SPEAKER: All right, I want to welcome you to Professor Nima Arkani-Hamed's second lecture on the philosophy of fundamental physics. I assume most of you were here on Tuesday for the first lecture, so I will not give you an extensive introduction.
But for those of you who missed Tuesday, Nima is the most influential theoretical physicist of his generation. He got many prizes and deserves many more, [INAUDIBLE]. And among many amazing, awesome things he's done in his life is the messenger lectures that he gave here in this room three years ago. Those are online so you should watch them. So without much ado.
NIMA ARKANI-HAMED: Thanks so much, Maxim. So last time we ended with making this remarkable point that the structure of the laws of nature at sufficiently long distances, the kinds of particles and interactions that we can have and their properties, are almost completely dictated by the general principles of relativity and quantum mechanics that, as I said a number of times, shockingly constrain what those theories can look like.
In fact, they remove all the freedom we have for what physics can look like at sufficiently long distances to the following. We get to choose a finite set of particles with spins taken from this list. 0, 1/2, one, 3/2, and two. The spin two guy is unique if it's there. It has all the properties of gravity and we know that ahead of time from completely general principles.
And these particles interact or the only interactions that matter at sufficiently long distances or low energies are these three particles. The most basic interaction you can possibly have between three of them, the properties of those interactions are completely dictated by the principles of relativity and quantum mechanics up to some numbers that just specify the strength of the interaction. So that is a remarkable fact that wasn't true before we knew about both relativity and quantum mechanics, but we now know to be true.
All right, so what's left? So what's some of the more recent fuss about, for example? So what about the Higgs? What's going on with the story of the Higgs? So I want to start this lecture by explaining how the Higgs fits into the structure.
And if you remember from last time, to even get into the world that we were just talking about, we made the simplifying assumption. We said we're going to imagine going to energies that are very, very high compared to the masses of the particles. And therefore, in that approximation, we should be able to ignore the masses of the particles and we basically treat them as massless.
So imagine that mass is some dirt at very low energies, but at very, very high energies we're talking about just massless particles. Everything that I told you about the restriction was on the interactions of massless particles and that should be a good approximation in the regime where we're at very, very high energies.
But there is one little bit of fine print with that. Which is something else important about life that we can discover by counting on our hands. There is an important difference between massless and massive particles that have spin. So let's say we talk about a massive spin one particle. Remember the W boson is a massive particle, mediates the weak interactions, it's massive, it has spin one. Because it's spin one, imagine you have a W particle. So it has spin.
So you think of it as a little top. As the W particle is zipping around, I can always go to a frame of reference where it's at rest. And let's say I find it in that frame spinning up. Then if I tilt my head, I can see it spinning sideways in all three possible directions. So the fact that I see it spinning up immediately tells me that all the other spin possibilities also exist. And so how many degrees of freedom are there? There are three degrees of freedom associated with a massive spin one particle.
A massless spin one particle, on the other hand, as we already said on Tuesday, it's different. There's no frame where you can go to catch up with it. And therefore if you find a massless particle whizzing around, moving that way with its spin, let's say, spinning in this direction of motion, I can't conclude that it has all the other spins as well. Because I can never go to a reference frame where it's at rest.
And therefore, massless spin one particles only have two degrees of freedom. So two is not equal to three. And that's a problem. That means that when we said that we want to go to a very, very high energies and ignore the mass, yes it's true, we can ignore the mass compared to the energy. But the mass of spin particle has one extra degree of freedom that we have to account for as we go to very high energies.
Now, here I'm cheating a little bit. So this is slightly metaphorical. But it would take me maybe another five minutes to say this perfectly accurately. But it basically captures the correct spirit of the argument. Something that you're all familiar with probably is that in special relativity, things can contract and expand when you go to frames of reference that are boosted relative to each other. So you can have length contraction, time dilation, and things like that.
But something that you also know is that, so let's say you have a rod like this and the rod is moving in this direction. Then it's said to suffer length contraction. But if the rod is like this and is moving in that direction, then the length of the rod doesn't change. It's only the things that are aligned in the direction in which you're moving in that suffer these contractions and dilations.
So let's imagine that we have our W particle. Here's a W particle. It could be spinning up or it could be spinning sideways. But let's say I boost it a lot. I boost it a lot. And let's say there is some probability defined it spinning up or spinning sideways. Well, that probably defined it spinning up or spinning sideways. Or the amplitude to find it spinning up or spinning sideways doesn't change.
But things are different if I have the W particle spinning in the direction in which it's going to be boosted. Here there is one of these dilation contraction kinds of effects. And in fact, the amplitude to find it in that configuration grows with energy. And it grows with energy roughly as the energy divided by the mass of the particle.
Now, strictly speaking, this isn't an accurate statement because it's not a statement about a single W particle. Because obviously then that would allow me, if this was true, it would allow me to determine whether I am in the rest frame of the W or in some other frame.
But the more correct statement is if you have two of them that are banging into each other, then if you bang them into each other at a higher and higher energies, the relative amplitude to find them in that state grows relative to the other ones where they're aligned transverse to the direction in which the beams are moving. Because of one of these special relativistic effects.
Now, that has a very important consequence. If you remember from an early part of Tuesday's lecture, we talked about these basic interaction strengths, these basic three point interaction strengths that we have between the elementary particles. And we said that the strengths of those interactions were dimensionless and that those dimensionless strengths of the interactions also gave us a measure for the probability amplitude, when I banged these things into each other.
So remember we said when electrons bang into each other, the chances that anything happens at all, that the electrons come zipping off at 20 degrees compared to the initial angle that they were colliding in was a small number. It was this number q squared all squared. So it was around 10 to the minus 4 was the probability for anything to happen at all.
The same thing is true for these W's. Exactly the same thing is true for these W's naively. All the strengths of these interactions are still dimensionless numbers. They're about the same two squared. The amplitude is still about a percent. But that's only true for those components where for those spin components of the W's, which don't suffer this boost enhancement effect, this relativity boost enhancement effect.
If instead we collide those components of these W's, which have a spin in the direction in which they're moving, then we find that while the amplitude has this piece that looks like it's just the same old q squared as before, there is an additional enhancement due to just the relative probability of finding them in that state. Just this sort of basic kinematical fact.
And now we're in trouble. Because this is no longer something that's just a small number. It grows with energy. And since this number is about a percent, you can even estimate the energy at which this whole thing starts getting us in trouble. Remember, we talked about something analogous for gravity. That the difficulty with quantum gravity really, or the first difficulty with the quantum gravity is that the probability amplitude for particles that collide with each other gravitationally starts exceeding one when the energies become comparable to these huge Planckian energies. And that's nonsense, because probabilities can't be bigger than one.
Well, we run into exactly the same problem, except much, much closer to home. We can ask, when does this thing become of order one? Well, this q squared is about a percent. The mass of the W is around 100 times the mass of the proton, 100 GEV. And so this thing becomes a large and nonsensical when the energy becomes larger than, when you put the correct numbers in around 1,200 GEV. So that's rather startling.
So if we just took the particles we'd seen in nature before July 4, 2012 and we just asked, can I take the set of particles with the rules that I know, can I make consistent predictions for any in principle physical experiment I could imagine doing? The answer is no. And not because it was contradicting any experiment. There was not one experiment that it was contradicting.
No experiments had been done with sufficient precision to see this effect. But the theory itself was crying out that there was a problem. The theory itself was saying that you can't trust me anymore. And you can't trust me not very far away. You can't trust me just a factor of five or six or seven higher energies than the energies that it had been probed in other ways.
Now, there are other problems of this sort. Exactly the same difficulties with the same spin components of the W and other particles that you may or may not have heard of. The top quark, which is the sort of heaviest particle we know in the standard model. It's kind of a partner of the up quark that we find in protons and neutrons. Exactly the same kind of amplitude by the very dint of the heaviness of this particle becomes nonsense and bigger than one around 1,000 GEVs, around again, roughly 10 times higher energies than the mass of the particles themselves.
You see, when we started our discussion before, we said let's ignore the mass of the particles. We ignore the mass of the particle, we limit where all the particles are massless, everything is beautiful, we have this constrained universe. All its properties are dictated. And you might think we're done. We understand sort of everything about it. And mass is just a little detail that sort of kicks in at very low energies.
But mass is not quite a detail because of this degree of freedom problem. And so while we might have a naive expectation that we take the sort of mass of particles that we have, we ignore the masses, we go to very high energies, and everything is hunky dory until we have these humongous energies that are Planckian where the gravitational processes start becoming strong and uncontrolled. But the actual expectation was that something broke down much, much closer to home. Only a factor of 10 above the energy scales that we had been to. All right.
So what could it be? So that means that that despite, again I emphasize, despite not a single disagreement with experiment, just the structure of the theory itself suggests that there is something wrong and we need something new. Now something rather remarkable happens. Because while I don't have time to explain it, this sort of phenomenon occurs all over the place elsewhere in physics. The sort of phenomenon where you have some degrees of freedom that appear to be elementary degrees of freedom.
But you look at how they interact with each other and the interaction gets stronger and stronger as you go to shorter and shorter distances. This happens rather frequently. And normally what happens is that indeed, the interactions just become so strong that at some point if you go to even higher energies, you find that the sort of particles that you're talking about melts into totally other kinds of degrees of freedom.
For example, this is true of the protons and the neutrons themselves. Remember, the protons and neutrons are made out of up and down quarks. But we don't see that until we go to very short distances compared to the size of the proton and neutron at around 10 to the minus 14 centimeters.
And indeed, for analogous kinds of particles, protons, neutrons, other things called pylons, which are the progenitor, before people really understood how the strong interactions worked in terms of quarks and gluons were thought to be responsible for holding the protons and neutrons together and nuclei. The interactions between these particles start weak. They get stronger as you go to higher and higher energies. And then what happens isn't that they just kind of stay there. They just melt to completely different stuff. They turn into quarks and gluons and completely different sorts of degrees of freedom.
That's typically the way it happens. That's the way it happens all the time elsewhere in physics. And if what was going on with this physics, the problems with the W's and the top quarks that we just talked about. If what was going on with this physics was similar to things that happen all over the place elsewhere in physics, you would expect the same thing would happen. The interactions would really become strong somewhere. They'd melt in some totally different degrees of freedom. And we'd want to know what that looks like.
Now, it was interesting. People explored this idea quite vigorously in the early 1980s, even really through today, although fewer and fewer people are exploring it for the reason that I'll allude to in a second. So it's a very natural idea. People explored it. But they kept finding theoretical difficulties with it. So while it works all sorts of places elsewhere in physics, it did not seem to work as nicely here. It had all sorts of internal theoretical difficulties.
Nonetheless, ignoring that, people tried to find some evidence for this kind of explanation already by the late 1980s and the early 1990s. So let's say we have these. Here are our friends, the dangerous guys, these longitudinal, these funny spin components of the W's. The W's and the Z particles interact with each other in some prescribed way.
And if the interactions between these guys do become strong somewhere, if nothing new happens until you hit this scale of around 1,000 GEV where this stuff finally becomes strongly interacting, you can make a good estimate for quantum mechanical corrections to what to most of you would be completely abstruse and sort of irrelevant things like the detailed coupling between a Z particle and an electron and a positron, for example.
But theorists did some heroic calculations to actually compute what you would expect. And they found that you would expect corrections, deviations from the simplest expectations of order about a percent. I'm going through so much detail just so you have an idea of the precision with which we can understand the world and how seemingly small differences can have huge ramifications.
Here people did a calculation and they found that they should expect 1% level corrections. This was already heroic calculations by theorists. But then there were even more heroic measurements done by experimentalists. So experimentalists went out and they produced at laboratories in CERN and at Stanford, they produced roughly five million of these Z particles.
Five million is so many Z particles that you can actually measure the strength of the interaction between the Z's and all these particles not just to 1% accuracy, but to one tenth of a percent accuracy. So that's what people did. You don't need to understand that plot that I've put up in detail. But people measured it to a tenth of a percent accuracy and they found no deviation even at the tenth of a percent accuracy.
So this difference between the measured non deviation at the tenth of a percent. And the robust prediction at the percent level of what was going on here was the same as what was going on all the time elsewhere and physics was an early indication already by the late '80s and early '90s that something new was up with this physics. Something unlike anything we'd seen elsewhere in nature.
And not only was this given as an indication that that canonical picture was wrong, but it also gave us an understanding for where the new physics really should come in. The new physics couldn't be delayed up to 1,000 GEV. The new physics had to be somewhere between 60 and 200 GEV. That's what came out of both the theoretical calculations and these very precise experimental observations. All right.
But the fact that we don't just have these particles become enormously strongly interacting and melting to different degrees of freedom means that we need something simpler. But we need something new. So here we are. Here are the usual ways that these W's interact with each other. So that gives us the trouble. This is getting big as we go to higher and higher energies. So what are we going to do about it?
Well, there has to be something new. There has to be some other process that competes with this such that the sum of them stops getting big as we go to higher and higher energies. And that new thing had better come in beneath around 200 GEV and more than about 60 GEV. Just from these very general arguments we had before. Now, who could that new guy be?
So now you see we're in an interesting spot. Because we need something new. And now if you're the radically conservative quantum field theorist, if you're conservatively radical, you say, oh, everything has gone to hell again. We need to start all over. God knows what it means. The mists of time, the beginning of the universe. And it might be that all these things are related to each other and the sort of philosophy of how we think about things.
Or you just follow your nose a little bit and you say, OK, we need something new. Let's look at what we're allowed to do. So we go back to the previous lecture. What are we allowed to do? Well we have our menu of spins of particles that we're allowed to have. This x can't be some random leprechaun. The x has got to fit into the theoretical structure that we talked about before.
So the first thing we should decide is what spin can x have? Well, for good reasons, it can only be 0, 1, 2, 3, 4, and so on. We can immediately x out 3, 4 and higher because those weren't even allowed. We can x out 2 because we have the spin two guy already. It's the graviton.
Now, spin one is kind of interesting. This massive spin guy is kind of interesting. Because what we'd then be doing is trying to solve the problems of our massive W particle by adding a new massive W particle, some W prime particle or something. But then there's a chicken and egg problem. So who solves the problem of the W prime particle? And then OK, there's an x double prime. Is that another W triple prime and you keep going?
So this is not impossible. And there is, in fact, lots of work that was done. People like Chava right here and spent some time thinking about these possibilities. It's not stupid at all. But it feels like there's a Russian doll structure there that doesn't feel like the most simplest possibility. So while it's not excluded theoretically, and it was worth looking for, it was clearly not as simple as the simplest possibility, which is it's just a single particle of spin 0.
All right. Now, so we're going to add a single particle of spin 0. And that single particle of spin 0, when x is a particle of spin 0, it's called the Higgs. That's the Higgs particle.
Now, you'll notice what we'll come back to in a second. I have not mentioned the word molasses once. I have not drawn a little picture like that. I have not talked about symmetry and symmetry breaking. And that's because all of those things, as I said last time, some of them are just really terrible metaphors. Some of them aren't terrible metaphors, but they're often mistaken as being a fundamental fact about nature, where they're merely a linguistic convenience of certain kinds of theoretical physicists who like to talk about things in a certain way.
And I've talked about them in a different way where I've never said those things. And the fact that you never have to say those things is actually important. It tells you that they're in the head of the theorists who are talking to you about them and they're not actually a feature of the real world.
All right. Now, there's something else very important that we learn from this picture. That whatever this x is, since its job in life is to solve the problem with the W's banging into each other. And with the W's banging into the top quarks, the strength with which this x interacts with the W's has to be correlated with that strength. The strength in which it interacts with the T's has going to be correlated with that strength. So these strengths are totally nailed by the job that this x has to do.
So we know those couplings ahead of time. The only thing we don't know about x in detail is its mass. But we already know from these indirect observations from the late '80s and early '90s it has to be between 60 and 200. So far so good. I have caught you up to the year in 1992 in this quest.
Oh, and I should say that with the addition of this one extra particle now, what happens, now we get to go to very high energies. Now we can ignore the masses, because we've taken care of the one extra degree of freedom that we're talking about.
Now we go to very high energies and we actually find that this x, the Higgs, as well as those funny extra spin components of the W and the Z, they all beautifully unite together into one object at very, very high energies. And this one object at very high energies, now we can ignore all the masses, now all the degrees of freedom are taken care of, and now these guys are described by precisely the rules that we got to by the end of last time.
So what this Higgs particle has done is allow us to smoothly merge the world of the large and massive with the world of the small and massless. You would naively think that it's a smooth merge from one to the other. That was our naive expectation from our discussion on Tuesday where we said when we go to higher energies we should be able to ignore the masses of the particles.
But because of this interesting subtlety with degrees of freedom, it's not true in precisely that naive way. In fact, the distinction between massive and massless starts growing as you go to higher and higher energies. But all we have to do is add a single particle in order to erase the distinction and smoothly connect between massive and low energies and massless and high energies. That's what the Higgs particle does for us.
Now, if you remember, when we made our list of all the particles that had been seen, we hadn't seen the spin 0 particle. We'd only seen from the allowed list 0, 1/2, one, 3/2, and two, we'd only the particles of spin 1/2, one, and two. So spin 0 is allowed by the rules, but had never been seen before.
And that's why this was a brave proposal. You're saying there has to be a particle there between 60 and 200 GEV with exactly these properties, even though no one had ever seen such a thing before. And you can do it because nature can do it. And it solves the problem. And you can predict its properties ahead of time and you can set it up for experimentalists to look for. Despite its tremendous simplicity, it was a brave proposal. Nothing like this had ever been seen before in physics.
So of course, we go to the LHC. I won't tell you all about the marvels of the Large Hadron Collider. Other than just to remind you that at the LHC, we bang protons into each other at energies of thousands of GEV. And in each proton, we have quarks and gluons. And what we're really interested in are the head on collisions of the quarks and gluons inside the protons, those head on collisions at these humongous energies.
We know they're head on because the product of the decays that come out of these collisions come off at big angles corresponding compared to the angle with which the initial protons are beamed. Those head on collisions probe what's going on at very short distances. And we capture the debris of those collisions. The we here is the royal we. It's the experimentalists capture of the debris of these collisions of these enormous defectors.
But for the purposes of this story, what I want to emphasize is that every single thing about the way the Higgs in particular was produced at the LHC was completely nailed ahead of time by its job, by what the Higgs had to do. So the way the Higgs is produced at the LHC. The Higgs, it only has sizable couplings, so things like the W the top quarks. And there are not too many top quarks inside the proton.
But nonetheless, when we bang the protons into each other at very high energies, a gluon from one proton, the gluon from the other one can produce out of the vacuum temporarily a top quark empty top pair. And that in turn can produce a Higgs, because the Higgs has a big coupling to these guys. But I remind you, that interaction between the Higgs and the tops, we know ahead of time by one of the Higgs's jobs. So that's how you make the Higgs. We know everything about the rate at which it's made.
Now, the Higgs then also decays very rapidly. Most of the time it decays, for example, to, again, particles most of you haven't heard of. The bottom quark, an anti bottom quark for example. That's the way the Higgs decays a good fraction of the time. And when the Higgs was back in July 2012, there were maybe half a million Higgs's produced by the LHC. Something like that.
You think that's lots and lots and lots of them. So you have half a million extra bottoms and anti bottoms produced. But unfortunately billions of bottoms and anti bottoms are produced from completely ordinary processes. So it totally swamps out these few hundred thousand extra guys that you added after all this work.
So do you take your ball and go home? Not if you're an experimental particle physicist. You say, look, some tiny fraction of the time, the Higgs must decay in some other way. And here's one that must happen. Because you remember, the Higgs must have this coupling to W's. That's another interaction that's totally nailed by its job. It must couple to W's. And when it couples to these W's, they're charged particles. So they can be virtually excited and eventually decay to two photons.
0.1% of the time the Higgs decays into two photons. So 0.1% of those 500,000 Higgs's will decay to two photons. And that you have a chance of seeing because the rate from other processes for make two photons is not as big. It's still way bigger than this. It's still like 50 times bigger than this. But you have a fighting chance because the photons that come from the Higgs, roughly speaking, their energies have got to add up to the mass of the Higgs particle. Whereas if they're just coming willy nilly, they would sort of be all over the place.
So what you should find is if you take the photons, you sum their energies to something slightly more accurate, you look at the invariant mass. The so-called invariant mass of the two photons, you should see a bump. You should see a bump somewhere. And that bump should be in the neighborhood of what the mass of the Higgs is. Everything about this process other than the mass of the Higgs we know ahead of time. So there is just one number that we don't know. And of course, that bump was exactly what was found.
So I want you to take a second to appreciate how remarkable this is. There's a problem. It was a problem that had nothing to do with a conflict between theory and experiment. There was no conflict between theory and experiment. There was a purely internal theoretical problem in our understanding of the world. We go about trying to solve that problem.
We find it's extremely restrictive what we can do. In many ways, the nicest possibility is just adding this one spin 0 particle. From these exquisite theoretical calculations from 20 years ago and those observations, we even narrowed down the range in which it could exist to be between 60 and 200.
Now, accelerators in the interim kept looking for Higgs's, kept looking for Higgs's, not finding them, not finding them. I would talk to colleagues from other parts of physics as the window narrowed. And they would say, you really think the Higgs is going to be there now? Right. We ruled out almost all of them. What are the chances it's going to be just in that little window?
And I would have to tell them, look, unfortunately we knew ahead of time that this is where it's going to be. It just so happens the place that it ended up, it had to be, was difficult experimentally. So we just have to be patient. Yeah right, the window's going to close and no one's going to find it.
I had all sorts of interesting bets with all sorts of people. I bet a year of salary that the Higgs would be discovered, jokingly to a reporter, who then published it in some newspaper and my inbox was filled with e-mails from crackpots taking me up on this offer. Amusingly, none of them have paid up since.
But we knew ahead of time that it had to be there. It was just tough luck that where it had to be was more difficult experimentally, so we had to wait a long time in order to find it. But the fact that this showed up in the range it was supposed to show up with a rate that was compatible with what we expected. Because everything about the rate was nailed once we know what the mass is. It could have easily been a factor of a million off in either direction. And on the nose out of the box to within a 50% and well within experimental accuracy, it came out correct.
So the discovery of the Higgs, it's first of all a triumph for experiment. But it's also a triumph for theory. It's a triumph for relativity and quantum mechanics and this incredible theoretical structure that's been thrust on us by those developments of the early part of the 20th century and a testament to how well we've understood them that we can make a prediction for something like this, for an experiment no one has done involving a sort of particle no one has seen and getting everything about it right, essentially, out of the box. This part of the talk is supposed to convince you that we know what we're doing. And that belief in principles paid off, in this case. And we got to turn that red 0 into a black 0.
So now all of the things nature can do, we get to add, for the first time, a spin 0 particle. And that's one of the reasons why the Higgs is actually novel and exciting. It's not just hype. We're actually ourselves really excited that the Higgs has been discovered because of this.
And it's actually the first time that you see by the time theorists had realized that the menu of possibilities was 0, 1/2, one, 3/2 and two, the 1/2, one, and two had already been known about for a long time. So this is the first time you got to say that something that hadn't been seen before must be seen and it was seen. So now we have that guy that's left. And we'll come back and talk about him momentarily.
All right. Now, let me just take one second and talk about Higgs metaphors. So the first metaphor is this one, which as I said, I used even in, I'm ashamed to say, I used in my messenger lectures. I've used all over the place for the sake of time. But you do something often enough, you feel dirty.
So this is just wrong. The picture that the Higgs is something filling the universe and particles bump into it and thereby slow down and get mass. Now, the problem every time I use this metaphor in a public talk, is there is some smart 12-year-old kid in the back of the room would say, isn't that just like the ether?
And the subtext of the question is, didn't you idiots learn anything? Don't go about filling the universe with crap. It's not going to be there and then you look like an idiot 50 years later. And the best I could do when using this metaphor, that's what it looks like. That's just what it looks like.
And the best I could do when the smart aleck asked the question is I said, yes, something that fills the universe that's not like the ether. So that's not very satisfying. But it's just wrong. It's just terrible, it's misleading, it's wrong. So just forget about that one.
But one often also sees pictures like this, pictures that there's beautiful symmetries, the world is governed by beautiful cemeteries. And sometimes the symmetries are good and sometimes the symmetries are broken and maybe you'll see pictures of what are called Mexican hat potentials and things like that. And this is correct. It's perfectly correct. It reflects one way of talking about the physics. But as I said already, it's just a language. And there are other ways of talking about the physics, as I just did, which never use the language.
Not just laypeople, but physicists themselves tend to suffer from a problem of mistaking the formalism with which they describe nature with what's going on in nature itself. And of course, it's very natural to do it when you have a very powerful formalism. And particularly natural to do when you only have one formalism, because it's the only way you can go about doing things.
But it's dangerous because you might one day discover other ways of thinking about it, other formalisms, other pictures for what's going on, which give you exactly the same answers but with totally different words. And if you're too wedded to the picture that's behind the first set of words, it might obstruct you from discovering the second set of words.
I actually suspect that in 100 years, nobody will be talking about the Higgs like this anymore. I mean, it will be correct, it will be done like we talk about the way people describe other parts of old classical physics using old notation and everything is perfectly correct. But there are more powerful ways of thinking about things that the language won't be as particularly useful in the future as it is now.
But even ignoring that, that may or may not turn out to be true, the real difficulty with this picture is it gives you a sense that where did all of these things come from? If I'm describing this to you, and you're not a physicist, where do we pull these things out of a hat? We draw these pictures like this. I mean, why are we doing it?
It gives you the feeling that we're inventing nymphs and leprechauns and dryads around every tree and in every brook. And OK, maybe sometimes it works, sometimes it doesn't. But you might get a sense that we stumble into some pictures. And then maybe there are totally other ones no one else knows about. It doesn't do justice to what's actually going on.
What's really actually going on is what I told you. And what the Higgs actually does is what I told you. What the real purpose of the Higgs is to allow us to take the description of massive particles at low energies and continuously connect it to massless particles at high energies. That's what it really does. That will never change. That will be as true today as it is in 500 years. The rest of the language that we use to wrap around it may well change. And this picture is part of the language, not part of what's actually going on.
But let me now end this part of the discussion by saying that with the discovery of the Higgs, the discovery of the Higgs really marks a transition point between physics before this point and afterwards. Up to now, at no point in the history of our subject were we in a situation that we had a theory that on the one hand we knew correctly described nature at the distances we were probing, or at the energies that we were probing, and on the other hand, could be self consistently extrapolated to much, much higher energies. We always have some phenomena we just didn't understand.
Or even if we understood it so that it explained things at energies that were available, it predicted its own demise like within a factor of 10 higher energies. Just like we talked about with the story of the W and the Z and the top. So we're always in the situation that we knew something had to be around the corner. Something just had to be around the corner, either from direct hints from experiments or from the internally the theoretical structure telling us something was breaking down. Right around the corner.
With the discovery of the Higgs for the first time in our history, we have a self-consistent theory that can be extrapolated not to infinite energies. Eventually we hit these Planckian energies and we don't know what's going on there and there's mysteries of gravity and we'll talk about that in a moment. But it can be extrapolated to exponential higher energies than we have.
So that means that we can now ask, and we're forced to ask, to make progress different sorts of questions about physics. The questions that are left on the table have two qualitative characters that we'll spend a little bit of time talking about. One of them has to do with the fact that eventually when we go through enormously high energies close to these Planckian energies, the idea of space and time breaks down, quantum gravity becomes important. And there are some essentially new ideas are needed. That's one set of questions.
And another one is closer to home. While we do have this theory which if you take now what we've seen, including the Higgs, makes sense. It can be extrapolated to exponentially high energies. You can ask, since we understand the theoretical structure so well, how plausible does it look that this theory would have actually arisen from some more fundamental theory at short distances?
In other words, we're not asking a mechanical question about how the world works. We understand that, at least until we get to Planckian energies. We're asking a more meta question now, a structural question. How plausible is the world that we actually got? I mean, is it reasonable that we had a world that looks like this?
And well, the answer to the second question is, it doesn't look plausible at all. It looks ludicrous that we got this world. And it's very closely connected with the fact that we discovered the Higgs after all. So those are the two sorts of questions that we have to confront. And I want to tell you a little bit about them. So I'll go relatively rapidly through some of these subjects.
But first, let's talk about the first issue that has to do with the end of the idea of space time. So we talked about various aspects of this. I think this is another slide from my old messenger lectures. One way of describing the difficulties with quantum mechanics and gravity has to do with taking a little magnifying glass and trying to probe shorter and shorter distances.
Look at what's going on in the vacuum at shorter and shorter distances. Because of the uncertainty principle, we have to put more and more energy into smaller and smaller distances until at some point we put so much energy into such a small region that we collapse the region into a black hole. If there was no gravity, this would never happen. But because of gravity, eventually at short enough distances, there is so much energy in such a small region of space that you collapse it into a black hole.
And when does that happen? You guessed it. It's at the Planck length around 10 to the minus 33 centimeters. And that means that there's no operational way of talking about space and time separations down at these minuscule scales. Every time this has happened to us before in physics, every time there are some ideas that we can't even give operational meaning to, it means that they're ill-defined and they have to emerge from something else.
In this case, the thing that's ill defined has to emerge from more primitive building blocks as space time itself. And that's a pretty startling thing, because while physics has changed an enormous amount in the last 400 years, the one thing that hasn't changed is that we atempt to give a description of what's going on in space as things move through time. So losing space time is not a small thing.
Now, last time we talked about another aspect. These are all related. Another aspect is that if we scatter particles, including the gravitational interaction, at very, very high energies, the amplitude grows. And when the energy becomes Planckian, the amplitude gets bigger than one and we have nonsense.
And there are places where these issues just become important. What's happening close to the Big Bang is when the curvatures of space time are becoming Planckian. Or inside black holes exactly the same kind of thing happens.
Now, the problem of quantum gravity, it's another problem that's often portrayed as a sort of La La Land for theoretical physicists. There is this very difficult problem. But all of the issues and features show up at the Planck scale. It's hopeless to get the Planckian energies. We can't do experiments there any time in the near future. So all we can do is speculate about what's going on up there. But since we're speculating and since all that matters in science is agreement with experiment, that's the popperian philosophy, all that matters is agreement with experiment.
Before you agree with experiment, everything is equally probable. Then experiment comes along and tells you which one of the ideas was right. Then you're just full of crap. If you're a theoretical physicist working on this problem, you might like something, someone else might like something. It's also sociological what you like or don't like.
And then experiment, which will never happen anywhere in your lifetime anyway, will never get there. So why are we wasting our time doing this? So that's a caricature, an extreme caricature of an attitude. But it's not far from what some people think.
And that's missing something very fundamental. So I want to spend a few minutes telling you how you can make a breakthrough contribution to quantum gravity. Any of you. And this is another reflection of the tremendous straight jacket we're in.
So here's the first challenge. The first challenge in a quantum gravity is that these amplitudes that we talked about, as we said, they're getting big as we go to higher and higher energies. Let me write down a specific amplitude. You don't need to know what the formula actually means. I just want you to see. It's a very concrete formula. The symbols mean something very simple. They just have to do with the momentum of the particles.
So let's say I have four gravitons, one and two, three and four going out. One and two have negative helicity. Three and four have positive helicity. Everything is great. And you can calculate what this is by the rules that I told you about before. This is the computation we can trust at low energies. And here is the answer. There's a nice answer sitting there.
This answer is correct that low energies means when these parameters, S, T, and U, that correspond centrally to the momenta of these particles are related to the momenta of the particles. When the energies are very small compared to Planckian energies, we can trust this answer. But as we've said already a number of times, it's just wrong at high energies. This amplitude becomes bigger than one. It's nonsense and we have to fix it.
Now, that means out there somewhere in the world of formulas, there is a formula for that process which depends on those variables. Which when S, T, and U are small, looks like that. When S, T, and U are large looks like something different and which doesn't get big. It can't get bigger than one. It's nonsense for it to get bigger than one, so it doesn't get big. All right.
So here's a first challenge in quantum gravity. There are many other challenges. But one challenge is write that formula down. What is that formula? OK, now you might think this is easy. So easy. Here. Here, I'm going to do it. Here we go.
So there it is. It's getting big. That's the standard piece. What I'm going to do is multiply it by this factor. 1 over 1 minus S. 1 over 1 minus T. 1 over 1 minus U. There's some parameter there that's maybe that's somewhere near the Planck scale or something like that.
But anyway, look, you don't need to know much math to notice that if S and T and U are small, these are all 1, so I haven't changed anything. If S and T and U are large, then I can ignore the 1. And so these things get much, much smaller. And so I've put enough of them in there such that this actually makes the amplitude as we go to very high energies and makes it smaller and smaller, in fact.
Great. I'm done. What's all the fuss? There's a candidate theory of quantum gravity, at least for that problem. Not so fast. Not so fast. Because this amplitude that you've written down has to respect which laws? The laws of relativity and the laws of quantum mechanics.
The laws of relativity we've made respect sort of manifestly by using these variables. But the laws of quantum mechanics tell us something very specific. They tell us something specific about what the structure of this function has to be. Technically speaking, it tells us about some properties of the poles of this function.
And as you approach the places where this function becomes singular, the whole function has to have a certain property. It has to be positive in some specific sense. If it isn't positive, the probabilities are negative. It has to be positive for probabilities to be positive, to be consistent with the laws of quantum mechanics.
So you check this naive attempt and you find it fails it badly. So you tried. So there's a try. That one didn't work. Surely I'll screw around a little more, I'll find another one that works. Right? No. You keep screwing around and actually quickly you convince yourself that as many guys downstairs and upstairs that you put here, if it's a finite number of them, it cannot possibly work. So this starts getting a little more interesting.
Now remember, just like the story of the Higgs, forget about comparing with experiment. Just like with the Higgs, we didn't have the conflict with experiment. Here we have even less of a conflict with experiment. The concept of experiment would hit us on the Planck scale. But just as with the Higgs, we can begin to explore how we can solve the problem. And here too we can to begin to explore how we can solve the problem.
And something interesting happens. Even candidate solutions are difficult to come by. Forget about whether they agree with experiment. Even a candidate formula, which agrees with quantum mechanics, agrees with relativity and solves the problem is hard to find. And in fact, you can stare at this problem for ages and ages and ages and not find the answer.
What's needed sometimes in situations like this is a bolt from the blue. And such a bolt from the blue was a discovery of a truly magical formula. Now, this is a concrete formula. You don't need to know. Those of you who know, know. This is the gamma function. It's like a factorial. For those of you who don't, it doesn't matter. But it's a function that you can know and you should know and love if you love functions. It's a very standard, nice mathematical function.
Anyway, it turns out this function, it's not a ratio of polynomials. This function was discovered. I'm bending the history a little bit. The history was slightly more complicated than this. But roughly speaking, this function was just discovered out of the blue by people thinking about analogous problems. And people noticed, holy crap, you look at this thing and it solves the problem. It makes the amplitude small.
It's compatible with relativity. It's compatible with quantum mechanics. And all the naive attempts fail. But this one miraculously worked. And when you look at how it works, it's a complete miracle about some properties of these gamma functions and all sorts of abstruse things about them have got to be true in order for this thing to work.
All right. So now we have one candidate for what this function can look like. Maybe there are others. But we have one candidate for what this function can look like. It's a very low bar. We're only talking about two gravitons go to two gravitons. Never mind more complicated things. A very low bar. Even this low bar, it's very hard to find even one that works. This was found to work. And incredibly, it had no negative probabilities, as I said.
Now, in physics when we run into miracles, they have an explanation. And so you don't just sort of sit around looking permanently at how amazing it is that this thing exists. You start wondering where it came from. So even though it was guessed, even though it was accidentally discovered, you start wondering, where could formulas like this come from? And in a few years people realized the origin of this formula and similarly remarkable formulas.
These formulas actually arise from a picture where the graviton instead of being a point-like particle is a little closed loop of string. loop. So if you start with a picture that is a closed loop of string, you would develop what we developed for particles in our lectures, in the last lecture. If instead of starting with particles as a starting point, where you begin with the loops of string as a starting point, you figure out how they can consistently interact with each other.
And then you ask the question for how those loops can bang into each other, two in and two out, amazingly these gamma functions just come out of that calculation. So what was in the beginning a miracle started finding an explanation in this picture that things are fundamentally strings.
So as I said, if you want to join the quantum gravity club, find a function like this. Find another function. Find another function which satisfies the properties that it needs to satisfy. And it's not a question of aesthetics. It's not because people like little loops. People were impressed because this miracle was pulled off.
And I also encourage you, if anyone is telling you about their favorite theory of quantum gravity and all the deep principles involved and all this stuff, that's great. You can listen and say, that's wonderful. But what is your formula for two gravitons in, two gravitons out? I mean, surely you have a formula for that, right? You'll find it amusing how many people who spend decades thinking about quantum gravity do not have such a formula.
And the fact that such a formula exists is not a sociological statement. It's a mathematical statement. But it's a mathematical statement about a deeply physical question. And this is the sense in which we can make progress. Because it's so hard to find even one of them that if you find one, maybe you find two, maybe you find another one, it's still going to be so few of them that you restrict your attention to those possibilities that even have a chance of passing the bar of comparison with experiment, ultimately.
So the fact that this formula exists doesn't mean that the formula is correct. But it shows you that it's possible to make progress even without having experiments up there. The progress isn't necessarily getting to the correct theory, but whittling down the set of possibilities to the tiny set that have a chance of being correct.
Now, in this picture. And the picture is string theorists themselves arrived at by the mid 1980s that the sort of deep thing that was going on is that we replace the picture of points by loops. Something remarkable happened as people started developing this picture more and more into the mid 1990s.
So you see that the group of string theorists was sort of completely convinced that the deep thing was that things weren't points, but they were little loops. That was the whole point. That was the point of the name of the subject.
But then they discovered something. After a long, long chain of purely internal theoretical developments, they discovered something. I won't have time to describe it in detail. But they discovered that if you wanted to talk about a theory of strings and gravity and extra dimensions and all the other stuff you have to have in string theory, on the interior of some region of space time, that it was actually completely equivalent to a theory without strings, without gravity, of good old fashioned particles like quarks and gluons that lived on the walls of that space time.
So this is a very deep and very long story. I don't have time to explain in any detail here. But I want you to appreciate the irony that the same group of people who got into a subject trying to understand what was happening with particles interacting inside space time and saying, no, it doesn't work as particles, we have to turn them into loops, eventually realized that this picture of loops in the middle of the space time was actually equivalent to a picture of particles not in the same space time, but even in a lower dimensional space time. The lower dimensional space that lived on the edges of the universe in which the strings and all the rest of it lived.
So that means that the particles are back. Such is the richness of this structure of putting quantum mechanics and special relativity together. So great is its capability to surprise us that even being led by the nose away from the picture that particles are fundamental things, we return after 15 years to the best description we have for this physics in terms of a different kind of theory, but still of interacting particles in one lower dimension even.
So that's another sort of thing that happens in theoretical physics when you're on the right track. All sorts of different remarkable structures that at first seem utterly disconnected to each other become, upon further reflection, to be seen to be different aspects of the same thing in highly non-trivial ways. If I had more time, I would tell you a little bit about the dictionary about how this remarkable correspondence works.
But I do want to mention that there is still something that we really don't understand. The thing that we really don't understand. So this picture, one thing that this picture accomplishes is it gets us part of the way to a picture of this idea of emergent space time. So here we know because of gravity and quantum mechanics, we have to have emergent space time.
And this picture is getting us partially there. The gravity and some extra space is emerging from some intricate dynamics of good, old fashioned particles like quarks and gluons interacting strongly with each other on its boundaries. But time is not emergent. The time of the good, old fashioned theory of particles is the same as the time of this gravitational theory and the interior of this space.
And that means that there's a lot that's still missing. And almost everything that's still missing has to do with mysteries involving quantum mechanics and cosmology. For example we have learned that our universe is accelerating. And because our universe is accelerating, there are regions of space time, light from which we will never, ever see again. So what we see out there in the universe today is what we're ever going to see.
And this opens up a huge number of conceptual paradoxes. Because we have an essentially finite universe. Before we knew about the accelerating universe, we could imagine that the universe would expand and expand and eventually everything that was out there, we would see. So we'd have access to an infinite amount of stuff if we waited long enough.
But in this accelerating universe, it's not true. We only have access to a finite, albeit enormous, amount of stuff. And in a world with a finite amount of stuff, it becomes very difficult to think about what quantum mechanics is supposed to precisely mean.
Quantum mechanics makes precise statements about situations where you observe like a little system here. But you have to observe it with a measuring apparatus which, in principle, to say perfectly sharp things about it, the measuring apparatus should be infinitely large. If the measuring apparatus itself is finite, that measuring apparatus itself suffers quantum mechanical fluctuations eventually. And that limits the precision to which you can say anything, make any observation.
So quantum mechanics, in order to be perfectly sensible, perfectly sharp and sensible, needs to divide the world into two pieces. An infinite detector and the finite system that it looks at. That's a basic feature of the quantum mechanical world that was well appreciated by its founders. But the accelerating universe for the first time seems to make that impossible. Because we just have access to a finite, albeit enormous, number of stuff. Amount of stuff.
So certainly whatever is going to make sense of this picture, time is playing an absolutely crucial role, because it's cosmology and the universe is expanding. Even going back in time there was a big bang, which is also confusing. And perhaps even quantum mechanics itself is not telling us how to think about this. And we might even need some extension to our usual picture of quantum mechanics to deal with this situation. That's a speculation. But trying to make sense of the accelerating universe in a sense opens up the biggest number of conceptual paradoxes we have to deal with in physics today.
All right. So there is a large number of questions having to do with putting quantum mechanics and gravity together. Some of them involve physics at very high energies, like the question of the graviton scattering that we talked about. There, as I mentioned, all of you were invited to join in the party and try to find formulas of that sort.
But many theoretical physicists have moved on from that set of questions to other sets of questions, still involving quantum mechanics and gravity, and in particular cosmology. And these questions we have no good answers for right now. We don't even have a good way of starting to think about them yet. But they are on the table.
So another qualitative thing that we don't know about the world is why it's big and it has big things in it. And the fact that the universe is big already is not obvious. Everything is quantum mechanical. And because of quantum mechanical fluctuations, we would expect even the vacuum to have some energy. Because everything is jiggling around all the time. And that energy in the vacuum or the energy density in the vacuum, because energy is like mass and everything gravitates, that energy in the vacuum should by all rights curve the universe.
The curvature that we would naively expect it to have would be Planckian, as gravity has to do with the Planck scale. Everything is Planckian. And that's ridiculous. A Planckian curvature would curl things up to the 10 to the minus 33 centimeters, which looks nothing like our beautiful 10 billion light year sized world. So there is something really terribly wrong there. We don't even understand why the world is big.
Having the world be big requires that this energy density in the vacuum is 120 orders of magnitude smaller than any back of the envelope estimate would suggest. We have no idea or we have no sharp idea why that's true. There's lots of pictures for why it might be true, but we have no really good ideas, well understood theoretical ideas for why it's true. Lots of attempts, but nothing with the sharpness and concreteness that we've come to expect in our subject.
Now even granted that the world is big, there's a second question of why it has big things in it. Why is the Earth big? Why are elephants big? And this is related to something else, as why gravity is so weak compared to all the other forces. The size of the Earth is determined by the gravitational pressure that wants to crush rock being supported by essentially atomic pressures that are resisting it.
And you can do a little back of the estimate for how big is the Earth compared to the size of an atom? And you discover that the Earth is bigger than the atom by precisely the factor that gravity is weaker than the electric force. So the size of the Earth being big compared to atoms is a direct reflection of the weakness of gravity.
Actually we can even understand the size of elephants because for elephants, we have to do more than just resist be liquefied by gravity. It's already bad if you just break one layer of bonds in their bones, because then they'd just fall over. That wouldn't be good either. And so you actually get the 2/3 power of exactly that same ratio that we talked about. And that explains the size of elephants relative to the size of atoms.
So the fact that the world has big things in it is ultimately related to the weakness of gravity compared to everything else. Why is gravity so we compared to everything else? It's because the masses of all the particles are so much smaller than the Planck scale. Masses of all the particles, force of particles we care about like electrons and up quarks and down quarks. But all those masses are links to each other and ultimately are linked to the mass of W and the top quark and all those masses that we were talking about earlier in the talk.
But remember that where the W is is pegged to where the Higgs is. Because the Higgs couldn't be too far away from that, otherwise we had all the disasters that we avoided by having the Higgs there. So all of those questions of why there is a big universe get transmuted to one, which is, why is the Higgs mass so much smaller than these Planckian energies, or these much, much, much higher energies that we could imagine it being?
Well, that doesn't sound like much of a problem. It is what it is. We just measured it and that's life. But once again, quantum mechanics makes the situation much more confusing. Because of quantum mechanical fluctuations, there's something you might call the totalitarian principle. That says that everything that can happen must happen. Now, why is that a problem?
Well, let's say we talked about a particle like the photon. We say the photon is exactly massless. The photon is exactly massless. That's good. So why is the photon exactly massless? You might think that the photon is barreling around. But it can produce virtual electrons and positrons and they can interact with each other in complicated ways.
And you remember that there is this here, I'll have to return to this caricature picture of the vacuum of our world as being this seething cauldron of particles and anti-particles. And unfortunately I'll have to return to the molasses metaphor momentarily. But if you even roughly think about that metaphor, then you might wonder, why even the photon? Why is it barreling through this stuff? Why doesn't it slow down? Why doesn't it pick up a mass? Why is it exactly massless?
And if you stare at this picture, you get very scared. There's no reason for it to massless. The totalitarian principle. Everything that can happen will. If it can slow down, it'll slow down. But it doesn't. And it doesn't for a very simple and deep reason. It's because 2 is not equal to 3.
Let's say in some approximation where the photon didn't interact with the electrons and the positrons at all, then of course we'd have no problem. It's barreling along at the speed of light. But then it has only two degrees of freedom. Now, if it became massive, it would have three degrees of freedom. That's just accounting we talked about before. But you can't just spontaneously turn two into three degrees of freedom. You can't invent one out of thin air where one didn't exist before.
So if I turn on these interactions that allow me to have all these complicated interactions, this can do all sorts of interesting things. But one thing it cannot do is make the photon massive. It's because 2 is not equal to 3.
But you see the problem. Exactly the thing that makes the Higgs so interesting, exactly the thing that makes it so novel, the fact that it's a particle of spin 0 now means I can't make the same argument. I can draw exactly the same kind of process here is one thing that could happen.
And now why doesn't it have an enormous mass? The massless Higgs particle would have one degree of freedom. Doesn't spin at all. The massive particle still has one degree of freedom. There's no difference in the degrees of freedom between the massless and a massive spin 0 particle.
And so this is our question. So the Higgs by all rights should be up at the Planck scale. If it was off of the Planck scale, it would drag everything else up with it. The masses of all the other elementary particles would slide up to the Planck scale. You and I would all be black holes. It would be a very unpleasant world. But the world doesn't look like that.
Now what do we actually do? What do we do? I mean, we discovered the Higgs, we discovered all these things about it. So what do we actually do? Well, we say something which we're perfectly allowed to say. We're perfectly allowed to say it's consistent, it's mathematically consistent. And once we do it, we get exactly the structure that we talked about that allows us to make any predictions we like to exponentially higher energies. And I remind you, we actually also discovered the Higgs. So it's there.
But what we have to do is say, yes, there seem to be no reason why the Higgs isn't go all the way up to a Planckian energies by having all these fluctuations. So by some accident, it turns out that that doesn't happen.
More specifically, it's these quantum mechanical corrections so the mass of the Higgs might be enormous. But we say yes, they're enormous, but they're just counterbalanced by a non quantum mechanical piece that was sitting there already. And one of them is opposite to the other one by 30 decimal places. They agree, they keep going, and they disagree in the 30th decimal place, in the 31st, 32nd second decimal place.
No one stops us from doing this. But for obvious reasons, it's called fine tuning. And it's like walking into a room and seeing a pencil balanced on its tip to within 10 to the minus 30 degrees of vertical. Or for the case and even more severe problem of the vacuum energy I alluded to before, which is the even more basic question of why the universe is big, it's as if the pencil is standing on its tip to 10 to the minus 120 degrees of vertical.
It's possible. It's possible to find pencils like that. But it's also likely that if you walked into a room and you saw that situation, that you would think there is something up. You might look for a string hanging from the ceiling. You might look for a little hand holding up the pencil. You might want to find some mechanism, something that explained the situation, and not just say it is the way it is.
All right, so what could it be? Let's say you wanted to do that. Let's say you wanted to find a mechanism that could explain why this happened. Now we have a new puzzle, just like we had. There's always some problems left to solve in physics.
So before the Higgs, we had the most pressing problem of the difficulty with the scattering of the W's. That was a real problem. This problem may or may not be real. This problem is a little bit more in our head. As you see, and as I said already, now that we understand mechanically how the universe works, we're led to ask a new set of questions about how plausible it is we got the world that we have.
And the difficulty is not about the mechanical one. If I just say the Higgs is where it is, everything is great. I can calculate everything else. And not until I get to the Planck scale do I have any difficulty. But it's remarkable that exactly that situation seems so wildly implausible as something to arise from some more fundamental microscopic theory.
So it's a different character problem. I want to stress it's a different character problem. So it's not guaranteed that it has a solution. The difficulty with the W's was guaranteed to have a solution, because the theory just broke down and gave us nonsensical predictions when you try to calculate a process that energy is even 10 times bigger than we'd been. This is not like that. But still, it appears to be a serious problem and we can go about trying to see what might solve it.
So let's go back. So what else can we do compared to what we've done already? We can go back to our toolkit. What are we allowed to do? What is nature allowed to do at long distances? And remember that was the guy that was missing.
So it turns out that it is possible to have massless spin 3/2 particles. But that can only happen under very special circumstances. But it's possible. It's possible. And it occurs if the universe has a symmetry known as supersymmetry.
So remember the graviton was this massless spin two particle, and it had completely universal interactions with all ordinary matter. That's the universal law of gravitation. But this massless spin 3/2 particle, it turns out these theorists locked in the room doing the calculations to see what's consistent with relativity and quantum mechanics would tell you, yeah, you can have the spin 3/2 guy, but it's extraordinarily constrained and special. These are the properties that it needs to have.
And one thing that has to be the case is that every single particle out there that has, let's say, spin 1/2 needs to have a partner that has spin 0. And the interaction between the gravitons, this interaction and that interaction have to have absolutely identical strengths. So all particles have to be partnered up with other particles which have a spin that differ by 1/2 in order for this to even have a chance of being consistent.
Now, that's why theoretical physicists are excited about supersymmetry. It's because it's the last thing nature can do that we haven't seen it do. The last thing nature can do consistent with general principles at long distances that we haven't seen it do yet. Doesn't mean that we will see it. Doesn't mean that we'll see it tomorrow.
But again, that's a non sociological reason to find supersymmetry interesting. It's the answer to the question. It's one of the consistent possibilities. In fact, the last one that we haven't seen happen. I don't have time to explain why it also in a sense extends our picture of space time.
But now here's something else that didn't have to happen. This possibility exists. But supersymmetry could exist, but it may not have anything to do with this problem, the Higgs that we were just thinking about a second ago. But in fact, if nature is supersymmetric, it would solve the problem with the Higgs. And the reason it would solve the problem with the Higgs is because the Higgs has spin 0.
Remember it's a spin 0, guys. The guys that have no spin that have this degree of freedom difficulty, this difference between massless and massive. But if nature is supersymmetric, the Higgs particle has to have a partner. The partner has spin. So the partner has a good reason to be massless compared to the Planck scale. And therefore, if there is supersymmetry, the Higgs also must be massless compared to the Planck scale. Dragged along for the ride by its partner. This
Didn't have to be the case that supersymmetry solved the problem. But we have this new problem and supersymmetry can solve it. And in fact, something else that didn't have to happen is the following.
So if supersymmetry does solve the problem, then the supersymmetric particles can't show up at energies that are 50,000 times bigger than the mass of the Higgs. If they did show up at energies that were 50,000 times bigger than the mass of the Higgs, then we wouldn't understand why the things didn't go up there. So whatever this physics is has got to happen soon, right around where the Higgs mass is itself.
Now, already back in the early 1980s, people who realized that supersymmetry could solve this problem said, great, now as we go to shorter distances that energy is much higher than the mass of the Higgs, in addition to the particles that we know, there should be all these supersymmetric particles around.
And something that I didn't have time to explain in these lectures is that the strengths of the interactions of the strong and the weak and the electromagnetic forces, which we measure at the energies that we're at now, we can theoretically extrapolate and see what they do as we go to shorter and shorter distances or higher and higher energies. The nature of that extrapolation depends on the kinds of particles that are there as we go to higher and higher energies.
And if we put in supersymmetric particles, where we expect them to be to solve this problem, then people found this amazing thing. That the strengths of all the interactions, while they start off roughly a factor of 10 different from each other at low energies, as we extrapolate the high energies, all three of these lines converge and meet at a point to percent level accuracy. All three of them converge. And where do they converge? Within a few orders of magnitude of the Planck scale.
So this didn't have to happen. Maybe two lines meeting at a point. Three lines meeting the same point doesn't have to happen. The fact that it meets at a point close to the Planck scale didn't have to happen. And this is some indication that if supersymmetry is correct that these interactions are unified with each other, the strengths are becoming the same. They otherwise can become unified with each other at these enormously high energies, which is not far away from where gravity catches up with everything else too.
So these are all sort of spectacular things about supersymmetry. It's the last possibility that nature can do. That alone doesn't tell you the energy scale at which we should see it. We should just see it somewhere beneath Planckian energies, perhaps. Because that's where physics changes really radically.
But if you assume that it's there around the corner, right around where the Higgs is, then it solves the problem with the Higgs that we identified. That didn't have to happen. And in doing so, it also gives us a picture that the strength of all the interactions becomes unified at a scale close to where gravity catches up, which also didn't have to happen.
None of this means that supersymmetry is going to be correct. And in fact the difficulty with supersymmetry and the growing difficulty with supersymmetry is that many of us would have expected to have discovered it already. And even before the LHC and the fact that we haven't seen it so far at the LHC is making the situation even somewhat more comfortable. It's not remotely a killer yet, but it's also true that it could have been discovered in the '90s. At machines in the '90s.
So it's confusing. It's a little confusing. It's at least a little bit overdue. Of course, we could turn on the LHC in 2015 and they'll all be there and then that will be that. And the fact that we're worried all this time will be relegated to the dustbin of history. So there's no guarantees. There are some things that argue that it's on the right track.
But once again, supersymmetry, like string theory and supersymmetry is an important part of string theory, is an example of this radically conservative philosophy for how to proceed in physics. The reason people are excited is not a sociological one. It's not because they think it's pretty. It's because of that. It's because it's the last thing that nature can do. It's the answer to this mathematical problem, which is motivated by this deeply physical question of what's consistent with relativity and quantum mechanics at long enough distances.
And of course, if supersymmetry is correct, there is very specific kinds of predictions for what we should see at the LHC. And people are looking for it. So I don't have time to talk about that in detail.
Another thing which is worrisome about all of these ideas is that while all of these ideas are wonderful for explaining the difficulty with the Higgs, none of them work for explaining the much, much bigger difficulty associated with why the universe is big to begin with.
So if you have in your head that you want to solve these two seemingly similar problems in the same way, it's not going to be done with supersymmetry. And some other ideas are needed. They're not solved in the same way. We never know. That's why we have to try things and find out, both theoretically and experimentally.
But I think by the end of the next run of the LHC in 2018 or 2019 or something like this, what we're going to get is some strong push in one direction or the other for this basic question. Not if supersymmetry is right or not. An even more basic question of whether the lightness of the Higgs is associated with the fine adjustment of parameters or with some new dynamics.
If there's new dynamics, you really, really have to be seeing it. Now we can't make even mild excuses for it. Or we'd have to make more extreme excuses if we don't see anything by the end of 2018. So many people just assumed obviously we have to find some new dynamics of the sort. Many times previously in physics, these sort of issue has arisen and people have found new dynamics associated with it. That's what's called natural. And if we see it, we have to see some major new physics. For example, something like supersymmetry.
But if we don't see it, if we continue to see nothing but the Higgs, that's by far the most surprising and interesting thing that could actually happen. It's the most shocking. It's the most unlike anything we've seen anywhere else in physics. The Higgs already we've never seen anything like it before. It's already the first spin 0 particle that we've seen before. And if it's going to be natural, it has to come along with a ton of other stuff. Supersymmetry was the one example that we talked about.
But even more surprising would be if it was sort of naked and alone to much, much higher energies. And then we would have some sharp, direct evidence that there's some fine tuning hardwired into the laws of nature at some level. If we see nothing but the Higgs at the LHC, we will have direct evidence for this parameter, for the parameter that controls the mass of the Higgs of a fine tuning at the 1% level.
And maybe you think 1% is not so bad. It's not so good. It is what it is. But we wouldn't be having this discussion had we discovered all sorts of new particles coming along with the Higgs. So this bifurcation is a very crucial fork in the road and we're going to have to get some feeling for it experimentally.
All right, so given the time, I think I won't have time to talk about this subject in detail. So we started off talking about the difficulty with the end of space time. As I said, that's the question of what's going on when we finally get the Planckian energies. The second set of questions is a structural one for why we have the kinds of theories that we have, why the Higgs has a small mass, why the cosmological constant is small.
But the really overarching question that we have to deal with is trying to find possibly this deeper structure underlying what space time and quantum mechanics really are. And here too this basic philosophy, this basic radically conservative philosophy, tells us a way to go. Here it's very unlikely we're going to get direct clues from experiments in a while. It's unlikely we're going to go up to Planck energies. But it's possible that there are lots of clues to what's coming next in the next level of physical theory by trying to look at things that look like funny features of the existing theoretical framework.
Sometimes these most crucial clues are hiding in plain sight as funny features of the existing theoretical framework. There is a famous example. Einstein realized that the mass that gravitates and the mass that appears in F equals MA is exactly the same mass. This fact bothered Newton a lot. Didn't seem to bother anyone in between.
So it's good, he was in good company. It bothered Newton and it bothered Einstein. And he used that fact that any high school student in 1850 would have also known about to realize there had to be a remarkable new picture for what gravity really is. So there the clue to what came next was lying in a structure of the old theory.
And it's even true that clues to quantum mechanics were lying concealed in funny features of classical physics. The classical physics was originally, as described by Newton, told you where a particle went next given where it was now, how fast it was moving, and the forces that were acting on it. But people in the course of time found totally different ways of thinking about what classical mechanics was.
In terms of imagining that a particle moves from A to B by imagining all the possible paths that it could take and picking the one that minimized the average value of the kinetic energy minus the potential energy is something called the action. It's a totally different picture for what classical physics is. It didn't look like it was deterministic, even though it ended up being deterministic.
And it really bothered people. It bothered the Frenchman, the [INAUDIBLE] and Euler and Lagrange and people who discovered this fact. They wondered, why is it true? There is such a radically different sort of very philosophically different seeming way of talking about classical mechanics. And today we know why. The second way exists. It's because the world isn't deterministic. It's quantum mechanical. And determinism is something that happens in a limit.
But because the underlying fundamental theory doesn't have determinism, there must be some way of talking about the limiting theory goes to in a way that determinism isn't the star of the show. And indeed, that was the case here. So sitting there in the structure of classical physics were clues to what came next.
And we could be in the similar situation today, that sitting in the structure of this remarkable marriage of relativity and quantum mechanics, in quantum field theory that we've spent all these lectures talking about, the straight jacket they put us in, that actually staring at what it gives us is going to give us clues as to what's coming next.
And I won't have time to talk about some of the concrete things along these lines. I give a colloquium here back on Monday on some of these topics. You can call this kind of development sort of mining theoretical data. That as we actually stare at what comes out of quantum field theory, even when we theoretically study it, we start seeing that the structure of the answers that we get out seem to reflect, in many cases, some completely alternate starting point that might arrive at exactly the same answers, that might be giving us a hint about where to go next. In much the same way as the principle of least action gave just an alternate way of talking about Newton's laws, but simultaneously gave an important clue as to what was coming next.
All right, so let me end not like that. Let me just conclude by saying that I hope to have conveyed to you that this is a singular time in the development of fundamental physics from a number of points of view. And the 20th century started with these revolutions. Much of the rest of the 20th century was understanding how those ideas could actually be used to describe the world we live in, which is, of course, extremely important.
But we've done that. We've basically done that. And now the questions that are left are in a sense we've plowed through the mechanical ones, the mechanical questions. And we can confront and ask a new sort of question, a sort of deeper sort of question, more structural questions. In fact, there are these very deep ones that have to do. The questions that are left are these very deep ones, having to do the underpinnings of space time and the origins and fate of our big universe.
And what I think is remarkable is that you can talk about these things in college bull sessions. You could have talked about these things in college bull sessions in 1800, in 1900, in 1930, in 1950, in 1980 even. And it would have been useless. It would have never gone beyond college bull sessions. But today we can articulate them sharply enough that we can barely start working on them meaningfully.
And it's because we have so much constraints on what the answer can look like that help from experiments is fantastic and will illuminate many aspects of this. And even without help from experiment in some of the other questions, we can meaningfully make progress because of the tremendously tight straight jacket we find ourselves in.
So that's just what I said a moment ago. We do have absolute fantastic experimental probes in a number of fronts that might give us some important clues. But we can also make progress on some of the other questions, purely theoretically. Or have a sense at least of winnowing down the set of possibilities to the small number that will eventually have to make contact with experiment.
Now, since these problems are of a different nature and in a sense more difficult than the problems that we had to deal with for the last 50 years, it's difficult to know when the really big breakthroughs will happen. And I'm not being hyperbolic here. Sometimes I'm very optimistic. And maybe many of us are very optimistic and think at least key pieces of the puzzle, certainly not the whole puzzle, but sort of key pieces might fall into place as quickly as 10 years from now.
Certainly experimental information of the LHC is going to come in and weigh in in a very significant way in one way or the other on some of the questions about the big universe that we're talking about. But even theoretically in my most wildly optimistic moments, I think we might know at least a little bit about some of these things on the 10 year time scale. Other times, depending on how well my calculations are going and those of my colleagues, I think 100 or 200 is more appropriate. So it's hard to tell.
But what I think is fair to say and it's true is that at any given moment in time, you have to work on the next hardest problem. Not the hardest problem there is. That's stupid. But you have to work on the problem that's hard, that's important, but that is the next most accessible thing that you can do. And one way or another, the problems that we're talking about are those problems. So that's what's on the table now.
So solving these problems are the next things that we have to confront. So whether the pace of progress is gradual or rapid, those are the things that we're sinking our teeth into and that we'll hopefully be able to report some progress on on the shorter rather than longer time scale in the coming years. Thank you very much.
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In the second 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.