyperbola here, to get between them you would be sliding down the hyperbola at fixed invariant mass. So that's a little different. But in general in SCET2, there are also 0 bins. So in general, you would have something like, met me denote it this way, CN minus CN soft. And what I mean by this, is this is my original amplitude, where P mu was scaling like Q lambda squared 1 lambda. This was the original. And this would be a subtraction where you take that amplitude, and you'd make it scale like in the soft regime. So it would be P mu, lambda, lambda, lambda. So that's different than the example of SET1. In The SET1 case, we would really just be scaling down the 1 in the lambda, so that they would be both of order lambda squared. Here we're actually scaling down the 1 and scaling up the lambda squared to a lambda. And that's the right thing to do to go from here to here. So we're just taking the amplitudes in this region and subtracting them in this region. And in general, we do have that. But actually that's not the real complication that shows up in the SET2. The real complication has to do with whether there's any divergences associated to that. If this amplitude here didn't have any divergences, it wasn't kind of log singular, then you wouldn't really care about doing this subtractions, because then there would be no infrared singularities that you're double counting and it would just be effectively a constant. And the constants are always ambiguous. So whatever mistake you make in constants here you just make up by changing your hard matching. So you don't have to worry if when the colinear goes down into the soft region there's no divergences. And that is actually what's happened in all the examples we've treated so far. That when the colinear goes into some region where it's not supposed to have singularities, that you just end up with no singularities. There's no log singularities. OK, so, so far there's no log singularities from the overlapped regions. But that's not in general the situation. And we'll do an example in a minute where there are singularities that overlap. And the true difficulty here is the following. If you think about what's separating these modes, you might draw lines like this. Just to draw some straight lines separating the modes. And remember that we're plotting here in the P minus, P plus plain. And that fixed P squared is like fixed product of P minus P plus. So P squared is P plus P minus, up to the P perp squared piece, which we're ignoring. So you can think of these lines as lines of constant P plus over P minus. And if I-- this is the-- so something orthogonal to P squared. All right. So that would be one way of thinking about-- so you need something that's orthogonal to P squared in order to distinguish these modes. And the real issue with that is related to the regulators. When you use dimensional regularization it turns out the dimensional regularization is not sufficient to regulate a divergence that would happen when the CN comes down on top of the S. And the reason is, because dimensional regularization regulates P squared. It regulates-- remember, it's a Lorentz invariant regulator. So it's regulating Lorentz invariant things like P squared, not something like the rapidity, which is this P plus over P minus that you would need to distinguish these modes. So invariant mass does not distinguish the low energy modes. So rapidity, you could define-- is usually defined this way. So exponent of 2Y, where Y is the rapidity, is P minus over P plus. And if you look at the scaling of that, that scaling either lambda minus 2, lambda 0, or lambda squared for the different cases for CN, S, and CN bar. So it's this variable that's really distinguishing the different modes. OK, and that's these lines-- these orange lines are just putting dividing lines between these in rapidity. All right. OK. So there's a complication that dimensional regularization doesn't suffice. So you can think of it as regulating P Euclidean squared once you do the Wick rotation, for example. So it regulates-- it separates hyperbolas, but it does not separate modes along a hyperbola. It's a way of regulating singularities between hyperbolas, but not along hyperbola. So that's one complication. We'll need an additional regulator. And we'll see that that regulator will eventually lead to a new type of a normalization group flow, which is flow along a hyperbola. It's not a flow in invariant mass, but a flow in rapidity. OK. So let's explore what can happen in an example where there are these divergences in sort of the simplest possible example. So there's enough going on that we want to make our lives as simple as possible. So what I'll talk about is something called the massive Sudakov form factor. So you should think of it set up as follows. We're going to consider a form factor, and it's going to be a space-like form factor. So it's a space-like quark-quark form factor. Q of the photon here is space-like. And we're going to think about, rather than having photons or gluons, we're going to think about massive gauge bosons. So this is going to be some kind of Z, if you like. It could be a Z boson. And I'll just call the mass M. OK. So the thing that I'm going to want to iterate is the mass-- rather than doing QCD, I'm doing electroweak corrections-- electroweak corrections from a massive gauge boson. So this is relevant if we have electroweak corrections in a situation where Q squared is much greater than at M squared. And we're not going to be having gluons. Instead we'll be talking about these massive gauge bosons-- multiple Z bosons, if you like. We could also put the W's in, but let's make it simple and just talk about Z's. OK, so let's do this example. So in the full theory, you would start with a vector current, say, and you'd want to match that onto SET. Before we do that, let's just-- let me just write down a kind of full theory object using Lorentz invariance. So you could think about the quark form factor. And four massive gauge bosons, this is just some form factor that you can calculate that's a function of Q squared and M squared, and then there's some spinners. And so, really kind of the dependence is encoded in this F, which is a function of Q squared and M squared. M squared is acting kind of like an infrared regulator. So this is Z boson, there's not a soft singularity associated to it. And so, what you'd like to do-- and in this process-- is factorize Q squared and M squared, i.e. expand this thing in Q squared and M squared, and maybe some logs of Q squared and M squared. So we want to factorize some logs, et cetera. OK, so what are the type of degrees of freedom that we could have here? So lambda is going to be M over Q, massive Z boson over the energy scale of the collision, of the gamma star. And if you just look at the Z boson, then it could be colinear or it could be soft. So it could be actually three different possibilities. And I could think about doing this effectively in a bright frame. Just like we did earlier. And then what you would have is that the colinear guy is like the quark, and then the anti-colinear guys is like the outgoing quark. [INAUDIBLE] So you'd be making a transition in this diagram from N colinear of objects to N bar colinear of objects. And you could likewise have a Z boson which could be colinear. Or you could have a Z boson that's soft. And you have a soft rather than ultra soft because of the mass. If Q times lambda is M, and so if we want a propagator that's like P squared minus M squared, P squared better be of order M squared. That happens for softs, not for ultra softs. So that's why we have softs. And the same thing for colinears, P squared of order M squared for these colinear. So it's really an SET2 type situation, where the hyperbola is just set by P squared of order M squared. We have our mode sitting on that hyperbola. OK, so it's exactly of this type over there. And the thing that's new in this example is that we're going to encounter these rapidity divergences. STUDENT: You mentioned there's the only form for vertical in your theories, is that? PROFESSOR: Yeah. Make things as simple as possible. So you could talk about mixed sort of electroweak in QCD, but yeah, we don't. Let's make it-- in some sense that's kind of just like mixing a problem that we already would know how to deal with, with this one. So let's just deal with this one. All right. So in terms of the external court momenta, we can therefore kind of treat them as follows. Let's just let them be of a large component. They're massless particles. So this is P, and this is PR. And these guys are massless-- should have said that. And if you go through the kinematics, Q squared, which is minus P, minus P prime squared. If you square that, you just find it to P minus P plus. And we're just effectively, if we pick the bright frame-- which is what we're going to do-- each one of those is separately keep going at prime. Call it bar. Each one of those is separately Q. The large-- these guys are just fixed-- both would be Q. All right. So we could factor is this current with these degrees of freedom. The quarks are colinear. So at lowest order they've just become a CN and a CN bar. And then we have to address that with Wilson lines. And we know how to do that. So let me just break down the answer. We could again follow our procedure of going through SCET1, but it's now so familiar we just know what to write down. OK. So the current would look like that. And I'm not to worry too much about the Dirac structure. So I won't worry, for example, about gamma fives. And we could put that in, it's easy. So that's the leading order current, and then we'd have leading order Lagrangians, and we need to start calculating. And if we calculate what you would expect from a major from that is, you'd expect that F of Q squared and M squared is going to split up-- given the degrees of freedom we have into some kind of hard function-- and then some kind of amplitude for the colinear parts, and then some kind of amplitude for the soft part. OK, so you'd expect some hard times colinear factorization of the form factor. And this is what we'll be after. So it's always good to sort of have an idea where you're going. And that's where we're going. So let's consider just one loop diagrams. And it suffices, in order to make the point, just to consider the most singular one. So I'm going to consider-- so there's various loop intervals that you could have to do when you're doing the diagrams if they're fermions. Let's just take the simplest, which is a scalar loop interval. And I'm going to contrast how that scalar loop interval would look if you were doing the full theory calculation with how it would look in the effective theory. And then we'll see where the divergences come from. So you can think about this as kind of-- the piece where the numerator is independent of the loop momenta, so the numerator just factors out. OK, so if I took our vertex triangle diagram over there, then a piece of it-- where the numerator is trivial and factors out-- would look like this interval. So let's just study this guy. If we did this interval in the full theory, this would be both UV and IR finite. So this is just giving us some result that involves logs of Q squared over M squared. So it does have double logs of Q squared over M squared, and single logs of Q squared over M squared. But it's perfectly-- there's no 1 over epsilons. So now let's see what-- let's think about what would happen in the effective theory. So we have a kind of analogous loop interval for colinear, where the gauge boson is colinear. There's some numerator that again I'm not going to worry about. If this numerator is constant, it doesn't-- it's effectively the same constant. In the case of the-- if you take the leading order numerator in the full theory, it'll be the leading order numerator the effective theory as well. But the denominators do change. And so, if we took the N colinear, then yeah, so this guy doesn't change. Because that's just like saying P minus and K are-- P minus and K minus are the same size. But this guy does change. OK, so this guy would be K minus P bar plus, because K minus is big and B plus is big. Both of these are big. So both of those are big in this diagram. And so that's effectively the Wilson line diagram. OK, where the propagator here was off shell, got integrated out, and just became iconal. And the K squared is smaller, so we don't keep it in a leading order term. And then analogously, for IN bar, it's the other way. So both of these are big, and this one remains. And then they're soft. And in the soft case, what happens is that both of the propagators end up being iconal. And in our SET operator, that's a diagram where we have our colinear lines, and then we have kind of a self contraction of the S. But we're taking an SN with an SN-- we have a contraction that's like this. We have two of the Wilson lines-- soft Wilson lines that are sitting in that operator. That's a non-zero contraction. That would lead to a diagram like-- that would lead to this amplitude. All right. I'm going to leave a little space here, because I'm going to add something in a minute. All right. So how do we see that there's a problem with these intervals that they're not regulated by dim reg? Well, you could look at the soft integral and you could just do the perp. The perp is only showing up in this K squared minus M squared. So you would get something by doing that. And so, if we do the perp with dim reg and IS, we would end up proportional to something that's DK plus DK minus K plus K minus, minus M squared to the some power of epsilon, divided still by the factors of K plus and K minus. So you see that the invariant mass is being regulated. We just did the perp. Perp is gone. Plus times minus is being regulated, because plus times minus-- if plus times minus grows larger, or goes small, and regulated by this epsilon-- but either one, plus going large or minus going large, or plus going small minus going small, with plus and times minus fixed is not regulated. And that's the rapidity divergence. If the invariant mass is fixed and K plus over K minus goes large or small. So let me write it as K minus over K plus going to 0, or going to infinity. Let me say, with K plus times K minus fixed, then it diverges as these things happen. And if you think about what's happening in our picture over there within these limits, it's exactly a situation where this X here would be sliding up or sliding down. So in one of these limits, this one's going towards the CN bar, and this one would be going towards CN. So it's exactly a region where you would be overlapping-- sliding down the hyperbola. And the interval has log singularities. So this is exactly a situation where we can't ignore the overlaps and we have to worry about them. OK so we need another regulator. Dim reg is not enough. We have to do something else. So what could we do? So there's lots of different things that you could do. One thing is, you could just sort of put it in some plus something in these denominators-- that's called the delta regulated, K plus, plus delta-- that's one choice. We'll do something a little bit more dim reg-like. Which makes it sort of easier to think about the renormalization group. So one choice for an additional regulator is the following. So if you think about where these divergences came from, they came from the Wilson lines. So what you really need to do is regulate the Wilson lines. And you can do that as follows. Let me write out the Wilson lines in our kind of momentum space notation. So we have some N dot P type momentum for the soft Wilson line, and an N dot AS field. And really, it's this one over this iconal denominator that's giving rise to these denominators here that are giving rise to the singularity. So if we want to regulate that singularity we need to add something, and we could do that as follows. So this is the regulator we'll pick. And I'll just write everything as kind of a momentum operator. So I've just tucked the Z momentum in and raised it to some power. So PZ is the difference between P minus and P plus. And that seems kind of arbitrary, but that'll do the job for us. You can motivate why you want to do PZ-- so this is 2PZ actually. You can motivate why you want to do PZ rather than something else in the following way. And it's a true fact, that once you have enough experience you realize it's good to use PZ, because PZ doesn't involve P0. And the softs don't really make a distinction between any of the different components. And if you put in P0, something that involved P0, that would be dangerous. So this is nice, because there is no P0. So it's the combination of P plus and P minus that you can form that doesn't have the P0, which is energy. And remember that the polls in P0 are related to things like quarks and anti-quarks. They're related to unitarity. So not messing up the structure in P0 means that you'll be fine with unitarity, fine with causality, you're not messing up a lot of nice things about the theory. So if you do put P0 in, then you have to be careful about those things. So if you just arbitrarily put in some power of P0, then you'd have more trouble. And so that's kind of why we're avoiding and just putting in PZ. For the colinears, we can do something similar. But for the colinears we also have a power counting between the minus and the plus. So for the colinear, we can still make the power counting OK by thinking about putting in PZ, but then just expanding it to be a P minus. And that's true up to power corrections. And we don't really need to worry about power corrections when we're regulating these divergences. So it's just putting in the large momentum. And so W written in a similar notation. So I'll explain what the other things in this formula are in a minute. But the important thing for regulating is that we have some factor. In this case, it would be a factor of N bar dot P. STUDENT: Dot [INAUDIBLE]? PROFESSOR: Sorry? STUDENT: Dot eta to bar dot P? PROFESSOR: No. It's supposed to be an N. Looks like eta. Too many variables. There's the eta. So there's some factor raising of the-- again the iconal propagator sort of mixing up with the iconal propagator. In this case, it's even more obvious that it's just regulating that iconal propagator. OK. So if we were to do that, and go back over here, and put the regulators into these integrals, what would happen? So here, we'd get an extra factor-- K minus to the eta. And so, that would regulate this K minus. And these integrals will also have polls from the 1 over K minus. You could think about-- well, OK. If we did those integrals, we would also have rapidity divergences that are kind of the analog ones, and the colinear SECT are still the soft ones. Here we have two propagators. And so if I have two soft Wilson lines, and so I get two factors. But I've conveniently chose it to be the square root. So it comes out kind of looking the same here. So one thing that is just a part of this regulator-- which actually I don't know a good argument for-- is kind of a priority from the symmetries of the theory, you might like to argue that that should be eta over 2, and this should be eta. But it's really just part of-- it's just a choice, a convention that we've made, as far as I know. There probably is some nice deep argument for it, but I don't know it. So what are these other factors? So nu is going to play the role of mu. We've changed the dimension of the operator. We've compensated it back with nu, just like we were doing with mu. We're going to get 1 over eta divergences, which are like our 1 over epsilon divergences. And we're going to get logs of nu, which are the analogs of logs of mu. And that's the sense in which there's kind of an analog of this M up with our usual dim reg setup. In order to have a full analog, we should think about having a coupling. And so, that's what this W factor is. You can think about it like there was some bare pseudo coupling, which is really just 1. But just imagine that you're switching from bare to renormalized, in order to set up a renormalization group equation. And then this guy here, which is in eta dimensions, if you like, would have a renormalization group which would say nu D by D nu of this W of eta and nu is minus eta over 2. Sorry, this is eta over 2. So that's the analog of saying mu by D mu of alpha is minus 2 epsilon alpha. So an analog statement. I think this is OK. So this guy here is like a dummy coupling. And the boundary condition for it after you've carried out these-- this is just to set it to back to 1. So it's identically 1, it's really just a bookkeeping device. It's just--e it's a dummy coupling once you go to the eta dimensions. But you just set it always the renormalized coupling is just identically set to 1. And identically setting it to 1 is what you need to keep gauge invariance in these Wilson lines. It turns out actually that this regulator here is gauge invariant, though it doesn't look like it. We've modified the structure of the Wilson line in some kind of way that looks like it might be drastic. But actually these factors here are gauge-- still leave a gauge invariant object. So-- STUDENT: Can you write [INAUDIBLE] space, I assume? PROFESSOR: Not that I know of. Yeah. STUDENT: I think [INAUDIBLE]. PROFESSOR: Maybe you can. Yeah. But it's not-- since it's not-- yeah, I don't know how to-- I don't know what it would look like. You could probably transform that power, and it-- STUDENT: [INAUDIBLE]. PROFESSOR: Yeah. I'm sure you can probably just try out before you [? transplant ?] that. I'm just not sure if it would look nice. Yeah. It might not look too bad. Yeah, and it might actually be a nice way of saying what I'm about to say in a less nice way, which is, if you look at the gauge symmetry, why is this not messing it up? So one way of thinking about that is just to look at general covariant gauge. So note, the 1 over eta and eta 0 terms are gauge invariant. And you can think about that by just going to a general covariant gauge and seeing the parameter dependence drop out. So for example, at 1 loop, you would take [? g mu ?] nu in the contractions and replace it in general covariant gauge by some gauge parameter-- of general covariant gauge, K mu, K nu over K squared. And you'd like to see it independent of this. But this eta to the 0 piece is kind of independent of that for the usual reasons. And the 1 over eta term is independent of that, because this guy actually doesn't deuce any rapidity singularities. What happens is that if you have an N dot K, then you have a corresponding N mu in the numerator. And so, basically what happens is, you get an extra N dot K in the numerator. So any time you have 1 over N dot K, you would get for this piece multiplied by an N dot K upstairs. And so this is cancelling, you don't have a rapidity divergence in the C-dependent part. So that's why this is invariant under the gauge symmetry. And then, because of this boundary condition, the kind of cancellation of the C-dependence in the order [? A ?] to the 0 piece, this kind of works out in the standard way. So it gives you an idea of why it's gauge invariant without giving you a kind of full proof or anything. So we have both 1 over epsilon polls and 1 over eta polls in general, and we have to understand what to do with them. So here's what we're going to do. For any fixed invariant mass, it turns out that we can have these one over eta polls. And the right procedure for dealing with them is as follows. First you take eta goes to 0 and deal with these new polls that you have introduced in your amplitude. In order to deal with them, because you can have them for any invariant mass, you actually have to add counter terms that can be a whole function of epsilon, where you have an expanded in epsilon and then divide it by eta. So let me abbreviate counterterm as CT dot. Then, after you've done that, you take epsilon goes to 0, and you find your 1 over epsilon counterterms. And this is the correct way of doing it. And we'll see how that works in practice in a minute. So let's go back to our integrals that I've now erased and just write out the answers. We're doing those integrals with this regulator. And I'll also make them fermions, so I'm putting in the numerators. We wrote them down for scalars. The scalars where the most divergent integrals actually. I can include the numerators, that doesn't really change the story. And I can include the pre-factors as well. And I'll kind of write things in a QCD type notation, even we can imagine that it's a non-abelian group, just so CF is the whatever group it is, it's the Casimir of the fundamental. Whatever group our gauge boson's in. So here's the eta poll. It has a whole function of epsilon in the numerator, and it's even divergent. So this is 2 eta. And then the rest of it I can expand. So there's going to be 1 over epsilon times the log. When the log replaces that 1 over epsilon then I can start to expand, and I get another 1 over-- when the log nu replaces the one over eta, I can expand this gamma, and it gives me 1 over epsilon. And there's also some other pieces. So over 2 epsilon there's a log mu over M. And there's a constant. And I'm never going to write the constants. So let me read all the results and then we'll talk about them. So ICN bar is the same. The only difference between this is that P minus close to P bar plus. It was really symmetric. And then IS is different. So the 1 over epsilon-- 1 over eta poll comes with the opposite sign, and it also comes to a factor of 2 different. And in this case, there's actually 1 over 2 epsilons squared term. So there's also a double log of mu or M. And then there's plus constant. OK? And so, you could think about adding them up. And what happens when you add them up is, you have 1 over 2 eta, 1 over 2 eta, minus 1 over eta, and so that 1 over eta polls cancel. And that's exactly what you'd expect, because in the full theory the eta was something we introduced in order to distinguish these effective theory modes. It wasn't something that was there needed for the full theory integral. And so you don't really-- you'd expect that it's sort of-- that there's a corresponding regulator between the two sectors. So that when you add them together, that the dependence on that parameter is canceling away, because it was just an artificial separation, if you like, or separation that we're doing. So if I add them up, 1 over eta is cancelled, and so do all the logs of nu. We have alpha-- I'm left with a log of mu over Q, 1 over epsilon poll, double log, some types of single logs, and some other type of double log. So it would look like that. All the nu dependence is canceling away. So sort of various things which we'll start talking about now, and we'll continue talking about next time. So the rapidity 1 over eta divergence, which we can call a rapidity divergence, cancels in sum. And of course, so does the log nu's. And that's as expected. And if you add an overall counterterm, for the entire thing it just involves the hard scale log mu over Q. So if you were to think about there being some Wilson coefficient, which is sort of C bare is ZC minus 1, Z bare is ZC, C renormalized, then ZC and the C renormalized only involve logs of mu over Q, which is the hard scale. OK? And that means that our hard function, which is the Wilson coefficient squared, or just the Wilson coefficient in this case, is only a function of Q and mu. OK, so integrating out the hard scale physics didn't know about the separation. The separation was really something that we needed to do in the effective theory to distinguish the CN and S modes. And you can see why we needed to do it if you look at these answers, because if you look at the types of logs that are showing up here, in this case, we have a nu over P minus. And in this case, we have a-- is it nu over mu? Just make sure I got that right. I guess it is. In this case, we have a nu over M. And we also have a mu over nu. And so, the sort of right scale to-- in order to minimize the logarithms here we're going to have to, again, as usual take different values of mu and nu. Well, it's the same value of mu. All of them are M. But it's a different value of nu, because it's the nu that would need to be of order Q here. P minus is Q. And the nu would need to be of order M here. So it's the nu that distinguishes the modes. So the logs NCN are minimized. Or mu of order M, which says being on the hyperbola, but the nu should be of order P minus, which is Q. And that's precisely actually where we put the X in our picture, if you think about it. OK, so that's saying that you have a large P minus momentum, and we have-- and we're on this hyperbola where P squared is an order M squared. So this is-- and it's likewise for the other pieces. So for the soft piece-- so for the-- say for the anti- for the other colinear piece, we need the same thing. And then for the soft we need a different value for this new parameter. So having this regulator is behaving like dim reg, where we needed different mu's, when we had different hyperbolas. Now we have different places on the hyperbola, and we're tracking that with the new parameter, and that's showing up in the logarithms. And if you think about what the logarithms are doing, you can see that when you combine terms, let's see-- if you look at the 1 over epsilon, and the mu's are canceling out, you're getting a mu over Q. Yeah, that's maybe not the best example. Look over here. You have this log of M over Q In this kind of complete decomposition. The way that that logarithm here gets made up is by having m over nu and nu over Q. All right? So in order to get this log that doesn't have any mu's in it, mu is not telling you that there's that large log. But there is a large log. So there's large logs associated to these rapidity divergences. And what we'll talk about next time is how you do the renormalization group with the diagrams like this. How you write down in almost dimension equations. There'll be an almost dimension equations in both mu and nu space. So we'll have to move around in that space and see how it works to sum of the large logarithms. But I'll postpone that to next time. So any questions? So the general idea is really, as usual, it's just that now we're dealing with a situation where there's two regulators. And they're actually independent regulators. One's, if you like, is regulating invariant mass, and the other is regulating these extra divergences. And so, we'll be able to move around in the space without any worrying about path dependence, for example. We'll talk a little bit about that next time-- in this two dimensional space of mu and nu. But you can see just from looking at the logs, as usual, you can see where you need to be. And you can see that you need to be in different places for the different modes in order to minimize the logs of these amplitudes. And if you do that, then there should be some renormalization group that would connect these guys. So there should do something RGE that goes between these guys, and it'll be an RGE in this new parameter.
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https://ocw.mit.edu/courses/8-03sc-physics-iii-vibrations-and-waves-fall-2016/8.03sc-fall-2016.zip
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. YEN-JIE LEE: So welcome back, everybody. This is the final exam checklist. For the single oscillator, we need to make sure that you know how to write down the Equation of Motion. We have discussed about damped, under-damped, critically damped, and over-damped. We did that. Oscillators, and we have tried to drive oscillators. We observed transient behavior in steady state solution. Resonance, right, so which we actually demonstrated that by breaking the glass. And then we moved on and tried to couple multiple objects together. And that brings us to the coupled system. What are the normal modes? And how to actually solve M minus 1 K matrix, the eigenvalue problem. What is actually the full solution for the description of coupled systems. And can we actually drive the coupled system, and we found out we can. So the system would respond as well similar to what we have seen in the single oscillator case. We see resonance as well. We can excite one of the normal modes by driving the coupled system. Then we put more and more objects until at some point, we have infinite number of coupled objects. What is actually the solution of refraction and the transmission-- refraction and the translation symmetric system. That is actually the discussion of symmetry. We go to the continuum then, and we actually found wave and wave equations. So we found that finally, we made the phase transition from single object vibration to waves, and that is actually an achievement we have done in 8.03. We have discussed about different systems, massive string, massive spring, sound wave, electromagnetic waves, and we have discussed a progressive wave and standing waves. For the bound system, we have also normal modes. We discussed about how to actually do Fourier decomposition, and what is actually the physical meaning of Fourier decomposition in 8.03. For the infinite system, we also learned about Fourier transform and uncertainty principles. And we learned to apply boundary conditions so that we constrain the possible wavelengths of the normal modes. Therefore, we also learned about how to put a system all together. Finally, how to determine the dispersion relation, which is omega as a function of K, the wave number. Until now, we discussed idealized systems, and we also moved on to discuss dispersive medium. We have learned some more, even more about dispersion relation for the dispersive medium and signal transmission, how to send signal through a highly dispersive medium. The solution we were proposing is to use an amplitude modulation radio and also the pattern of dispersion. The group velocity and phase velocity, we covered that. As I mentioned before, the uncertainty principle. A 2D/3D system. We have bound system, which we have normal modes for two-dimensional and three-dimensional systems, as well. Because we're all over the place, so just make sure that you know how to actually dewrap all those standing waves for different dimensions of systems. We showed and approved geometrical optics, which essentially is the direct consequence of waves. Wave function, a continuation of the wave function and boundary condition. We learned about the refraction rule and also Snell's law. We talked about polarized waves, linear, circularly polarized, elliptically polarized, and the polarizer and quarter-wave plate. At the end of the discussion of 2D/3D systems, we discussed about how to generate electromagnetic waves by accelerated charge. Finally, we went on and talked about how those EM waves propagate in dielectrics and again, boundary conditions, which leads to interesting phenomena, which belongs only to electromagnetic waves. For example, Brewster's angle. So the refraction amplitude-- refraction-- the wave amplitude is governed by the property of the electromagnetic waves, which is coming from the laws which governs electromagnetic waves, which is Maxwell's equations in matter. We were trying to also manipulate those waves by adding them together, and we see constructive and destructive interference and diffraction phenomena. Then we connect that to quantum mechanics by showing you a single electron interference experiment. That connects us to the beginning of the quantum mechanics, which is the probability waves, which behave very different from other waves we have been discussing. But you are going to learn a lot more in 8.04. OK, so don't worry. All right. So that is the checklist. You can see that I can write it in two pages, so it's not that bad, probably. I hope that there was nothing really sounds like new to you by now. If you find anything is new, you have to review that part. That means you missed a class. All right. So what I'm going to do now is to go through all the material faster than the speed of light. All right. So that you will get nauseated. No, you are going to get a list of the topics. You just have to feel it. If you feel good, like when you are having a cupcake, right, then you are good for the final. If you don't feel good, what is Professor Lee talking about? He's talking about nonsense now. Then you are in trouble, and you have to review that part. All right? OK. So that's what we'll do. So let's start. All right. Why 8.03? We started a discussion-- welcome, We started a discussion of 8.03 and it's vibrations and wave systems, is name of this, 8.03. And the motivation is really simple, because we cannot even recognize the universe without using waves and vibration. You cannot see me, and you cannot hear anything, and you cannot feel the vibrations-- sorry, the rotation of a black hole by your body anymore, then it's not very cool. Therefore, we study 8.03 to understand the basic ideas about waves and vibrations. And we found that waves and vibrations are interesting phenomena. Waves are connected to vibrations. Because if you look at only, for example, a single object on these waves, you see that it is actually a single object which is oscillating up and down, oscillating up and down, and this is your vibration. So there's a close connection between single particle vibration and the waves. And that is the first thing that you learned. Therefore, we need to first understand the evolution of a single particle system. And we make use of this opportunity to start the discussion of scientific matter. So using this opportunity, basically, what we have been doing for the whole class is the following. So the first step is always to translate the physical situation which we are interested into mathematics, right? Because mathematics is the only language which we know which describes the nature. If you come out with a new language, and that is going to be a super duper breakthrough, it cannot be estimated by Nobel Prize. But the problem we are facing now is that this is the only language we know which works. Therefore, we really follow this recipe, which is similar to many, many other physics classes. And we have a physical situation, we use laws of nature or models, and we have a mathematical description, which is the Equation of Motion. And this is actually the hardest part, because you need to first define a coordinate system so that we can express everything in a system in that system, then you can make use of the physical laws you have learned from the previous 8.01 and 8.02 to write down the Equation of Motion. And most of the mistakes, and also most of the problems or difficulties you are facing is always in this step. Then we can solve the Equation of Motion, which is, strictly speaking, not my problem. It's the math department's problem. Yeah, that's their problem. Then we solve the Equation of Motion, and you will be given the formula. Then we use initial conditions and then make predictions. And then we would like to compare that to experimental results. And that is the general thing which we have been doing for physics. So let's take a look at those examples. Those are examples of simple harmonics motions. And you can see that these, all these systems have one object, which is oscillating. And you can see that their Equations of Motion are really similar to each other. It's theta double dot plus omega, zero squared, theta equal to zero for those idealized simple harmonic motions. And we learned that the solution of those equations are the same, which is a cosine function. Then we went ahead and added more craziness to the system. So basically, what we tried to do is to add a drag force into the game. And we were wondering if this more realistic description can match with experimental data. So this is the Equation of Motion, and the additional turn is the one in the middle, the gamma theta dot turn. And after entering these turns, not only is this an interesting model to describe the physical system we are talking about, but the mathematical solution is far more richer than what we talked about in the single harmonic oscillator case. Basically, you see that a general solution depends on the size of the gamma compared to omega dot zero, which is the oscillation-- the natural frequency of the system. OK. And then you can see there are three distinct different kinds of solutions. They have different mathematical forms. And we call them under damped solution, critically damped solution, and over damped solution. So those equations will be given to you. And the excitement is the following. So you can see that those solutions, if you plot the solution as a function of time, they look completely different as a function of time. So in the case of no damping, the amplitude is actually the constant, it's not actually reducing as a function of time. But when the damped system, the damping is turned on, then in the under damped situation, you can see that they end up reducing as a function of time. And if you have too much damping, you put the whole oscillator into some liquid, for example, and you see that oscillator disappear. The cool thing is the following. The excitement from-- as a physicist is that all of those crazy mathematical solutions actually match with experimental results. Wow. That is really cool. Because there is nobody saying that these should match and how, naturally, I should learn that OK, when should I change the behavior of the system. So this is really a miracle that this complicated mathematical description is useful and that it is super useful to describe the nature. Once we have learned that, we can now add a driving force into ligand. From the equation here, we can see that there is a natural frequency, omega dot zero, of this system, and there is a drag force turn, which is actually to quantify how much drag we have, we have a gamma there. We are driving it at a driving frequency omega t. So what we have learned from here is that if you are driving this system, you are-- for example, I am shaking that student, shaking you. OK, in the beginning, this student is going to resist. No, don't shake me. Come on. But at some point, he knows that Professor Lee is really determined. Therefore, he is going to be shaked at the frequency I like. OK. So that is actually what is happening here. This is so-called transient behavior. So in the beginning, the system doesn't like it. So this is making use of the superposition principle. So you can solve that homogeneous solution, which is on the right-hand side. It depends on the physical situation you are talking about. You choose the corresponding homogeneous solution. And lamba and psi is the driving force from E and G, right, and that is going to win at the end of the experiment, because I'm going to shake it forever, until the end of the universe. So you can see that at the end, you-- what is left over is really the steady state solution. And it has this structure, A omega d, depends on omega. And you get resonance behavior. Don't forget to review that. So you have a delay in phase because when I shake the student, the student needs some time to respond. Therefore, the delta is non-zero if the student is damped. All right. So now we have learned all the secrets about a single object system. Then we can now go ahead and study coupled oscillators. There are a few examples here, which is coupled pendulums or coupled spring-mass systems. And we found that a very useful description of this kind of system is to make use of the matrix language. So originally, if you have n objects in a system, you have n Equations of Motion, and that looks horrible. But what is done in 8.03 is that we introduce a notation with a matrix. Basically, if you write everything in terms of matrix, then it looks really friendly, and it looks really like a single oscillator. OK? Although solving this equation is still a little bit more work. And basically, you can see that from this example, we can actually derive M minus 1 K matrix, and the whole equation won't be-- the Equation of Motion problem solving problem becomes an M minus 1 K matrix eigenvalue problem. What is an M minus 1 K matrix? This is describing how each component in the system interacts with each other. Once we have this, we can solve the eigenvalue problem, and we are going to be able to figure out the normal modes of those systems. So what is a normal mode? Normal modes is a situation where all the components in the system are oscillating at the same frequency and they are also at the same phase. So that is the definition of normal mode. And those are what is used in a deviation, also, which leads us to the eigenvalue problem. We define Z equal to X 1 H or I omega t plus 5. Everybody is oscillating at omega and also at phase 5, right? So that is what we actually learned. And what is actually the physical meaning of those normal modes? So if we plot the locus of the two coupled pendulum problem, what we see is the following. So basically, you will see that the locus looks like really complicated as a function of time if you plot X1 X2 versus time. But if we rotate this system a bit, then we find that there's a really interesting projection, which is the principal coordinate. You see that all those crazy strange phenomena we see with coupled systems are just illusions. Actually, you can understand then by really using the right projection. To one-- to the right coordinate system. Then you will see that actually the system is doing still simple harmonic motion. So that is actually the core thing which we learned from coupled systems. So we learned about how to solve the coupled system, and we also learned about going to an infinite number of coupled systems. So then this is an example here. So for example, I can have pendulum and springs, and we connect them all together, and I need to hire many, many students so that they plays it, plays until it fills up the whole universe. So this is the idea of an infinite system. You can see that that means my M minus 1 K matrix is going to be an infinite times infinite long matrix. It's two dimensional. And the A is infinitely long. And that sounds really scary. And in general, we don't know how to deal with this, really. And it can be as arbitrarily crazy as you can imagine. What we discuss 8.03 is a special case. Basically, we are discussing about systems which are having a spatial kind of symmetry, which is translation symmetry, as you can see from all those figures. And you can see that all those figures will have to all have the same normal modes because of this base translation symmetry. What we discussed about is that we introduce an S matrix, which is used to describe the kind of symmetry that this system satisfies. And if we calculate the commutator S and M minus 1 K matrix, if this commutator shows that the evaluate-- if you evaluate this commutator and you get zero, now it means they commute. And the consequence is that the S matrix and the S M minus 1 K matrix will share the same eigenvectors. So you don't really need to know how to derive this-- to arrive at this conclusion, but it is a very useful conclusion. So that means instead of solving M minus 1 K matrix eigenvalue problem, I can now go ahead to solve the S matrix eigenvalue problem. And usually, that's much easier. So for the exam, you need to know how to write down S matrix. You need to know how to solve eigenvalue problems, including M minus 1 K matrix and the S matrix. And then we can get to normal mode frequency, omega squared, and we can also solve the corresponding normal modes. And here is telling you what would be the solution for space translation of the matrix system. And basically what we will see is that making use of the S matrix should be-- brings you to the conclusion that A, j must be proportional to exponential i, j, k, a, where this A is the length scale of this system, the distance between all those little mass. And the j is a label which tells you which little mass I am talking about. And k is the-- some arbitrary constant. But by now, you should have the idea basically that's-- that's what? That, essentially, is the wave number, right? So that is really cool. So that's all planned in advance. And basically, you can see that we can also write down the A, k because we know that A, j will be proportional to exponential i, j, k, A, after solving the eigenvalue problem for S matrix. Then we actually went one step forward to make it continuous. So basically, we made the space between particles very, very small. And also, at the same time, we make sure that the string doesn't become supermassive. And we concluded that we get some kind of equation popping out from this exercise. M minus 1 K matrix becomes minus T over rho L partial square, partial x squared. You don't have to really derive this for the exam, but you would need to know the conclusion and that psi j becomes psi as a function of x and t. And the magical function appeared, which is the wave equation. Oh my god, this is the whole craziness we have been dealing with the whole 8.03. This is actually really remarkable that we can come from single object oscillation, putting it all together, making it continuous, then this equation really popped out. And this equation really describes multiple systems. Then we went ahead to actually discuss the property of the wave equation. It looks like this. Basically, I replaced the t over rho L by v, p squared. By now, you know the meaning of v, p is actually the phase velocity. And we discussed two kinds of solutions, special kinds of solutions. The first kind is normal modes. The second one is progressive wave solution, or traveling wave solution, whatever name you want to call it. Let's take a look at the normal modes, what have we learned. So if you have a bound system, a bound continuous system, the normal mode is your distending waves for the wave equation we discussed. And basically, the functional form is A, m, sine, k, m, x plus alpha, m and sine, omega, m, t plus beta, m. So what we actually learned from the previous lecture is the following. So basically, you can decide the k, m and alpha, m by just boundary conditions. So before you introduce boundary conditions, which are the conditions allow you to describe multiple nearby systems consistently. So that is the meaning of boundary condition. Before you introduce that, k, m and alpha, m are arbitrary numbers. Whatever number you choose is the-- can satisfy the wave equation. But after you introduce the boundary condition, you figure that out from the problem you are given, then k, m and alpha, m cannot be arbitrary anymore. And they usually become discrete numbers. OK. So that is what we learned from the previous lectures. And finally, we also see that omega, m is determined by the property of the system, by a so-called dispersion relation. In this case, it's linear, it's proportional to k, m, because we are talking about non-dispersive medium for the moment. And we have this beta, m, which is related to the initial condition. And the a, m, which can be determined by a Fourier decomposition. So if you are not familiar with this, you have to really review how to do Fourier decomposition. I know most of did very well on the midterm, but maybe some of you forgot how to determine a, m and it will be very, very important to review that for the preparation for the final. Now the second set of solutions is the following. So you have progress-- progressing waves. And the functional form is really interesting. So you can see this can be written as F, F is some arbitrary function, x plus-minus v, p, t. Basically that is that you're describing a wave which is traveling to the positive-- to a negative or positive direction in the x direction. Or you can actually write it down as G function k, x plus-minus omega, t. Actually, they all work for wave equations. Now we went ahead and applied approach which we learned from the general solution of wave equation to massive strings, and we discussed about sound waves. For the sound waves, it will be important to review what are the boundary conditions for the displacement of the molecules in the sound wave, compare that to the pressure deviation from the room pressure. So I think it's important to make sure that you understand the difference between these two, what are the boundary conditions and basically it should be very similar to the solution-- the boundary condition for the massive strings. And we also talked about electromagnetic waves. And that is another topic which you will really have to review. Several things which are especially interesting is that an electric field cannot be without a magnetic field. They are always together, no matter what. So if you have trouble with the electric field, then there must be trouble in the magnetic field. And that is governed by the Maxwell's equation. Before we go into the detail of those, we also discussed about dispersive medium. So in the case of dispersive medium, we used a special kind of example, which is strings with stiffness. So basically, what we found is that if you have a certain kind of wave equation, like this one, I am writing this one here. Basically, if I add the additional term to describe the stiffness, then what is going to happen is that the dispersion relation, when I ask you to plot the dispersion relation, you will be-- I am requesting you to find the relation between omega and K. And I'm going over this in more detail because I see so many similar mistakes on the midterm. So basically what I'm asking is omega versus K. And in the-- if we don't have this turn, then basically, you have a straight line. Straight line means you have a non-dispersive medium. And if you add this turn, you need to know how to evaluate the dispersion relation. The quickest way to evaluate the dispersion relation is to just simply plug in the progresssing wave solution for the G function or harmonic progressing wave solution, find omega-- K, x plus-minus omega t, into this equation, then you will be able to figure out the dispersion relation. And what we figure out is the following. If we include stiffness, then you can see that the dispersion relation is not a line anymore and is actually some kind of curve, and the slope is actually changing. And there are dramatic consequence from this thing. That means if I have a traveling wave with different wavelengths, that means the phase velocity v, p equal to omega over K is going to be different for waves with different frequencies, or different wavelengths. So that is how you clear the problem. Because if I have initially produced a signal which is a triangle and I let it propagate, what is going to happen is that the slow component will be lagging behind. Those are the slow components. And the fast components will go ahead of the nominal speed. So there will be a spread of the signal. Originally, maybe you have some kind of a square wave, and this thing will become something which is actually smeared out in space, and then you lose the information. And we are going to talk about that later. And we also learned about group velocity. So what is your group velocity? Group velocity v, p-- oh, sorry, v, g is actually partial omega, partial K, which is the slope of a tangential line here. And where the phase velocity is connecting this point to that point, and the slope of this line is the phase velocity, and the slope of the line cutting through this point, which is giving you the group velocity. And we actually learned the definition of-- the consequence of group velocity and phase velocity by introducing you a bit phenomena. Basically, we add two waves with similar wavelengths, or wave numbers. Basically, what we see is the following. So basically, you see some behavior like this. We see this-- the superposition of these two waves which produce a bit phenomena can be understood by something which is oscillating really fast modulated by a much slower more variating envelope. Basically, you can actually understand the bit phenomena by actually identifying these two interesting structures. And the speed of all those little peaks is traveling at phase velocity. And the speed of the envelope is found to be traveling at group velocity. So that is what we have learned. And we can have group velocity and the phase velocity traveling in the same direction. And we can also have a negative group velocity. So that is a technique which is really, really very difficult. And I'm still trying to practice and make sure they I can demo that in 8.03. Basically, it's like the whole system, the whole detailed structure moving in a positive direction. But the body, or say the envelope, is actually moving in the negative x direction. So that is also possible. And you can actually construct a system which has a negative group velocity. So once we have done that, we also tried to understand further the description of the solution for the dispersive medium. So basically, what we actually went over during the class is that OK, now, if the f function f of t is describing Yen-Jie's hand, and I'm holding an infinitely long string and I shake it as a function of time, and that essentially, this motion, is actually described by this f function. What we know is that this oscillation, OK, I can do one, but I won't, but all kinds of f functions can be described as superpositions of many, many, many waves with different angular frequencies. So that's a miracle which we borrowed from the math department again. And you can see that f function can be written as the sum of all kinds of different waves with different angular frequencies with population c omega. This is the weight which makes that become the f function. And we can figure out the c omega by doing a Fourier transform. And finally, what will be the resulting wave function, psi, x, t, which is the wave function generated by the oscillation of my hand. And those are governed by the wave equation, which gives you the relation between omega and the k can be returned in that functional form. So the good news is that with the help of Fourier transform, we can also describe and predict what is going to happen no matter if this system is dispersive or not dispersive using this approach. OK. So that is really cool. And you can of course can do a cross-check just to-- assuring that this is a non-dispersive medium. And you are also going to get back to what you should expect the solution to non-dispersive medium for the psi, x, t. So that is one thing which is really remarkable. And I think what is needed to know is not a deviation of all those formulas, but how the plotting and the derived c omega by using the formula you are given and how to then put together all the solutions and it becomes the resulting solution for the psi, x, t, which is really the solution we really care. So for that, you need to know how to do the integration. You need to know how to derive the dispersion relation. Then one thing left over is to put the problem into that equation, which is also given to you in a formula. And we will not ask you to do a very, very complicated integration for sure on the final. So what is the consequence? Basically, one thing which is interesting to know is that if you have a wave in a coordinate space, which is really widely spread out, and you can do a Fourier transform to get the wave population in the frequency space, what we find is that when this wave is really, really wide in the space, then what we find is that the wave population in the frequency space is very narrow by using a Fourier transform. And that just gives you the result. And on the other hand, if you have a really-- a very narrow pulse in the coordinate space, for example, I do this-- shwhew --very, very-- really quickly. I create a very narrow pulse. And then what is actually happening is that I will have to use a very wide range of frequency space to describe this very narrow pulse. So that leads to-- direct consequences of that is uncertainty principle. And this is closely connected to the uncertainty principle we talk about in quantum mechanics. Delta, p times delta, x greater or equal to h bar over 2. All right. So we have done with the one-dimensional case. And we also talked about a two-dimensional and a three-dimensional case. And this is the example of two-dimensional membranes, and they actually are constrained so that their boundary condition at-- the boundary is equal-- no, the wave function is equal to zero. And you can identify all those normal modes. And we went ahead also to talk about geometrical optics laws. Basically, how we derive that is to have a plane wave. First, you have a plane wave propagating toward the boundary of two different mediums, and we were wondering well, what is the refracted wave and the transmitting wave. By using the-- by making sure just one point, which is that the membranes don't break, the wave function is continuous at this boundary. That's the only assumption which you use. We went through the mathematics, which you don't really need to remember all of them. But you really need to remember the consequence. The consequence is the following. Basically, what we see is that if you have incident plane wave with incident angle theta 1, the refractive wave will be having an angle of theta 1 as well. So that's the first law of refraction, refraction law. And then the second one which we learned is that the transmitting wave will satisfy Snell's law, n, 1, sine, theta, 1 equal to n, 2, sine, theta, 2. And that is very interesting because this, Snell's law has also nothing to do with Maxwell's equation. You see? Right? That's actually what you can learn from here. We usually use electromagnetic waves to demonstrate Snell's law. But from 8.03, we learned that it has nothing to do with Maxwell's equation. It applies to all kinds of different systems, which you can-- which can be described by wave functions. So that is actually the very important consequence. But on the other hand, as we all discussed later, the relative amplitude of the incident wave, refracted wave, and transmitting wave, the relative amplitude is governed by Maxwell's equation. So I would like to make that really crystal clear. So the relative amplitude is governed by really the physical laws, which actually governs the propagation of those plane waves. OK. So I think we can take a five minute break to have some air. And of course, you can-- you are welcome to continue to use all this juice and coffee. And coming back at 38. OK. So welcome back, everybody, from the break. AUDIENCE: [INAUDIBLE] YEN-JIE LEE: So we are going to continue the discussion. We have learned about the two important laws for the geometrical optics. And we also went ahead to discuss the polarization that's solved in greater detail. So for example, we can have linear depolarized wave. So basically, the wave is essentially moving up and down, up and down. But the direction of the background field doesn't change. It's always, for example, initially, if it's in x direction, then it is x direction forever. And in that case, I call it linearly polarized. Of course, I can also have the case that I can have a superposition of two waves. One is having the electric field in the x direction. And the other one is in the y direction. And they are off by a phase of pi over 2. If that happens, then basically, you will see that it produces something really interesting. That direction of the electric field is going to be rotating as a function of time-- as a function of the space these waves travel. And we call it circularly polarized waves. And we can also have elliptically polarized wave. Then we learned about how to do a filtering, which is the polarizer. So suppose I have a perfect conductor here, where I have the easy axis, which is described by the green arrow there. And you can see that easy axis means that if you have electric field parallel to the easy axis, and then since that's the easy axis, so it is supposed to be easy, therefore, this electric field is going to be passing through the polarizer. On the other hand, if the electric field is perpendicular to the direction of the easy axis, that means it's taking the perfect conductor in the hard way. Therefore, when it pass through-- when it is trying to pass through with the perfect conductor, the electrons in those conductors are going to be working like crazy to deflect this wave when the direction of the electric field is perpendicular to the direction of the easy axis. So that is how this works. For example, in the first example, you can see that in this case, you have an easy axis which is perpendicular to the direction of the electric field, which is the red field, then this wave actually got refracted. There will be no transmission-- sorry, no electromagnetic field passing the perfect conductor. And on the other hand, if you have another perfect conductor, in which you have easy axis which is parallel to the electric field, then you can-- you will see that it will pass through the perfect conductor. So that is the polarizer. And also, we discussed about quarter-weight plate, which I would suggest you to have a review about the concept which we have learned about polarizer and quarter-wave plate so that you make sure that you understand how to calculate the electric field after passing through a polarizer and quarter-wave plate and how the secondary, or the elliptically depolarized waves are created using all those wave plates, et cetera. All right. So the next thing which we discussed during the class is how do we produce electromagnetic waves. I think by now, you should know that a stationary charge doesn't produce electromagnetic waves. Even a moving charge at constant speed doesn't create electromagnetic waves. So how do we create an electromagnetic wave which propagates to the edge of the universe? That is-- the trick is to create a kink in the fuel line. So you have to accelerate and stop it. Accelerate and then try to actually stop the acceleration. So then you can create a kink. And this kink is going to be propagating out of the-- as a function of time. And this kink is creating the so-called radiation from this accelerated charge. So you don't really need to remember all the deviations, but you really need to know the conclusion. So what is the conclusion is the following. The radiated electric field is equal to minus-- very important that there's a minus sign in front of it, which is a common mistake to drop it, and the q is the charge of the oscillating-- the accelerated charge, proportional to the charge. If the particle is more charged, then you have more radiation. Aperp is the acceleration projected to, which is-- the perpendicular projection of the acceleration of the particle with respect to the direction of propagation is so-called the Aperp. And only the perpendicular direction acceleration counts. The one which is parallel to the direction of propagation doesn't really count, as you can see from this equation. And the t prime what is t prime? t prime is t minus r divided by c. So t prime is the retarded time, so that is telling you that it takes some time for the information to propagate from the origin, which is the position of the moving charge to the observer, which is r, this distance, away from the moving charge. So the information takes some time to propagate, and you cannot know what is really happening, for example, 100 light years away from Earth. You have no idea about what is happening. Maybe a black hole is created there and is going to suck everybody up in a few years. But nobody knows, and we don't care because we cannot control it. All right so that is very important. And also very important to know the magnetic field must be there. You can see the relation between magnetic field and the electric field. And the Poynting vector is also its joint field. And when we went ahead, given all the knowledge we have learned, we discussed about how to take very beautiful photos using a polarizer filter. And we discussed about how to filter out the scattered light from the sun. And it would be nice to figure out why this is the case, how these polarizer lines, scatter lines are created. It's purely geometrical. And also, we discussed about Brewster's angle and also how it leads to the explanation of the filtering of the light, the refracted light from the, for example, window of a car or from the water. And this is the demonstration of-- the summary of Brewster's angle. So somebody reminded me that the amplitude should be given. So I think, this is the amplitude formula for Brewster's angle will be given to you. If not, it's asked in the final exam. So don't be worried about it, and you don't have to remember this formula. And I'm not going to ask you to derive that just in such a short time, the three hours in the final exam. But what is very important is to know how this Brewster's angle, why there's no refracted light polarizing in a way that the polarization should be-- why the refracted light is polarized, for example. And also why the transmitting wave is slightly polarized. And I think the conclusions you need to remember, and you need to know how to calculate the angle, at least. Because for this purely polarized light to be produced in a refracted light, you need to have normal angle between the direction of the refracted light and the direction of the transmitted light. And that, you should be able to remember. And you should be able to derive that also from your mind as well, because that means the direction of the oscillation of the molecule at the boundary will be in the direction of propagation of the refracted wave. Therefore, that cannot be the solution to the progressing electromagnetic wave. Therefore, the refracted waves are polarized. So if you follow this logic, then you don't really need to memorize all those formulas. All right. So finally, in the last part of the course, we focused on the superposition of many, many electromagnetic waves so you can produce constructive interference. Or that means all those waves are in phase. And you can have destructive interference when they are out of phase. And that is a very important topic, so you should review that for the preparation of the final. And you can see that there are three concrete examples which we used during the class. A laser beam. We talked about a water ripple in a demo. And we also studied how it make use of this phenomena to design a phased radar. So to detect this unknown object in the sky, what we really need to have is electromagnetic waves pointing to a specific direction. And that can be achieved by using multi-slit interference. And this is the property of the two-slit interference pattern. And you are going to have many, many peaks. They have equal height for two-slit interference If you ignore any effect coming from diffraction. So we've assume that the slit is infinitely small. The slit is super narrow. And then we can ignore the diffraction-- single-slit diffraction. In fact, then all the peaks due to this two-slit interference will have the same height. On the other hand, when we start to increase the number of slits, for example, unequal to 3, unequal to 4, unequal to 5, unequal to 6, as you can see that, the structure of the intensity as a function of delta, which is the phase difference, is actually changing. And you can see that the general structure is the following. So if you have unequal to 3, then basically, you have 2 of adult, and between them, you have 1 child. And if you have unequal to 6, then basically, you have 2 adults and somehow there are 4 children in this collection. So basically, that is what we learned from the solution of the multi-slit interference. And in this way, we can actually make the width of the principal maxima as narrow as you want. So that is why phased radar works. And then we discussed about diffraction. So that is related, again, to the explanation of laser beams. And we discussed about the design of a Star Trek ship, the gun for the ship. And we also talked about resolution. And what is actually happening here is the following. A single-slit diffraction essentially can be viewed as an infinite number of source interference. And you just need to integrate over all the point-like sources between the two walls. And all of them are acting like a spherical wave source. So basically, for every point-- continuously, every point between these two walls are a point source of spherical waves. And that is Huygens' principle. And we can see that the structures is-- of the intensity as a function of position is the following. So basically, you have a principal maxima, which is a peak in the middle. And at some angle, basically, you have destructive interference such that if you integrate over all the contributions from an infinite number of sources in this window, basically, you would see that they completely cancel each other. So that is the origin of all those deep structure minima. And then, after the minima, actually, you will see another peak, but the height of the peak is suppressed by 1 over beta squared. And it would be good to review that. And what is the consequence? So if you shoot a laser beam to the moon, the size of the laser beam will be very large. After you learn 8.03, you know that the size of the laser beam is going to be very, very large due to interference between all the point-like sources from the laser beam. And finally, we can put them all together. So the single-slit diffraction and the multi-slit interference, you can put them all together, and basically, what you get is the following. So basically, you have a multi-slit interference pattern, which is showing there. But now the intensity of the multi-slit pattern is modulated by the single-slit diffraction pattern. And of course, the full formula will be given to you. But on the other hand, you are also requested to know how to calculate, just to add the contribution from multi-slit together in case if we change the amplitude of the incident light or we change the phase, like what we did in the homework. And I think that is one important point, and you should review that. And if you are not sure about how to proceed with that, it would be good to review Lecture 22, Lecture 23. So finally, we talk about the connection to quantum mechanics. Einstein already told us that "I have said so many times, God doesn't play dice with the world." But what we actually find is that there are two very interesting things which we found. The first thing is that if we have a single photon source, and basically, if we don't play dice, we cannot explain the intensity of the-- after this single photon source passes through two polarizers. And what happens is the following. Basically, the result of a single photon source tells you that you really need to play dice so that you can get the resulting polarized light intensity. And also, the second pseudo-experiment we discussed is that if you have billiard balls, basically, you have them pass through the two-slit experiment, what you are going to get is two piles, Gaussian-like distribution. And if you have a single electron source, what it does is that it interferes with itself. An electron, a single electron, can interfere with itself and produce a pattern which is very similar to what we see in the double-slit interference pattern. So that is really remarkable. And also, we talked about a single-slit-- single electron experiment. That gives you also a diffraction pattern. We have to use the wave function to describe the position-- the probability density of the position of the electron on the screen. And know this issue closely connected to the uncertainty principle, which we discussed earlier, delta, p, delta, x is greater than or equal to h bar over 2. So if you have a very narrow window, that means you have very similar delta x, so you have very, very good confidence about the location of the electron. And then the momentum is in the x-- in the momentum in the x direction, you have large uncertainty, according to this equation. And that can be seen from this single-slit diffraction pattern and it is closely connected to what we have learned before. So where is this-- how to actually describe what this is really the dispersion relation of the probability density wave is actually coming from Schrodinger's Equation. And this is given here. We briefly talked about that. And the consequence is the following. You can describe the evolution of the wave function as a function of time by using this wave equation. And this wave equation is slightly different from what we have learned before. And we also can use what we have learned from 8.03 to solve a particle in a box problem, which is covered in lecture number 23. And I just wanted to say that you need to know the general principle, but I'm not teaching 8.04, so I'm not expecting you to solve a quantum mechanics problem. But I would like to say that OK, from this point, it's motivating you to take 8.04, right? Because there can be a lot of fun there as well. And it is closely related to what we have learned from 8.03. So I just want to say, the last point is that this is really not the end of the vibrations and waves. It's just the beginning. And that there is a path toward the peak. And it may take a long time to reach the peak. All right. And I would like to let you know that I'm really, really very happy to be your lecturer this semester. And I really enjoyed teaching this class and getting your responses when I asked questions. Thank you for the support. And I would like to say good luck with the final exam. And we have 800 contributions on Piazza, many thanks to Yinan, who is actually doing all the hard work, day and night. And thank you very much, and see you around MIT in the future. Thank you.
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https://ocw.mit.edu/courses/7-012-introduction-to-biology-fall-2004/7.012-fall-2004.zip
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Today is my last class with you. Awe, I'm sorry, too. You guys are a lot of fun. This has actually been the most interactive 7. 1 I've ever had. Usually there are a couple of people who perk up and say things, but you guys are great because all sorts of people are willing to contribute. So, I've had a wonderful time and it certainly seems like you guys have learned a lot. What I'd like to do for my last lecture is pick up again a little bit like I did with genomics and try to give you a sense of where things are going. I always like doing this because I get to talk about things that are in none of the textbooks that, well, I mean, it's just stuff that many people working in the field don't necessarily know. And that's what's so much fun about teaching introductory biology is because it only takes a semester for you guys to get up to the point of at least being able to understand what's getting done on the cutting-edge. Even if you might not yet be able to go off and practice it, you might need a little more experience for that, but you'd be surprised, it's not that much more. Take maybe Project Lab and you'll be able to start doing it already. It's really wonderful that it's possible to grasp what's going on. And, in many ways, you guys may have an advantage in grasping what's going on because, as I've already hinted, biology's undergoing this remarkable transformation from being a purely laboratory-based science where each individual works on his or her own project to being an information-based science that involves an integration of vast amounts of data across the whole world and trying to learn things from this tremendous dataset. And, in that sense, I think the new students coming into the field have a distinct advantage over those who have been in it. And certainly the students who know mathematical and physical and chemical and other sorts of things, and aren't scared to write computer code when they need to write computer code have a really great advantage. So, anyway, all that by way of introduction. I want to talk about two subjects today of great interest to me. One is DNA variation and one is RNA variation. The variation of DNA sequence between individuals within a population, and in particular our population, and the other is RNA variation, the variation in RNA expression between different cell types, different tissues. And the work I'm going to talk about today is work that I, and my colleagues, have all been involved in. And it's stuff I know and love. So, feel free to ask questions about it. I may know the answers, but what's reasonably fun about these lectures is if I don't know the answers it's probably the case that the answers aren't known. So, that's good fun because it's stuff I really do know well, and I love. So, anyway, here's some DNA sequence. It's pretty boring. This is a chunk of sequence from, let's say, the human genome. How much does this differ between any two individuals? If I were to sequence any two chromosomes, any two copies of the chromosome from an individual in this class or two individuals on this planet, how much would they differ? The answer is that much. That's the average amount of difference between any two people on this planet. Not a lot. If you counted up, it is on average one nucleotide difference out of 1, 00 nucleotides on average, somewhat less than one part in 1, 00 or better than 99.9% identity between any two individuals. Now, that is a very small amount, not just in absolute terms, 99.9% identity is a lot, but in comparative terms with other species. If I take two chimpanzees in Africa, on average they will differ by about twice as much as any two random humans. And if I take two orangutans in Southeast Asia, they will on average differ by about eight times as much as any two humans on this planet. You guys think the orangutans all look the same. They think you all look the same, and they're right. So, why is this? Why are humans amongst mammalian species relatively limited in the amount of variation? Well, it's a direct result of our population history. It turns out that the amount of variation that can be sustained in a population depends on two things. At equilibrium, if population has constant size N for a very long time and a certain mutation rate, Mu, you can just write a piece of arithmetic that says, well, mutations are always arising due to new mutations in the population and mutations are being lost by genetic drift, just by random sampling from generation to generation. And those two processes, the creation of new mutations and the loss of mutations just due to random sampling in each generation, sets up an equilibrium, and the equilibrium defines an equation there, Pi equals one over one plus four and Mu reciprocal which equation you have no need to memorize whatsoever and possibly even no need to write down. The important point is the concept, that if you know the number of organisms in the population and you know the mutation rate, those set up the bounds of mutation and drift, and you can write down how polymorphic, how heterozygous random individuals should be at equilibrium. That is if the population has been at size N for a very long time. Well, the expected amount of heterozygosity for the human population -- Sorry. For a population of size 10, 00 would be about one nucleotide in 1300. We have exactly the amount of heterozygosity you would expect for a population of about 10, 00 individuals. Yeah, but wait, we're not a population of 10,000 individuals. Why do we have the heterozygosity you would expect from a population of 10, 00 individuals? We're six billion. It's a reflection of our history. Because remember I said that was the statement about what the population heterozygosity should be at equilibrium? We haven't been six billion people except very recently. The human population has undergone an exponential expansion. It used to be a relatively small size, and then it very recently underwent this huge exponential expansion. If you actually write down the equations, the amount of variation in our population was determined by that constant size for a very long time. And then a rapid exponential expansion that's basically taken place in a mere 3, 00 generations, it's much too rapid to have any affect on the real variation in our population. What do I mean by that? What's the mutation rate per nucleotide in the human genome? It's on the order of two times ten to the minus eighth per generation. In a mere 3,000 generations, a tiny mutation rate like two times ten to the minus eighth is not going to be able to build up much more variation. So you might as well ignore the last 100,000 years or so. They're irrelevant to how much variation we have. The variation we have was set by our ancestral population size. Now, don't get me wrong. Eventually it will equilibrate. A couple million years from now we will have a much higher variation in the human population as a function of our size, but the population variation we have today is set by the fact that humans derive from a founding population of about 10, 00 individuals or so. And that means that the variation that you see in the human population is mostly ancestral variations, the variation that we all walked around with in Africa. And, in fact, that makes a prediction. That would say that if most of the variation in the human population is from the ancestral African founding population then if I go to any two villages around this world, in Japan or in Sweden or in Nigeria, the variance that I see will largely be identical. And that prediction has been well satisfied. Because when you go and look and you collect variation in Japan or Sweden or Africa and you compare it, 90% of the variance are common across the entire world. Most variation is common ancestral variation around the world, and only a minority of the variance are new local mutations restricted to individual populations. This is so contrary to what people think because there's a natural tendency to kind of xenophobia, to imagine that world populations are very different in their genetic background. But, in point of fact, they're extremely similar. So, anyway, there's a limited amount of variation. That's why we have such little variation in the human population. Now, that variation, humans have a low rate of genetic variation. Most of the variance that are out there are due to common genetic variance, not rare variance. If I take your genome and I find a site of genetic variation at the point of heterozygosity in your genome, what's the probability that somebody else in this class also is heterozygous for that spot? It turns out that the odds are about 95% that someone else in this class will also share that variance. So that the variance are not mostly rare, they're mostly common. And it turns out that some of this common variation, that is most of this variation is likely to be important in the risk of human genetic diseases. So human geneticists have gotten very excited about the following paradigm. If there's only a limited amount of genetic variation in the human population, actually, if you do the arithmetic, there are only about ten million sites of common variation in the human population, where common might be defined as more than about 1% in the population. There are only ten million sites. Folks are saying, well, why not enumerate them all? Let's just know them all, and then let's test each one for its risk of, say, confirming susceptibility of diabetes or heart disease or whatever? After all, ten million is not as big a number as it used to be. We now have the whole sequence of the human genome. Why not layer on the sequence of the human genome all common human genetic polymorphism? Now, that's a fairly outrageous idea but could be a very useful one. Some of these variance are important, by the way. We know that there are two nucleotides that vary in the gene apolipoprotein E on chromosome number 19. Apolipoprotein E is also an apolipoprotein like we talked about before with familiar hypercholesterolemia. But, in fact, it turns out that apolipoprotein E is expressed in the brain. And it turns out, amongst other tissues, that it comes in three variances, the spelling T-T, T-C and C-C at those two particular spots. And if you happen to be homozygous for the E4 variant, homozygous for the E4 variant, you have about a 60% to 70% lifetime risk of Alzheimer's disease. In this class 13 of you are homozygous for E4 and have a high lifetime risk of Alzheimer's. And it would be fairly trivial to go across the street to anybody's lab and test that. Now, I don't particular recommend it, and I haven't tested myself for this variant because there happens to be no particular therapy available today to delay the onset of Alzheimer's disease. And, therefore, I don't recommend finding out about that. But a number of pharmaceutical companies, knowing that this is a very important gene in the pathogenesis of Alzheimer's disease, are working on drugs to try to delay the pathogenesis using this information. And it may be the case that five or ten years from now people will begin to offer drugs that will delay the onset of Alzheimer's disease by delaying the interaction of apolipoprotein E with a target protein called towe, etc. So, this is an example of where a common variant in the population points us to the basis of a common disease and has important therapeutic implications. There are some other ones, for example. 5% of you carry a particular variant in your factor 5 gene which is the clotting cascade. It's called the leiden variant. Those 5% of you are going to account for 50% of the admissions to emergency rooms for deep venous clots, for example. The much higher risk of deep venous clots. And, in particular, there are significant issues if you have that variant and you are a woman with taking birth control pills. Some of you were at higher risk for diabetes, type 2 adult onset diabetes. There's a particular variant in the population that increased your risk for type 2 diabetes by about 30%. 85% of you have the high-risk factor, so you might as well figure you do. 15% of you have a lower risk, et cetera. And one I'm particularly interested in here, it turns out that HIV virus gets into cells with a co-receptor encoded by a gene called CCR5. Well, it turns out that if we go across the European population, 10% of all chromosomes of European ancestry have a deletion within the CCR5 gene. If 10% of all chromosomes have that deletion then 10% times 10%, 1% of all individuals are homozygous for that deletion. Those individuals are essentially immune to infection from HIV. They are not susceptible. It's not through immunity, it's through lack of a receptor. Yes? You certainly can. It's not hard. It's a specific known variant. You could test for it. Absolutely. Now, of course, that only helps the 1% of people who have that variant. But what it did do was point to the pharmaceutical industry that the interaction between the virus and that variant is essential. And now companies are developing drugs to block the interaction with that particular protein. And that tells you that it's an important protein. Yes? Over the whole world? I just specified European population for that one. That one, interestingly, is not found at as high a frequency outside of Europe, and no one knows why, whether that might have been due to an ancient selective event or a genetic drift. By contrast, the apolipoprotein E variant, at that frequency of about 3% of people being homozygous and being at risk for Alzheimer's, is about the same frequency everywhere in the world. So, there's a little bit of population variation in frequency. Now, the HIV variant is found elsewhere but at considerably lower frequencies there. And that's an interesting question as to what causes that variation. So the notion would be, I've given you a couple of interesting examples, but, look, there's only ten million variants. Just write them all down. Make one big Excel spreadsheet with ten million variants along the top and all the diseases along the rows, and let's just fill in the matrix and then we'll really, you know, this is the way people think in a post-genomic era. Now, could you do something like that? You would have to enumerate all of the single nucleotide polymorphisms, or SNPs we call them, single nucleotide polymorphisms. Now, to give you an idea of the magnitude of this problem, as recently as 1998, the number of SNPs that were known in the human genome was a couple hundred. But then a project has taken off. In 1998 an initial SNP map of the human genome was built here at MIT that had about 4, 00 of these variants. Then within the next year or so an international consortium was organized here and elsewhere to begin to collect more of these genetic variants. The goal was going to be to find 300, 00 of them within a period of two years. In fact, that goal was blown away and within three years two million of the SNPs in the human population were found. And as of today, if you go on the Web, you'll find the database with about 7.8 million of the roughly ten million SNPs in the human population already known. Now, that isn't all ten million. And it takes a while to collect the last ones, you know, collecting the last ones are hard, but we're already the hump of knowing the majority of common variation in the human population. Not just a sequence of the genome, but a database that already contains more than half of all common variation in the population. So, we could start building that Excel spreadsheet. Now, it turns out that it's even a little bit better than that because if we look at many chromosomes in the population, here are chromosomes in the population, it turns out that the common variance on each of those chromosomes tend to be correlated with each other. If I know your genotype at one variant, like over at this locus, I know your genotype at the next locus with reasonably high probability. There's a lot of local correlation. So, instead of looking like a scattered picture like that, it's more like this. If I know that you're red, red, red you're probably red, red, red over here. In other words, these variations occur in blocks that we called haplotypes. Here's real data. Across 111 kilobases of DNA there's a bunch of variants, but it turns out that the variants come in two basic flavors. 98% of all chromosomes are either this, this, this, this, this or this, this, this, this, this. Then there tends to be sites of recombination that are actually hotspots of recombination where most of the recombination of the population is concentrated. And you get a couple of possibilities here. So, the human genome can kind of be broken up into these haplotypes. Blocks that might be 20, 30, 40, sometimes 100 kilobases long in which within the block you tend to have a small number of haplotypes, or flavors as you might think of them, that define most of the chromosomes in the population. So, in fact, I don't actually need to know all the variants. If they're so well correlated within a block, if I knew this block structure I would be able to pick a small number of SNPs that would serve as a proxy for that entire block of inheritance in the population. So, what you might want to do is determine that entire haplotype block structure of hwo they're related to each other, and pick out tag snips. And it turns out that in theory, a mere 300,000 or so of them would suffice to proxy for most of the genome. So, you might want to declare an international project, and international haplotype map project to create a haplotype map of the human genome. And indeed, such a project was declared about a year and a half ago through some instigation of scientists and a number of places, including here. And this is $100 million project involving six different countries. And, it is already more than halfway done with the task, and it's very likely that by the middle of next year, we will have a pretty good haplotype map, not just knowing all the variation, but knowing the correlation between that variation, being able to break up the genome into these blocks. By the next time I teach 701, I should be able to show a haplotype map of the whole human genome already. That will allow you to start undertaking systematic studies of inheritance for different diseases across populations. And in fact, people are already doing things like that. Here's an example of a study done here at MIT like this, where to study inflammatory bowel disease, there was evidence that there might be a particular region of the genome that contained it, and haplotypes were determined across this, and blah, blah, blah, blah, blah, blah, blah. And this red haplotype here turns out to confer high risk, about a two and a half or higher risk of inflammatory bowel disease. And it sits over some genes involved in immune responses, certain cytokine genes and all that. And, things like this have been done for type 2 diabetes, schizophrenia, cardiovascular disease, just right now at the moment, a dozen or two examples. But I think we're set for an explosion in this kind of work. In addition, you can use this information to do things beyond medical genetics. You can use it for history and anthropology as well. It turns out rather interestingly, that since the human population originated in Africa and spread out from Africa all the way around the world arriving at different places in different times, you can trace those migrations by virtue of rare genetic variants that arose along the way, and let you, like a trail of break crumbs, see the migrations. So, for example, there are certain rare genetic variants that we can see in a South American Indian tribe, and we can actually see that they came along this route because we can see that residual of that. In fact, we can do things with this like take a look at Native American individuals and determine that they cluster into three distinct genetic groups that represent three distinct migrations over the land bridge. And, you can assign them to these different migrations. You can do this on the basis of mitochondrial genotype, etc. You can also, for example, determine when people talk about the out of Africa migration, there's now increasing evidence that there really were two, one that went this way over the land, and one that went this way following along the coast into southeast Asia. And, it looks like we're now beginning to get enough evidence of these two separate migrations by virtue of the genetic breadcrumbs that they have left along the way. So, it's really a very fascinating thing of how much you can reconstruct from looking at genetic variation, both the common variation that allows us to recognize medical risk, and the rare genetic variation that provides much more individual trails of things. None of this is perfect yet. There's lots to learn. But I think anthropologists are finding that the existing human population has a tremendous amount of its own history embedded in pattern of genetic variation across the world. You can do other things. I won't spend much time on this. Well, I'll take a moment on this, right? There's some very interesting work of a post-doctoral fellow here at MIT named Pardese Sebetti who has been trying to ask, can we see in the genetic variation in the population, signatures, patterns of ancient selection, or even recent selection in the human population? Now, hang onto your seats, because this will get just slightly tricky. But, hang on. It's only a couple of slides. Here was her idea. You see, when a mutation arises in the population, it usually dies out, right? Any new mutation just typically dies out. But, sometimes by chance it drifts up to a high frequency. Random events happen. But it usually takes a long time to do that. If some random mutation happens, and it happens to drift up to high frequency with no selection on it, then on average it takes a long time to do so. If you want, I could write a stochastic differential equation that would say that, but just take your gut feeling that if something has no selection on it and it's a rare event that'll drift up, when it drifts up it's kind of a slow process. It was a slow process. Then over the course of time that it took to drift to high frequency, a lot of genetic recombination would have had to have occurred many generations. And the correlation between the genotype at that spot and genotypes at other loci would break down. And there would only be short-range correlation. So, in other words, the amount of correlation between knowing the genotype here and the genotype here, maybe allele A here and a C here. That is an indication of time. It's a clock almost. It's like radioactive decay, right, that genetic recombination scrambles up the correlations. And, if something's old, the correlations go over short distances. But suppose that something happened. Some mutation happened that was very advantageous. Then, it would have risen to high frequency quickly because it was under selection. If it did so quickly, then the long-range correlations would not have had time to break down, and we'd have a smoking gun. A smoking gun would be that there would be a long-range correlation around that locus, much longer than you would expect across the genome. Things even out of this distance would show correlation with that, indicating that this was a recent event. So, we just measure across the genome, and look for this telltale sign of common variance that have very long range correlation that indicate that they're very recent. So, a plot of the allele frequency, common variance, sorry, if something has a common high frequency and long-range correlation, you wouldn't expect that by chance. So, something that was common in its frequency and had long-range correlation would be a signature of positive selection. So anyway, Pardise had this idea, and she tried it out with some interesting mutations, some mutations that confer resistance to malaria, one well-known mutation causing resistance to malaria called G6 PD and another one that she herself had proposed as a mutation causing resistance to malaria, variants in the CD4 ligand gene. And to make a long story short, both the known and her newly predicted variant showed this telltale property of having a high frequency and very long range correlation. Well that's very interesting because she was able to show that each of these mutations probably were the result of positive selection. But what you could do in principle is test every variant in the human genome this way: take any variant, look at its frequency, and compare it to the long range correlation around it, and test every single variant in the human population to see which ones might be the result of long range correlation. Now, when she proposed this, this was about a year and a half ago or two years ago, this was a pretty nutty idea because you would need all the variants in the human population, and you would need all this correlation information. But in fact, as I say, that information's almost upon us, and I believed that this experiment, this analysis to look for all strong positive selection in the human genome will in fact be done in the course of the next 12 months. So, I'm hoping by next year I can actually report on a genome-wide search for all the signatures of positive selection. Now, this doesn't detect all positive selection. It will detect sufficiently strong positive selection going back pretty much only over the 10, 00 years. When you do the arithmetic, that's how much power you have. Of course, 10,000 years has been a pretty interesting time for the human population, right? The time of civilization and population density, and infectious diseases, and all that, and I think we'll have an interesting window into the change in diet. All of that should come out of something like this. So, there's a lot of really cool information in DNA variation to be had. All right, that's one half. The other half of what I would like to talk about is totally different. It's not about inherited DNA variation. It's about somatic differences between tissues in RNA variation. So, let's shift gears. RNA variation: let me start by giving you an example here. These are cells from two different patients with acute leukemia. Can you spot the difference between these? Yep? More like bunches of grapes and all that. Yeah, it turns out that's just a reflection of the field of view you have if you move over to look like that. But I mean, that's good. It's just that it turns out that that isn't actually a distinction when you look at more fields. Anything else? Yep? White blood cells like different. They look broken. There's more of them in this field of view. But you look at 100 fields of view and it turns out that's not either. Well, the reason you're having trouble spotting any difference is that highly trained pathologists can't find any difference either. I generally agree there's no difference between these two if you look at enough fields of view. But you can convince yourself if you look that you see things there. But these actually are two very different kinds of leukemia. And, these patients have to be treated very differently. But, pathologists cannot determine which leukemia it is just by looking at the microscope, it turns out. This is the work of this man, Sydney Farber, namesake of the Dana Farber Cancer Institute here in Boston, who in the 1950s began noticing that patients with leukemias, some of them seemed different in the way they responded to a certain treatment, and he said, look, I think there's some underlying classification of these leukemias, but I can't get any reliable way to tell it in the microscope. And he put many years into working this out, first by noticing certain difference in enzymes in the cells, and then people noticed certain things in cell surface markers, and some chromosomal rearrangements. And nowadays, there are a bunch of test that can be done by a pathologist when a patient comes in with acute leukemia to determine whether they have AML or ALL. But it turns out that you can't do it by looking. You have to do some kind of immunohystochemical test of some sort in order to do that. So this is a triumph of diagnosis. After 40 years of work, we can now correctly classify patients as AML or ALL. And they get the appropriate treatment. And if they don't get the right treatment, they have a much higher chance of dying. And if they do get the right treatment, they have a much higher chance of living. So, this is great. There's only one problem with the story. It took 40 years, 40 years to sort this out. That's a long time. Couldn't we do better? Surely these cells know what they are. Surely we could just ask them if they are. Well, here's the idea. Suppose we could ask each cell, please tell us every gene that you have turned on, and the level to which you have that gene expressed. In other words, let us summarize each cell, each tumor by a description of its complete pattern of gene expression to 22,000 genes on the human genome. Let's write down the level of expression, X1 up to X22, 00 for each of the 22,000 genes of the genome. So, ever tumor becomes a point in 22, 00 dimensional space, right? Now clearly, if we had every tumor described as a point in 22, 00 dimensional space, we ought to be able to sort out which tumors are similar to each other, right? Well, it turns out you can do that now. These are gene chips, one of several technologies by which on a piece of glass are put little spots, each of which contains a piece of DNA, a unique DNA sequence. Actually, many copies of that DNA sequence are there. Each of these is a 25 base long DNA sequence, and I can design this so whatever DNA sequence you want is in each spot. The way that's done is with the same photolithographic techniques that are used to make microprocessors. People have worked out a chemistry where through a mask, you shine a light, photodeprotect certain pixels; the pixels that are photodeprotected you can chemically attach an A, then re-protect the surface. Use a light. Chemically photodeprotect certain spots. Wash on a C. And in this fashion, since you can randomly address the spots by light, and then chemically add bases to whatever spots are deprotected, you can simultaneously construct hundreds of thousands of spots each containing its own unique specified oligonucleotide sequence. And you can get them in little plastic chips. And then if you want, all you do is you take a tumor. You grind it up. You prepare RNA. You fluorescently label the RNA with some appropriate fluorescent dye. You squirt it into the chip. You wash it back and forth. You rock it back and forth, wash it out, and stick it in a laser scanner. And it'll see how much fluorescence is stuck to each spot. And bingo: you get a readout of the level of gene expression. I guess each spot, you should design it so that this spot has an oligonucleotide complementary to gene number one. And the next one, an oligonucleotide matching by Crick-Watson base pairing complementary to gene number two and gene number three. So, if I knew all the genes in the genome, I could make a detector spot for each gene in the genome. And of course we know essentially all the genes in the genome. So you can make those detector spots and you can buy them. So, you can now get a readout of all the, I mean, this is like so cool because when I started teaching 701, which wasn't that long ago because I ain't (sic) that old still, the way people did an analysis of gene expression is they used primitive technologies where they would analyze one gene at a time, certain things called northern blots and things like that, right? And, you know, you'd put in a lot of work and you get the expression level of a gene, whereas now you can get the expression of all the genes simultaneously, and it's pretty mind boggling that you can do that. How do you analyze data like that? So, we still use northern blots. It's true. So, every tumor becomes a vector, and we get a vector corresponding to each tumor. So, this line here is the first tumor, the second tumor, the third tumor, the fourth tumor. The columns here correspond to genes. There are 22, 00 columns in this matrix, and I've shown a certain subset of the columns because these genes here have the interesting property that they tend to be high red in the ALL tumors, and they tend to be low blue in the AML tumors, whereas these genes here have the opposite property. They tend to be low blue in the ALL tumors and high red in the AML tumors. These genes do a pretty good job of telling apart these tumors. So, here's a new tumor. Patient came in. We analyzed the RNA, squirted it on the chip. Can somebody classify that? Louder? AML. Next? Next? Congratulations, you're pathologists. Very good. That's right, you can do that. It works. And in fact, in the study that was done that was published about this, the computer was able to get it right 100% of the time. Not bad. So now you say, wait, wait, wait, but you're cheating. You're giving it a whole bunch of knowns. Once I have a whole bunch of knowns it's not so hard to classify a new tumor. What Sydney Farber did was he discovered in the first place that there existed two subtypes. Surely that's harder than classifying when you're given a bunch of knowns. And that's true. So, suppose instead, I didn't tell you in advance which were AML's and which were ALL's, and I just gave you vectors corresponding to a large number of tumors, do you think you would be able to sort out that they actually fell into two clusters? Could you by computer tell that there's one class and the other class? Turns out that you can. Now, I've made it a little easier by not listing most of the 22,000 columns here. But think about it. Every tumor is a point in 22, 00 dimensional space. If some of the tumors are similar, what can you say about those points in 22,000 dimensional space? They're going to be clumped together. They're near each other. So, just plot every tumor as a point in 22,000 dimensional space, and your question is, do the points tend to lie in two clumps up in 22, 00 dimensional space? And there's simple arithmetic you can learn using linear algebra to get some separating hyperplane and ask, do tumors lie on one side or the other? And, it turns out the procedures like that will quickly tell you that these tumors clump into two very clear clumps. They're not randomly distributed. And so, if you get these tumors, and you do gene expression on them and put the data into a computer, the amount of time it takes the computer to discover that there were actually two types of acute leukemia is about three seconds marked down from 40 years. That's good. So, you can reproduce the discovery of AML and ALL in three seconds. Now you know what the pathologists say about this. They say, oh, give me a break. It's shooting fish in a barrel. We know there was a distinction. Big deal that the computer can find the distinction. We knew that there was distinction there. I know the computer didn't know it and all that. Tell us something we don't know. That's a fair question. So it turns out that you can ask some more questions. You can say, suppose I take now just the ALL's. Are they a homogeneous class, or did they fall into two classes? It turns out that extending this work, folks here were able to show that we can further split that ALL class. There was a hint that you might be able to do so because there's some ALL patients who have disruptions of a gene called MLL. And this tends to be a little more common in infants, and tends to be associated with a poor prognosis. But it was really very unclear whether this was simply one of a zillion factoids about some leukemia patients, whether this was a fundamental distinction. So, what happened was folks took a lot of ALL patients, got their expression profiles, and lo and behold it turned out that ALL itself broke into two very different clusters. This is an artist's rendition of a 22,000 dimensional space. We can't afford a 22,000 dimensional projector here, so we're just using two dimensions. But, the two forms of ALL were quite distinct from each other, and so actually ALL itself should be split up into two classes, ALL plus and minus, or ALL one and two, or MLL and ALL. And it turns out that these forms are quite different. They have different outcomes and should be treated differently. It also turns out that a particularly good distinction between these two subtypes of ALL is found by looking at this particular gene called the flit-3 kinase. The flit-3 kinase gene, whatever that is, was of great interest because people know that they can make inhibitors against certain kinases. And so, it turned out that an inhibitor against flit-3 kinases, against this flit-3 kinase gene product. If you treat cells with that inhibitor, cells of this type die, and cells of this type are not affected. So in fact, there's a potential drug use of flit-3 kinases in the MLL class of these leukemias, and folks are trying some clinical trials now. So, not only did the analysis of the gene expression point to two important sub-types of leukemias, but the analysis of the gene expression even suggested potential targets for therapy. So, I'll give you a bunch more examples. I have a bunch more examples like that there. They are examples of taking lymphomas and showing that they can be split into two different categories, examples of taking breast cancers into several categories, colon cancers. Basically what's going on right now is an attempt to reclassify cancers based not on what they look like in the microscope, and based not on what organ in the body they affect, but based on, molecularly, what their description is, because the molecular description, as Bob talked to you about with CML and with Gleveck, turns out to be a tremendously powerful way of classifying cancers because you're able to see what is the molecular defect and can make a molecular targeted therapy. So, these sorts of tools are quite cool, and I've got to say, in the last year we've begun using these expression tools not just to classify cancers, but to classify drugs. We've begun an interesting and somewhat crazy project to take all the FDA approved drugs, put them onto cell types, and see what they do, that is, get a signature, a fingerprint, a gene expression description of the action of a drug. And then we hope, here's the nutty idea, that we can look up in the computer which drugs do which things and might be useful for which diseases, because we'd put the diseases and the drugs on an equal footing. All of them would be described in terms of their gene expression patterns. So, I'll tell you one interesting example, OK? This is an interesting enough example. I don't even have slides for it yet. It turns out that these patients with ALL that I've been talking about, some of the patients with ALL will respond to the drug dexamethasone. Some won't. If you take patients who respond to dexamethasone, and patients who are resistant to dexamethasone, and you get their gene expression patterns, you can ask are there some genes that explain the difference? And you can get a certain gene signature, a list of, say, a dozen or so genes that do a pretty good job of classifying who's sensitive and who's resistant. Then you can go to this database I was telling you about of the action of many drugs and say, do we see any drugs whose effect would be to produce a signature of sensitivity? If we found a drug X, which when we put it on cells turned on those genes that correlate with being sensitive to dexamethasone, you could hallucinate the following really happy possibility that when you added that drug together with dexamethasone, you might be able to treat resistant patients because that drug could make them sensitive to dexamethasone, and that you could find that drug just by looking it up in a computer database. So, we tried it and we hit a drug. There was a certain drug that came up on the screen, yes? That's very much in the idea too. We found a drug that produced the signature sensitivity, and tested it in vitro. In vitro, if you take cells that are resistant and you add dexamethasone, nothing happens because they're resistant. If you add drug X, nothing happens. But if you add both drug X plus dexamethasone, the cells drop dead. It's now going into clinical trials in human patients. It turns out drug X is already a well FDA approved drug, so it can be tested in human patients right away, so it's going to be tested. So, the gene expression pattern was able to tell us to use a drug which actually had nothing to do with cancer uses in a cancer setting because it might do something helpful. Now, what's the point of all this? We can turn up the lights because I think I'm going to stop the slides there. The point of all of this, which is what I've made again, and I will make again, because you are the generation that's going to really live this, is that biology is becoming information. Now, don't get me wrong. It's not stopping being biochemistry. It's going to be biochemistry. It's not stopping being molecular biology. It's not stopping any of the things it was before. 45:57 But it is also becoming information, that for the first time we're entering a world where we can collect vast amounts of information: all the genetic variants in a patient, all of the gene expression pattern in a cell, or all of the gene expression pattern induced by a drug, and that whatever question you're asking will be informed by being able to access that whole database. In no way does it decrease the role of the individual smart scientist working on his or her problem. To the contrary, the goal is to empower the individual smart scientist so that you have all of that information at your fingertips. There are databases scattered around the web that have sequences from different species, variations from the human population, all of these drug database, etc., etc., etc., etc. It's a time of tremendous ferment, a little bit of chaos. You talk to people in the field, they say, we're getting deluged by data. We're getting crushed by the amount of data. I don't' know what to do with all the data. There's only one solution for a field in that condition, and that is young scientists because the young scientists who come into the field are the ones who take for granted, of course we're going to have all these data. We love having all these data. This is just great, couldn't be happier to have all these data. We're not put off by it in the least. That's what's going on. That's what's so important about your generation, and that's why I think it's really important that even though it's 701 and we're supposed to be teaching you the basics, it's important that you see this stuff because this is the change that's going on, and we're counting on this very much to drive a revolution in health, a revolution in biomedical research, and we're counting on you guys very much to drive that revolution. It has been a pleasure to teach you this term. I hope many of you will stay in touch, and some of you will go into biology, and even those of you who don't will know lots about it and enjoy it. Thank you very much. [APPLAUSE]
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