Please don't be terrified by the numbers on the slides, okay? The numbers are no longer what they were before. There's a good reason for that, which I won't go into. But we don't have 164 slides to go through and we have not gone through 107 slides. But I want to keep you on your toes. So today I want to discuss the second qualitative big set of open problems that we have to confront and solve. Yesterday we had to deal with the drama of the end of space time. And today we have a very different seeming set of questions, although there might be some relationships between these problems. The very different seeming set of questions is about a very startling, well, a very zeroth order obvious fact about the world, which is that it's big and the bigness of the world, the fact that we have a macroscopic universe is actually very, very mysterious. And trying to give a good answer to such a zeroth order question is very likely another one of the big challenges that we're gonna have to confront and solve in the century. So if we go back to the pictures of the scales that we're talking about before, once again, we have the Hubble scale, the WEEK scale and the Planck scale and the salient feature that we're going to be talking about. Credibly obvious feature of this plot are these gigantic separations of scales, 16 orders of magnitude between the WEEK scale and the Planck scale and this much larger 60 orders of magnitude or so between the Hubble scale and the Planck scale. And you might not think that this is such a big pressing problem why there are these enormous separations between these scales, but as we'll see, it's actually a very, very major puzzle. And once again, it all goes back to the excitingness of the vacuum that we've mentioned a number of times. But in this case, the first incarnation of the problem, the first way that you realize that there is a real issue here is that all these quantum fluctuations in the vacuum mean that even the vacuum should have energy. So you all know from undergraduate physics that in a classical system, you might imagine having the pendulum hanging on its end, hanging at the bottom and not moving and so having zero energy. But quantum mechanically, we can't both know that it's at the bottom and that it's not moving so it has a certain minimum energy, it's Planck's constant times the frequency. The fact that it's Planck's constant times the frequency means that it's bigger and bigger fluctuations as we go to shorter and shorter distances. And so there is larger and larger energy in quantum fluctuations in smaller and smaller boxes and that's going to lead to a real puzzle that we're gonna talk about in a second. But first let's actually just try to estimate the energy in the vacuum. So we're gonna do a little calculation here. So I want to compute the total energy in the vacuum due to quantum fluctuations. So first let's put the universe in a big box. Here's a big box, the side length of the box is L. And let's imagine we have any particles that we know and love, let's say photons. There are particle and a box modes for photons of this wavelength, that wavelength, that wavelength, all sorts of wavelengths. Each one of the frequencies associated with these guys is the square root of P squared plus M squared, the particle is a mass M. The momentum is given by N over L. The momentum are quantized in units of one over L in each one of the directions. So this N is just a bunch of integers, Nx, Ny, and Z for the particle in a box mode in each one of the directions, X, Y, Z. So that's the frequency. This is the square root of N squared over L squared plus M squared. I remind you that the mass of a lot of the particles that we're talking about, let's just focus on particles like electrons and W's and Z's and so on. They get a mass that comes from interactions with the Higgs field that we talked about in the first lecture so that a fermion will have a mass that's some coupling constant times whatever the size of the Higgs field is in the universe. The boson will also have the mass which is another coupling constant times the size of the Higgs field in the universe. So this M parameter actually secretly depends on what the overall mass scale of the Higgs is, what the overall scale of the Higgs is in the universe. Okay, but anyway, so let's now go back and compute the vacuum energy. Each one of these particles on a box mode is our harmonic oscillator, and so it has a vacuum energy which is a half h bar omega. I'm just gonna add them all up, okay? So just a sum over all these modes of a half h bar omega. It turns out for the fermions, for good reasons that we could explain if we took another seven minutes, very simple and good reasons that many of you know if you've taken a field theory course and if not, you'll learn it soon. The zero point energy for a fermionic oscillator is negative a half h bar omega, not positive a half h bar omega. Okay, so that's what we do. We add the fermions and the bosons, a half h bar omega with a plus sign for one and a minus sign for the other one. That's the total energy in the vacuum. All right, so let's do this calculation. You're very familiar with the fact that the sum over n, you can replace as an integral over all the momenta, right? The sum over n, you can approximate as an integral d cubed k, where k is all the momenta, multiplied by the volume of the box. That has to be because the sum over n is dimensionless, so it has to turn into the volume times the integral over momentum space. So the energy in the vacuum is the volume of the box times the integral over k, d cubed k, the sum over bosons and fermions, and now we have a half root k squared plus some boson squared minus a half root k squared plus some fermions squared. Okay, so let's do this integral. It's very, very easy to do this integral. Let's first look at what happens. Where is this integral dominated? If we go to big k, okay, this measure piece is getting bigger. It goes like dk times k squared. So we go to big k, that piece is getting bigger. This piece is getting bigger. So at large k, I can ignore the masses compared to k, and it's the integral d cubed k times k, the magnitude of k. So what is that integral? What is the integral d cubed k times magnitude of k? It's infinity, it's just divergent as you go to higher and higher and higher momentum. That's a short distance divergence, high momentum, short distance divergence. Now, we just went through this whole song and dance last time that there's no such thing as space time beneath the Planck length, et cetera, et cetera. So this whole calculation clearly makes no sense when the k gets big, let's say, compared to the Planck scale.  So let's cut it off somewhere. Let's cut it off at some k max and just estimate what this is. So this leading piece that we just talked about just goes like k max to the fourth. So the, sorry, I should have divided by the, here or now I'm talking about the energy density. So the vacuum energy divided by the volume, scale's like the volume, so the energy density has a piece that goes like k max to the fourth. That's all it can be by dimensional analysis, okay? Now underneath that is another piece that diverges, actually. If you expand the square root out, the leading piece goes like magnitude of k and then there's a subleading piece that goes like m squared over magnitude of k, okay? That subleading piece, when you multiply it by the d cubed k, still diverges as you go to high k. Again, this is guaranteed by dimensional analysis. The piece underneath this, there's a piece that can go like k max squared and it can go like mass squared, mass squared of the boson or the mass squared of the fermion.  And something that you'll learn if you study quantum field theory is every time you have a constant in quantum field theory there's actually a logarithm there. So there's a mass to the fourth times the log, okay? But let's never mind that, we're gonna be concentrating on these really big pieces, k max to the fourth and k max squared mass squared. And remember that the mass squareds themselves are being set by the value of the Higgs field everywhere in the universe.  This little computation illustrates these two major massive paradoxes we have in trying to understand where there's a, why there's a macroscopic universe. In fact, we get an estimate for how big these are. For example, if we imagine there's nothing other than what we've seen in the standard model, nothing else when we extrapolate up to the Planck scale we get an estimate for the vacuum energy which is k max to the fourth, let's say roughly m Planck to the fourth here. And also this sub leading piece gives us a new contribution to what you might think of as the potential for the Higgs, see? So this quantum mechanical energy depends on what the value of the Higgs is. And so it's giving us, as I now change the value of the overall scale of the Higgs it's getting a correction that goes like k max squared. Both of these things are enormous numbers. Okay, you remember the Planck scale as an energy scale is massive compared to all the other energy scales that we've seen in nature. We've probed experimentally. And yet the energy density in the vacuum seems to be, want to be comparable to the Planck scale. And also the typical mass scale, associated with the Higgs field seems to be from this quantum mechanical vacuum energy to want to be associated with the Planck scale. These are two big problems. One of them is called the cosmological concept problem and the other one's called the hierarchy problem and let's talk about them in turn. So let's just stare again. I'm just repeating what I said before and saying it even more simply here, a little more qualitatively. If we want to estimate the energy density in the vacuum we just did it, it goes like in Planck to the fourth. That's because there's larger and larger fluctuations in smaller and smaller boxes. That's the basic problem. And so the kind of energy density we have is a Planck energy per Planck volume. Now, in a world with gravity this energy density in the vacuum does something. Without gravity we all learn in high school the overall energy doesn't matter. You can subtract the constant from the energy and it doesn't matter. But with gravity it matters because everything gravitates. So this energy density in the vacuum, if it was as big as this Planck scale it would curve spacetime and give some typical curvature scale. Now what do you think that typical curvature scale is?  Only one word makes an appearance on this slide. One relevant word, Planck, Planck, Planck. Planck energy, Planck density, gravity associated with the Planck scale, the only curvature that could possibly arise from this would be Planckian. But this is a complete disaster because the world doesn't look like that. Planckian curvature would mean that it's curled up to a tiny ball on scales of order 10 to the minus 33 centimeters. Or maybe it's exploding and accelerating and doubling in size every 10 to the minus 43 seconds. That looks absolutely nothing like the universe that we see. Now, as we mentioned before, the universe is accelerating, of course, but it doubles in size every 10 billion years or so, not every 10 to the minus 43 seconds. And the simplest possible explanation of this constant doubling rate is that there is a non-zero vacuum energy. But its size that it's needed in order to make the universe double every 15, 10, 15 billion years rather than 10 to the minus 43 seconds is 120 orders of magnitude smaller than this back-of-the-envelope estimate that we just did. When I used to talk about this, I used to call this the biggest error in the history of physics, and I realized that's implicitly insulting other sciences, like someone else made a comparable or worse error. I don't think anyone has made an error so big as this. This is, I think, the biggest disagreement. It's not an error, really, as we'll see. It's the biggest disagreement between the back-of-the-envelope estimate and reality in the history of science. And as I mentioned in the first lecture, if you're a decent theoretical physicist, just a decent theoretical physicist, eventually, you should be able to estimate anything, zero authority thing about nature anywhere, and get it right to a factor of 10 level. If you're pretty good, you should be able to estimate everything to a factor of three level.  But certainly, anyone can get it to a factor of 10. So we really take this personally. This is not, you know, we're not a bunch of yahoos. We make predictions, they work to part of 1,000, G-minus two works to 12 decimal places. This is the kind of precision and the sort of exquisite control of nature that we're getting used to. This is a kick in the teeth from that point of view. Now, what do we do about this? Clearly, if we're going to, we have this fantastic theory that predicts very detailed, wonderful things like the G-minus two of the electron to 12 decimal places. Clearly, before getting started to predict the G-minus two to 12 decimal places, we should start off with this nice big universe that's not exploding in size every 10 to the minus 43 seconds, right? So we must do something about this before even getting started. So what is it that we do? This is literally what we do. What we do is we say it's not a problem. It's not a problem because what we just calculated is one contribution to the energy density of the vacuum coming from these quantum mechanical vacuum energy. And we can estimate it or we can try to get a big army of graduate students to work very, very hard and calculate what this is. It's actually not, well, this is a bit of a lie because it's hard to compute this reliably, precisely, because it's so sensitive to what's going on at very, very short distances. But anyway, let's continue with this fiction for a little bit. So these army of graduate students does this computation and let's say working in plot units, they find that this vacuum energy is 2.6493781 dot, dot, dot, dot. They come back up to 40 decimal places, they say go back, keep working, come back to the 70 decimal places, go back, keep working. And then they come back completely exhausted, they've computed it to 120 decimal places, you say go back, keep working. And they compute the 121st and the 122nd and they collapse. Okay, great. So they say thank you very much. That's the quantum mechanical contribution. So I know what's going on. What's going on is that there just so happens to be a classical piece that was sitting there already. And its value, guess what, is negative 2.6493781 dot, dot, dot, and this agrees with that to 120 decimal places, but they start disagreeing in the 121st decimal place. That's what we actually do. Now, we don't actually take grad students to do this because we say whatever this result is, we're gonna imagine that there was another contribution that was sitting there that's just finally canceling it to 121 decimal places. Now this looks completely ludicrous, right? But it's what we actually do and nothing stops us from doing it. We do it and we proceed. And we calculate the g minus of the electron to 12 decimal places. So this is not some kind of problem that just tells you you are wrong. There's something absolutely definitely wrong with this. You're not punished for doing this, right? Everything else works great. It's not like some of the problems that we talked about last time. Last time when the curvature of the universe gets plonky and near the big bang, we just really don't know what's going on. It's not a question of whether we like it or not. There's just something missing and wrong. That's just clearly a physical problem. There's clearly something wrong with it, okay? This is not a problem of that nature. We can solve it. We can just imagine that these two things cancel to 120 decimal places and proceed. And everything else is fine. But it's such a major, huge piece of craziness that it seems that we have to do, that it suggests that this is maybe not the real explanation for what's going on. And one should think at least about how this is coming about a little more carefully. It's like walking into a room and seeing a pencil standing on its tip on a table.  But it's just standing on its tip. Now, it's possible, it's possible. There's a consistent solution of Newton's laws that has the pencil sitting on its tip to within 10 to the minus 120 degrees of vertical. But if you saw something like that, if you walked into such a room and saw just a pencil, you would probably think there's something up, right? And it's worth figuring out why the pencil is doing that. Like maybe you'd look for a string hanging from the ceiling. Oh, I see it hanging from the ceiling, okay? We'll talk about some other things that it might be. But you would think that it's worth trying to understand this. It's so dramatic that you think it's worth trying to understand this. What we actually do, canceling these two things to a part in 10 to the 120, is for good reasons called fine tuning. Because we really have to imagine that there's adjustments in the parameters of the theory. But things that seem to have absolutely nothing to do with each other that have to cancel not perfectly even. Maybe if they canceled perfectly, you would think there's some mysterious deep mechanism that we don't understand about gravity and cosmology and so on that just makes this vacuum energy zero exactly, for some reason. But it's not even that. It's not even that. They don't cancel exactly. They disagree in the 121st decimal place. And I remind you, this is our answer. Fine tuning is our answer to a very basic question. Why is the universe big, okay? If we did have this gigantic vacuum energy, as we said, all the curvatures would be over Planckian and incredibly unpleasant world. Well, we'll come back to the incredible unpleasantness a little bit later. Now, remember this K max to the fourth was the leading term of the vacuum energy. There was a sub-leading term that went like K max squared H squared. And that sub-leading term is associated with another problem. Because that would seem to want to make all of the mass scales associated with the Higgs of order of the Planck scale. Well, that's 10 to the minus 33 centimeters. Whereas we seem to know that the length scale associated with the weak interactions is more like 10 to the minus 17 centimeters. The mass scale associated with the Higgs wants to be 250 GeV, not M Planck. So once again, what we do is imagine that there's a cancellation between parameters. Now a measly part in 10 to the 32, so you can estimate by 250 GeV squared over M Planck squared, that kind of accuracy is needed. And that's what we do in order to have a separation between the Planck scale and the scale of the Higgs. If this didn't happen, if the Higgs scale was really up and near the Planck scale, we would all be, all the electrons in your body would be 17 orders of magnitude heavier. We'd all collapse into black holes. It would be a very unpleasant world again. And remember, we saw very explicitly in the first lecture, the reason the world is big, even though it's made out of small things, it's made out of atoms, but the reason planets are big and there's a macroscopic world at all is precisely the weakness of gravity. The size of the earth and units of the size of the atom was precisely the ratio of the strength of the electromagnetic force to the gravitational force. So this is directly in answer to the question again, why is the universe big? But in a more precise sense, why is gravity weak? So this fine tuning of parameters of a part in 10 to the 32 is what we have to do to accommodate that basic fact. So if I go back to this picture, this separation needs a part in 10 to the 30 about tuning and this separation needs a part in 10 to the 120 tuning. That's what we have to do to explain this incredibly basic fact about the world. If these tunings didn't happen, another way of saying it is all these scales would have ended up smack on top of each other. If you handed another graduate student the standard model of particle physics on a napkin, you put them in a spaceship. I always give this example and I always realize it doesn't make any sense because, well anyway, you put them in a spaceship, you close the window so they can't see the world outside and you ask them to predict what the world would look like. It doesn't make any sense because then the spaceship wouldn't make sense anyway. But anyway, if you did that, they would not remotely predict the world that we see. They would think that all the scales are on top of each other.  There's this gigantic world out there. It seems totally astonishing. And of course, this is an incredibly unpleasant universe. So this is the big question. What is it that controls the violent fluctuations of the vacuum? Why is there a macroscopic universe? Why is there a macroscopic universe in a world which has gigantic quantum fluctuations which get bigger and bigger at shorter and shorter distances? Now, this is such a big problem. And I emphasize this way of thinking about it that it's so tightly tied to the existence of these vacuum fluctuations. Because just so you see that it's incredibly, it's almost inexorably tied to the structure of physics that we know works. Quantum mechanics, relativity, they both work. These big fluctuations are there. Quantum fluctuations of the vacuum are there. It's these fluctuations that give us these incredible 12 decimal place predictions for g-2 that work in countless other situations. So it's not like somehow we should be scared of quantum mechanical fluctuations. They're definitely there. But they seem at a level which is far larger than these tiny logarithmic effects that we were talking about before to make it impossible for there to be a macroscopic universe. So if something is gonna solve this problem, but I mean solve this problem, I mean just the simplest way to solve it would be to just change the calculus of the discussion. Just to say, no, it actually isn't fine tuned at all. Everything is fine. There just aren't these gigantic fluctuations as we go to short distances. That would be a very nice way of solving the problem. So let's say we wanna do that. So let's say we're talking about this problem of why gravity is weak. We'll come back to the more bigger problem in a second. But if we talk about the problem of why gravity is weak, we have these fluctuations are getting bigger and bigger as we go to shorter and shorter distances. Something new has got to happen around 10 to the minus 17 centimeters in order to change the discussion. Change this estimate that we did for the size of the vacuum energy. And it has to happen right around 10 to the minus 17 centimeters. It can't happen at 10 to the minus 25 centimeters. Let's say something new does happen at 10 to the minus 25 centimeters. Great, you go sailing past 10 to the minus 17 centimeters. The fluctuations are enormous, bigger and bigger, down to 10 to the minus 25 centimeters. That's still no good. That would still make it so that all the scales associated with the massive particles in our bodies and all of that would go up, not 17 orders of magnitude, but it's still down to 10 to the minus 25 centimeters. If there's a real solution to this problem that changes the discussion, changes this estimate so we don't get this horrendous tuning, it has to show up around the corner, right around 10 to the minus 17 centimeters. That happens to be the place we're about to probe experimentally. Now, if you're suspicious of this sort of argument, it might soothe you a little to know that arguments like this and problems like this have come up in the history of physics, just the relatively recent history of physics in the last 100 years, a number of times. There's at least three examples I can think of, two of them which are incredibly close to the problem at hand, actually. This one is not literally the same, but it's very, very close, but it's a little more dramatic, so I'll mention it. Three times in the last century, this kind of issue arose, different degrees of how quantitatively dramatic it was, but this basic kind of issue arose. And all three times, the kind of logic that we just used that something new has got to happen in order to change the discussion, this ended up being the correct answer. So let's talk about a famous example which bothered people a lot in the late 1800s and early 1900s. People like Abraham and Lorentz were really bothered by this point-like picture of the classical electron because there was this electric field that surrounds the electron. There's energy in that electric field, a half electric field squared, and so if you just look at the total energy that's contained inside that electric field, it diverges. It diverges as you go closer and closer in towards the electron. And that really bothered them because how can it drag this infinite energy around with it everywhere it goes? It should bother you even more after it equals mc squared because that would mean the electron should be infinitely massive, and it's not infinitely massive. So, maybe I'll do this just quickly on the board because I didn't put it there for some reason. So, so this is what they argued. So let's say you cut this off at some scale of order a. The energy that's stored in the electric field is of order e squared over four pi. So this is the classical calculation. So it's e squared over four pi, one over a. And so what you can do is say, well, something has got to happen such that when a gets too small, once this energy starts getting much bigger than m electron c squared, something new has got to happen by then. Otherwise, I'm gonna have to imagine finally balancing this electric energy contribution against some other contribution in order to come up with the mass of the electron the way it is, right? So let's estimate where that should happen. That should happen at some size, let's call it ae, where e squared over four pi, one over a becomes comparable to m electron c squared. I'm putting in the C's and stuff for a second here. So we get this ae, which is e squared over four pi m electron c squared. This is called the classical radius of the electron. And that's what they thought. They thought something new has got to happen by that distance scale. Now, the new thing that they were trying were things like the electron of the shell. It's a little spherical shell. And they tried all sorts of concrete theories like that and they had horrendous difficulties making them work. To any aficionados in the audience since we're making an analog to the hierarchy problem, Abraham and Lorenz were the technicolor model builders of the early 1900s. They were literally trying to do the analog of technicolor and then as now it failed miserably. Now, why does the right answer? What is the right answer? The right answer is that in fact something new does happen as you go closer and closer to the electron. We have this cloud of electrons and positrons that surround it. This cloud of electrons and positrons has the effect that you can no longer tell where the central electron is anymore. So it's not like this electric field line going down towards the center. There's now just a cloud surrounding it. And that effectively acts as the scale that cuts off this enormously increasing divergent energy as you get closer and closer to the electron. What is this typical scale? What is this typical scale? This typical scale is around h bar divided by m electron c. That's the Compton wavelength. Now this is really cool. Let's compare, here's the electron. Here's the electron classical radius which is e squared over four pi m electron c squared. And let's compare it to the actual scale where something new happened which was the electrons and positrons coming in which was h bar over m electron c. And you see that the scale where something new happened over the scale where they expected is one over e squared, the m electrons cancel is one over e squared over four pi h bar c which is exactly one over alpha which is around 1.37. So they were right. They were right that something new does have to happen by the scale of this classical radius. In fact it happens 137 times earlier than it had to. At a much larger distance scale something new happens. In this case a new thing was really dramatic. It was quantum mechanics and relativity and this whole picture of particles and antiparticles things they couldn't even anticipate but just because they couldn't anticipate it doesn't mean that their basic logic wasn't right. Something new did have to happen and not only did it happen, it happened earlier than it had to happen. This is very analogous to the problem that we're talking about. And one of the solutions that we're gonna, potential solution that we're gonna talk about a supersymmetry is essentially the modern retelling of this story. Almost word for word, the modern retelling of this story. Maybe I'll skip this for now. So what could it be? What could it be that might change this discussion? Since the problem is very closely tied in to the quantum fluctuations in the vacuum, it's closely tied into this whole picture of particles and antiparticles which are there because of quantum mechanics and relativity and so on, you might expect that it's got to be something fairly dramatic. It can't just be a random dinky particle here or there. It has a job to do. We have to remove huge quantum fluctuations in the vacuum. So it can't be some random modification of the theory. And it's not unreasonable that it might involve some kind of extension to our notion of space time. In fact, this is a good problem. This problem of wondering what happens to these quantum fluctuations is a good problem in the sense that it's at least a hard problem. Decent problems in physics tend to fight back. So they don't, you can't immediately think of 7,000 solutions for them. And in this case, people thought about these problems for 25, 30 years and they basically came up with two kinds of solution to this problem. And both of these solutions in one language or another involve some extension to our notion of space time. One of the classes of extensions is in one geyser or another, looks like extra dimensions of space. Extra dimensions of space are fun and they're interesting and they're great. They'll be incredibly cool if the world had them. And everything is great about them. But I think, despite the fact that I've even worked on the subject, I think they get way too much press. I mean, certainly out there in the general world. And they're not all that conceptually interesting.  You have some ordinary dimensions, you have some other dimensions. It doesn't take the, it doesn't, it's not a big challenge to the theoretical imagination to imagine that they're there. They're not a particularly deep idea. And one illustration of the fact they're not a particularly deep idea is that we actually see that what you call an extra dimension can be in the eye of a beholder. That we gave you an example last time that you even think you have gravity in space and all this extra stuff in there and these anti-decider spaces. And in fact, it's just the dynamics of an ordinary four-dimensional theory. So what space isn't even an invariant concept. It depends on how you look at it, whether you think the extra dimensions are there and really fundamental or they're just some property of the dynamics of a lower-dimensional theory. So there's nothing wrong with the idea of extra dimensions but they're not a particularly deep theoretical idea. However, there is another idea which is much deeper, which is what I will talk about in a second. But anyway, both of these ideas involve some extension to our notion of spacetime. And if I go back to my pencil analogy, it's like saying that if you look close to the pencil, instead of seeing it as hanging from the ceiling, you see a tiny hand holding it up. It's literally like saying there's a short distance mechanism that actually removes the fluctuations and stabilizes it, which you see if you look at short enough distances. So we're looking for this little hand that's holding up the pencil. Now, this much deeper extension of our notion of spacetime compared to ordinary extra dimensions is supersymmetry. And it's again like a kind of picture of extra dimensions that you may have seen before. But what's so interesting about it is that here we imagine there's our usual four dimensions and again we're gonna imagine that somehow in some orthogonal directions, there are some extra dimensions. But what's exciting about these extra dimensions is that the distance in these extra dimensions are not measured with the same kind of numbers that we measured distance in our ordinary dimensions. So in our ordinary dimensions, distances are measured in meters and they're commuting numbers. Five meters times two meters equals two meters times five meters. But the distances in these quantum dimensions are not commuting. In fact, there's only one other natural thing that they could be, which is anti-commuting. So the quote unquote distance in the x direction in the quantum direction and the y direction has a property that dx dy equals minus dy dx. And in particular, dx dx is equal to zero. That's extremely interesting. Because dx dx is zero, in a specific sense, you can't take more than one step into the quantum direction. Imagine any old function of position in our ordinary space. You get a measure for how far you can go because you can take that function and just tailor expand it around any given point. And you just keep going. You want to know what's going on further and further out? Well, you just got to keep going more and more and more tailor expansion coefficients and you learn what's going on further and further away. But let's say you try to do the same thing in the quantum dimension. Let's say our own physical world is at the places where all these d's vanish. And then I'm going to try to do a tailor expansion away from us to find out how far what's going on far away in the quantum dimension. This tailor expansion terminates pretty damn quickly. Because dx squared is equal to zero, dy squared equals zero, you only get a few terms like dx dy, dx dz, dy dz. The tailor expansion terminates very rapidly and so you just can't probe what's going on. And in fact, there's nothing going on. There's no, you can only take one step in each one of these quantum dimensions at a time. So when you have a particle like the electron wandering around and if it chooses to step off into this quantum dimension, then it can only do so in one step. And what we see four dimensionally is a partner of the electron with exactly the same properties. If the symmetry was perfectly exact, it would be with exactly the same properties, but with a different spin. That would be spin zero, for example, okay? And that's it. Let's contrast that with an ordinary dimension. You see, with an ordinary dimension, if I'm moving along here, then I could imagine moving with any tiny momentum, any amount I want in some extra direction, right? And so what that would look like since the total p squared is equal, let's say I have a massless particle. Let's say I have a massless particle which is moving along and it has some components of momentum, p zero, one, two, three and also in some fourth direction, four. Let's say that the total p squared is equal to zero. Well, that means that the energy squared minus p one squared plus p two squared plus p three squared and then minus p four squared is equal to zero. So let me put the p four squared on the other side. And so you see, as far as the four dimensional quantities are concerned, the momentum that's taking off in the extra direction looks like a mass for the ordinary four dimensional particles. So if you have some extra space, it would look like you have a whole variety of possible different kinds of particles that you would have. They would also look similar to the particles that we see with different masses, but there would be a whole continuum of them there. Here by contrast, there's only one guy. So it's very, very different. Now what makes this interesting is not just that we can have this sort of structure, but that there can be remarkably a perfect symmetry between what's going on in the ordinary directions and what's going on in the quantum directions. That's the symmetry part of supersymmetry. So that means that if we have an electron which has a spin, we have a selectron that doesn't have a spin. Quark, which has spin a half as a squark super partner, which doesn't have any spin. Photon, which is a Helicity two would have a partner which is a spin one would have a partner which has been a half called the Fotino and so on. But the important point is here that the interaction strength between two electrons and a photon is the same as that between an electron, a selectron, and a Fotino. This also makes sense because as the electron is coming along, it can split into a selectron and a Fotino. They're a picturesquely balancing momentum in the quantum directions.  But these couplings are exactly the same. So you have an electron. It can either stay ordinary and go electron photon or it can pop off and turn into a selectron and the photon has got to pop off and turn into a Fotino. But those couplings have to be exactly the same. So that's the story. If this symmetry is perfectly good, perfect symmetry of nature. But now that would also mean that for every electron, we should have seen a selectron already with exactly the same mass, every same properties except for having a different spin, which we haven't seen. So that means that, again, some long-distance accident is hiding the symmetry between the ordinary dimensions and the quantum dimensions. Just like some long-distance accidents prevented us from seeing that the strong interactions looked like, the weak interactions looked like electromagneticism, that the weak interactions have a short range and so on. This is the same idea over and over again. The basic simplicity shows up at short distances. But now you've got to get down to, let's say, less than 10 to the minus 17 centimeters or so, then you can see that these quantum dimensions are there and all of this stuff is going on. Okay, now, why does this solve the problem? I'll show it both graphically and in a formula. Graphically it solves the problem because here we are, we're getting bigger and bigger fluctuations as we go to shorter and shorter distances normally. But then when you get to around 10 to the minus 17 centimeters, you see these quantum dimensions. So now, now you want to keep fluctuating more, bigger and bigger fluctuations as you keep going down, but that starts competing against another desire, which is to have perfect symmetry between the quantum dimensions and the ordinary dimensions. If you're going to have that perfect symmetry, then it's hard to imagine having big fluctuations in the ordinary dimensions because you can't have hardly any fluctuations in the quantum dimensions. You can't even take more than one step. Never mind go back and forth and back and forth and back and forth, okay? So something has got to give because the dynamics, this is why it's important that there is a symmetry. Because the dynamics has perfect symmetry between the quantum dimensions and the ordinary dimensions, you can't have huge fluctuations in the ordinary dimensions because they're not tolerated in the quantum dimensions. The only solution is then, as the box gets small enough, there's just no quantum fluctuations at all. These gigantic ones are just not there at all. That's extraordinary, right? We need this extension to a picture of space-time. We needed something to change the calculus. This changes the calculus and it does it in this way. Now this sounds like a qualitative hand-waving argument, but it's actually very close to literally the technically correct argument for why power divergences are absent in supersymmetric theories. So it's really, this intuition is 10% away from the correct formal proof when you actually understand what's going on. Now, here's another way of understanding it. You see, one of the features of the symmetry again is that for every particle, there should be another one with spin differed by half. So for every boson, there should be a fermion. For every fermion, there should be a boson. Again, at large distances, there's some splitting between the masses of these guys. But as we go to short distances, that splitting should be irrelevant. And so, remember in our formula where the vacuum energy was a sum over all these modes of half H bar omega of the bosons minus half H bar omega for the fermions, this big problem that we found at short distances was that there was no cancellation between these guys all the way up to the Planck scale. But if we have supersymmetry in between, as soon as we get above 10 to the minus 17 centimeters or K max of order, hundreds of GEV or TEV, then they start beautifully canceling. There is no huge contribution that survives so much shorter distances. So in this way of talking about it, it's a consequence of both fermi cancellation. But the both fermi cancellation is there and makes sense. You can't do it, just to stress the point.  So the number of bosons is equal to the number of fermions. It's much deeper and more interesting than just the numbers matching. It's really all the details of the interactions and everything has got to make sense because we have to make sure that this cancellation is perfect at short distances. And you can't just randomly monkey around and have that happen. It's really the symmetry which is responsible for it. All right, so that's one possible solution to this problem. It's pretty dramatic. As you get down to energies of order, hundreds of GEV, 1,000 GEV or so, you should see all of these supersymmetric partners of the ordinary particles and all of these interactions and everything should have this property. Sometimes people say that these supersymmetric theories, when you study them in detail, the ones that might describe the world, people often complain these theories of lots of parameters are not very predictive and part of the complaint is correct. But part of it is wildly incorrect. Supersymmetric theories are incredibly predictive. They predict that the interaction strength between an electron, a fotino, and this electron is exactly the interaction strength between an electron, electron, and a photon. All of these dimensionless couplings, the ones that control the strengths of the interactions, they should all be identical between the ordinary particles and the new ones. If you found a 30% deviation between one or the other, it's not supersymmetric, no matter what else it looks like. They make an enormous number of predictions. The part that's not predictive is precisely the part that doesn't respect the symmetry. It's the part that's not supersymmetric that's not predictive. Now, the part that's not supersymmetric is very important to us for practical reasons because we're stuck down here at low energies. We don't even know if this stuff is up there. So we certainly have to barrel through this non-supersymmetric mock in order to start seeing what's actually up there where all the parameters and all these things are sharply predictive. So it's all the stuff in between that are parametrized by lots of parameters and it's true that there isn't an agreed upon theory for how all that works. But it is really a conceptual big mistake to say that supersymmetry itself is not predictive. It's wildly predictive. It makes an infinite number of predictions for an infinite number of processes so long as you go to energies that are high enough that you can ignore the supersymmetry breaking effects. It's a symmetry. So it's a very sharply predictive conception. One of the reasons people are excited about supersymmetry is that first, as I said, it's a deep theoretical idea. It's an idea that theorists have run into over and over and over again from many different angles for 40 years. It's an idea that keeps giving and does surprising things and has taught us an enormous amount about the structure of quantum field theory. It's just one of those things that once you, at least once many people encounter it, it's hard not to fall in love with it. Just purely theoretically, that there's something very remarkable about this theoretical structure. But of course, that's not the only reason people have been excited by it. In the 30 years that we've been thinking about what might solve this hierarchy problem, supersymmetric theories are the one class of theories that have something circumstantial going for them. Some circumstantial piece of experimental evidence going for them. And for that, I want to talk a little bit about this famous picture of what happens to the coupling constants, the strength of the interactions of the coupling constants in of the strong electric weak, of the strong and the electric weak forces as we go to a shorter and shorter distances. So here I'm plotting one over the strength of the coupling as we scale from 10 to the minus 16 centimeters down to very short distances. And as I think I mentioned in the first lecture, these vacuum polarization effects, due to all the particles and antiparticles popping it in and out of the vacuum, cause the strength of these interactions to change very slowly, logarithmically, with scale as you go to shorter distances. So the strong force is the strongest of all because it's interaction strength, the one over strength is around 10. E and M is the weakest of all because this is up there. But as we start looking at what it does as we go to shorter and shorter distances due to these virtual corrections, we find that it's what I was mentioning before.  I'm plotting one over the strength so this is what the plot looks like. And if all we put in are the particles that we know and love in the standard model and nothing else, this is what it looks like. And they have this kind of sort of remarkable property that they seem to be converging to some value. They seem to be converging to some value at very, very short distances. So that's a picture that qualitatively looks really nice, but that's really what it looks like quantitatively. And they don't really meet. These strengths don't really become equal. There's something kind of interesting about the fact that they're converging, but it doesn't quite become equal. I should say that in the late 1970s and early 1980s there were much bigger error bars, I haven't put any error bars on here, but there were bigger error bars on these quantities. And this picture seemed to work quite well in the sense that within the error bars all three of these lines converged to a point. So people were very excited about the general idea of unification. Not just qualitatively, which we've talked about already, all the forces are described, all the interactions are described by the same mathematical structure. That in a sense is the biggest triumph of all. And that's the big qualitative triumph that we're talking about in the same way. But now even quantitatively that the interaction strengths are becoming identical as we go to short distances. And that appeared to be the case within error bars in the late 70s and early 80s. But by now we know that with much more precise measurements it just doesn't look like that. Okay. But now we say, but this theory, which is just the standard theory, the standard model and nothing else to very short distances, that had this terrible problem, this hierarchy problem. Anyway, so maybe we don't like that theory anyway. Let's imagine that we have supersymmetry right around the TEV scale, right around the thousands of GV scale to solve the hierarchy problem in just the way we talked about. Well, that modifies how these couplings evolve as we go to shorter distances. And that's the incredible thing that happens. This did not have to happen. The supersymmetry is put in at these distances for a completely different theoretical reason to solve the hierarchy problem. And yet you put them in at the TEV scale and you find when you extrapolate the couplings to short distances that the percent level accuracy they all cross at a point. This is really remarkable. No one guaranteed that three lines converge to meet at a point to percent level accuracy. And just to give one small historical story here, because I've made this point several times, good theories do what they do regardless of what theorists want, what experimentalists want, but when anything wants. Back when this prediction was originally made, the prediction didn't change. The prediction was what it was, but the experimental values were such that this looked much worse than the prediction made from the standard model. So these poor guys who were making the prediction were like, well, we're trying to solve this theoretical problem. We doubled the spectrum. We add all these particles. Everyone's laughing at us because we say that everything is a super partner. And on top of it, we make a prediction for how the couplings unify, that's worse. Than the standard theory, okay? But guess what happened? As the experiments improved, you know, the numbers moved such that in each later iteration, they're within the error bars of the previous one, but by the time it all settled down, this was the picture, okay? The theory didn't change. The experiment changed when the data became better. And now the theory that was sitting there with good theoretical reasons, with good motivation to solve a problem, was all of a sudden looking pretty damn good. The kind of thing happens all the time in this subject. This is just one little example. Now, another amazing thing you'll notice is that this unification, another thing that didn't have to happen, which happens, is the scale where they converge is around 10 to the minus 30 centimeters. That's also remarkable, right? That's pretty close to another scale that we know and love. The Planck scale, which is 10 to the minus 33 centimeters. In fact, you probably don't remember, but you might that in the first lecture, we actually did a slightly more detailed estimate of the relative strength of the forces. Gravity compared to the other forces. Remember, gravity becomes super strong near the Planck scale around 10 to the minus 33 centimeters, but it becomes comparable in strength to the other forces around 10 to the minus 31 centimeters. That's because the other forces are a little bit weak. They're not, so they're a little bit weak. So the real scale where gravity catches up qualitatively in strength with the other forces around 10 to the minus 31 centimeters. And here we're finding that within an order of magnitude of there, just within the shouting distance of there, all the other forces are converging and becoming and having a unified value, okay? None of this had to happen. It didn't happen to happen to happen that the three lines converge to a point and it didn't have to happen that the place where they converge to a point was so close to the scale where gravity is catching up with all the other interactions anyway. So this is the best possible, rosiest possible picture that I could paint for what supersymmetry is and a solution to one of these big problems, a big solution to the hierarchy problem. It's a big theoretical idea. If we discover it, it should show up at the LHC if it solves this problem because it has to show up right around the scale we're about to be at the argument that we had at the beginning. If we discover it, it would be the first extension to our notion of space time since Einstein. We'd have discovered quantum dimensions. It's hard to imagine something. It's definitely front page of every major newspaper in the world. It'll be in all the high school textbooks within 10 years. And by the way, it would literally be a repetition of this story, right? Because once again, here, here we found a situation. This is really useful. Let's use this with a blackboard. Here we had a situation where as we went to very short distances, we had to have something really dramatic and new happen. There was an extension to our notion of space time and quantum mechanics. We doubled the world and not solve this problem. And it'll just be the same again. We doubled the world in a new way by finding these additional quantum dimensions. And instead of having for every particle, there's an antiparticle. Now for all the particles and antiparticles, there is their corresponding supersymmetric partners. But it'll be even more interesting because in this case, the symmetry that was discovered was an exact symmetry of nature, exact spacetime symmetry and its quantum continuation. The rent symmetry is exact and quantum mechanics and relativity force that particles and antiparticles are there and they have exactly the same mass. We'll have discovered not just an extension to our notion of space time, but the first example of a broken spacetime symmetry. So there'd be really an enormous amount to understand and explore if that's the correct picture of the world. But it might be bothering you, justifiably if it is, it might be bothering you that we spent so long talking about this problem, which was the subdominance problem to the much bigger problem, this 10 to the minus 120 order of magnitude problem of why the vacuum energy is as small as it is. Well, why can't we try the same strategy? Let's try the same strategy. The strategy would be that there has to be some new physics at some scale, such that actually it's natural. So we have to have some K max, such that K max to the fourth is comparable to the vacuum energy of the universe. That's what we did here for the hierarchy problem. We said K max should be such that K max is comparable to hundreds of GB scale so that there's actually no tuning. We're just gonna have to do the same thing. The K max should be comparable to the scales that would give us the vacuum energy of the universe. Now, the vacuum energy of the universe is around 10 to the minus three electron volts per millimeter, cubed. The scale associated with the vacuum energy is a millimeter. So that means that K max would have to be, one over K max would have to be about a millimeter. And that should make you nervous because let's do an experiment to confirm that there's no new physics at a millimeter. Done, okay? No new physics at a millimeter. There are, you might turn into a lawyer and say, no, maybe the new physics only affects gravity at around a millimeter so just doing this isn't enough, all right? You could be a lawyer and do that. And you get what you deserve. If you're a lawyer, you make good people go spend lots of hard work to actually measure gravity at distances of a millimeter and then down to 200 microns or 100 microns and still no sign of any kind of modification even if something purely gravitational as we go down to the millimeter scale and below. Purely experimentally, it isn't the case that's open and shut yet. It's not completely ruling out the idea. There's some modification like that. But I'm just telling you there is not a single decent theoretical idea that makes any sense at all that involves some kind of, I mean, some kind of modification of the properties of the graviton at the millimeter scales in order to be able to solve this problem. Okay, so I'm happy to talk about that in discussions later but let's just for the sake of argument say that this experiment just tells us there's nothing new happening at a millimeter. So this is disturbing. It's worked before, it worked here. It worked in these two other examples that we talked about. If we apply it to the hierarchy problem it leads us to think about supersymmetry which seems to be gloriously on the right track and yet we get stopped dead in our steps when we try to apply the same logic for the cosmological constant. So for many years, and I still, I think it's fair to say, the cosmological constant problem is probably the hardest problem in fundamental physics. The hardest problem in the sense that it's such a massively huge problem and unlike many of the other problems that we talked about where you can get started, you can work on it, you can work on maybe toy problems, you can solve closely related problems, you can sort of feel like you're making any progress on it at all. The cosmological constant problem for 30 years has resisted any such feeling, any such feeling that you're making progress towards solving it along the lines of the kinds of things we're looking for, dynamical mechanisms, little hands holding it up. So much so that many of us spent many years of our life thinking about the cosmological constant problem I can tell you for myself, I spent eight years with the majority of my time, maybe 85, 90% of my time thinking about the cosmological constant problem, and I got three completely lousy papers out of it. So it's a very, very tough problem, it's absolutely dominated my thinking, and I got really nowhere. I wish I could take some of those papers back, but anyway, the archive is what it is, so permanently on record. So that has led many people, and interestingly, it leads the people who tend to have banged their head on this problem the hardest, are the people who start thinking along these lines the most, to think that there might be a radically different solution to this problem, and in fact, there's a possibility for a radically different way of addressing these sort of fine-tuning problems altogether. In order to set the stage a little bit, I want to just back up for a second, and say something about, something that was learned about string theory in, well, it's really something that was there in some senses all along, but became appreciated more and more in the early part of the 2000s.  Well, one qualitative thing that we started learning is that while the theory might be unique in the sense that there's one structure there, there's one theory, it can have lots and lots of different solutions. Now, ordinarily, a theory like GR is also unique, but has lots and lots of different solutions.  But we thought, especially in high energy physics, we thought that there was a very special sort of solution. There are vacuum solutions, basically empty universes, and these are the solutions which are what we study. We study the vacuum, it's a state with the most amount of symmetry, the lowest energy state. By the way, why is it that we're so interested in the lowest energy state? That really has something to do with cosmology. It has to do with the fact that no matter what was going on in the universe, it expands, things cool, moving particles slowly start coming to rest, and just the expansion of the universe drives us to be closer and closer to the vacuum state as we go into the future. So no matter what's going on early on, in the end of the day, at sufficiently late times, we'll be close to the vacuum state, and that's why we're so interested in the vacuum state. So there was a feeling that this vacuum state might be unique. Even though the theory, GR isn't unique, there's planets, galaxies, everything, the vacuum state might be unique. And the realization is that there's an analog in string theory for the solutions that are black holes and galaxies and people and all the rest of this, which is just different vacua. You can have zillions of different vacua. Now, before 1995, it was already pretty clear that string theory had zillions of vacua. But before 1995, zillions meant millions. That's what people thought. Lots and lots, roughly millions. And in fact, there was a hope before the big developments in the mid-90s that allowed people to understand the non-perturbative aspects of this theory that there would be some kind of subtle inconsistency in these millions of vacua, and only one of them would survive or only the one that looked like our world would survive. But the number that had to be killed, it was pretty huge, it was millions, but it was millions. It's like the population of a small country. Post-95, it's clear that zillions means infinity. First of all, actually infinity. Before we even talk about the number 10 to the 500, there's literally an infinite number of supersymmetric vacua of string theory. So before even having any discussion about a finite number, there's first a continuous infinity of them. They're all supersymmetric. So let's say, oh, I want to forget about the ones that are supersymmetric. Why? I don't know. There's a continuous infinity of them, but let's put them aside. Then there are ones that are not supersymmetric, and you can estimate how many of them there are. And zillions is zillions. Zillions is 10 to the zillions, okay? 10 to the 500, 10 to the 1,000. Call it any number you want. It's bigger than any number you can imagine. Okay, well, not that you can imagine. We just imagined it, but any relevant number. I remind you, the number of atoms in the universe is like 10 to the 80, okay? So we have, these numbers are enormous compared to even the number of atoms in the universe, okay? So this enormous number of possible vacua is called the string landscape. There's one theory with zillions of solution. Each one of these vacuums, or vacua, looks, has some low energy dynamics.  A lot of them can look qualitatively just like the world we see around us, the same kind of interactions. They all have gravity. It's not true that they all have their own private laws of physics. Absolutely not. There's one set of laws. They're just different solutions. It's not like, you know, apples fall up in some of these other vacuums. All the same. It's the same basic structure of gauge fields, fermions, matter, gravity. But the detailed menu changes as you go from one vacuum to another to another. Okay, now how can this impact our problem? And here there's a nice little point. So instead of going through the complexities of string theory, let me give you a little toy model for a landscape. Very, very simple. Imagine that we have, I don't know, a thousand different scalar fields in nature. Thousand of things like the Higgs. Thousand might seem like a big number, but if you just count the total number of degrees of freedom in the standard model, it's still, it's pretty big already. Like the, you know, there's like 45 species of fermion, all set. If you take colors and everything else into account. So the number of degrees of freedom in the standard model that we've seen already is of order hundreds, okay? So I was saying a thousand is a big, I could also call it a hundred. I'm choosing it to be a thousand just to make the numbers work out a little better, okay? So imagine each one of them has its own little potential like this, you know? Each one can be in, it has a potential energy for each one. Each one can be in one of the two states, in one of two vacuums. And let's just imagine, just to make it simple to think about it, they're all decoupled from each other, okay? Of course it's not realistic, but let's just imagine that some approximations are all decoupled from each other. All right. So I put in a relatively modestly large number of order a thousand. And the first thing that happens is that it gets two to the thousand vacua. So the number of vacua is exponentially large in the number of moving parts, right? Each field can either be in the top or the bottom of its potential. So there's two to the power of a thousand different vacua. Very simple. That's why it's so easy to get numbers like 10 to the 500 out of relatively modest input numbers, okay? Now let's imagine what do the vacuum energies look like? Well, I don't know. There is two to the thousand different vacua. I wouldn't wanna go work out in detail the energy in each one since there are so many of them, but the total amount of, the total energy, the distance between the top and the bottom is something of order one. Probably something of order a thousand, right? Because maybe a thousand can be up or a thousand can be down, but there's a relatively small distance between the top and the bottom, but I have to pile in between them two to the thousand different states. So because there's such an enormous number of states, this is clearly a good place to think about it statistically, right? That's what it looks like. There's a bajillion different states here. And now, now you see it doesn't take a miracle in order to find some state that has tiny vacuum energy. Just statistically, there will be in this big set some guy that has tiny vacuum energy. How tiny would that splitting be? Oh, the splitting will be of order a half to the thousand, roughly speaking, two to the negative a thousand, okay? So just statistically, you might imagine that there's some vacuum with energy of order two to the negative a thousand, incredibly small. That's, I picked it, so that's like 10 to the minus 120 is no problem, right? So without anything else, this just shows that if you just want to know, is it possible? You see, beforehand, if we imagine there's a single vacuum, a single theory, you just ask, is it reasonable? Is it possible for the vacuum energy to be close to zero? It just seems like in that single theory, with a single parameter, some ridiculous conspiracy have to occur in the parameters to fine tune things to a part in 120. The second we have a thousand fields, two to the thousand vacua, just for the question of, is it possible to find a vacuum with a small vacuum energy? That is now possible. Just statistically, it's possible. So that doesn't take a ridiculous conspiracy. It's at least possible. You might still think, though, I've just translated the problem from one place to the other, right? Because now, what are the chances that I'm going to find myself living in the one vacuum that has an energy that's that tiny? Why don't I live in any one of the other vacua? Then from this point of view, that's what the question becomes translated to. You know, how come it is that of all the ones that could be there, I'm conspiratorially at the one that has this tiniest possible energy or one of these very tiny energies? Okay, but that has a potentially interesting answer which occurs, which really becomes possible when we take the dynamics of gravity into account here. Without gravity, I should have said this, without gravity, actually, it doesn't even really make sense to talk about all these vacua, right? Because if I put the field in the upper vacuum, it'll eventually tunnel to the lower one. So if I wait long enough, everyone will end up, all the fields end up in the bottom of the potential and it'll just be one vacuum. So all these other guys are metastable, but everything would eventually go down to the one, one bottom vacuum. But something very interesting happens with gravity now. Imagine that here we have this picture again and I have a vacuum for the whole thing, which looks like this. This guy's up here, that guy's down there, up there, down there, up there, and such that the whole thing has a net positive energy density. There's a net positive energy density. This positive vacuum energy causes the universe to accelerate as we've discussed a number of times. It doubles in size at a uniform rate. So it's inflating, doubling in size at a uniform rate. Great, that's what it looks like. Nothing will stop, let's say that guy, it's a pretty shallow potential, that guy might tunnel from there down. So from the green region to the blue region, okay? So what would that look like? Now how does this sort of tunneling happen? This isn't like just a particle sitting there that goes to the other side of the well, right? There's some field here that's filling all of space and time. In order for it to a tunnel, it can't be that everywhere in space, it all jumps from one place to the other. That would be exponentially suppressed by the volume of the entire universe, okay? What actually happens is in some region of space it jumps from one place to another, okay? And in that region of space, it can locally gain some energy because it's going to a lower energy region. But it has to fight the fact that there's some gradient energy in the fact that you're now, that the field has gotta change across a sort of a wall from the inside of the new region to the outside. So in order for this bubble that you make by this quantum fluctuation to grow, it has to be bigger than some critical size. But if it's bigger than some critical size, then it'll just grow. There's a pressure on the inside that just pushes it out. And if we had nothing else, if there's no gravity and nothing, we just had a tunneling process like this happen. Then what would happen is somewhere in space you'd nucleate a bubble and this bubble would grow and expand and eat up the entire universe eventually at the speed of light, okay? This is exactly what happens when you boil water, okay? When you boil water, what happens is that you have little nucleations occur inside the water where you get a bubble of steam which then opens up and eventually you make one here, you make one there, you make one there and they all bang into each other and you get steam, okay? And you start evaporating everything. The same thing happens with quantum fluctuations. There are thermal fluctuations, here it's quantum fluctuations, okay? But now something remarkable happens which has to do with this inflation of the underlying of the space around it, okay? The inflation means that without gravity what would happen is that would make this bubble, it would expand and it would eat everything up, okay? But with gravity what's happening is this bubble is expanding, its outer edges reach the speed of light, but everything around it is continuing to inflate away, double in size at a uniform rate. If you wait a little while later if this tunneling time is even a little bit long, if the tunneling time is long compared to this doubling time, then somewhere else, somewhere over there, the bubble might form and would expand out so its bubble walls would reach the speed of light. But these walls never hit each other. They never hit each other and completely convert the space from the old vacuum to the new vacuum. The reason is that between them all the time there's continuous to be inflation, continuous to be this doubling of the size of everything all the time. Despite the fact that we have all these different vacua and you would think if you give it long enough everything will settle down to the bottom because of gravity that's not what it looks like. You just keep inflating forever and there's always some region which is going to be inflating and not going to be covered up by these new vacua, okay? I won't be much longer here, so. So this gives the so-called picture of eternal inflation, where if we imagine going out, this was everything happening before, here is another bubble, there's a bubble, there's bubbles inside bubbles. In all the regions where the energy is positive you keep inflating forever. Sometimes you might tunnel into a region where the energy is negative and if you study that problem in detail you find that you get a big crunch in those regions of space time. So there are some places here which are dead, deathly regions which end up crunching. But there's also an enormous and an infinite volume of places which just keep on expanding and inflating forever, okay? And so even if you start in one vacuum if you just let this go, right? We're just being completely conservative. We put in these, we put in these vacua, now we just ask what does the universe look like and we get this remarkable picture. It's a kind of a fractal where everywhere, somewhere one of these vacua is realized, okay? And it just never ends. It just keeps on going and going and going. This now makes, now comes the final observation which is an observation that goes back to Steve Weinberg in the late 80s. So often this kind of picture is drawn sort of picturesquely like this.  This, it's called this multiverse. It's kind of a dumb name because there's one universe but it's a multiverse in the sense that all these things are happening place, are happening outside, all the stuff is going on outside the cosmological horizon of any one of these observers. But we might imagine, we might imagine just picturesquely taking a sort of bird's eye view of all of this stuff from outside the spacetime and it would look like this big branching tree of one vacuum and another vacuum and another vacuum, all these things happening everywhere. In this region, they're all different vacuum of the theory. So in this region there might be 17 kinds of electron. There it might be 10-dimensional. Here there might be no strong force, okay? They're all the same basic physics, same standard kind of physics that we talk about but realized with different menu of particular particles and interactions. And here is the interesting observation. Let's now ask what even roughly qualitatively would it look like in this multiverse, okay? In most of the places, the vacuum energy is huge. Most of them it's close to the Planck scale or maybe some other gigantic scale. But in those precise regions, either it's crunched to tiny scales or it's just blowing itself apart an enormous expansion rate, okay? Doubling rate. In all those regions where it's either crunched or it's blowing apart enormous expansion rate, we can't have any structure of any sort. In our own universe, I reminded you, I told you in the first lecture, the reason we have structure like galaxies is ultimately because we have these tiny differences in energy density between that part of the sky and that part of the sky and then the universe cools and then slowly by slowly, the regions that are a little bit over dense, they attract each other by gravity. It's very gentle and eventually we make structures like galaxies and you and me and everything else and we have a nice universe with stuff in it. Now let's just imagine what would happen in our universe if the vacuum energy was a little bit bigger than what it is now. Just for the sake of argument, imagine that it's 10,000 times bigger. Then what's going on is that these, everything is the same for a long time. We're not talking about the Planck scale, just like 10,000 times bigger, right? Everything is fine, everything is going along. The galaxies start clumping, everything is fine, things start clumping, but before they get to really finally close and make this nice held together structure that's held together by gravity, before they do that, while they're sort of a tenth of the way there, the universe starts accelerating underneath it and it just starts blowing everything apart again. So it's trying to make structure, but even this minuscule cosmological constant will blow it apart and it'll be left with an empty universe. Never mind if it's much bigger than that, then forget it, you don't even get started. So Weinberg did this analysis and found that you couldn't make the cosmological constant more than one or two orders of magnitude bigger than what we actually ultimately observed it to be without having an empty universe. So we're not talking about some very detailed property of the world, we're just talking about something like, is it empty, is it not empty? And so that's sort of qualitatively what it looks like out in this multiverse. In most places, it's empty and lethal. There's nothing there. Only in a few places, just accidentally, because we have so many vacua, only in a few places has it become possible, just statistically, accidentally, is it possible to find places with small vacuum energy and in those places, we'll have structure. So now ask yourself the question, now let's go back and ask, in a picture like this, would it seem strange to an observer to see that the cosmological constant is tiny? Would it seem strange to an observer to see that the vacuum energy they find is so minuscule? And the answer is finally, no, that's what they would expect to see. In all the regions, that's true, the vast majority of the vacua have much, much bigger vacuum energy, but those places are empty. The only places where anyone could possibly be to observe and measure the value of the vacuum energy is going to be one of these ones with minuscule cosmological constants that are there possible and statistically possible for this reason. Now, this kind of reasoning is often called anthropic reasoning. And I hope at least in the context of this example, you see there's nothing anthropic about it. It doesn't, it's not like people matter, giraffes matter, right? Where it's a very basic thing, empty, not empty. And so we're not making very detailed use of what it takes to make life or anything like that. Let me give you an analogous problem that almost exactly the same kind of problem that you don't obsess about in our own universe.  We live in this universe, we live on the earth. The earth is a tiny rock. The volume of the earth takes up compared to the volume out there in our hobble patch is 10 to the minus 60. Isn't this a big crisis for theoretical physics? Why have all the places we could live, why do we live in this place that occupies a volume 10 to the minus 60 compared to the volume of the entire universe? Well, maybe you think this is a problem. If you do work on it, I mean, I don't think it's a problem. No one I know thinks it's a problem. We live on the earth, on this rock, because where else are we gonna live? We're not gonna live in the middle of nowhere, okay? We're gonna have to live where there's any structure at all. It's exactly the same argument here. In this big multiverse, the places that aren't empty will have apparently minuscule values of the vacuum energy that seem completely unnatural, that seem like they might be very, very finely tuned and adjusted if all you thought of was a single vacuum but make much more sense in this bigger global picture. And a lot of things have to go and work nicely for this picture to make sense. Not only do we have to have all these vacua, but we saw that it's natural to have exponentially large number of vacua, but there also has to be a mechanism to at least imagine that they might all be realized somewhere in the universe and there is that mechanism. It's just inflation and it's natural continuation to this idea of eternal inflation. Okay, so this is obviously a radically different picture that might explain the fine-tuning problems. It also involves an extension to our notion of spacetime, not at very short distances, but at enormous distances. And it's completely different in nature than the first one, which was some kind of mechanism, the hand that that's holding up the pencil. Let me just make one comment about this picture. If this picture is correct, one thing that will for sure happen to our universe is that we have a very sad and catastrophic end, okay? If this is what it looks like, then the value of our vacuum energy is just accidentally small because of some, because just because it needs to be because that's where a structure can take place. But that means that we're surrounded by negative energy vacuums all around us in field space. And so what's gonna happen is eventually we're gonna tunnel into one of these guys. And if we tunnel into a region of negative energy, as I mentioned, it immediately gives rise to a crunching collapse of our entire universe. Don't worry, we'd be dead before any of us knew it, but that is our end, okay?  So this is a second possibility, completely different than the first. And the proposal is that we're a tiny part of a vast multiverse. If this picture is true, then it's the modern Copernican Revolution, modern version of the Copernican Revolution.  On top of all of that, our entire universe is a little speck of nothingness in this vast multiverse. 10 of the 500 other things out there. Mostly the universe is a horrible, unfriendly, lethal place and in a few teeny, teeny, tiny places, accidents happen that allow things to exist. If it's true, you have even more than the usual amount of powerful responsibility, feeling of responsibility and guilt for making the most out of the life that you have, but anyway. So if it's true, it's a modern Copernican Revolution. Now, many people have lots of philosophical discussions about whether this is good or bad or it makes sense or it doesn't make sense, but and a lot of the discussion goes around whether the anthropic principle is good or bad and we should discuss it or not.  We drew this picture for what this eternal internally slating universe looks like taking semi-classical physics at face value, right? The semi-classical physics that gives us inflation we took at face value, this whole multiverse in picture, it's a consequence of that, we're following our nose, we are solving the equations, we're led to a conclusion, it all seems fine, but in the second lecture I told you about another situation where people followed their nose, we're led to a conclusion and that conclusion was false and that's what happened in the context of the black hole information problem where you follow semi-classical physics, you keep going and you conclude that that information is lost and that ended up being wrong in a very subtle way. So here too, it could be that this picture of this infinite multiverse that stretches out as far as the eye can see is not precisely correct. It's a semi-classical picture, but it's not precisely correct. Remember I gave you at least a diagnostic for when to start worrying. The whole notion of locality on which this picture is also based, this whole notion of locality, remember is violated by effects that are of order e to the minus s where s is some entropy associated with the system and I told you that you should worry if in the problem you find competing e to the plus s somewhere. That's at least, there's a chance to start worrying. In the context of this eternally inflating picture, there is exactly such a cause for concern of an e to the plus s. There's this infinite volume that's accumulating as the universe continues to eternally inflate and expand. So it's really conceivable that the violations of locality due to quantum mechanics and gravity are playing a role at these exponentially large distances outside even our observed horizon, okay? And may dramatically change this picture in some way or another. It could be that in some way very analogous to black hole formation and evaporation that we shouldn't actually talk about what's going on infinitely far away in the multiverse, but that in some sense what's going on out there is encoded in subtle entanglements and interesting information that can in principle be understood and measured by a single observer. Just as a wild fantasy, that's one kind of possibility. There are others that you might entertain, but I just want to make it clear that it's a physics problem now. It's not a philosophy problem. If someone comes up with a conceptual way of making sense of eternal inflation, a mathematically precise way of making sense of it, one way or another or prove that the picture doesn't make sense, that would go a long way to settling the discussion. It's not a philosophy problem, it's a physics problem. A very, very hard physics problem, but a physics problem. And that's a big question is how do we know these other things are there because we can't see them? They're all outside our cosmological horizon, we can't see them. The other regions, especially because our universe is accelerating, all those other regions are out here away from what we can actually see from our cosmological horizon. So if the second picture is true, once again, in order to make progress, it's really what I said at the end of last time, we have to learn to deal with thinking about quantum mechanics and cosmology at the same time. And once again, the essential issue is time and how to make sense of what observables are and all these sort of deep questions of quantum mechanics and gravity playing together with when you have finitely many degrees of freedom, all of these issues are collide in the same place in trying to make sense of this question, okay? So these are the things that we're led into if we try to understand why there's a macroscopic universe. I should say that it could be that the hierarchy problem has a solution like this as well. It could be that if we change the value of the, if we change the value of the electric scales that's set by the Higgs, then we'd also get a universe that's relatively different than the one that we see. It's not as dramatic as empty or full, but it can still do pretty dramatic things. If we change the value of the overall size of the Higgs by a factor of three, we can completely change atomic structure. We can totally change the fact that we have all these complicated and interesting nuclei. That turns out to rely on various accidents that would be gone if the overall value of the Higgs was a little different. So it's conceivable that the hierarchy problem has a solution along these lines. There's a much less good argument for it than there is for the cosmological concept, but it's conceivable, okay? So this is the kind of dichotomy that we're left with. It's a picture of order versus chaos. Either new symmetries as we go to shorter distances or this very radically different picture of a multiverse at very, very large distances. But in both cases, the physics involved is dramatic and important, and it will be a tremendous progress if we figure out what's going on one way or the other. All right, thanks a lot. Okay, thank you very much again, Nima. We've had a run a little bit out of time, but any questions? The question about supersymmetry, not at the last part of your lecture, but somewhere in the middle. You told about cancellation of normal bisonic stuff and the one which comes from extra dimensions. And I understand it so that if there is a perfect symmetry, this cancellation is perfect and you have zero energy by supersymmetry, like zero solution for supersymmetric quantum mechanics. Yeah. I should say actually, if you don't mind me interrupting for a second, this is one reason why supersymmetry is so ubiquitous in people who concretely study quantum gravity. One of the reasons you run into it over and over again is if you wanna do any kind of reasonable theory that includes gravity. Before anything else, it would be nice to have a solution that looks like a flat universe, okay? If you're even gonna get started, it would be nice to have a solution that looks something like a flat universe, but as I told you, it's completely mysterious that there is any flat universe at all. So supersymmetry is, and I should have stressed this, this is one of the really deep things about it, is it's the only mechanism that we know that solves the cosmological constant problem perfectly. It's the only mechanism we know that guarantees vanishing vacuum energy, okay? Now, it guarantees vanishing vacuum energy if it's exact and that's one reason even theoretically people run into it over and over again because if you're even gonna get started talking about something that has a chance to be called gravity, that has a big, big space and planets in it that might orbit each other or gravitons that might interact with each other, you have to, from the get-go, have a vanishing vacuum energy. So supersymmetry is the only mechanism we know that does that and that's why we run into it over and over again. Like in simple quantum mechanics, zero energy solutions. The difficulty is that in our real world we don't see superpartners up to 100 GeV and so that gives us a size for the vacuum energy that we can estimate. That's not 120 orders of magnitude too big, it's way better than that so supersymmetry makes it way, way better than that but it's still 60 orders of magnitude too big. So it's vastly better than it was but it's still way, way too big. So supersymmetry had a chance to solve the cosmological constant problem. You might imagine that we are creatures, other creatures made out of particles whose Compton wavelength was like 10 meters, okay? But they're living in our universe, somewhere else in our universe. They also would be very confused that the universe isn't exploding in size, they would think there was a cosmological constant problem but they would not have been to the millimeter scale. So they would build a large millimeter collider for them to probe the short millimeter scale and all the theorists in that sector would have a brilliant theory, in fact a unique theory for what solved the cosmological constant problem which would be supersymmetry at the millimeter scale and those theorists would be sorely disappointed. But still, even though supersymmetry doesn't appear to solve the cosmological constant problem in our world this is one of the things that makes it such a deep theoretical idea. It's the only thing that we know of that at least at very short distances completely removes these zero point fluctuations. Sorry, I didn't mean to interrupt. I just wanted to stress that because I didn't. I just come to the second part of the question. Yes. So then at the end of the lecture you are talking about the supersymmetry in some weak sense. You say that it is essentially the reason why there could be very close, energy very close to zero, but not exact. So we are talking about violation of the supersymmetry. That part had nothing to do with supersymmetry in fact. Okay. That part is just purely statistical. Okay, but in any case, what I didn't understand well is it means that the solution where the cancellation of 120 digits is needed, the supersymmetry, the trace of the supersymmetry is there, but it is not exact. Did I get it correctly or not? No, no, no, no.  It's merely a consequence of the fact that you have 10 to the thousand vacuums. See, if you have 10 to the thousand vacuums, right, right, that's right. If you have 10 to the thousand vacuums, they all have energy between minus m-plunk and m-plunk, and you're gonna stack 10 to the thousand things between minus m-plunk to the fourth and m-plunk to the fourth, and they will run into something that's tiny along the way. Right. The fact that you've got so many vacuums. Yes. It loosely has to do with the string theory. That's right. It loosely has to do with the fact that string theory has extra dimensions. So here in my little toy example, I didn't talk about string theory extra dimensions. I just said that there's a thousand scalar fields. In the case of string theory, the scalar fields are there. If we have an effective four dimensional theory, there are a lot of extra scalar fields that parametrize the size of the extra dimensions. They parametrize little handles and other things that might be there. And it's true because we have to get from 10 dimensions to four dimensions with a relatively complicated internal space, then you tend to have many of these scalar fields. So having 40 or 50 scalar fields is not complicated, and each one of them will have a potential that doesn't have something as dumb as just that, but has more bumps and wiggles in it, and when you count all of them, you get numbers of order 10 to the 500, 10 to the 1,000 easily. As I said, that's 10 to the 1,000 on top of a continuous infinity of things that are exactly supersymmetric. So there is no, but anyway, but the ones that are not supersymmetric, you easily get these very large numbers. The important point is that a relatively little amount of complexity, numbers of order 50 and 100 are getting exponentiated to give the huge number of vacuum. And that's this basic effect that I was illustrating in this simpler toy example with the scalar fields. Now, it is nice that there are other features of our world. String theory in 10 dimensions is incredibly simple. It's incredibly simple, very supersymmetric theory. Our world is four dimensional for one thing, and it's also complicated. We have three gauge groups, we have three generations, we have these weird pattern of fermion masses. It's another circumstantial hint, I think certainly is a hint that I take. Again, some other people disagree, but the structure of even the parameters that describe our world do not, to me, remotely look like the unique solution for some underlying theory. It would seem like madness to me if you thought it looked like a unique solution to some underlying theory. It looks a lot like some garden variety solution of some perhaps underlying unique structure, but just one of many solutions. And one of the ways that all the complexity comes out in effective four dimensional theories, even though the underlying theory is so simple, is that you compactify on internal spaces that have a little bit of structure. And that little bit of structure is what allows things to look sort of interestingly complicated at large distances. So it's nice that the same little bit of structure that's needed to do that also gives you these 30, 40, 50 fields that you need that exponentiate it, gives you enough vacua to be able to find ones where structure can form with these nice, relatively complicated theories that you get at large distances. I think this reasoning shouldn't be called enthropic so much as it should be called environmental. The idea is that some of the parameters in the standard model, like the vacuum energy, parameters that have a huge impact on long distance physics, as the cosmological constant does. These parameters that very sensitively control long distance physics might be environmentally selected in the sense that the places in the multiverse where they have the wrong values, they're natural values, but too large, positive or negative will just be empty and so it will be zoomed in only on those regions where that doesn't happen. I just wanna make a little historic comment because you, Nima, you seem to appreciate a lot the unification of Susie couplings, right? One of the main reasons we like supersymmetry. I just wanna add that this is not emphasized enough that not only that this was predicted 10 years before left. It's a comment for people in the field, high energy. It's just that at the time when it didn't work, some of us decided to stick our neck out so in particular, Marciano and myself, when it didn't work, ask ourselves how in the world would I make, this was science-created at W, compatible with the experiment and it had to do with the row parameters so we decided we had to do the row parameter which would be the way to do the row parameter if there was a heavy top work of a mass around 200 GV.  I think it's worth emphasizing alone, behold, the top work mass is around 200 GV and science-created at W turned out to be okay. So there was a prophecy in unification itself, okay, which is somehow, not a prophecy. It's maybe a general comment which is that things in this field work well and when ideas work, it tends to have the following flavor over and over again. There's some big zero-thorter idea, fantastic. You need a big idea. Nothing in our subject is solved by some dinky technicality. There's some big idea behind it on the one hand but they don't just stay at the little big airy-fairy ideas. Somewhere along the way, there's some very sharp calculation that someone has to do to some pretty decent accuracy, okay, because you have this big idea and there's a structure that's powerful enough to allow you to do a calculation. You do a sharp calculation, wham, right? Then you have a sharp calculation associated with the big idea. It either agrees with the experiment immediately or experiment changes and it agrees with the experiment after the experiment gets it right. More questions? Well, let me ask two questions myself so that I have the much chance. In the landscape pictures, so you have many, of course, and the picture you have is essentially four-dimensional, but in principle we know that there will be a vacuum of any dimensions from 10 down all the way. And is there any different, do you see any different treatment of the fact that you can go from one vacuum, one dimension to another vacuum, another dimension? With your whole picture of inflating. No, no, I think that the picture is essentially multi-dimensional. What is true is that it's not so easy to find higher dimensional solutions with all the modular stabilized. So if you want to imagine finding the sitter solutions with, or approximately the sitter solutions with stabilized moduli, already you need to get down to low enough supersymmetry to make it possible. Well, of course there might be wildly non-super symmetric theories where that isn't true. So we might be looking under the lampposts. There might be whole classes of very non-super symmetric theories. But if it's gonna be somewhere in the neighborhood of supersymmetric theories, there is something a little special about four dimensions to get down to low enough Susie's where it's possible to start controlling Susie breaking to parametrically large low scales compared to the plonk scale and find all these factors. But there could be lower dimensional solutions, also higher dimensional solutions, as you said.  But there will be some regions in the landscape that will tunnel out into 10 dimensional flat space. So that'll just open up a nice 10 dimensional flat space eventually. And I should also comment just, it's a cool fact that a number of us noticed some years ago. If you just take the standard model, our theory, the standard model, you see I just told you string theory is a unique theory and it has this big landscape of vacua. And you can love it or hate it, but it seems plausible that it's true. But the standard model has the same feature. In other words, string theory is a unique theory in higher dimensions, but it has a bazillion lower dimensional solutions. Well, in a sense, the same thing is true of the standard model itself. The standard model is a unique theory in four dimensions as a unique four dimensional vacuum. But it turns out that the standard model itself has lower dimensional vacua. There are solutions of the standard model where some of our dimensions are compactified. For example, one of our dimensions is compactified. It turns out it's stabilized. You can calculate that it's stabilized. The size of the compactification radius, so the z direction will be curled up to a circle. The compactification scale turns out to be around 60 microns. But number 60 microns turns out to be determined by detailed properties of the neutrino masses and so on. So the 60 microns and the rest of the cosmology, sorry, the rest of the solution is anti-dissuiter space, three dimensional anti-dissuiter space. So the standard model has other vacua that look like ADS-3 across a circle that's 60 microns big. And actually, when you look at it in detail, there's a near continuous infinity of vacua corresponding to turning on some parameter, which is roughly the Wilson line or the Aronoff-Bohm phase going around for electromagnetism, going around this circle. So it's amusing. People often say ADS-CFT doesn't apply to the real world because we don't have ADS space. It's true, it's not in our solution. But the standard model has another solution, which looks like ADS-3. And so there might be some one plus one dimensional conformal field theory out there which contains everything. This world and our world are identical at distances short compared to 60 microns. You and I and giraffes can't live there, but amoeba can live there happily. And so I'm just saying this idea that you can have a unique theory with lots of vacua is not just a property of string theory, it's even a property of the standard model itself. In the standard model, we happen to live in the top vacuum, which is great. In string theory, though, the top vacuum, higher-subventional vacuum is incredibly simple, incredibly supersymmetric. And it's these lower-dimensional vacua that have a chance to have enough complexity to maybe do interesting things. And at the same time, that complexity makes it possible to have so many vacua so that it's possible to find this whole picture and have it run. It doesn't mean that it's right. Definitely doesn't mean that it's right, but it's not an insane picture. And I should say, at the moment, it's the best idea we have, I didn't stress it. It's the only idea we have for solving the cosmological constant problem. So this is something I encourage you to do, maybe not right now, as you're diploma students, you still need to learn things and calculate things and become grown-ups. But early on, after your grown-ups, you should spend a lot of time thinking about the cosmological constant problem. Convince yourself, either find a better solution or come to peace with this one. But this continues to be the most important problem, I think, one way or another, for fundamental physics. And it's a fantastic problem to think about. Let me continue my question. Oh, sorry, you had another question. Yeah, sorry. So let me just finish it, because at the end you can put it in your way. So there are two parts of it. One is that you find new ways of experimentally ruling out this picture. And the second one is to have, what is your opinion about the recent world of people trying to relate this multiverse with the Everett multiverse and the quantum mechanics? Oh, well, we can talk about this in more detail later. I think, so Fernando is referring to the fact that there are papers that claim that this multiverse and the kind of old multiverse of the many worlds interpretation of quantum mechanics are the same thing. And I don't find these papers very deep. I mean, anyone who's thought about quantum mechanics and understands decoherence, there's very little in these papers that's novel. But there are some nice points, but they're not earth-shattering by any means. I mean, I was not surprised by one page of any of these arguments. So maybe, well, I won't say that. No, there are some nice things, but it's not some big earth-shattering thing. So we can talk about it more in detail. And the first part of the quiz? Yeah. What was the first part of the quiz? Experimentally to rule out. Oh, experimental ways. Well, experimental ways to rule, well, okay. Let me say it in increasing or decreasing levels of increasing levels of plausibility, okay. Could be, could be that gravity really is modified at 100 microns, and it shuts off at 100 microns, okay. There's not, I mean, this is this idea that's sometimes called the fat-graviton idea, right? And that maybe actually the cosmot... It really is a natural solution, and gravity is modified at the millimeter scale in some crazy way. And our friends in Washington who are doing these experiments just have missed it, and they go another factor of five down-shortened scales, and incredibly gravity will shut off at 100 microns, okay. Then I think everyone would believe there's a natural solution. And even if we don't understand the mechanism, you'd have to understand it. So that's an experimental way that would definitely shake our confidence, or would shake this picture to the core. I mean, it would be the biggest thing that's happened in physics in, you know, 500 years if you see the gravity shuts off at a millimeter. And I will bet any unborn child I have that it won't happen, okay. So there's absolutely no way it's gonna happen, because there's no decent theoretical structure for it at all. Okay, it's just some very good friends of mine have worked on these ideas, and they've worked on it. As everyone does when they work on the cosmological concept problem, you lower your standards so much compared to when you work on any other problem. And even with these much, much lower standards than usual, it's really low standards for this class of ideas. There isn't a single consistent complete theory that even toy theory that makes it go like this. Now, okay, so that's one thing. Now, so that's an experimental thing. However, that would convince us that it's wrong. Absent that, oh, something else that would of course demolish this picture right away, little more plausible, I still think very implausible, but a little more plausible, if we discover that our accelerating universe is actually not a cosmological constant, okay? So let's say precise observations tell us that actually the expansion rate isn't doubling inside the constant rate, but it's changing slowly. Now, part of the problem is it's very hard to imagine a future experiment actually convincing us of this, because the rate of the doubling is known to be a constant already to very, very good accuracy, okay? And it's hard to imagine a future experiment finding a value that was not a constant in any way that would be convincing. The error bars are so close that even if a future experiment said, ah, it's not a constant, we already know it's so close to being a constant that I would more likely believe that they made a mistake or they underestimated their errors and they actually convincingly proved that it wasn't the case, okay? But it's so conceivable, maybe even more futuristic experiments somehow, I'd have to think about it more. Again, it's not very plausible, but if we see that it's not a cosmological constant, game over, right? Completely game over. Notice that, however, people talk about theories like this. They were called quintessence a long time ago, many years ago. And so the idea of the cosmological constant isn't the constant, it's gradually changing, maybe it's getting smaller and smaller. Okay, any such theory, it's not like it's better. Any theory has the cosmological constant problem. Any theory has the problem of why there isn't the big vacuum energy sitting there to begin with. The problem isn't what's making us accelerate now. You see, this is what bugs me about people who say dark energy is such a mystery. The mystery isn't that the universe can be accelerating. Making it accelerate is no problem, wants to accelerate, completes triviality. Give it a positive vacuum energy, wants to accelerate. The problem is it wants to accelerate way too much. And any theory is gonna have to understand why it's not accelerating incredibly much. Now, on top of that, you have to explain why somehow you got rid of this big vacuum energy, God knows how, but then on top of that, you have to have something that very slowly varies. So that's why these theories are also very, very implausible. Something else that might have happened, though, along these lines, is if you might imagine there's some kind of mechanism that's screened the cosmological constant dynamically somehow. The universe expands, something goes on, and you're sort of constantly dynamically screening and adjusting the cosmological constant. That's a nice qualitative idea, but that nice qualitative idea runs into an experimental problem, which is that today the cosmological constant dominates the universe. But if there's any kind of solution that's sort of tracking, whatever it is, I don't have a theory, but it's somehow tracking the vacuum energy, there's comparable to radiation and matter now, but we know it was not comparable to radiation and matter in earlier epochs. During nucleosynthesis, for example, it was not there. So that's also very hard to imagine any kind of tracking solution like this, sort of screening or dynamically canceling the cosmological constant as you go. So that, sorry, that wasn't an experiment, that was just a mechanism that probably doesn't work. Another mechanism I should say that probably doesn't work is the general idea that you somehow modify gravity at very large distances in such a way that even though there is a cosmological constant, it doesn't gravitate. And that idea doesn't work because the difficulty is not at very large distances. If we had a vacuum energy that was the TEV scale, then we would have a curvature of the universe around a millimeter. It is very hard to imagine some modification that takes place up at distances of order Hubble that's gonna do anything about the very, very huge curvature scale that you would induce if you did have a big vacuum energy. The final thing, maybe the most plausible thing that might push against this is if at the weak scale we did discover an extremely natural theory like supersymmetry, okay? If we did discover a very natural theory like that, it would be a very natural solution to one problem. It wouldn't prove that the other problem has a natural solution, but you would start thinking about that again. Something I didn't have any time to emphasize in these lectures because from the point of view of these lectures, it's a second order point. It's a detailed point, although it's a detailed point that all of us in the business have obsessed a lot about over the last decade, is that in detail, in a little bit more detail, supersymmetric theories are not looking as good as they looked in the early 90s. Supersymmetric theories aren't looking like the most natural versions of them actually work. The most natural versions of them could have given us super partners that we saw in the 90s or we saw 10 years ago. And so there's already some quiet concern that there's some pressure on the idea of naturalness, on the idea that these fine tunings or problems are solved at a much more, much, much smaller level than the cosmological constant, but still some quiet unease associated with what we're learning about particle physics. So that's why if what we end up seeing at the LHC is no hint for a natural solution of electro-eximetry breaking, then that also doesn't prove that the cosmological constant isn't natural, but it makes you take the unnatural option, the sort of multiversitarian option more seriously, not because it proves it scientifically, but because it just tells you that that's probably where you should devote your energies to try to make some scientific sense of it. But if it goes the other way, then I think it would also take us the other way. One little comment that before you finish is that I think you forgot to mention that the work of Weinberg was more than 10 years before the observation came. Yes, yes, yes, sorry, I should have said that. That Weinberg's paper was in 1987, and Weinberg did not write the paper like I love the anthropic principle and this is his prediction. Weinberg actually, his purpose in the paper, people had started talking about this idea that perhaps the vacuum energy is small for those regions that we said, and they were talking about it at some kind of qualitative level, and Weinberg did one of the things Weinberg does so well, which is to take some idea like this and really turn it into something very sharp and useful and meaningful, and his paper was written in the following way, he said, look, if these anthropic ideas are right, then there's a prediction. We should expect to see a non-zero value of the vacuum energy in the next round of experiments. And it was the only person who made that prediction. Not that he made it, but he said the anthropic ideas would lead you to suspect that we'll see a vacuum energy in the next round of experiments. And the way the paper was written was that if we don't see it, we can put this idea to rest. That is probably wrong. I can say that I ran into Weinberg's paper when I was a graduate student and it really threw me for a loop. It really, a lot of people struggle in thinking about anthropic thinking, a lot of people viscerally hate this kind of explanation. I was certainly one of those people, I viscerally hated it as a graduate student, but it really worried me because I was already thinking a little bit about the cosmological concept problem. Certainly, his explanation, this explanation was way better than anything I had thought of then and I have thought of since. And it really made me worried a lot. But I also thought, okay, it's gonna be okay, the next round of experiments will come out, it'll just kill this idea, we don't have to think about it anymore. My reaction when the accelerating universe was discovered was crap, argument of Weinberg's looking good. And even that wasn't enough to make me, when you hate something, it takes a long time to come to terms with it. And it took at least me three or four years after that to start at least taking this picture really seriously. But it's a possibility and one shouldn't reject it out of hand. What I really wanna emphasize though because it's gotten so much out into the public discussion of these things that it seems to be a problem of philosophy and what you like and don't like. It's just not like that. There is a wonderful, very hard, but physics problem behind it. If you make mathematical sense of eternal inflation so you can predict probabilities from an ab initio principled way and you make a prediction for what you'd expect the cosmological constant to be in a way that follows from some well-defined mathematics, people, I think that would be it. That would really be something. If you prove that it's impossible, that will also be something. But it isn't some question of philosophy. It's a very hard but poseable and workable on question of physics. One thing that keeps bugging me about the cosmological constant, which is clearly the most natural way of saying that this is behind the acceleration that you don't know for sure unless you see another phenomenon related to the same cosmological constant, right? It could be even smaller than what we think it is. It could be zero and the problem remains equally. Why it's not as big as it is. So strictly speaking, you said you don't like that but it could be also true that that bloody thing is zero and that there is some modified gravity that has some other consequences, right? I mean, I think it's worth looking at other solutions as long you don't know for sure. I entirely agree. That cosmological constant is behind it, right? So that's why I encouraged our wonderful young people here to ambitiously try to attack the problem that bested all their elders. That's what normally happens. Some day someone might solve it in a completely different way. Let me actually, just let me interrupt you for a second by just giving one concrete example along these lines. Just so you see that it's not like, there are ideas, there are thoughts. There was an absolutely fantastic qualitative idea by Whitten in the mid-90s for solving the cosmological constant problem. This is one of those electrifying, sort of fantastic ideas that's just awesome and then in the next step it fails, okay? So but it's worth talking about. So what was his idea? Whitten's idea was that nature is exactly supersymmetric. Sounds completely crazy. Now it's exactly supersymmetric and that's why the vacuum energy is zero. Sounds nuts. So what is he talking about? Well, if you're in three dimensions and not in four dimensions, if you're in three dimensions, then life is a little bit strange, okay? In two plus one dimensions. If you have any mass at all in three dimensions, it gives rise to so much curvature of space that actually what the geometry looks like, it's not like a Schwarzschild solution. It really makes a geometry look like a cone, okay? So any mass at all sort of warps space so that it turns things into a cone at infinity and there's a little conical singularity where the particle is, okay? But because there is this big modification of what the geometry looks like at infinity, one consequence is that the vacuum can be supersymmetric but if you put a particle here that's a boson, let's say and then you try to prove that it has a superpartner which is a fermion, you find that you can't prove it. In fact, it doesn't have to have both fermions splitting. Supersymmetry in three dimensions does not imply that bosons and fermions are degenerate, okay? That's a special feature of supersymmetric theories in three dimensions. You say, great, why should I care? We don't live in three dimensions. She says, okay. Well, let's still imagine that we're in three dimensions. The splitting between bosons and fermions is really small. It's gravitational in size, right? Because the whole effect that I told you is about gravity. So not only is it in three dimensions but we should care even less because the splitting is minuscule even if we were in three dimensions. But then the idea was, imagine we start cranking up the size of the coupling. If we start cranking up the size of the coupling, this boson fermion splitting starts increasing, okay? And also at some point the interaction's becoming incredibly strong and perhaps we don't know what's going on, right? Because the interactions are so strong. But this idea came out right as the string duality revolution was happening and Witten had already observed that in the context of 10 dimensional theories when you make the coupling constant very strong and you thought you didn't know what's going on, in fact what happens is that there's a dual description where there's a radius that's growing in size and the theory becomes 11 dimensional. So the idea was exactly the same thing would happen here. You make the coupling stronger and stronger. At some point these boson fermion splittings are getting bigger and bigger. The coupling gets so strong that the three dimensional description isn't any good but there's a dual four dimensional description which becomes weakly coupled and as the coupling gets bigger and bigger the fourth dimension becomes very large. So what was exciting about this idea is that supersymmetry guarantees that the vacuum energy is zero in the strongly coupled description but everything else is mysterious. The dynamics and everything is mysterious. But the dual description makes it clear that there's four dimensions there and of course there's no boson fermion splitting but the vacuum energy is exactly zero. So the zeroness of the vacuum energy is obvious in one description and the interactions are obvious in another dual description. That was a qualitative idea. Sort of duality might solve the cosmological constant problem that the cosmological constant might be obviously zero in one description and the interactions might be obviously four dimensional in another description. That's a fantastic zero-thorner idea. It makes an immediate prediction which is that the vacuum energy shouldn't just vanish in four dimensions it should vanish on any circle of any size because it's supersymmetric the whole way and that's something which fails because if you imagine taking our world and putting it on a circle it has a non-zero vacuum energy the chasmier energy we can calculate the chasmier energy and it's not exactly zero. So that's a good example. It's a fantastic idea then in the next level of approximation there's something a little bit wrong with it. Well of course Witton knew this and says okay that means that the geometry has got to be a little more complicated than just a circle needs to have some interesting interactions and somehow not only should the cosmological constant vanish but the chasmier energy should vanish at all values of the radius as well. But then it's hard to engineer such a theory and it's hard to come up with concrete examples. So that's an example of an idea that probably doesn't work in detail but it's still out there. That sort of idea is still out there. Now that general approach is what seems disfavored now because the vacuum energy isn't exactly zero but is this tiny, tiny value. So there are ideas out there. There are other things that it might be but all the other things just start seeming like they're, at the moment they're less possible. Sorry, I interrupted you. No, no you didn't, you didn't. I just wanted to make one more sociological comment. I think you're gonna agree with me. I just wanna try. This is for the young people. For those of you who like supersymmetry, I just wanna say a little preaching. I wouldn't give up yet because it's starting to look bad. As there was a reminder today, in 1981 it looked very bad from the point of unification as well. In 1977 the standard model to many people was ruled out because of neutral current in atomic physics, even Weinberg. Who wrote the paper trying to modify it. And I think the moral is don't give up if you believe in the theory. Okay, it may survive the bad moments. Don't join the red school, leaving the thinking shifts. One more example along those lines that Nazi Cyberg emphasizes to me all the time is a discovery of the cosmic microwave background. When people first started looking for the CMB most theorists thought that it would be there at the 10 to the minus three level for the fluctuations in the CMB. Then the experimentalists, that's good. They could give them a kind of a reachable number to like head four. And so okay, they started designing experiments to get at 10 to the minus three. Then they started doing the measurement and the theory started changing too. They didn't see it at 10 to the minus three. And the theorists like, oh, maybe it's not 10 to the minus three, maybe it's 10 to the minus four. Okay, not see it at 10 to the minus four, maybe it's not 10 to the minus four, it's 10 to the minus four and a half. But it wasn't the theorists were cheating. This is really a bad, it's a bad impression sometimes I'll get that the theorists are just sitting there and just twiddling the dials underneath to constantly change and evade experiment. Bad theorists do that. But good theorists, there's a theory that makes a prediction. Maybe the theory changes for some reason. Then you have to look at what it does. You just have to honestly see what's going on at any time. In this case, what happened is people started believing in dark matter more. And as you started believing in dark matter more, what you needed in order to make the observed fluctuations we see in galaxies was honestly smaller value of the underlying fluctuations. So the ultimate theory settled down to around 10 to the minus five, two orders of magnitude smaller than the kind of promised initial size fluctuations that people might see. It was really the 11th and a half hour as far as the experimentalists were concerned. And they saw it in the 11th and a half hour at around the 10 to the minus five level. So that's again along the lines of Goran's comment. Thank you very much. It's a good place to finish. And thank you very much. Thank you again to Nima.