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SPEAKER 1: It is my great pleasure to introduce Professor Nima Arkani-Hamed for the first of his two public talks this week. He's a renowned particle theorist at the Institute for Advanced Study at Princeton. He got his PhD in '97 from Berkeley and then he was a postdoc at SLAC. And he was a junior faculty later on at Berkeley and he was a professor at Harvard from 2001 to 2008. He joined the faculty at the IAS in Princeton in 2008.
He has won numerous prizes and fellowships, including [INAUDIBLE] American Academy of Arts and Sciences, the Sackler Prize from Tel Aviv. And most importantly, the one with the big money, the Milner Prize for Fundamental Physics. He was one of the [INAUDIBLE] group of scientists winning that prize two years ago, I guess.
He's the father of many really novel and important particle physics theories, including large extra dimensions, little Higgs, dimensional deconstruction, ghost condensation, split supersymmetry, light, dark matter. And over the last five years he is working mostly on the subject of the novel approach to scattering amplitudes, the subject on which he's giving a separate series of lectures all week here.
He gave the messenger lectures here in 2010 and this fall he was appointed to be an AD White professor at large here at Cornell. This is his inaugural visit, as I mentioned. He will give two public talks in this visit on the philosophy of fundamental physics. So let's welcome Nima.
[APPLAUSE]
NIMA ARKANI-HAMED: It's a tremendous pleasure to be back at Cornell. I had a absolutely wonderful time here three years ago giving the messenger lectures. And as many such public lectures about physics tend to be, we spend a lot of time talking about things we know about the world and things that may be true about the world that we're exploring and thinking about, both theoretically and experimentally.
And we spend a lot of time on the things and the topics. And I don't think we spend quite as much time explaining why exactly it is we're doing what we're doing in the way we're going about doing it. And I think if I were someone who didn't know much about what was going on in theoretical physics especially and I only had some picture of the subject offered by what you read in magazines, newspapers, blogs, whatever, that I might have a sort of sense that these guys, especially these theoretical physicists, they're just making up stuff all the time.
There's all this weird stuff they're making up all the time. They give them strange names. There's supersymmetry, extra dimensions, black holes, Higgs's. What is all this stuff? And of course, you have some sense that you've got to go to school for a long time, you've got to study, so they can't be all idiots, probably. Also they all admit that experiment is ultimately what decides the validity of ideas. This is, after all, science.
So we all agree about that. So OK, OK, it's all right. Whatever they're doing, they can make up any fairy tale stories they want. In the end, experiments can decide who's right and who's wrong. Until experiment renders its verdict, maybe it's sort of irrelevant. Let them have their fun. It's harmless. They're not triggering the financial crisis or something like that.
But then if you think a little more, you realize that in the subject progress and experiment takes a long time. So the Large Hadron Collider was already conceived in the mid 1980s. It took 20, 25 years to start bringing it to a point where we're actually getting interesting data out of it. We finally discover the Higgs, all this incredible stuff. But it takes 25 or 30 years.
And then that might start making you really skeptical. So wait a minute, these guys sit in their offices, they dream up all this fairy stuff, and it takes 30 years to decide if they're right or wrong. So how do they know? What are they doing all day? They're all just spinning more and more and more crazy yarns until 30 years later, wham, comes down the hammer of experiments and decides who's right and who's wrong.
But before that, how do they decide to do what they're doing? Why doesn't it just proliferate into a million branches of irrelevant stuff which is finally cut down to size by experiment? And in between, what is it that's constraining them? Are they just every day saying, I don't know, today I'll invent another particle, give it a crazy name and continue like that.
So you might get a sense that that's how things work, at least if you're sort of fairly somewhat cynically looking at the way our subject is sometimes portrayed. And that's really the purpose of these lectures. It's not so much to tell you so many of the details of things that we know about the world or that we hope we might learn about the world. We're going to do some of that. In fact, we're going to do some of that at a very fundamental level, in this lecture especially.
But I also want to give you a sense for why the picture is very much not like what you might naively expect. And that in fact, to those of us who are in this business get up every day attempting to use our imaginations in an incredible straight jacket. And it's incredibly difficult to do anything. It's incredibly hard to make even the tiniest progress.
And that there are purely internal signs, purely internal to the structure of our subject itself, even before confirmation from experiment, which is the ultimate arbiter, of course. But even before that, it's not true that there is this exploding tree of infinite possibilities. That there is, in fact, incredibly tiny set of possibilities for what might actually come beyond what we know.
And that's a function of a rather extraordinary period that we find ourselves now in the development of our subject that what we know already about the universe, both from experiments for 400 years and progress in theory for 400 years, has brought us to a point that the structure of the world we see around us is almost completely inevitable. It's almost completely dictated by basic principles.
And it's therefore extraordinarily difficult to monkey with those principles in any way and not ruin everything, not destroy everything. And we have an incredible guide to progress by finding those very, very few ideas that managed to thread 100 needles to even have a chance of being right, to even have a chance of making confrontation with experiment. And that gives us both a very exciting set of things to do as theoretical physicists and a sharper and more focused set of things to be looking for experimentally as well.
So that's what I want to really tell you about. So I apologize if there happen to be any philosophers in the audience. What I mean by philosophy is I'm sure not what you mean by philosophy. What I mean by philosophy is a very rough and ready notion of philosophy. But it guides us nonetheless. It's the set of 0th order of thoughts we have in the morning when we get up and go to work and think about what it is we're doing and why we're doing it. And that's what I want to give you a sense for in these two lectures.
So let me begin by putting things in a little bit of context. The 20th century started with these big revolutions of relativity and quantum mechanics. And in a sense, we can summarize in a slogan the accomplishment of fundamental physics in the 20th century by saying that the discovery of these two broad principles of relativity and quantum mechanics put such incredible constraints on what the laws of nature can look like that, as I said a moment ago, in a specific sense, we'll talk about at length in this lecture the structure of our universe at least at long distances is almost inevitable.
There are no choices. There are some choices, but there are very, very few choices and we know exactly what those choices are. And the situation, we couldn't have said this 100 years ago. We couldn't have even really said it 50 years ago. It's something that we've understood deeply in the last 50, 40, 30 years. And it's something we can say now that we couldn't have said before. It's something qualitatively new that has happened in the development of our subject. So that is a triumph. And I think that's the real triumph of the 20th century.
But at the same time, that has exposed a number of really deep difficulties and paradoxes in our understanding of nature. This is also true that exactly the same principles taken to their logical extreme, taken, pushed to their limits, which we can explore and think about theoretically in various situations, exactly the same principles seem to at an even deeper level be in conflict with each other. And in some sense, appear to make the universe impossible. The universe we see impossible.
So on the one hand, they make the universe inevitable. On the other hand, the thought of more deeply, they seem to make the universe impossible. And that's the conundrum that we have to deal with. We have the spectacularly successful theoretical framework that agrees with experiments beautifully well. It rests on solid foundations. And yet we know there's something up. There's something up at an even deeper level. And those are the questions that we're going to have to confront in this century.
So how do we deal with this? Well, let me first tell you what the cause of these problems are. And we'll talk about the 21st century difficulties a lot in the next lecture. This lecture is mostly about explaining that first slide and telling you about the way it leads us to think about things. But just peeking ahead for a second, when I say that these principles are in conflict, the idea of relativity is embodied in Einstein's picture of space time.
But we have very good reasons to suspect that space time is an approximate notion because of effects due to quantum mechanics and gravity. And it's even conceivable that quantum mechanics itself has limitations eventually when we try to apply quantum mechanical rules to describing the entire universe. So that's one set of difficulties with the very principles themselves.
Another set of difficulties is that if we take those principles and say, well, if I take various things about our world for granted, can I predict what happens next in any experiment? The answer is yes. So we understand beautifully mechanically how the world works. But if we can ask the question up a level in structure, why do we get the sort of particular universe we get?
It seems wildly implausible given the laws that we know that we get the universe that we have. In particular, that we get a universe that's big. One of the most basic features of the world is that it's big with big things in it. And it's very difficult for us to understand that extraordinarily simple fact about the world.
So these are the sort of central questions that are driving us in the 21st century. And as theoretical physicists, we have to think about ways in which we might attack them. But there's a wonderful strategy that we now have. In a sense, it's the strategy that we've always had. It's a strategy that the best work in theoretical physics going back to Newton, Galileo and Newton and Maxwell and Einstein and Dirac and everyone else since has pursued. But it's so sharp now, this strategy, that even lesser mortals of us can pursue it effectively.
And the reason is what I alluded to already. Something new about theoretical physics today and any attempt we make to address these difficulties is that anything we do is shockingly constrained by what we already know in a way unlike ever before. Just to give you one little example, we'll talk about many others in a moment.
But if you know about Newtonian physics, you know that Newton tells you there is an inverse square law for the gravitational force between two bodies. So the force between two bodies goes like one over the distance squared. But if you're Newton, you have no idea why it went like one over distance squared. That's just a fact. It looks very nice. It fits the data. Everything is great.
But you could equally well have, just from a theoretical point of view, chosen an inverse 1.9999 power law or an inverse 2.7 law or an inverse pi law. Or any number that you want would have been fine. There is a continuous infinity of choices that you can make even for that one force, never mind everything else. There is enormous number of choices that you have to make.
Of course, you make the ones that work and describe nature and you proceed. But there is an infinite freedom in the structure of classical physics. And that infinite freedom is reflected in all the other parts of classical physics as things proceeded and went forward.
But what we discovered relativity and then when we discovered quantum mechanics and we had both of them, those choices collapsed incredibly. Until today, the set of choices we have instead of being a continuous infinity for a continuous infinity of possible interactions we can even imagine have boiled down to a choice of a finite, small number of numbers that specify the interaction strengths between a few basic elementary particles, all of whose essential features we know about already.
And because things are so shockingly constrained, it is extremely difficult to try to take what you have and just hid it in some random way. You hid it in some random way. You destroy this entire beautiful structure. So you can't just do that. You can't just proceed by willy-nilly taking the principles we have right now and throwing this one out and throwing that one out and crossing that one out.
Well, you can do that. And if you do that, and incidentally if you have a sort of a caricature of the way big developments in physics have happened, one often talks about big revolutions in physics. So the sort of languages that, well, we had Newtonian mechanics and relativity overthrew Newtonian mechanics. And quantum mechanics also in a much more radical way overthrew Newtonian mechanics. It makes it sound like the bloody awful messes revolutions in society are. But revolutions in physics are never actually like that.
Now, if you're slightly more sophisticated, you know that, well, the newer theories have to contain the old theories in some limit. So if you're talking about Newtonian mechanics, if you ignore the fact that the speed of light is finite, the effects of relativity can be neglected. But if you've got very, very high velocities compared to the speed of light, you see that Newtonian mechanics is modified and things become relativistic.
So it's very different. The ideas are very different, space and time are unified and so on. But relativity has got to reduce to Newtonian mechanics and some limit. That's certainly true. So the old theories are never overthrown in that sense. They're always valid in some new regime of validity when you discover a new phenomenon that you didn't know about before. But still, whatever you learn is contained in the new physics in some way.
But there's an even deeper sense in which the revolutionary ideas are not as different from what came before them as you might think. And the reason is that over and over again when we've had these, even when we've had these big conceptual shifts, the seed of those conceptual shift, the origin of the shifts, can already be seen as funny theoretical features of the theories that came before them.
So that there is a way of preparing yourself for what's coming next by taking what you have already and pushing it to its extremes. If you discover the laws of quantum mechanics by studying the hydrogen atom, then if you're strictly empirical about the way the world works, in every new situation you can say, oh, I don't know if quantum mechanics works now. Maybe I shouldn't trust it.
But we never do that. We always say, well, let's keep going. Let's keep going and apply it to new situations, new situations. We keep going. And if it seems like the picture that we have, seem that the equations make sense, if the theory isn't crying uncle, if there isn't some internal contradiction, we keep going. And of course, in the best case scenario, we keep going, we make predictions for new experiments. The experiments come along and confirm that everything is OK. And we keep going and going further.
But the essential philosophy is not to be afraid, not to constantly hem yourself in by the degree to which you've tested the theories you have already, but to just assume that things are going to keep working. You take what you have and you keep pushing them into new arenas and new places.
Until, most excitingly, something starts going wrong. And the something that goes wrong can be theoretical. It can be experimental, of course. But even more strangely, the things that can go wrong can be purely internally theoretical difficulties. And when one or the other those things happen is when you know it may be time for another jump.
So the progress in physics, the old things are never destroyed, even when they're modified in radical seeming ways. The way of understanding what's coming next is almost always sitting there in studying more deeply the structure of the theories that we have at the present.
So this way of perceiving, this philosophy that you take the laws you have and you keep pushing them way beyond the first places they were discovered into new territory and you keep pushing them, exploring what they say, exploring the kinds of phenomenon they can in principle explain, exploring their theoretical structure and preparing yourself for contact with new experiments, this philosophy is known as radical conservatism. I think the first person who coined this phrase I think was John Wheeler.
And again, I think it's a response to the sort of caricature that especially when physics has deep difficulties, you need some revolutionary to come out of left field and the establishment has no idea what's going on. Some Messiah will come in from some completely different point of view and figure out what is going on. And that's half of the story. Or that style is half of the story. That's the radical part of the story.
But the really good ideas in physics are never just radical. They have an element of radical and an element of conservative. This is actually a sort of a caricature. There's two extremes in new ideas and thinking about what might come next. On one extreme, you have what you might call the crackpot extreme. This is what fills my inbox.
[LAUGHTER]
All right. The uncertainty principle wrong. Einstein was an idiot. You guys all believe in a deep authority and no one is brave enough to challenge Einstein, et cetera, et cetera. By the way, these people are very happy. They're supposed to look really happy. And in a funny way, I envy them.
Because nothing is more fun than sitting around and having crazy ideas. You just dream absolutely crazy ideas. Wouldn't it be great? That's the crackpot direction. And the slogan of the crackpot is, everything old is wrong. Everything that came before me is wrong, this is the right way of thinking about things. You guys, you don't know anything.
In the other direction, these people don't tend to be professional academics. These people tend to be professional academics.
[LAUGHTER]
In the other direction, you have the curmudgeons. And you don't have to be old to be a curmudgeon. There are lots of young curmudgeons. And these people tend not to be very happy. And their attitude is basically the opposite. It's that everything new is wrong. And this is the just radical and this is the just conservative.
And good physics is always in between. And the art is to know how much of each you need at any given time. And you don't know ahead of time how much of each you need. You don't know ahead of time whether the problems that are confronting you need more of this or more of that. The 20th century started with a lot of this.
And there is a whole group of people, people who were 25, 30 years old in 1930, who missed out on it completely. So they just started getting into physics when quantum mechanics was just set and so they thought oh, these guys five years older than me, they just completely revolutionized physics. That's the way things are going to work. They were totally wrong. They were disappointed over and over and over again.
The next 70 years were a lot more like that. The next seven years were a lot more about the conservative development of these ideas. Every time there seemed to be some strange phenomenon. Parity is violated, some strange particles in the cosmic rays. This interesting group of people who just missed discovering quantum mechanics hoped for another revolution. They were wrong every time, disappointed every time.
Ironically, the generation of people like Richard Feynman, Freeman Dyson, Julian Schwinger and others, who reinvigorated this union of relativity and quantum mechanics called quantum field theory after the Second World War, especially since some of these people like Feynman have such a public reputation as being wild and crazy guys. Despite being young, they were the conservatives.
Unlike their elders, they said, OK, thanks for all the revolutions. That's just not the way it's working anymore. We're just going to take what you handed us and calculate with it and understand it and continue pushing it much, much further. You see people like Oppenheimer was one of these people who missed the revolution. Oppenheimer kept saying at every sign of difficulty, we have to, all the principles, everything's got to go. We've got to start over again.
It's always in fog and mist of fantastic new principles. And he was wrong. He was wrong over and over again. People like Feynman said, no thanks. The theory works. We'll calculate with it. Seems hard, we'll work harder, we'll calculate, figure out how to do it. And it's right. And it's that second style, which has worked over and over again for most of the rest of the 20th century.
And it's conceivable that the problems that we're dealing with now are going to force us to start heading back more in this direction. Because they seem more structural. We seem to hit the difficulties purely internally theoretically in trying to make sense of these ideas. But it's a constant seesaw. It's a constant seesaw back and forth. You don't know which way is going to be the best. It's definitely not this extreme and it's definitely not that extreme.
So again, this is the thing I want to give you a sense for in these talks. Radically conservative is supposed to be in contrast with conservatively radical. If you're conservatively radical, what you do is do what I said a moment ago is a really stupid thing to do. Make a list of all the known assumptions you're making and cross them out one by one. I know relativity is wrong. Look how brave I am challenging Einstein. Or I know, let's throw out the laws of quantum mechanics.
Of course, any idiot can do that. The difficulty is what I said. For some reason, these laws that we've been given worked so spectacularly well in such subtle interconnected ways that you can't just do that. Or if you do it, you immediately run into a disaster. This conservatively radical way of proceeding, often they don't tend to realize what the disasters are. They fool themselves into thinking there isn't a disaster.
But in any case, it doesn't work. It's never worked. And the thing that does work is this much more difficult and subtle business of taking what you have, pushing it to its limits until it tells you it's breaking, and then figuring out what you're supposed to do about it.
And because of this rather new thing that I just alluded to, because of this great constraint on what the laws of nature look like following from relativity and quantum mechanics, which is much more powerful than what we had before, following this philosophy of radical conservatism does give us theoretical physicists a very interesting and quite unambiguous metric for theoretical progress. This is free of aesthetics and it's free of sociology.
Sometimes you hear that well, xy physicists like this kind of theory and the other people like that kind of theory. And it seems like it's a matter of aesthetics. You think these ideas are beautiful, I think those ideas are beautiful. This word beauty is tossed around in a very misleading way, as we'll talk about in a little bit. The theoretical physicist sense of beauty is radically different than many other senses of beauty. We don't mean the same thing by beautiful as at least the very common usage of the word suggests.
And in fact, there is an unambiguous metric for theoretical progress, as I said, which is free of aesthetics in sociology. And that's because finding even candidates for solving the problems we're facing, which have the feature that they actually solved the problem at least in principle theoretically, while being consistent with everything else we know is extraordinarily difficult. It can't be done willy-nilly.
Whenever it does happen, it requires not just one, but a whole slew of deep, new ideas. And associated with that, this is also not something that happens every day. In fact, these kinds of new theoretical ideas have entered our subject a number of times in the last 50 years. But they come around on a sort of a decade time scale or so.
So it's almost the opposite of the picture that you walk into your office and you dream up something new and then the next day you dream up something else and then experiment comes along and tells you which one of these hundreds of thousands of choices is right. It's rather the other way around. Every decade or so you might come up with one new idea and it takes three decades for the experiments to come online.
And by that point, there's maybe if you're lucky two or three ideas on the table that even have a chance of being right. And then the experiment comes along and tells you which one of them. So there is a gradation. There's an interesting kind of gray zone between 0, which is not confirmed by experiment, and one, which is confirmed by experiment. There are ideas that are not one confirmed by experiment, but have passed the bar, have passed this bar. And there are few and far between.
And that's what we do. That's what we do in our offices. That's what we work on. We try to find these ideas that even have a chance of being right.
One reason for doing that, of course, is they have a chance of being right. Another reason for doing it is that because it's so rare to find them, finding these classes of theories then also gives our experimental colleagues a suite of interesting things to look for. Of course, our experimental colleges just look at all the weird and wonderful stuff they make when they run their experiments. But it's also a good idea to give them a few targets for the kind of experimental signals you might expect from these broad classes of theories that pass this gauntlet, that have a chance of being right.
So in the rest of this lecture, I want to illustrate all of these ideas. So everything I said was quite abstract. In the rest of this lecture, I want to illustrate how these ideas work in the context of 20th century physics. Telling you how it is that relativity and quantum mechanics greatly constrain the way the world works. And the whole story will culminate with why the Higgs really had to exist.
So the Higgs is an example of one of these particles that theorists always talk about, Higgs, Higgs, Higgs. And before it was discovered, there was all sorts of people who would say, well, how do you know it's there? It's just a figment of your imagination. But we'd say, no, no, no. We're sure it's there. Trust us, it's there. And we know even roughly what mass it's going to have, what properties it's going to have.
How did we know that? How did we know ahead of time? No one had seen anything like it before. Literally no one had seen anything like it before. So how are we so sure that it had to be there?
The story of the Higgs is intimately connected to this philosophy of radical conservatism. It's in fact the sort of latest success of the philosophy of radical conservatism. And so I want to illustrate these ideas culminating in the theoretical prediction of and the experimental discovery of the Higgs.
All right, so just a few quick reminders about basic facts about the world. All matter that we know consists of atoms. We have the atoms as electrons going around the proton, there's protons and neutrons, and the nuclei of more complicated atoms. And if we stare inside these nuclei, the protons and neutrons, we see that they're made out of still other forms of matter, the up and down quarks. So we have particles like electrons and up and down quarks and heavier cousins of them. These are particles of matter.
And the particles of matter are acted on by four basic interactions, gravity, electromagnetism, and these weak and strong nuclear forces that we discovered in the 20th century. And these forces, of course, hold sway over an enormous range of distances, all the way from the size of the observable universe, kind of the 28 centimeters, down to around a 10th of the minus 17 centimeters, which is what we're probing right now at the Large Hadron Collider.
So there's a large array of there's many kinds of matter, many kinds of interaction. But one of the remarkable things that we learned in the 20th century is that all of this slew of particles and interactions are actually described in a uniform way. And we can think of all of them, all of these interactions, as arising from these little basic stick figures. These are the famous Feynman diagrams.
And so the electromagnetism, for example, is associated with this little stick figure, two electrons interacting with a photon. Or gravity, the force of gravity on an electron is a similar stick figure with three guys. Two electrons interacting with a graviton. And if we now glue these stick figures together in every possible way, concatenating these interactions one after another after another, that's what gives us all the other interactions and everything else that can take place in nature.
So two electrons bouncing off of each other is given by this kind of process and more complicated things can happen. And the remarkable statement is that everything that can happen is just the gluing together of the concatenations of these basic three particle interactions of this sort.
Now, that was a picture for the very familiar forces of electromagnetism and gravity. But it turns out, and this was not at all obvious. This was, in a sense, the big accomplishment of physics from the 1950s to the 1970s, culminating in the so-called standard model of particle physics, is to realize that all of the interactions are actually described by these stick figures.
So for example, the strong interactions are associated with exactly the same kind of stick figure coupling between two quarks and a gluon. These gluon particles are what, if you like, holds the quarks together inside the protons and the neutrons. And the weak interactions, which are ultimately responsible for radioactivity, making a neutron decay to a proton in outer space in 15 minutes, that happens because we can have an interaction that allows a down quark to change into an up quark while omitting this what we often call a virtual W particle.
But it's the same basic stick figure interaction. Again, for the purpose of these talks, thinking about the details of how that works isn't going to be crucial. We're going to tell you something structural about this picture instead.
Now, why did it take us so long to understand that the strong force and the weak force were described in much the same way as the other interactions? It's because that basic picture, the fact that they do look like that, that the basic interactions are those stick figure three point interactions, is not obvious at long distances. Accidents at long distances hide this basic similarity between the different particles.
For example, for subtle but well understood quantum mechanical reasons, the force between two quarks that's mediated by gluon gets gradually stronger as you go to longer and longer distances. Until finally at a humongous distance scale, where humongous is to a particle physicist. So this humongous distance scale is around 10 to the minus 14 centimeters. So at the enormous distance scale of 10 to the minus 14 centimeters, the force becomes so strong that it actually traps that the quarks are permanently confined inside the protons and the neutrons.
And for a similar reason, the weak interactions took a long time to be understood to have the same character as the others. And that's, in some sense for an even simpler reason, these W particles and the Z particle, unlike the photon they have a mass. I should have said this before. This putting together of the two three point interactions between electrons to give the force between two electrons coming from electromagnetism, this gives us an inverse square law force, as you know.
The electrostatic force goes like an inverse square law, just like gravity. And a way of talking about that is that that photon isn't real. There's a linguistic sense in which this is true. We call it a virtual photon. It's a quantum mechanical excitation that can take place as a consequence of the uncertainty principle out of the vacuum.
But the fact is the force goes like an inverse square law tells you something interesting. This will be important for us later. It tells you that it's harder and harder to make fluctuations at longer and longer distances. And that's why the force gets smaller and smaller. So the force eventually goes like the inverse square law for that reason. But for W particles, it's even harder still because they have a mass. Because they have a mass, getting this fluctuation out of the vacuum actually shuts off exponentially as you go to longer and longer distances. And so you don't notice it at all.
In order to see this interaction, you have to get to short distances. And going to short distances by the uncertainty principle also requires you to go to high energies. So we couldn't have known that the weak contractions looked like the other ones before going to very high energies. So this accident that they happen to have a mass was removed. And we could see the essential similarity between these forces.
This is the reason why we go to short distances in particle physics. It's because what we've learned from all of these examples that long distance accidents can obscure what is otherwise a fundamental similarity between all the interactions. And that perhaps there are other interactions, other particles that are telling us about further laws that we don't know about yet that are just being hidden because we don't see them at long distances.
All right. So much for what the matter particles and the interactions are. But the particles have one more feature that I want to talk about. They have what's called a spin. So you can think of the electron loosely as a little spinning top. And the fact that it's spinning around, it's charge and it's spinning around, is the origin of its magnetic properties. Ultimately origin of all the magnetism of the magnetic properties of a bar magnet, for example.
Now, for good quantum mechanical reasons, in quantum mechanics things are quantized. And the amount of angular momentum that you can have actually comes in some package, in a unit of Planck's constant. That's H with a slash through it. And for good quantum mechanical reasons, the allowed values, if you just knew the basic principles of quantum mechanics, are any multiple of a half. So you could have 0, 1/2, one, 3/2, two, 5/2, and so on. So that's the kind of angular momentum that you're allowed to have.
And what we see in nature for the elementary particles are a small subset of everything that's allowed. The electron of the spin a half. The W that goes on has a spin one. The photon has also a spin one. The graviton has spin two and so on. All right, so that's interesting.
It's funny. What we see in nature is extremely simple. We don't see these particles of spin 17/2 or larger crazy things. All we see are this 1/2, one, and two. I should say that the rest of this talk is going to be perhaps slightly more demanding than typical public talks on physics. So I'm going to ask you to keep these numbers, like 1/2 and one and two, in mind as we go along. And we might even count on our fingers every now and then. But I promise you, it will be worth it.
So the particles that we see are extremely simple. They have this very simple menu of spins. So in fact, in addition to the electrons and up quarks and down quarks and the other particles, there's actually what we know of in nature in the standard model of particle physics has a bunch of these guys. And I don't need to even tell you what this menu looks like in detail. You just need to know that it's the same stick figure interactions.
We just have more of these particles. We have a bunch of them. We have sort of three copies of them. And anyway, it's not the simplest thing in the world, it's not the most complicated thing in the world. We just have some finite menu of things that have exactly that character with exactly the spins that we just talked about. Spin 1/2, spin one, and a single particle of spin two, which is the graviton.
So you can ask, why is it so simple. Why don't we see something a lot more complicated? There is two aspects of the simplicity that we could talk about. One of them is, why is it that this little miracle happened? And all of the amazing complexity of the world around us can be reduced to just gluing together basic three particle interactions.
Why didn't we have to worry about other basic interactions? For example, why wasn't there a however many that is? Fundamental nine particle interaction. That we have to glue those guys together to get more and more complicated things. Why did it go all the way down to the simplest possibility of three?
Now, you see that would have been a complete disaster if we had all these other guys to keep track of. Then there would be an enormous number of things we didn't know. But that's not the way it is. We managed to reduce everything down just to these basic three particle interactions. So that's interesting.
And the second question is why do we get such a tiny menu of spins? Why don't we get anything else? And you might add many more questions along these lines, which will all have answers in a second. So why do we get something like gravity? Why do we get this interaction that's universal? It's the same for everybody. It gives you this attractive force between massive particles at long distances.
We see gravity in the world, but who ordered gravity? I mean, could there be two kinds of gravity? Could there be three kinds of graviton? Is that possible? How much of what we see in our world is sort of contingent? How much is hardwired? That's the question that we're going to explore now. And of course, the answer, as I've said a number of times, is an awful lot is actually hardwired and we can't do anything about it.
So I want to now give you the answer to these two questions. So the answer is that due to a chain of reasoning that we'll go through momentarily, it turns out we don't know all of the laws of physics. But we do know some of them. We know the principles of relativity and quantum mechanics. And this alone, without knowing everything, just the principles of relativity and quantum mechanics alone, have a remarkable consequence.
That at sufficiently long distances, since it'll become clear in a second, at sufficiently long distances, it's guaranteed that the universe will be described first by putting together elementary three particle interactions. So that has a good answer. So we have to make them out of these building blocks. And second, the only spins that are allowed are spin 0, 1/2, one, 3/2, and two. And in fact, the spin two guy is unique. You can't have even more than one of them. You can have exactly one of that spin two guy.
So let me say what I mean by this in more detail. Let's say you took a bunch of theoretical physicists and you lock them up in a room. Good ones. You locked them up in the room and you didn't tell them what the world actually looked like outside, which is a little counterfactual, because there are no rooms in the universe, but never mind.
So you close the windows. They can't look outside. But you hand them the laws of relativity and quantum mechanics. And you say, go. What kind of world can you make compatible with these laws? And you give them a long time and food and graduate students.
[LAUGHTER]
If they're very good, they'll need equal amounts of both. And you come back and you ask, what did you find? And they'll tell you something like, if they're good, they'll tell you something like, at first I thought it would be really easy to build worlds compatible with these rules. But it's really, really hard. And actually, there's only a tiny set of things that are possible. Here's what's possible. And they'll tell you. You'll have to have particles interacting in this way and the spins of the particles got to be pulled from this list.
And that spin two thing, that thing is really weird. It gives this universal attraction between all particles and it gives orbits that can go around each other and things like that. That guy's interesting. This guy, the spin one guy, that's actually pretty interesting. Sometimes it pushes, sometimes it pulls, and so on and so forth. And so you start recognizing the interactions we know and love in nature. But those properties are just a consequence of these basic principles of relativity and quantum mechanics.
So I want to give you a sense for why this statement is true. But before getting there, I just want to pause and talk a little about how remarkable this is. Because for this statement to be true, you need both the principles of relativity and those of quantum mechanics. If you just add the laws of quantum mechanics, it wouldn't be true. You could imagine zillions of more universes just compatible with quantum mechanics. If it's just relativity, it wouldn't be true. You could have zillions of universes, even more zillions, compatible with the laws of relativity. It's both that make it so amazingly constrained.
And this set of words more accurately reflects. And this kind of argument and this fact is the sort of thing that we love the most in theoretical physics. The words associated with it are inevitability. You start with a few basic principles and nothing stops. The machine churns.
It keeps going and it does what it wants to do and it gets to the end. Nowhere can you make choices. Nowhere does some arbitrariness sneak in. You just see what it can do and what it spits out is this amazingly beautiful sort of crystalline structure. The fact that parts of that crystalline structure describe our world is amazing, of course.
This also highlights this feature about the laws of nature that I've alluded to many times. They have simultaneously a rigidity and a fragility to them. They're very rigid in the sense that because the basic properties of the world we see around us are almost completely dictated by these principles. As we've said many times, you can't just go mucking around with them without bringing down the entire structure and smashing this incredible architecture that we've succeeded in building for 400 years.
On the other hand, so it's very rigid. Things follow completely inevitably from the starting points. On the other a hand, that tells you that it's also extremely fragile. So you can't go back to the underlying principles of quantum mechanics and relativity and muck around with them a little bit without, again, smashing the whole thing to bits. So the rigidity and the fragility of ideas are a hallmark of really good ideas, really important ideas. Not just in physics and in many fields, but certainly in physics.
And finally, when we talk about beauty in the laws of nature, this is what we mean. It's not a subjective criterion. It's not like pretty or cute. Those are also adjectives that we use in describing our ongoing work. So if you have a clever idea or a colleague has had a clever idea, you'll say, that's a really cute idea, or that's a really pretty idea.
And that more or less means what you mean by cute or pretty in real life. It's good. It's a compliment. But it's an intermediate thing. It's not this. It's not this kind of beauty. This is much more rare. It's much more austere and much less subjective. Much, much less subjective. You know it when you see it. Almost everyone agrees when they see it.
Something else that I wanted to mention before proceeding is notice that this is the list produced by our competent theoretical physicists. The allowed spins are 0, 1/2, one, 3/2, and two. And if you remember from a few slides ago what the spins of the particles that we've actually observed in nature are so far, remember we had particles like the electrons and the quarks that were spin 1/2, particles like the photon and the W and so on that were spin one, and the graviton that's spin two.
That's what we've seen. So we're in an interesting situation. Actually we've also seen the Higgs, which we're coming to in a moment. But imagine it was before July 4, 2012. So we hadn't seen it yet. So all we'd seen was 1/2, one, and two.
So that's an interesting situation. What we've seen in nature, what we've seen nature do, seems to be a subset of what nature can do compatible with its basic principles. Whenever that happens, you have a very interesting set of opportunities as a theoretical physicist. Because you've done your work, you've figured out what the world can do. You figured out how to make predictions based on the menu of particles that's allowed. And if the world hasn't done all of them, then maybe it will.
And maybe it must in order to solve some of these theoretical problems that we have to deal with. And then when that kind of opportunity arises, you take it. You snatch it. And you see if you can enlarge the set of things that the world does by matching it to what it can do with these principles. Again, that's a critical part of this philosophy of radical conservatism.
By the way, what I've said very inarticulately in the last five minutes is said absolutely beautifully in maybe the greatest popular physics book ever written, which is by Steve Weinberg. This book, Dreams of a Final Theory where he devotes an entire chapter to talk about this idea of inevitability, rigidity, and beauty of the laws of physics. And if you haven't read it, he doesn't need the money, but it'll do your soul good.
All right. So I now want to spend a few minutes telling you about why this fact is true. And I just want to give you a feeling for why it's true. So we'll go over this pretty quickly. Some of these things, a few of the slides were actually taken from my messenger lectures. So if you want more details around some of these ideas, you can maybe find them there and many other places as well. But I just want to give you a sense for why this remarkable fact is true.
So I have to explain to you two things. First, why is it that the only interactions we see are these little stick figure interactions? ABC. And not much more complicated ones with 12 particles in, 17 particles out. And secondly, why are the menu of spins this tiny ones? 0, 1/2, one, 3/2, two. So let's do the first one first.
And to understand this first one, we're going to go and awfully long way by thinking a little bit about units. So units are something that's the bane of the existence of the high school physics student and certainly the bane of my high school physics experience.
But one of the reasons why units are so sort of complicated and messy and annoying when you learn them in school is that you're not learning something about the real world. You're learning something about the way human beings talk about the real world. Most of the units are human constructs. And it's a good idea not to deal with human construct units, but to deal with units as nature cares about them.
So for instance, when we say that someone is two meters tall, that meter, well now there's a fancier definition of meter. But a long time ago the meter was the length of a bar in Paris somewhere. So when we say someone is two meters tall, we say you put the bar next to them twice and you get the height of the person.
Now, that's great if you happen to have access to this bar in Paris and its replicates around the world. But if you're trying to tell an alien somewhere far away what two meters means, it's not very useful to them. They don't know how big that bar in Paris is. So it's much more useful to use natural units.
And this is greatly aided by the fact that because of relativity and quantum mechanics, we have these fundamental constants like the speed of light and Planck's constant that allow us to convert between time and distance, for example. So that's familiar. We can talk about one light second, which is the same as three times 10 to the 10 centimeters. That's the distance traversed by light in one second. So that's familiar.
Perhaps less familiarly, we can use Planck's constant to convert, for example, between energy and time. And energy is a Planck's constant over time. And we can actually use units where we put these two constants equal to one. I mean, it's just a choice of units.
Doing that allows us to reduce all units to one unit, which we can conventionally call the unit of energy, for example. So the mass of the proton and proton c squared happens to be close, it's not exactly, its 0.938 giga electron volt. It's lucky that it's close to a giga electron volt. Well, I think we should just define a volt to be such that this is true, actually. Yeah, that's more useful. All right. But the mass of a proton, that's something that the aliens on Alpha Centauri will know about. And now we can convert everything to this unit.
So for instance, my height is 10 to the 16th 1 over GEV. So 1 over GEV has units of length, has units of time, which is the same as units of length. Now that immediately tells you something and also the person on Alpha Centauri something that if they put 10 to 16th protons. Oh, I should have said 1 GEV inverse is around 10 to the minus 14 centimeters. And that's around the size of the proton.
So the unit has everything to do with the proton. The mass is 1 GEV. 1 over GEV is around the size of the proton. And when you say I'm around 10 to the 16 GEV inverse GEV high, that means if you stack 10 to the 16th protons one on top of the each other, you get something my height. So that's something that you can tell the alien and they'll know how tall I am. Because protons there are the same as the protons here.
My mass is 10 to the 29 GEV. That also tells you that I have around 10 to the 29 protons in me, because that's where most of my mass is. See, it's useful, it tells you things. The electro time as it is rapidly exceeding 10 to the 27th inverse GEV. The energies of the Large Hadron Collider are 7,000 GEV for example.
All right, now those units are useful because they immediately tell us something remarkable about very familiar facts about physics that you learn even in high school. So let's look at the force between two electrons. The force between two electrons, well you know it all goes like the inverse square law. Goes like 1 over r squared. So there are some coefficients sitting in front, which I'll just call the charge of one electron times the charge of the other one. That's called q. So the force goes like q squared over r squared.
Now, this is interesting because it's very easy for you to work out that the units of force in these natural units, force has the same units as 1 over length squared. Just work it out. It's mass times acceleration. And mass as units of 1 over length. Acceleration has units of velocity over time. Velocity has no units at all. So acceleration is units of 1 over time. So force is units of 1 over length squared.
So force is units of 1 over length squared. 1 over r squared already has units of 1 over length squared. So we learn amazingly the thing that multiplies it is just a number. It doesn't have any units. It's just a number. And that number is around 1 over 137. It's a famous number in physics called the fine structure constant. But the important thing is that you learn it's just a pure number. In a specific sense, the force of electromagnetism, the Selectric force, is weak. The dimensionless strength associated with it is a small number.
Now let's talk about gravity instead. Now, the force law is almost the same except it involves the mass of the particles. So it's the Newton constant, g Newton times the product of the two masses divided by r squared. And that tells you now something important, that g Newton times mass squared has got to be dimensionless. So g Newton itself has to have units of 1 over length squared. Sorry, g Newton has units of length squared.
So g Newton, the strength of gravity is not a pure number. The strength of gravity, in fact, in these units is 10 to the minus 33 centimeters squared. That's a famous Planck length. It's the Planck length squared.
So electromagnetism, the electric force has a dimensionless strength. It's weak. It's always weak. It's a small number. Gravity, it's not dimensionless. It has units. That number, 10 to the minus 33 centimeters, is so minuscule that it means if you go to very long distances compared to 10 of the minus 33 centimeters, which is everything we know in the universe, gravity is going to look astonishingly weak. But if you get the distances around 10 to the minus 33 centimeters, it'll look strong. There's no sense in which it's always weak or strong.
So the extreme tininess of that number compared to everything we know is reflecting the weakness of gravity in our world. And there's an associated energy scale with that. 1 over this Planck length is 10th of the 19th GEV. And I remind you that we're probing energies of around 1,000 GEV at the LHC. So this is 16 orders of magnitude removed from anything we can hope to probe in experiments anytime soon. So we learn this amazing difference between the two forces that are both inverse square laws.
Now, that fact shows up in a closely related context as well. Something that you know is that the world is quantum mechanical. It's not classical. And what that means is that if you do the same experiment over and over again, you'll get different outcomes. You can't predict what's going to happen ahead of time for a single experiment. Even if you do exactly the same thing, the initial conditions are the same every single time.
So for instance, if I collide two electrons together, maybe two electrons will come out. But maybe sometimes they'll come out at 10 degrees, sometimes they'll come out at 80 degrees, 130 degrees compared to the initial way that I collided them. So all we can do is predict the probabilities that something happens.
Now, in quantum mechanics that probability turns out to be the square of something more primitive that we call the probability amplitudes. So we talk about the amplitudes a lot. And so exactly these numbers that we talked about a moment ago also determine the amplitude for these interactions. So if I look at two electrons scattering off each other through exchanging a photon, then the amplitude for that process is just a number. It's in fact that same q squared that we talked about. It's around 1 over 137, it's about 1%, and it's small.
It means that if you want to know, now in detail it depends on the angle in some interesting way, and for that you've got to go to graduate school to figure it all out. But if you want to have some idea what is the probability you smash two electrons into each other, enormous energy is, what's the probability they come out at 40 degrees? That's about 10 to the minus 4. now it changes depending on angle, but roughly about 10 to the minus 4. It mostly doesn't happen. Mostly they don't interact with each other. The probability is small.
Now, this also immediately lets you appreciate in a sharp and important way what the real mystery is with the problems of quantum mechanics and gravity. And one of the earliest manifestations of this is that if we repeat this exercise for gravity, we get a very different conclusion. So if I scatter electrons against each other, then that amplitude now has to go like g Newton.
But the amplitude of a number, is dimensionless. So it can't just be g Newton. It's g Newton times the energy involved in that collision squared. That's what gives you the probability amplitude for the process. And so you learn that when the energies are tiny compared to Planck energies, which is most of the time, the amplitude is a small number.
Shockingly small number. But conversely, when the energies approached the Planck scale, something really terrible starts to happen. These amplitudes start becoming comparable to 1. And if we even extrapolate, they become bigger than 1.
Now, that can't be. Because if we square the amplitudes and add them all up, we have to get the total probability, which is 1. So you can't have 1 bigger than 1. And so there's a problem. And so this it. This is at an accurate, at a technically accurate level, the first difficulty with quantum gravity is that if we extrapolate, we do what we're supposed to do as radical conservatives, we take the theory, everything is great until we get the Planck scale.
At the Planck scale, it's not that we're disagreeing with the experiment. It's not that that's none of those things. But we know there's something wrong because the theory stops making sense. So the theory starts crying uncle. There's something structurally wrong with these ideas.
OK, so now we've gotten some experience with thinking about units. Now we can quickly understand why it is that the only interactions we see in the world are these little elementary stick figure three point interactions. Because remember, just those interactions we just talked about, let's say for electromagnetism, they were dimensionless. So there were numbers that didn't get big or small. They were the same at all energies.
But let's imagine there really were out there. There even might be out. There even might be other interactions, more complicated interactions which have not just one particle coming out, but two, three, four, many more particles coming out. They might exist. And if they're there, they're telling us about the amplitude for some process. Let's say the particles go in and I produce out of the vacuum not just one photon, but two, three, four, many photons.
But remember something basic that we learned much early on in the lecture, that quantum fluctuations are more and more difficult as you go to longer and longer distances. That's why we have an inverse square law. They're easier and easier as we go to shorter and shorter distances and higher and higher energies. So that means that these probability amplitudes for emitting more and more particles, they can't not depend on energy or on distance.
It becomes easier to make them at higher energies and therefore their strength can't be a pure number. Their strength has got to be a length or a length squared or a length cubed. So even if they're there, if we go to sufficiently low energies or sufficiently long distances, they will be irrelevant. The only things that will matter, the only things that we'll detect and see at long distances are the simplest ones, the elementary three particle interactions.
This basic bit of dimensional analysis, which is much more really understanding it properly and subtly is a very deep and very profound way of thinking about quantum field theory that was pioneered here at Cornell by Ken Wilson, who probably affected the way we think about quantum field theory more than anyone else in the second half of the 20th century and who died a few months ago.
This Wilsonian way of thinking about physics goes under the rubric of effective theories. And it tells us whatever the underlying laws are, whatever they are, we don't even know what they are, it doesn't matter. If we go to sufficiently long distances, they're guaranteed to be described by these very simple three particle stick figure interactions. All right, so that's the answer to question one.
Now, what about question two? Why do we have such a tiny menu of spins? And the first thing I'm going to do in approaching this question is we've just seen over and over again in various guises that the real place where the laws of nature look simple is we go to very, very high energies, very short distances, very, very high energies. And if you're just naive, and it's always a good idea in physics to be as naive as possible until you're forced not to be. If you're naive, you would think, OK, if I'm going to really high energies, the masses of the particle shouldn't matter.
I mean, if you're going to enormous energies, enormous losses and sort of kinetic energies are way, way larger than the MC squared rest mass energy that the particles have. So in some approximation, the real simple part of the laws of nature, we should be able to ignore the masses of the particle in some approximation. It should be true if we go to really high energies compared to their masses. And that's where the physics is simple anyway.
So let's first begin with an approximation where we ignore the masses. The entire story of the Higgs is going to have to do with a footnote and fine print to that statement that we'll come back to. We cannot completely ignore the masses for an important reason that we'll come back to.
But for the moment, let's ignore the masses. Everything I'm going to tell you is going to be completely correct. And in fact, it will tell us why the Higgs had to exist. But the entire story of why the Higgs has to exist has to do with more carefully thinking about the sense in which we can ignore the mass. But anyway, let's do it.
So we're imagining that all the particles are now massless. Enormous energies, all the particles are zipping around almost at the speed of light. Let's imagine they are massless. And now if they're massless, their notion of spin is actually a little more constrained even than you would think. Because the particle's moving around at the speed of light. We can see that maybe it's spinning in the direction in which it's moving or spinning opposite to the direction it's moving. But that's it. Those are the only kinds of angular momentum that it can have.
If the particle had a mass, then let's say we saw it spinning in this direction. Then I can always catch up with it. And if it's at rest, by tilting my head I can see it spinning in lots of other directions. So I can conclude that it has many different kinds of spin states. But if it's massless, I can never do that. I can't go to the frameworks at rest. That's relativity. I can't go to the frame where it's at rest. And so I can only conclude that it has spin in the direction of its motion or opposite to its direction of motion. That's called helicity. The spin of massless particles is called helicity for that reason.
All right. So now let's imagine that we want to look at the interactions between the elementary particles. And again, if you're most naive, you think, look, what's the first simplest process we could talk about? Two particles in, two particles out. And I want to know in detail what is the amplitude for it to come out at this angle, that angle. Depending on the helicities and the energy of the initial state.
That's something I should be able to do. And I want to see it, does it follow from the general principles of relativity and quantum mechanics? Doesn't seem to. Here I just write it down. And the amplitude looks like it could be an arbitrarily complicated function of the energy of the particles A and B and at the angle at which they come out, as well as whatever the helicities of all these guys are.
So it could be a complicated function of the energy and the angle. When I say energy, of course the energy of any one of these particles could change as you go to a different frame of reference. But there's a very natural frame of reference we could be talking about here where the particles are coming in sort of with equal and opposite-- are hitting each other head to head with exactly the same momentum, one this way and one that way.
So we're talking about the energy in that frame of reference. It's called the center of mass energy. So it's a function of the center of mass energy in the angle at which the particles come out. Anyway, I apparently don't need to convince you that this looks like a complicated function in general.
But fortunately, we know that these are not the elementary interactions we're supposed to talk about. The most elementary interactions of all are the three particle interactions. So let's think about what the three particle interactions could look like. And this is much more interesting and much more constrained. It all has to do with relativity again. If we have massless particles, massless particles screaming along in this direction, it's very easy to show that it cannot decay into two other massless particles that go off at some funny angle from the direction at which the particle is going.
You can't do that and conserve both energy and momentum. The only thing that can happen is to have a massless particle moving along. And all it can do is split into two other massless particles that are moving in exactly the same direction. That's all. Those are the only things that are compatible with energy and momentum conservation.
So now what could this amplitude depend on? Remember, this is the basic amplitude out of which we're going to build everything else. So what could this amplitude depend on? Is there choice? Is there arbitrariness? What could it depend on?
Well, it can't depend on any angle anymore because there is no angle. It also can't depend on energy anymore. The reason is that now the energy in the first particle, it depends on the frame of reference. Someone could say it has one energy. Someone else is going to say it has twice the energy. The amplitude shouldn't depend on the frame of reference. That's another feature of both quantum mechanics and relativity. So it can't depend on the energy. Can't depend on the energy. Can't depend on the angle.
What could it depend on? Nothing. In fact, it's technically really does depend on nothing. But it's really the only thing it can depend on is the helicities of the particles and nothing else. Once you specify the helicities, just the laws of relativity and quantum mechanics alone completely specify what that interaction amplitude must be. No choices.
One choice. The one choice is the overall strength of that interaction. Maybe it doesn't happen at all. Maybe it happens with some strength. But that's it. There's just one number that you get to choose. A single number, single interaction string. All right? So that's remarkable. That is already remarkable. It would not be true without both relativity and quantum mechanics. One or the other, could write down zillions of things. Both nailed up to a single constant.
All right, now let's return to the four particle amplitude. So is it now true that we've nailed the three particle amplitude just by the basic properties of the theory? Can we keep going? Are there any choices? The answer is there's no choices. Everything is now fixed. And it's fixed by another basic feature of quantum mechanics I don't have time to explain in detail. But that if you try to make a guess for what this four particle amplitude looks like. Remember, this is some amplitude that depends on energy and angle.
And I want to make a guess for this function of energy an angle. But it needs some an interesting property. This function can't just be some random function. It needs to have an interesting property that it's possible to tune what these initial particles look like such that they're meant to add up to make another real particle. If they make another real particle, the laws of quantum mechanics tell you that this amplitude has to get big in a certain specific sense.
Technically we say it has to develop a pole. And in the neighborhood of that pole, we interpret this four particle process as really being, first that happens, and then that happens. So we know everything about what this guy is supposed to look like in the neighborhood of where it's getting big, in the neighborhood of that pole. And remember, we know everything about these guys from our previous arguments.
So this is now the constraint. If our theoretical physicists locked up in their cabin were competent enough, they would discover all these things and they would set up the problem to this point. Now we're looking for functions of energy and angle that have this wonderful property that where they get big, they have to split up into these pre-determined three particle amplitudes. That has now reduced things to a math problem. You're always very happy in physics when you reduce things merely to a math problem. Because then someone can solve it, maybe a mathematician, or maybe you. But the physics has been taken care of.
And amazingly, now somewhere in this talk, I have to cheat. Here you need one year of grad school.
[LAUGHTER]
But the actual argument fits on a page. You can actually go through this whole argument in about a page of algebra. And what you discover is that it's almost impossible to do. It's almost impossible to write down functions of energy and angle that have the appropriate needed properties to match the laws of quantum mechanics and match what the basic three particle interactions have to look like as dictated by both relativity and quantum mechanics.
And you then discover this. That all the consistent theories you can have at long distances have to have these stick figure interactions. And the only menu of spins you're allowed to have is 0, 1/2, one, 3/2, and two. Again, I remind you that we hadn't seen the 0 before July 4. And we still haven't seen the 3/2, so those are in red. The ones that we had seen already in nature are 1/2, one, and two. And as I mentioned a number of times already, if you do this exercise, you discover that two guy's really special. It's unique. It has a universal interaction.
So all of these things that Einstein did. Einstein thought about these falling elevators and he discovered the principle of equivalence and all these deep facts about classical physics that led to thinking about general relativity. All of those things could have been discovered by much more mediocre theoretical physicists who knew about quantum mechanics 30 years later, just by following your nose.
You see, it has nothing to do. And if you think about Einstein, you think that there's this beautiful geometric picture of the universe and gravity is the way it is because of geometry and all of these pretty and wonderful things. But now we understand things more deeply.
And it's pretty and beautiful, sure, but it's actually much deeper than that. There's nothing you can do about it. There is no other way it could have been. There is no sort of second world that would have been uglier and Einstein's world was prettier and nature chose the prettier one. It's not like that. There's only one way it could be. It's completely inevitable. If you're going to have a spin two particle, it has to mediate gravity and it has to be described by general relativity and there's nothing else you can do about it. So that's an extraordinary fact.
This slide is telling you what I've stated a number of times, that the gross features, the broad features of the laws that describe the universe at long distances, are dictated by these general principles. The only thing we have left, the only arbitrariness we have left, is the menu. How many of this guy, how many of that guy, how many of that guy, how many of that guy.
That one's actually a little interesting. We'll come back to that a lot next time. You can have at most eight of him. So that's kind of interesting. Probably in our universe at most one of him. Definitely only one of that guy though. So first we've got to choose how many of each. And then you have to specify these interaction strengths. And then you're done. Everything else follows from general principles.
All right. So I think given the time constraints, I think I'll probably leave it at that today. And tomorrow we'll start by finishing the story of the Higgs. And then Thursday, we'll start by finishing this story with the caveat to all of the statements we made having to do with ignoring the masses of particles. So the entire story of the Higgs just goes back and thinks slightly more carefully about what it takes to ignore the masses of the particles and fills in a little gap in that logic.
By the way, in this picture of the Higgs, we will never use the word molasses. We'll never talk about things bumping into something that fills the universe. All of that stuff is pure metaphorical nonsense. And it's unfortunate, but that's the way the Higgs is discussed. I've done it myself. I did it in the messenger lectures. So I've done it myself. I'm not just criticizing my colleagues for doing it.
But the real reason the Higgs had to exist is much deeper, is much more closely related to these things that I'm telling you about and has nothing to do with that metaphor. In fact, there's ways of talking about it where you never talk about molasses, certainly never talk about molasses. You never even ever draw these little pictures of a Mexican hat. We never say the word symmetry breaking.
All of those things, well some of them are real garbage. Some of them are not garbage, but are just one of many languages for talking about what the physics is. And there's a really invariant hardcore fact sitting underneath it, which is much more closely related to these structural facts about the way the world works.
So that's what we'll start with next time to finish the story of the 20th century. And then after having hopefully convinced you that this philosophy has worked so spectacularly well for hundreds of years, I will then tell you how many of us are pursuing the philosophy of radical conservatism to try and attack the problems of the 21st century. All right, thanks.
[APPLAUSE]
SPEAKER 1: Thank you very much. Time for some questions for Nima.
AUDIENCE: Can you give the quantam gravity supposed to be an open issue? Like you present the graviton as something you think is a [? totally ?] sorted-out idea.
NIMA ARKANI-HAMED: We haven't seen a single graviton. We have not seen a single graviton. But we've seen large coherent collections of gravitons as gravitational waves indirectly. But you see, this is very much along the lines of what we're talking about.
It's true we have not experimentally seen a single graviton. So at that point, you could say why bother talking about quantum? We don't even know a graviton exists. Let's wait till the experiments discover a single graviton. You could wait a very, very, very long time before that ever happens. And yet there's no reason to think the graviton doesn't exist.
In fact, if we apply all of these rules and the laws of quantum mechanics and relativity beautifully are not only compatible with the idea of a graviton as I'm showing you, but in fact explain a huge amount about the properties of general relativity, which otherwise looks like it's some bolts from the blue of the genius of Einstein. But rather the other way around. If you didn't know there's gravitons, they obey the same kind of laws as everything else, you follow your nose, and you discover general relativity, even if you're a dumb guy.
That's the real sense of progress. Great ideas in physics are not for geniuses. Real progress happens when anyone can do it and anyone gets it and it follows inevitably from principles that are simple to state. So that's one of many reasons to suspect that the gravitons exist. You think they do exist, you apply quantum mechanical rules to them, you explain features of classical general relativity. You see how amazing that is, right?
But you could say, no, maybe it's not there. No, we don't know for sure. That's not the radically conservative philosophy.
AUDIENCE: [INAUDIBLE].
NIMA ARKANI-HAMED: No, absolutely not. No. And we saw where the contradictions are. There's no problem. There's no problem with quantum gravity so long as we're imagining these particles banging into each other at low energies. The difficulties arise when you get to , very, very high energies and Planckian distances, Planckian curvatures more properly speaking.
So we do that all the time. Theoretical physicists obsess about black holes. You can say, do we know there are black holes out there? I mean, yes we say there are these astrophysical black holes. We think there are black holes. But you say, have we been there to know for sure that there is a black hole horizon? Maybe there is something else that makes it look like a black hole. Sort of indirect saying it's a black hole. Much like it's sort of indirect saying we've seen gravitational radiation from the energy loss from pulsars and things like that.
So those are all indirect. Maybe there's no black holes. How do you know? You're extrapolating this crazy solution of Schwarzschild. How do you know there are black holes? Well, nothing's going wrong. The theory isn't crying uncle. Everything's fine. In fact, once again, you think more deeply about black holes, you start unifying aspects of thermodynamics and general relativity that are otherwise mysterious.
So there are all these hints as you go along that there are milestones that you pass purely internally, theoretically, that make you think you're on the right track. More things fit, more things acquire a simpler common description and so on. So one could have, as a sort of strict, strict empiricist, one could have the attitude that until I see an experiment, I'm not going to think about it. And that's a perfectly legitimate attitude to have, but it won't get you very far as a theoretical physicist.
But I'm glad you raised the question, because it's one of the most elementary, but most central features of the sort of attitude. You take what you have despite the fact that it was discovered in place x. You extrapolate as widely as you possibly can without apology, without fear, until either data or the theory itself tells you that something is going wrong.
And for example, in the case of quantum gravity here, it's the theory itself telling us something is wrong once we start approaching Planckian energies. And next time, pursuing these things, I will show you why, for example, why theoretical physicists are excited about string theory. And what you could do as anybody to make a proposal for a theory of quantum gravity and have people go gaga with excitement over it. There's a formula. It's not a sociological thing. It's not you have to be in the club of cool people who study the subject.
There is a formula. You have to produce a formula. I'll show you the formula you have to produce next time. But you have to produce a formula with certain properties and if you do it, people will go bananas with excitement. They went bananas with excitement when string theory did it 40 years ago. In fact, they didn't know it was strings when it did it. They found the formula first before they realized that the reason for it was strings.
But in this way of thinking, both the old ideas like relativity, quantum mechanics, the Higgs, the rest of them, and the newer ones, super symmetry, string theory, et cetera, they're all part of a continuum. They're all continuously connected to each other. And the same sorts of people study all of them. And it's for this reason. Some of them have been confirmed by experiments, some of them haven't. But all of them arise from the same basic bedrock of trying to build things compatible with these [? ocurrences. ?]
SPEAKER 1: More questions?
NIMA ARKANI-HAMED: Yes.
AUDIENCE: Hi there. As a freshman, this kind of goes my head mostly. My world is a block hitting another block and that--
NIMA ARKANI-HAMED: That's the way it should be. That's the way it should be.
[LAUGHTER]
Just avoid it hitting your head.
AUDIENCE: Out of curiosity, you kind of teased at this cliffhanger. What is that three halves spin multiple thing we've been looking for?
NIMA ARKANI-HAMED: Come next time.
[LAUGHTER]
It's supersymmetry. That's why people are excited about supersymmetry. To have that spin 3/2 particle, nature has to have a properly called supersymmetry. And that's why people are excited about supersymmetry. Theorists are excited about supersymmetry because it's the last thing nature can do compatible with these principles that we have not yet seen it do.
And in fact, the Higgs is also something nature can do compatible to these principles that we had not seen it do before July 4. So we've got to add 0. We got to turn the 0 from a red to a black. We finally found one. So the 3/2 is the only thing that's left.
Again, it's not a question of aesthetics. Do you think it's pretty? It's not pretty or cute. It's the answer to this question. It's an invariant answer. What can nature do? And 3/2, which is associated with super symmetry, is the last thing nature can do that we haven't seen it do. Doesn't mean that we'll see it tomorrow. Doesn't mean that we'll see it in our lifetimes.
There are some reasons to suspect we might see it tomorrow. There is some reason to suspect we should have seen it yesterday or in the past. So we don't understand everything. But the broad reason we're excited about it is that the last thing nature can do compatible with these general principles. And if we find it, it will reassure us that nature does everything it can do.
And then there is another set of questions we have to explore having to do with some of the other fine print in this talk. I told you when you go to sufficiently long distances everything has going to look like this. Well what happens when you go to high energies? What happens when you go to very short distances? Then a new world begins. And we have to think new thoughts and make a new explorations.
But keep working on those blocks. That's where it all comes from, you see. I mean it rather more seriously and specifically that not only is it not the case that the classical mechanics is more boring than quantum mechanics. Not only is it the case that it's the limiting case of the other theories. But in fact, the similarity and the commonality between old physics and new physics runs much deeper than that.
And there are features of classical mechanics that only make sense because the world is quantum mechanical. And so every single part of this edifice of physics is connected to every other part in deep ways. So you really need to understand every. There's this humongous castle that's been built for 400 years. And often people want to figure out what's going on.
Being obsessed with what's going on today at the cutting edge is like walking into this enormous castle and running to the top floor to the room under renovation and then saying, geez, there's plaster and the walls are peeling a little bit. This doesn't make much sense. But no, there's this enormous castle underneath it. There's this enormous castle underneath. And you need to see the whole castle before you can really understand what the what the point of the renovation and what the view might look like if you managed to knock out that wall and so on and so forth. So stay in school.
[LAUGHTER]
In the first of two public lectures as an A.D. White Professor-at-Large, theoretical physicist Nima Arkani-Hamed describes the different avenues being pursued in attacking the central problems of fundamental physics today, guided by the rough-and-ready philosophy of "radical conservatism," and speculates on where this philosophy might lead us in this century.
