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[APPLAUSE] HITOSHI MURAYAMA: I already feel kind of at home being at Cornell because I know so many people. Csaba used to a post-doc at Berkeley. We worked together. So did Maxine [? Pearlstein. ?] I also worked with [INAUDIBLE] on several occasions, and I also have met many of the [INAUDIBLE] physicists here in various occasions.
So this is a great place to be. I arrived yesterday. I was also greeted by snow.
[LAUGHTER]
But anyway, it's a great honor to be invited as a Bethe lecturer, and as Csaba described, he's really a giant in theoretical physics. And clearly, this tradition is living on at this department given this Sakurai prize award to Peter Lepage. So let's applaud Peter for this.
[APPLAUSE]
Hopefully, I live up to this. It's a requirement, being a Bethe lecturer. It's a major event with the department, as described, so we'll see how that goes.
I'd like to talk about this title, When A Symmetry Breaks. I'd like to get started with a quiz. What's common among a piece of magnet, ice cubes, halibut, and rack of laundry?
[LAUGHTER]
It's spontaneous symmetry breaking. This concept is so important, it appears everywhere in physics, even chemistry and biology, so that's the aspect I would like to talk about today. And part of the concept is attributed to Yoichiro Nambu, who got a Nobel Prize in 2008 for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics. Of course, this concept applies to many other areas of physics, and other disciplines, even, and I'd like to give this as a sort of tribute to Nambu because he just passed away earlier this year in July.
So let's get started with the symmetry. We know the laws of physics has large kind of symmetries. For example, if we happen to find a galaxy that actually spirals this way, you can also find a galaxy that spirals the other way because the laws of physics is more or less left-right symmetric. Of course, there's something called weak interaction we'll come back later and talk about which violates this symmetry, but more or less on large scales, there is a symmetry between left and right.
There's also a symmetry on the rotation of the system. Once you see a galaxy, then you can see all possible orientations of the galaxies because they're just random, because the laws of physics, again, allows any rotation of the disk of the galaxies.
And these days, the observation has advanced so much that we have millions of galaxies already measured in location and size and shapes. You put them in a computer, and you can pretend you can fly through this. And obviously, if you really would like to do this, you have to fly at a speed 100 trillion times faster than the speed of light, so don't try this at home.
[LAUGHTER]
But the basic message here is that no matter how far you go, it doesn't change very much, so that's the translational symmetry. So all we learn back in freshman physics is that the Newtons equation of motion is invariant under inversion of left and right, rotation, translation, because both sides of the equation change the same way, so we say there's a symmetry. The equation of motion is covariant under these symmetries.
But if the laws of physics have this high degree of symmetry, how come that most of us are right handed? Why is our heart on the left hand side of the body? So there is certain very high degrees of symmetry in laws of physics, but phenomena we see are very diverse. So where does it come from? And that's exactly where this concept of spontaneous symmetry breaking comes in.
The dictionary definition of this concept is that the system itself has a symmetry, but its ground state does not respect the symmetry, then there are multiple ground states that are degenerate in energy. So let's talk about that.
Then comes this rack of laundry. If I do a laundry, I'd like to hang it so that I can let them dry. The first shirt I hang on this rack, I can hang it either facing left or right. It doesn't matter.
But suppose I hang the first shirt facing left? Then when I bring my second shirt, I have this tendency that I want to have the shirt facing the same way as the first one. Second one is also facing left, third one, fourth one, and eventually I find myself hanging all the shirts facing left.
What happened? Symmetry is broken. So it really doesn't matter which way these shirts are facing, but due to some reason, once you actually put up a system, then they all end up facing one way or the other. Symmetry is then, we say, spontaneously broken.
Same with halibut. The halibut has two eyes on the same side, and the reason is clear. They wanted to evade the predators, so they wanted to bury themselves in the sand on ocean floor be not harmed by the predators. And for that purpose, they had to bring both eyes on the same side.
But it didn't have to be on a particular side. As a matter of fact, if you look at the flounder, they have eyes on the other side. It doesn't matter. So this must be also spontaneous symmetry breaking.
Another example I have seen is actually a beautiful experiment done by a biologist in Japan, Reiko Kuroda. If you take a conch, every conch winds in one particular direction, so there's a chirality to it. There's no symmetry between left and right.
But she did the following experiment. When you actually have the control sample coming from the embryo all the way up to adult conch, she decided at the stage of the first division of cells, then they start to wind around in one particular direction. But then she nudged it to turn in the opposite direction, and then she saw that they actually grow to a perfectly viable adult conch winding the other way. So again, it doesn't matter.
And as a matter of fact, which way it winds is written into the DNA. So the children of these strange adult conch are also born to wind around in the usual way, so it's written in the DNA. But clearly, in the course of evolution, at some point there was just a little notch more of the conch winding in one way over the other way, and once there is a symmetry broken that way, then it multiplies, because it reproduces, and eventually all of them are dominated by only one direction at the end of the day. So clearly that's what happened, so this is also an example of spontaneous symmetry breaking.
We have that, too. The kind of sugar we can digest has only one optical isomer, so there's only left handed ones we can digest. Then the other one, it still tastes sweet, but we can't digest that. So that actually makes a great sugar if you're on diet because it still tastes sweet, but you don't actually get any calories out of it. So that's another example that the left and right are not exactly the same. It's a spontaneously broken.
And originally, biological system could have chosen one type over the other. It didn't matter which one. But somehow, we chose only to digest left handed sugar, and there we are. And so that's another example of spontaneous symmetry breaking.
And we always depict this phenomena using this potential energy. So if you imagine randomly throwing a ball into this potential energy, then half the balls get stuck up on the left hand side of the minimum, the other half stack up on the right hand side of the minimum, but each ball chooses one way or another. It has to pick one way, now both. So that's the example of spontaneous symmetry breaking.
In this case, you're talking about two minima. That's the degeneracy in energy by a discrete symmetry. You can swap the two minima, but you can't change around in a continuous way. But there are many systems that also allow for this continuous symmetry that's spontaneously broken, and if you ask the Nobel Foundation, this is the way they explain it.
If you try to stand up a piece of pencil exactly straight up, then you release your hand. And as long as it's standing straight up, there's no preference in which way the pencil would eventually fall down to, but it does fall down in one particular direction. The original system is invariant under the axial rotation, but once it chooses one ground state, namely it falls down on the table, it chooses one particular direction, so rotational symmetry is not respected by the ground state. So that's also spontaneous symmetry breaking.
The other example Nobel Foundation used for Nambu's citation is this elastic rod. If you keep pressing the elastic rod just downwards, again, there's no preference in any particular direction. Everything is rotationally invariant. But at some point, the rod would not be able to resist this pressure, and it buckles, and when it buckles, it would choose one particular direction.
And once it does choose one particular direction, then as you keep the pressure on, it's very easy to rotate this thing around because there's no change in energy as you rotate this around by the symmetry. So you find a degree of motion that doesn't cause very much energy. That actually comes back in a later part of my talk on Goldstone boson.
Same thing with a piece of magnet. In this case, the magnet consists of many atoms with electrons with spins, and those spins can line up in one particular direction because they want to line up in the same direction. But the whole system, of course, is rotationally invariant and the rotation of all spins. But once they start to line up, they have to choose, again, one particular direction. So the system itself has the overall rotational invariance or symmetry, but they end up choosing one particular direction when you cool the piece of steel and go to this magnetize phase.
And that appears in many, many examples. Something that's frozen. I don't mean this kind of frozen here.
[LAUGHTER]
What I mean is that let's say you start with a vapor coming out of the volcano, and as everybody here knows, something hot means the microscopic particles some moving fast. That's the kinetic energy, and that makes things actually hot. So in this case, a vapor, the water molecules are zooming about at a very high velocity, so that's the hot stuff.
Once you cool it, eventually it becomes frozen, and that means they crystallize. In this case, all the molecules are lined up very neatly, and neatly is another way of saying it's order. You change from disorder to order. And once this kind of thing happens, then there is a particular location for individual molecules.
When they are all zooming around in chaotic way, then there is no preferred position, preferred direction. Namely, that there is a translational invariance, but also rotational invariance. Once it becomes a crystal, there is a particular position and particular orientation. Namely, both translational symmetry and rotational symmetry are spontaneously broken.
So we see tons of examples like this. A more interesting example quantum mechanically-- I just stole this video from BBC-- it's like superfluid.
[VIDEO PLAYBACK]
---with tiny pores can hold liquid helium, but the moment the helium turns superfluid, it leaks through.
[END PLAYBACK]
HITOSHI MURAYAMA: This is a little hard to understand, but what's actually spontaneously broken here is that macroscopic wave function has a value, but we normally don't care about the phase of the wave function because you always take the absolute square to find the probability density. But once it has a particular value, it also has a particular phase, and therefore, the symmetry to change the phase of the wave function is now lost, and that's another example of spontaneous symmetry breaking.
This also happens in a piece of superconductor, and the reason why the piece of superconductor floats on a magnet is because everybody knows the Meissner effect, which actually pins this magnetic flux that's going through the piece of superconductor, and in this case, also, this phase of the macroscopic wave function has a particular value. In this case, you can even locally change this. It's called gate symmetry, but it's also spontaneously broken here, and that actually makes a superconductor a superconductor. And Japan is actually building a maglev going from Tokyo to Osaka, and that's based on this niobium-titanium superconductivity.
So in these systems, you actually do have this continuous symmetry. Not just you have these two discrete minima you can swap around. You have infinite number of minima at the bottom of this potential, so they are all degenerate, the same in energies.
The form of the potential energy is also extended to two dimension, x squared, y squared. You can see that if you rotate x and y around, it doesn't change. That's the rotational symmetry. But once the system chooses one particular ground state, then it chooses the one particular direction in xy plane, so again, symmetry is spontaneously broken.
And these things happen many, many places in physics. For example, if you smash two heavy items against each other, what you're going to see is this. It's a huge mess. You produce so many particles.
But what's going on here is that when you smash these heavy items against each other, they actually melt once into a soup of quarks and gluons, and then as they fly apart, they start to cool down. And eventually, they address themselves into particles called pions, which are the bound states of quarks and anti-quarks. And when that happens, again, symmetry breaks. It's called the chiral symmetry.
That's what should have happened when the universe started with the Big Bang, and about when the universe was only like a few seconds old, this kind of phase transition is believed to have happened. So it happens everywhere, not just in laboratory, but also in the universe itself.
Another big event that happened in physics these days is the property of this weak interaction, and that really goes back to Hans Bethe. He told us how the sun shines, and of course, it is based on Einstein's equation E equals mc squared. And what happens at the core of the sun is that you're bringing four protons, and they group together to form helium. But then you have lost two units of charges, so you spit out two units of charges in the form of positron, the anti-matter of electron.
But then the electron number is no longer conserved, so you need to spit out two more things, called neutrinos. In this case, a particular type of neutrino called electron neutrinos. And what you start out with is heavier then what do you end up with, and therefore, mass is lost. And amount of mass that's lost is converted to energy thanks Einstein's equation, so that releases about 25 million electron volts of energy just by looking at the mass deficit.
As a matter of fact, because this is the way the sun is shining, it's always shedding its mass all the time, so the sun is actually getting lighter by 4 million tons every second. But she is so massive she doesn't really give a damn.
[LAUGHTER]
But nonetheless, she's getting lighter every second. I wish I could do that. If I could shed my mass into energy, I would love it. But anyway, this is how the sun shines according Hans Bethe.
But how do we know this is true? How do we know this is true? All you can do at looking at the sun is only the surface you can see. You can't tell what's going on inside because the photon would be scattered by constant scattering in the plasma state of the sun. You can't see through it. How do we know this is true?
Well, actually we do know because we can measure something called neutrinos here that can just get out of the sun without any hindrance. And as a matter of fact, the sun emits so many neutrinos, again, every second, that you are getting about 100 trillion neutrinos going through your body every second. We are baked in the wind of neutrinos coming from the sun. Does any of you feel this wind of neutrinos coming from the sun? No? When asked this question in Berkeley, there's always somebody raising their hands.
[LAUGHTER]
I guess Ithaca is more civilized place than Berkeley.
[LAUGHTER]
Anyway, so how do we know this is true? Well, you can actually see these neutrinos. The idea is very simple. Neutrinos can get out of the sun without any hindrance, which means they are very shy. They don't interact very much. So the only hope of seeing these neutrinos is to build a huge target and hope that maybe you get some of them causing some reactions, and that's what people did.
So this is a huge water tank in Japan. It's in a mountain area. It's 50,000 tons of water in this water tank. It's about 40 meters high, 40 meter wide, and it's a humongous tank. And using this huge tank, they have some chance of seeing neutrinos. It's still only like 10 a day, but they managed to do that. So if they get 10 a day, and if they keep sort of the exposure open for five years, then you can take a picture of the sun using neutrinos. No light. This detector is in pitch darkness 3,000 feet underground, but using neutrinos, they managed to take the picture of the sun.
Not just the sun. It's the picture of the core of the sun that is doing nuclear reaction. So this is how we know.
And these neutrinos have been very interesting. As a matter of fact, a Nobel Prize awarded this year was to this gentleman, Takaaki Kajita. He's actually a close friend of mine, so I'm very happy about this. And this is based on discovery called neutrino oscillation.
What they have seen is that if they detected a particular type of neutrino called muon neutrinos, and they come from the cosmic ray reactions in the atmosphere from above. And they detected about the same number as predicted, but the same reaction can happen on the other side of the Earth. And again, neutrinos don't care about the planet. They just pass through it without any problems. But it looks like half of them had been lost somehow compared to the predicted number. The neutrinos are lost, and that was the beginning of the discovery.
In the end, what we understood is like this. Just imagine you're working at an ice cream parlor, and you're throwing what Homer ordered, namely strawberry ice cream cone. But as you throw this ice cream cone towards Homer, then strawberry starts to turn into chocolate. Now it becomes completely chocolate. Starts to turn back into strawberry. Now it's completely strawberry. And as it keeps doing back and forth, eventually when Homer got hold of it, it's about half and half.
But just imagine that he cannot taste chocolate. Such a poor guy, but anyway. So suppose he cannot taste the chocolate, and the only thing he tastes is strawberry. Then he believes he has lost half of the ice cream and obviously complains to the ice cream parlor.
That's exactly what happened. The cosmic ray interaction produces a particular type of neutrino called muon neutrinos, but as they oscillated back and forth, eventually they found that half of them looked like they were lost because they converted into the different type called tau neutrinos. And that was the discovery that was worth Nobel Prize for.
So even though Hans Bethe predicted this mechanism of the stars shining which produces neutrinos, there have been also a long standing problem that we did not get enough of them coming from the sun, called the solar neutrino problem, and age-long problem starting in '60s. And that problem also got solved in 2002. That's the other [INAUDIBLE] Nobel Prize this year, Art MacDonald.
So again, the electron and neutrons produced at the core of the sun have transmutated. It morphed into other species of neutrinos on their way, and experiments couldn't see those other types of neutrinos. That's why we thought they were somehow lost. But it was just conversion from one type of neutrino to the other type, and so that's neutrinos actually do.
In this process, again, certain symmetry is broken because certain symmetry says the number of particles of definite kind should be conserved. But in this case, a particular particle turns into other particle, so that's again another phenomenon of symmetry breaking.
Weak force does many weird things like this. When you have this weak force that is actually causing this nuclear reaction in the sun, causes reaction for neutrinos [INAUDIBLE], there's actually a long standing puzzle about that, too. As people studied the nature of this weak force, it looked basically exactly the same thing as electromagnetism except for one big difference.
We know electromagnetism is a long range force. If you hold a compass, the compass points to the North Pole because the magnetic field for north pole would affect your compass in your hand. So it's a long range force.
But the weak force is incredibly short range. It doesn't [INAUDIBLE] over the size of atomic nucleus. How come that happens has been a big problem. It turned out that people modeled this after superconductivity in the condense matter physics, and that's the mechanism of the Higgs boson. And I'm sure all of you know that Higgs boson has been discovered at Large Hadron Collider where smashing two protons against each other at incredible energies. So this is basically trying to redo the Big Bang in the laboratory. You can't really do the Big Bang, but you can you do a little band, so that's the idea of these experiments. And this is a huge tunnel, 27 kilometers in circumference. Inside you have these high tech components all lined up and in amazing way. These are mostly dipole magnets trying to bend this high energy proton to stay inside this circular ring.
And people in this department are playing a major role in this one experiment called CSM. The CMS experiment is like 50 meters high, 20 meters long, but the "C" for CMS actually stands for Compact. It's meant to be a compact detector which is 20 meters long because it's competitor experiment, called ATLAS, which my colleagues in Berkeley and Tokyo participate in, is actually 40 meters long. So compared to that, this is the Compact Muon Solenoid. That's what CMS stands for.
The other experiment is called ATLAS. They've been actually very active on outreach, so they produced a video like this one.
It's pretty cool, huh? And you can tell what movie they had in their mind.
[LAUGHTER]
They're pretty serious about this.
See?
[LAUGHTER]
But anyway, using these humongous devices, they did manage to discover this Higgs boson that was long in the making. It was announced on the July 4th, 2012. But if you think about this, the initial theory by Peter Higgs and company was back in 1964, so it took actually 50 years since then. People started to design this experiment in the '80s, so that's 30 years ago. People started to build this machine like 15 years ago. This has been long time in the making.
And so when this got finally announced, especially because it happened to be on July 4th, all of us in the field got really excited and celebrated the Higgs Dependence Day. So that was an amazing day. And it got coverage all over the ward in the media, so it was a lot of excitement, not just in our community, but everybody else in the world.
And the CMS experiment produced the Higgs boson this way. Two protons come in, produces so many particles from this collision, but you see two clear yellow hits. They are photons in this case. So you make Higgs boson, it disintegrates right away, and then you're looking at the fragments, in this case in the form of two photons.
By measuring the energies and angle of these photons, you can reconstruct what the mass of the object that decayed into two photons. So that's how they managed to discover this new particle. This is what's really exciting, and what this means is that there was another important stage in the evolution of the early universe.
So the same slide as before. When things get cold, you start with this chaotic situation to this very ordered situation here, and that's what happened with the Higgs boson, too. In the early universe, Higgs boson was zooming around in this very, very hot and dense universe. But at some point, Higgs boson got frozen in empty space. That happened when the universe became cold enough for Higgs boson to be condensed in the universe's empty space, and when universe became as cold as 4 quadrillion degrees.
But anyway, so that's what happened. So our universe today is supposed to be filled with this Higgs boson, everywhere, densely packed. That's the idea. So when you think of the particles moving about in empty space, it's actually not empty. It's filled with this Higgs boson.
Photon just keeps going straight because the photon can interact with only things that have electric charges. Higgs boson doesn't, so photon doesn't see Higgs boson. It just keeps going at the speed of light.
But every other particles we know, like the carrier of the weak force, keeps bumping on the Higgs boson, so it slows down. And namely, that it doesn't go very far, and that's how the weak force has become short ranged.
And that's exactly the same thing that happens in the piece of superconductor in the previous video. In the piece of superconductor, what condenses there is the Cooper pair of electrons, and that actually makes the photon massive because photon does see the charged particles like electrons condensed in the, quote unquote, "vacuum." Then scatters against it and slows down, and that's how the magnetic field becomes short ranged. Inside a superconductor. So it's exactly the same mechanism happening in entire universe here.
And electron also bumps on this Higgs boson a little bit, so electron also does slow down, which means it acquires a mass. Because it's not going the speed of light anymore, which means it has a finite mass. And that's the only reason why electron can be happily circulating around in atoms, because it does have finite mass. It doesn't go with speed of light.
If for some reason if you manage you evaporate the Higgs boson stuck here, then your body would vaporize in about a nanosecond because the electrons start to move at speed of light. So it's actually what's keeping you together, and that's what Higgs boson is doing.
And even though this was discovered, we think we understood what's going on there, we are still very uncomfortable with this new particle because it doesn't have spin. Every elementary particle we have seen so far-- electrons, photons, quarks, muons, even neutrinos-- they have spins, but Higgs boson doesn't have spin. It's the only particle we know that doesn't have spin.
The way I characterize this is actually sort of a faceless particle like Jason. What I mean by this is that if something is spinning, it actually doesn't have rotational symmetry. Depending on how you look at it, it looks different to you, right? So it kind of has a face. But Higgs boson is totally faceless. Its kind of completely bland. No matter how you look at it, it looks exactly the same to you. It's a very spooky, strange kind of particle.
When I first learned of this theory back in grad school-- that was really in textbook-- I just couldn't believe it. Because you know, what we are doing here is to introduce the kind of particle nobody has seen before and let it do the most important job in the theory. It freezes into the universe, gives mass to everybody.
That sounded like a big cheat to me. I just didn't want to believe it. As a matter of fact, Csaba and I are the guilty people who eventually proposed what was later called Higgsless theory.
Well, I guess it got discovered.
[LAUGHTER]
So we lost the game, so we never go to international meetings these days. I pretend that I'm a proper Japanese, and I apologize like this.
[LAUGHTER]
But you can see this still begs a very important question. Is this the only one that doesn't have spin? That sounds like a very unusual thing to me. Maybe you got siblings and relatives, and that idea has a name called supersymmetry. So there may be just tons of other spinless particles. We have found the first one in this spinless stripe. Maybe so. We don't know that.
And we also wrote a paper where we claim that Higgs boson actually does have spin but it's spinning in extra dimensions of space. That's why we don't notice it. That's why it looks kind of spinless to us, but it may actually have secretly spin. And this theory is, I think, still alive, right? Yeah, so he's nodding. So that's another possible direction.
Or maybe it's not an elementary particle. Maybe Higg's boson's made up of smaller things. And smaller things have spins but they cancel against each other. Then we're talking about composite Higgs boson.
So we still don't quite understand this new particle. And the most important question, of course, is why did this particle-- particularly this particle-- freeze into empty space of the universe? And we still don't know the answer to that question. So we need to study this particular bit more, and our friends on CMS experiment are working very hard on this.
But it's actually pretty difficult because whenever you smash these protons against each other-- and this is a picture from the rival experiment with ATLAS-- what you get is this huge mess, and you need to dig out what you are looking for out of this huge mess. And it's actually not that easy. You're looking for something like tens of events out of a quadrillion collisions in this big, complicated mess, so that is a very difficult job to do. One of the reasons I myself and many of the colleagues here would like to see yet another accelerator built, called International Linear Collider, is precisely for the purpose of trying to understand this new particle better.
If I take an analogy, the LHC experiment is a little like smashing cherry pie against cherry pie. It produces a huge goo of mess, but what you are looking for is the collision of cherry pits inside of pies. That's what you're looking for. But if it has this huge goo, it's very difficult to pick that up.
It's a lot cleaner if you manage to throw cherry pits against cherry pits and study their reactions. But as you can easily imagine, cherry pie is kind of easy to throw, right? And also, it can easily meet with each other. It's much more difficult to throw tiny things, and all the more, it's more difficult to make them meet.
And as a matter of fact, if you would like to build this machine, you need to squeeze the beam of electrons down to a few nanometers and make sure you can control them so that they meet against each other. It's an incredible high tech you have to do, but after a lot of study, especially given people here studied this basic technology of building this e plus, e minus machine, now technology is at hand, so in principle, we can build this machine. And it's also curious to know that some of the main technology used in this kind of built accelerator is also based on superconductivity, which we are trying to use to study the superconductivity of the universe, which is the Higgs boson.
So you need to accelerate this beam of electrons over about 15 kilometers, and you see this structure. You have the cavity inside, and inside this cavity you put in basically the radio wave. So radio wave has wave. You put the beam of electrons a notch above the top of the wave. It keeps getting pushed by the electric field, gets accelerated, and eventually gets squeezed down to a few nanometer size.
And if they meet, they can cause a collision. It's based on cherry pits against cherry pits because the electron is an elementary particle. So you end up with a rather clean collision like this one without a huge goo. So hopefully this would happen.
But as you all know, building these things costs a lot of money, and I'm really envious that the condense matter people can also produce Higgs boson in the laboratory much, much more cheaply.
This is a very beautiful piece of data. You have a superconductor, and what you do is whack it with the high frequency laser. And then laser basically produces such a pressure on the system that what used to be the ground state here is now moved upwards like this, and then it starts to oscillate like cloud. And that's the same thing as the Higgs boson in the case of the universe. So this oscillatory behavior you see is that oscillating back and forth of this Higgs boson mode in a superconductor. That's what they can really do in the laboratory.
I talked about this concept of spontaneous symmetry breaking, which basically appears everywhere in physics, even chemistry and biology, and there's one important theorem for it. This is where I have made a little bit of a contribution to this business. And the theorem, which goes back to Jeffrey Goldstone at MIT, this theorem says the following.
"When a continuous symmetry is spontaneously broken, there appear the same number of massless particles, or gapless excitations in condense matter language, as the number of broken symmetries." For each symmetry that's spontaneously broken, there's supposed to be one gapless excitation, or massless particle. And that's the theorem.
And when there's no gap, energy should go to zero when momentum goes to zero. And then the question is, what is the power in momentum? And the answer is it is supposed to be linear. When momentum of this particle goes to zero, then it should also go to zero linearly with momentum, and thus the theorem. And we call these massless particles Nambu-Goldstone bosons.
And again, this idea seems to work pretty well in many different systems, so I mention crystal. When the liquid crystallizes, it breaks translational symmetry. Translation can go this way or that way, and both symmetries are broken. And what Goldstone theorem says is that if there is a symmetry that's spontaneously broken, you use the symmetry and provide a kick to the system.
And if you do the symmetry and provide a kick this way, then the crystal starts to vibrate and does a sound wave, and quantum version of that is a phonon. In this case, this is a longitudinal phonon. And indeed, you have one gapless excitation for one broken symmetry, and dispersion relation is linear.
If I use the other translation, again, I get a gapless excitation because the lattice starts to vibrate that way, and that's a transverse phonon. Again, has a linear dispersion. When momentum goes to zero, energy should go to zero, because the zero momentum limit means in quantum mechanics there is no spatial dependence, and no spatial dependence means the kick is on the entire system.
And entire system just moves from one position to another, that doesn't cost you any energy. It's the same idea of this buckled elastic rod. It doesn't cost you any energy to go [INAUDIBLE] it around. The same idea, that's the Goldstone boson.
In the case of superfluid, you also have this symmetry. I mentioned this already, that the microscopic wave function can change its phase, but once macroscopic wave function has a particular value, this phase symmetry, called U(1) symmetry, is also spontaneously broken, so there's one broken symmetry. And this is a textbook example of the spectrum of the liquid helium superfluid, measured by neutron scattering, and you see this linear rise in the energy here. So again, you have one massless particle per one broken symmetry.
A more recent example of this is the cold BEC, Bose-Einstein Condensate of cold atoms. Again, if you blow up the spectrum from Raman scattering experiment, you see the linear dispersion between energy and momentum, so this is a linear relation.
So it seems to work well, and this kind of thing is also supposed to be happening at the dense core of a supernova. It's very difficult to see directly, of course, but people have done a lot of calculations in nuclear physics and astrophysics. And that this is an example of the x-ray picture of a supernova remnant where supernova is still spurting out all kinds of material into space, and the dense core of the super neutron star-- it's supposed to be a neutron star-- is also a system with spontaneous symmetry breaking with its own phonon, just like in a superfluid.
But then there was a little problem I had to think about. We talked about magnets where spins are lined up in particular way. And you can have these Heisenberg models where you have the interaction of spin's nearest neighbor, and if you actually rotate the spins around, because it's the inner produce between two spin vectors, it doesn't change. The system has rotational symmetry.
And depending on the sign of this term here, if this sign is positive, then lining up spins would make the energy higher, so you rather make the spins anti-parallel to each other. Approximately, this is the ground state, called [INAUDIBLE] state, and everybody knows that.
If the sign is negative, then you'd rather line up spins in the same direction. Then the ground state becomes this one. So this is antiferromagnet, this is ferromagnet.
Either way, you have the same pattern of symmetry breaking. Originally, the system was invariant under all possible rotations, and you know there are three ways of rotating a system, around x-axis, y-axis, and z-axis, so they are three symmetries you start out with. But either case, you have spontaneously broken symmetries. The only remaining symmetry is the rotation around the z-axis.
This symmetry is broken, that symmetry is broken, but this symmetry is still unbroken. So you have broken two symmetries out of three possible rotations, and so the possible space of all possible ground states can be just described by specifying the direction. So it's actually a two-dimensional sphere. Surface of two-dimensional sphere is the possible space of all possible ground states.
So it has exactly the same pattern on symmetry breaking, but our condense matter colleagues have known for ages that in the case of antiferromagnet, there are two gapless excitations, but for the ferromagnetic case, there's only one. Somehow the theory is not working.
Also, dispersion relation is wrong. In the case of antiferromagnet, it looks right, consistent Goldstone theorem. But in the ferromagnet case, energy goes quadratically with momentum. Again, something is wrong. So what's going on here?
And this really defied my intuition because in the case of ferromagnet, because all the spins are lined up this way, just like in the case of crystal, you have this symmetry and that symmetry that's spontaneously broken. So I should be able to provide a little kick to cause a wave in the system. If I cause a kick sideways, it should behave something like that. That's my expectation. If I do a kick this way, then it should produce a wave like that one. At least, that's my intuition.
But it doesn't work that way. So it has been known for decades that the only possible way to create a wave in a ferromagnet is something like this. And that's it. No other mode. Something is missing here. I don't quite understand it.
And it turned out that this kind of issue was just not in ferromagnet, but more recently people actually got interested especially in nuclear physics community, and there had been many papers that says spontaneous symmetry breaking with abnormal number of Nambu-Goldstone bosons in kaon condensate.
Do people use the word "abnormal" in titles of papers? That sounds abnormal itself. But anyway, so people started to get worried about this because this kind of wrong projections from Goldstone theorem seem to appear in many different systems. Abnormal number of Nambu-Goldstone bosons. And here, the abnormal number of Nambu-Goldstone bosons. This is actually condense matter paper.
So some fundamental understanding is missing, and Nambu himself had a paper talking about the anomalous properties of Nambu-Goldstone bosons. After reviewing this basic theorem, he said, "There are, however, exceptions to the above theorem." So if there are exceptions, it's no longer a theorem. We can actually change that. So what is the right theorem?
And as a matter of fact, I actually started to think about this only because I was teaching a course in quantum field theory in Berkeley, and there was one guy sitting in my class who kept asking questions over questions and questions. I got irritated. Shut up!
It turned out he was asking a very important question. Goldstone theorem is not working. We need to do something about this. So remember, graduate students in this audience, you should ask questions. You know, this is very important because you end up writing four physical [INAUDIBLE] based on that idea.
So this is the answer, it turned out. After a lot of thinking, what's really different between the antiferromagnet and ferromagnet, there is one major difference between them. If you just look at the expectation value of spin in z-direction, because they're anti-parallel to each other, they cancel between nearest neighbors. So overall, there's no expectation value.
But in the case of ferromagnet, of course they all add up, so there is a big expectation value for speed in z-direction, and spin in z-direction can be written as a commutator spin in x-direction and y-direction, as you know from graduate quantum mechanics.
So this is the key. If you look at this equation, that looks awfully similar to the canonical commutation relation between x and p. Commutator of two operators have a finite value. In this case, h bar. In this case, the expectation value of the spin.
What that means is the following. My intuition, original intuition, was that if you use Sx-- that's this rotation-- that should create a wave like this one. And its y rotation, if you do the kick of this y, then it should create a wave like that one. So that it was my intuition.
But there is a finite commutator between Sx and Sy, just like x and p. And what do you know is that for x and p, x and p together describe a one degree of freedom. Not independently. You need to specify both x and p in classical mechanics to describe the motion of particle, position, and the velocity at every instant. In quantum mechanics, it's a little complicated. It's a wave function, but still the same idea. x and p together describe a single degree of freedom.
So what's going on here is that Sx and Sy are canonically conjugate to each other in a way. These two operators together describe one degree of freedom.
In the case of you antiferromagnet, then this commentator vanishes. So Sx and Sy are basically both analogues of x without p. They are both position operators in a way, so they produce different degrees of freedom because they are different operators.
But in a case of ferromagnet, because they have this commutator, they are canonically conjugate to each other. Sx and Sy together describe one degree of freedom, and that's what leads to only one Goldstone boson in ferromagnet. So that actually turned out to be the right answer, right way to look at the problem.
The way we prove this theorem is by using field theory. But it's only a single line equation you're supposed to write, and this contains everything in it. And the key is this first term. In the case of antiferromagnet, this first time is missing. Then you are supposed to find this time derivative squared, that's in quantum mechanics energy squared, balances against spatial derivative squared, that's in quantum mechanics momentum squared. And as you squared Goldstone momentum squared, energy goes with momentum, so that's a linear dispersion.
But in the case of ferromagnet, this term is there, and this looks indeed similar to this form of the Lagrangian, pq dot minus H, because this is the first power in time derivative. So you can tell that different degrees of freedom are now canonically conjugates to each other, so a pair describes one degree of freedom.
You also see that this is first time derivative that goes with energy should balance against momentum squared. That's the spatial derivative squared. So energy goes quadratically with momentum, so you understand that here, too.
So this single line ends up perfectly explaining what's going on in this system. And once you know what's going on here, then you can then go to the math book and completely classify on which occasion you are allowed to have this additional term that changes the number of Goldstone bosons and also dispersion relation between energy and the momentum. And once you have this Lagrangian, you can walk backwards and see that this term is really responsible for having this commutator between symmetry generators in expectation value, so there is a one-to-one correspondence between the presence of this term and this commutator.
And once you know that, here is the new theorem. Number of Goldstone bosons is not the same as the number of broken symmetries, but you need to subtract number of pairs which are canonically conjugate to each other because the pair will describe one particle instead of one particle each. So you need to subtract that number, and the number you need to subtract is by looking at this commutator, which can be a big matrix, so many different generators.
And the rank of this matrix tells you the number of pairs inside, and the half of that is the number of xp pairs you need to subtract. And that's the theorem. This way you can perfectly work out what is the number of these gapless degrees of freedom in all systems we have looked at in the literature.
We are very happy that this got published in Physical Review Letters. I got [INAUDIBLE] suggestion. Got even highlighted in Spotlight. And if you're interested in mathematics, there is some obscure mathematics which we didn't know about called presymplectic structure on homogeneous spaces that's relevant for this.
But never mind. Let me just give you one interesting example. It's called spinor BEC, and this is a system my colleague in Berkeley, Dan Stamper-Kurn, managed to create, and this is the BEC of atoms with the spin degrees of freedom not frozen. Original BEC was made with magnetic trapping or the spin's degree of freedom had been frozen, so that was not interesting. But he managed to create BEC with the laser trapping, so the spins are still dynamical. They can move about.
He particularly had a spin 1 atom, and if you have spin 1 atom, there is the large degree of symmetry because you have spin that can orient in that direction, this direction, and that direction. So you have a three component field here, x, y, z. So you can rotate them around. That's the three dimensional rotational symmetry. In addition, this macroscopic wave function can change its phase overall. That's yet another symmetry. So there are actually four symmetries you start out with.
And this wave function actually has an expectation value of this rather funny form, and once you actually do the math, you can still see there's one symmetry that's still unbroken. So out of four symmetries, one is unbroken, three had been broken in this Bose-Einstein condensate.
So there are three broken generators, and Goldstone theory will tell you there must be three gapless excitations with the linear dispersion. But if you work it out, you don't get that. There's one Goldstone boson with the linear dispersion. There is a second Goldstone boson with quadratic dispersion, but there's no third one.
But using theorem, this also works out just beautifully. What's going on is that there are three broken generators. You have three-dimensional space for the possible ground state. And out of these three dimensions, two of them pair up, and they become canonical conjugate pair between x and p. The third one is left alone. It's not paired.
I have never seen anything like this in any other context. Your space is partly a phase space of x and p, partly configuration space with x only but no p, and they're kind of mixed up. And that's the space you're looking at, and that's how you can understand this kind of spectrum. And we urged Dan's group to do the measurement of this quadratic dispersion, and they did that, and it is indeed quadratic. Ed Marti is the graduate student who led this experiment.
And so, it is true. And so Goldstone theorem doesn't work in many systems in condense matter atomic physics, but this revised theorem still works completely universally as far as I know.
Once you understand it, you can start applying it to different systems. For example, in some cases, you apply some external field that symmetry is no longer exact, and that creates a small gap to this Goldstone boson. Then it's called pseudo-Goldstone boson.
In some cases we found that you can come up with exact formula for the gap. You can compute, so that's probably useful in systems like a spin in external magnetic field or kaon condensation in supernovae. We also looked at some soliton solution to this, and soliton solution in the Heisenberg ferromagnet has a funny name called skyrmion, which actually originally was proposed by Witten in the strong interaction theory.
And once you have this kind of solution, then you can move things around. And when you actually move the whole thing around, momentum operator is supposed to push it in one direction or the other. We don't want to think momenta commute with this among themselves. Translating this way and that way, or doing the other way around, should give you the same answer.
But in this kind of system, once we know the correct Lagrangian, it turned out that these two momentum operators don't commute according to how many winding number of so-called soliton solutions. And this is actually a very curious effect.
What it means in practice is that if the momentum operations don't commute with each other, it's sort of like being in a magnetic field. So when you exert a force, the object would move in perpendicular direction. And people have been doing very big numerical simulations of macroscopic number of spins to actually simulate the motion of skyrmion.
It looks something like this. And when it comes close to the wall, wall tries to push it backwards, and it actually moves sideways. Wall is pushing it upwards, moves sideways. This object always moves perpendicular to the direction of the force.
And let me just play it once again because it's so cute. So when the wall pushes it backwards, it actually goes down. That's sideways relative to the force. In this here, wall is pushing upwards. It moves sideways again. So this is the result of [INAUDIBLE] simulation, but based on the Lagrangian, you can understand it in a single line. So it was also kind of a curious thing, and it's very fascinated by that.
Now that the Goldstone boson is kind of interesting from different aspects, we also thought of the idea that the dark matter of the universe may also be Nambu-Goldstone boson. And this is an idea I worked with a post-doc here, Eric Kuflik, together with the collaborators, Yonit Hochberg, Tomer Volansky, and Jay Wacker. And the idea is the following.
As I mentioned earlier, when you do this heavy ion collision, there is a symmetry breaking that leads to the pions as the particles that emerges from the symmetry breaking. So maybe we just make exact copy of that. In the real world of the strong interaction physics, we do have these pions. They are just another copy of it.
So here's the original pion where quarks are bound together by a particle called gluon, and here is the copy pion where you have dark quarks we haven't seen yet bound together with dark gluons, so this is the dark pion.
It turned out that if you actually do the math, now you know how to deal with the Goldstone bosons in a mathematical Lagrangian. You can compute the reactions and find that they actually do have some interactions among themself because they have strong interaction, after all. And you also get the right abundance of these dark matter particles if you actually compute the process.
So this is actually relatively like candidate for dark matter, about 100 MeV or so. It's lighter than even the mass of the proton. And maybe this particle is right here. The dark matter is everywhere in the universe, so maybe this is the particle that's floating around right here.
And you probably have seen these kind of pictures before. There are so many evidence for dark matter in the universe today. This is one such evidence. This is a picture taken with a telescope, and it's kind of cute. You see two eyes, nose, mouth, and the frame of a face. And of course, these eyes are galaxies. They're round, elliptical galaxies.
What's going on with the frame of the face and the mouth, it's also supposed to be a round galaxy, but the light rays coming form those round galaxies which happens to be in the background of the system get bent by the gravitational pull. It's called gravitational lensing effect, and that's why its image is totally distorted like this one. There are many kinds of systems like this one where this gravitational lensing exhibits itself and demonstrates the existence of dark matter in the universe today.
And this particular system is so cute that it actually has a name. It's named Cheshire Cat just like in Alice In Wonderland. So dark matter does exist in every single galaxy.
And according to numerical simulations, most people assume that dark matter would not interact with itself at all. When they get squeezed by the gravitational pull, then at the center of galaxy, you are supposed to see a huge spike in the number density of dark matter. But according to some observations of relatively small galaxies called dwarf spheroidals, there doesn't seem to be a sign of this big cusp in the distribution of the dark matter, so this is still a remaining puzzle.
But if you do take our kind of model, that dark matter would scatter against each other. As the gravity tries to pull them in, they scatter against each other, disperse themselves, and smooth things out, and then you don't produce a big cusp. You smooth things out. Maybe this is actually the better way of understanding distribution of dark matter.
In order to actually settle this issue, we need to do a much better measurement on the motion of stars in small galaxies, and I'm actually leading a collaboration to build an instrument for a Subaru telescope for the purpose. This is actually a multifiber spectrograph, and comparing the kind of observation people have done in the past, we can map out the velocity profile of the dwarf galaxies in a much bigger way. And once you understand velocity profile, you can translate that into the mass profile, because after all, it's just Newtonian mechanics. And that's what we try to do in the next five, seven years.
So we are right now building this rather complicated instrument. This is the wide field corrector lens system, and you put in these little robots called fiber positioner that controls the position of 2,400 optical fibers within 10 micron accuracy. And then fiber gets fed to the spectrograph where you actually look at the spectrum, and Doppler shift in a spectrum tells you the velocity profile.
The lens system by itself is this big, and then we have this finer focus instrument at the top to relieve the tension on the fibers. This is a test bench for this fiber positioner robots, and we put this microlens on top of each optical fiber to convert the F ratio to a manageable level to avoid light loss. This is the fiber connector, dichroic lens, corrector lens in the spectrograph. Mock up the spectrum itself in cryogenic system, and metrology camera. So this is the collaboration of us at U-Tokyo, and Princeton, Caltech, Johns Hopkins, Marseilles, Taiwan, Max Planck, and Brazil.
The original purpose for building this instrument is trying to understand dark energy of the universe. It can also test general relativity and can also measure the mass of the neutrinos. In some way, all of these things seem to be connected.
So this is end of my talk. I wanted to tell you is that even though they seem to be totally different phenomena in the world, they are all described by a single Lagrangian of spontaneous symmetry breaking, and that's exactly why physics is so much fun. Thank you for attention.
[APPLAUSE]
SPEAKER 1: Thank you very much, professor, for this very nice talk. Time for some questions.
SPEAKER 2: [INAUDIBLE]
HITOSHI MURAYAMA: I can't hear you at all.
SPEAKER 1: [INAUDIBLE]
[LAUGHTER]
SPEAKER 2: So when you have the Lorentz symmetry [INAUDIBLE]
[LAUGHTER]
HITOSHI MURAYAMA: That's right, right. So--
SPEAKER 2: [INAUDIBLE]
HITOSHI MURAYAMA: No. Yeah. So the question was the following. So the Goldstone wasn't wrong. He made an assumption that system is Lorentz invariant. But if you look at the piece of magnet, it's obviously not Lorentz invariant. That's why assumption is not true, and therefore, the conclusion is also not true.
So if you do have Lorentz invariant system, then Goldstone theorem is still supposed to be true, and you do get the same number of Goldstone bosons as number of broken symmetries. So you know, Goldstone wasn't wrong, but had to be generalized to be applicable to wider class of systems. Thank you for asking that question.
SPEAKER 1: OK, more questions? Yes?
SPEAKER 3: What is the difference between spin 0 and no spin at all?
HITOSHI MURAYAMA: What's the difference between spin 0 and no spin at all. So in the case of the ferromagnet, clearly you do need spin to have a ferromagnet. But you can also have the system where you don't have spins but you still have the phenomenon of spontaneous symmetry breaking. What people talked about in nuclear physics community was this possibility that yet another kind of particle called kaons, it's actually another Goldstone boson, sort of similar particle as a pion.
This kaon might condense, Bose-Einstein condense, in the core of the supernova. It doesn't have any spin but it still exhibits this abnormal behavior that number of Goldstone bosons is not the same as number of spontaneous broken symmetries. So that's where some of these discussions got started with.
So even particle without spin, you can still have this phenomenon that the Goldstone theorem apparently is violated, and you have a fewer number of these gapless excitations.
SPEAKER 4: Why do we have all these artificial sweeteners when we could just use L-glucose?
HITOSHI MURAYAMA: Yeah. It's actually a little bit more expensive to produce. I actually read up on the this. So artificial sweeteners, they're useful extensively partly because it's actually very cheap to produce. And if you like to produce the D-glucose, it's more expensive. But according to Wikipedia, at least, it had been marketed, so you can probably buy it. It's probably much safer than any other artificial sweeteners, because after all, it's a real sugar.
SPEAKER 5: So if glucose is more expensive--
[LAUGHTER]
Is there actually a symmetry between glucose and the chiral version?
HITOSHI MURAYAMA: Well, it's just a mirror symmetry, so it's a complete just inversion of space, left versus right. And let me just pull up the slide again.
You can look at this chemical formula, and you see at the top, it's just mirror, right? And that's it. These are called optical isomers, and they have the exact same chemical property except for the fact that they are mirrored.
Does that answer your question?
SPEAKER 5: But if it's more expensive, then there must be some difference?
HITOSHI MURAYAMA: Yeah. So there is difference because everything we can deal with in chemistry, you have to start with something, and that something already has a particular orientation. I don't know much about chemistry, so that's the best I can say.
SPEAKER 1: All right, last question.
SPEAKER 6: [INAUDIBLE] two-dimensional, one-dimensional [INAUDIBLE]
HITOSHI MURAYAMA: Right.
SPEAKER 6: [INAUDIBLE]
HITOSHI MURAYAMA: Yes, you can still use the same Lagrangian, but it turns out that in the say one dimensional spin chain for antiferromagnets, say, then the interactions among the pions actually become strong in infrared, and so they actually generate its own mass. So it turns out that description using this Lagrangian is still true at the sort of medium distances, but as you go to long distances, due to the effect called renormalizaton group, the interaction becomes so strong that this actually is no longer useful description at long distances. And it turns out that these Goldstone bosons become bound states and massive, and symmetry sort of restores itself.
And so that also you can tell, just by looking at the Lagrangian and do the power counting on which term becomes more important at what distance scales, and you can tell that it is actually growing more important interactions at long distances.
SPEAKER 1: All right. Let's thank Hitoshi again.
HITOSHI MURAYAMA: OK, thanks for attention.
[APPLAUSE]
As part of the Hans Bethe Lecture series, theoretical particle physicist Hitoshi Murayama presented the physics colloquium Oct. 19, 2015 at Schwartz Auditorium. Murayama is a professor of physics at the University of California, Berkeley and the founding director of the Kavli Institute for the Physics and Mathematics of the Universe at the University of Tokyo.
