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0:00
(gentle music)
0:04
- Thanks so much for having me here at the Ri. Very sad that I can't be there in person. I was there in person just a year ago for a talk celebrating volume one of this series, and now we're celebrating volume two. So I need to start, I hate it when people do this, but I'm gonna start with sort of an apology or at least an explanation for what you're about to hear for the next hour. The idea behind this book series, "The Biggest Ideas in the Universe," of which there will be three.
0:32
The last one was about classical mechanics, this one's about quantum mechanics, the next one will be about complexity and emergence. And the idea is I'm gonna teach you the biggest ideas, what I consider the biggest ideas in the universe. Really, they're the biggest ideas in physics, okay? I can't pretend to explain biology or psychology or politics or anything like that. But by the biggest ideas I mean ones that will really last, ones that will still be thought of as useful 500 years from now.
1:00
So we're not gonna speculate, maybe some of our current speculations about the multiverse or black holes or string theory will be very, very important 500 years from now, but we don't know which ones, okay? So I'm sticking to the stuff that is established and verified. And the secret is I'm showing you the equations, okay? So I am not just giving stories and metaphors about these big ideas, but I'm actually teaching you the math behind them even if you start from zero mathematical background.
1:30
And that's fun to do, it's not everyone's cup of tea, but the people who love it, love it. I love it and I hope that you do too. But here comes the apology part. A year ago, I was there at the Ri, one of my favourite places to give talks to broad audiences, talking about book one of the series, and book one was called "Space, Time and Motion" and its idea, like we said, was to discuss the classical world which goes from Isaac Newton up through Albert Einstein and the theory of relativity,
2:02
special relativity, general relativity, et cetera. And within that framework, given that the book shows you a whole bunch of equations, there is a natural sort of capstone equation, a triumphant conclusion to our story, which was this equation. This is Einstein's equation for general relativity. And if you read it out loud, it says R mu nu minus 1/2 R g mu nu equals 8 pi G T mu nu. It relates the curvature of space-time to the amount of energy and matter in the universe.
2:33
And these symbols are mildly off-putting. You know, there are Greek letters, there are subscripts, et cetera. But in the course of a single one-hour talk, I could get you there. We could go from you don't know what calculus is to understanding what this equation is. And I could tell you some fun background stories about the person who did all this work, who's of course Albert Einstein. A photograph of Albert Einstein when he was a younger man before he got into the sort of sweaters
3:01
and grey hair period of his life when he was writing down these equations. And it felt very good to sort of get that accomplishment that we finally understand this equation even though we couldn't solve it like various people have. But you know, the whole audience knew what this equation meant. So for volume two, which is quantum mechanics and quantum field theory, it seems very natural to try to do the same kind of thing. But you notice the picture of volume two here is smaller in the corner.
3:29
That's because there's more stuff I gotta fit on this slide. If I were to pick an equation to build up to, the equation would look something like this. This is what we call the Lagrangian for the Standard Model of particle physics. This is the quantum field theory that as far as we know describes the world in what we call the everyday life regime. So the atoms and particles and forces in you and me are completely described by this equation. But as you see, it's kind of less elegant.
4:00
There's a whole bunch of things going on there. I had the idea of maybe I would just have one slide where I showed you pictures of all the various physicists who contributed to this equation in one way, just like on the last slide we showed Eisenstein. So I got as far as writing down their names, and what you see is there are too many names.
4:20
Even though this equation is longer, it actually hides an enormous amount of work in very important ways by very, very smart people, experimenters and theorists and the whole bit, and it just becomes implausible to discuss the whole thing in any detail. So what I'm gonna do is I'm gonna pick out one idea that... I actually did a poll. I went online, I asked people on social media and listeners to my podcast, "Mindscape," you know, what should I talk about?
4:49
So they gave me a very definitive answer. I'll tell you what it is in a second. But first, let's not be too intimidated by this slide right here. It all comes down to this little graphic, okay? This graphic is basically telling you what are the particles and forces of which you are made. And the quantum field theory describes the behaviour of these particles and forces. So, roughly speaking, and this is very much a cartoon and we'll improve upon it in a minute,
5:18
but this is a picture of a hydrogen atom, okay? A hydrogen atom has a proton at the centre and an electron orbiting around it. And there's various particles going on here and various interactions mediated by forces. But we can list what they are, there's a finite number of them. So if you dig down into the proton, we know now that protons are made of quarks. Two up quarks, as we say, with charges +2/3 each, and one down quark with a charge of -1/3e, so the total charge of the proton is +1 one.
5:52
And those quarks are held together by what we call the strong nuclear force, which is carried by particles called gluons. And then orbiting the hydrogen atom is an electron with charge -1. So the total charge of the hydrogen atom adds up 0, +1 for the proton, -1 for the electron. The electron is held to the nucleus by a different force, the electromagnetic force carried by photons. And experimenters have shown that sometimes these quarks, the up and down quarks,
6:21
can convert into one or the other. And in the process of doing so, they spit out a different kind of particle called a neutrino, and that interaction is mediated by the weak nuclear force carried by W and Z bosons. And then of course we have gravity. Gravity is everything pulling on everything else through the gravitational force carried by gravitons. And as we now know, all of these particles are swimming in a background field called the Higgs field.
6:50
And the particle that is associated with that is the Higgs boson. So there's a bunch of ingredients, I admit, but it's still a finite number. We could understand all of these things. And what I wanna tell you in the talk is the physical principles that make this picture hang together. And a good way to start is actually to try to put ourselves in the mindset of a physicist from the late 19th century. So let's imagine we live in the 1890s, okay? There's a bit of a myth that goes around
7:20
that says that physicists of this era thought they were almost finished, thought that they just had to dot some Is and cross some Ts and they would understand everything there was to understand about the world. Now, in reality, almost nobody believed that. There were a few courageous people who did guess it, but most people knew that there were puzzles on the horizon. But you can still forgive the people who were tempted to believe that because it did seem like in the 1890s
7:48
we were honing in on a coherent story of the world at a fundamental level. And that story comes in two parts. There's matter which is made of particles, and there are forces which are carried by fields. So the idea of particles was still new in the 19th century. The chemists caught on to atoms before the physicists ever did. And they in indeed thought that atoms were indivisible. Now we know better. But still, the idea there are particles making up matter was a very powerful idea.
8:20
The different ways you can arrange the atoms tells you whether or not you have a solid, a liquid or a gas, for example. And then the idea of fields was also brand new there in the 1900s. In the 19th century I should say, sorry. It was electromagnetism that really made this idea of a field very vivid. So what is a field? A field is sort of the opposite of a particle. A particle has a location in space. An electron or a quark or whatever, you know where it is, you can say how fast it's moving.
8:50
A field has no location. A field is everywhere. A field exists at every location and it has a value. That value might just be a number or it might be a vector, like the magnetic field and the electric field. It might have a magnitude but also a direction at every point in space. So most of the time you think of empty space as empty, you don't see anything there. But in the modern way of thinking about the universe, even empty space is populated by these fields.
9:19
So here's a classic picture of a magnet with iron filings around it. If you just look at the magnet by itself, you don't see anything around it, but secretly there is a magnetic field all around that magnet, and these iron filings are basically telling you the direction in which the magnetic field is pointing. So in this 1890s view of the world, you had atoms and other kinds of particles. They knew about electrons, for example. And then you had forces like electromagnetism and gravity
9:48
carried by fields. And maybe, the hope was, all we had to do was figure out what all the particles are and figure out what all the fields are carrying the forces and we'd be done, all with under the umbrella of classical physics. Of course you know that in the beginning of the 20th century quantum mechanics came along and changed everything. Now there's a puzzle with quantum mechanics. Quantum mechanics is so profound that even though we've known about it for 100 years,
10:18
professional physicists still don't agree on what quantum mechanics actually says. And that's very embarrassing. I don't know, to me anyway. I think that we should know what our best theory of nature actually says. But the weird thing is, even though we don't exactly know what the theory says, we do know what it predicts. So it's pushed 20 and 21st century physicists into this sort of positivist instrumentalist mindset where they say, "Don't ask me what's really going on.
10:47
I can just tell you what you're gonna observe in your experiment." And you know what? I hate that attitude, this is very much not my attitude, but it is the attitude we're gonna take for this talk. I wrote a whole nother book called "Something Deeply Hidden" about the philosophical mysteries of quantum mechanics, but today we're gonna be hard-nosed physicists and ask what the theory actually predicts. But even at that level, we have to make some dramatic claims about the nature of reality.
11:14
You know that cartoon picture we have of the atom, it's reproduced here in this case with a helium atom. Two protons, two neutrons, surrounded by two electrons. That is not very accurate, okay? I know this is the standard symbol of physics, the atom. This is the Rutherford atom after Ernest Rutherford who was at Cambridge when he put this idea together. And it makes the atom look like a little solar system where the electrons look like little planets. But quantum mechanics says
11:43
that's not what is really going on. What's really going on is that electrons don't have positions. They don't have velocities. You can measure positions and velocities, but what the electron is actually described by is what's called a quantum mechanical wave function. A very, very boring name for a very, very important concept, okay? So the image on the right is a reconstruction from many, many experiments of the wave function of a certain kind of electron in a certain configuration in an atom.
12:17
And you see it's spread out, it's not a point, the electron does not have a location. But the rules of quantum mechanics tell you that when you observe the electron, you never see the wave function. You see a dot, you see the electron as if it were a point-like particle. So that's the profound thing about quantum mechanics, is that the way that we describe the world in terms of wave functions spread out all over the place is different from what we actually observe.
12:46
They're connected to each other, the connection is the wave function tells you the probability of getting different measurement outcomes. You cannot say exactly what you're going to see, but you can say the likelihood of seeing different things. Now, many people have ripped their hair out about this situation trying to say, well, okay, it's really particles, the wave function is just a crutch, or it's really the wave function, the particles are just an apparent thing, an illusion that you see.
13:12
Again, we're not going to get into that. What we're gonna do is put quantum mechanics to work to understand the world. And the first thing you notice, one thing you notice about this picture is, if you compare it to the point of view of our 1890s physicists, they said, well, there are particles, that's matter. There's also forces, those are fields. So we have described here a little bit, very, very briefly and casually, the quantum mechanics of a particle, right, the electron.
13:44
And we said it doesn't have a position as a wave function, et cetera. Well, what about the fields? And it turns out that not only is this a necessary thing to do to apply the rules of quantum mechanics to fields, but it solves a whole bunch of problems. It turns out to be a great unification of physics. What it turns out to say is there are not separate things called particles and fields. There are only fields. But because quantum mechanics says that what you see when you do an experiment
14:12
is not how we describe the world when we're not doing observations, you see particles under the right circumstances. This is the miracle of quantum mechanics, that even though everything is actually smooth and continuous, there's no pixels, there's no lattice that the universe is defined on. Everything is smooth. But when you observe things, you see discreet quantum-like behaviour. So what's going on with that? And the analogy I like to use is plucking a guitar string, okay?
14:42
Imagine you have a guitar string where you, it's an open string so you know exactly where one end of it is tied down and where the other end is tied down, you're moving your finger on the fretboard or anything like that. You pluck it, you get a note. And if you're an expert guitar player, you realise that if you pluck it carefully in just the right way, you can actually get different notes out of the same guitar string with the same boundary conditions tied down at the same two ends.
15:10
How can you do that? The note you ordinarily get is what's called the fundamental note of the string, and that means the string is vibrating up and down uniformly. But if you pluck it very carefully, that's like the top row of this image here, if you pluck it very carefully, you can set up a different kind of vibration with half the wavelength, which means twice the frequency, so it'll be a higher note, okay? Where half of the string is going up while the other half is going down and vice versa.
15:40
That is an allowed mode of vibration of that guitar string. You could even imagine doing it so you get an even higher mode where there's up then down then up vibrating at the same time, and so on. So the point is the string itself is a continuous thing, there's nothing discreet about the guitar string. What's discreet is the set of possible ways the string can vibrate. In physics terms, this is solutions to the equations of motion for the guitar string. So the string itself is smooth,
16:14
but the solutions to the equations are discreet. The discreetness comes out naturally, even though we didn't put it in. The same exact thing happens for electrons in an atom. If you took chemistry class, you know about orbitals, right? You know that there are different electrons with different energies and different shapes to their orbitals, and these are what physicists call wave functions. Here, this image is different orbitals that an electron can be in around an atom,
16:43
and you notice that there's a discreet set of them. There's actually infinitely more of them that we're not picturing here. But the discreetness, once again, is not put in by hand. We have an equation that governs the shape of the electron's wave function, and the solutions to that equation come in a discreet set. So the electrons that we see in the natural world have a discreet set of energies. That's why when these atoms give off light, you see lines in their spectra
17:11
corresponding to how much energy is being given off as an electron goes from one orbital shape to another one. So the discreetness comes out of the behaviour, and that turns out to be the origin of everything that you ever thought was a particle in the universe. Photons electrons quarks neutrinos. In the modern quantum field theory way of thinking about things, the real thing that exists are quantum fields. But when you set those fields vibrating, they have a discreet set of ways to vibrate,
17:46
and that shows up to you and me like particles. So this image is a particle detector. You're sending some particles through, you see lines, right? You see trajectories just as if these electrons and alpha particles were good old classical particles. The equations tell you that they're not, they're really quantum wave functions. But when you observe them, when you measure them, like putting them through a particle detector, they appear to us as discreet sets of particles.
18:15
So that's a wonderful kind of unification. You don't need matter and forces. You have one thing, quantum fields, and it gives you everything. So, like I said, I asked the people of the world listening, what is the one feature of quantum field theory you would most like to understand better? And the answer that I got was symmetries and what is called gauge theory, so that's what we're gonna talk about. It turns out to be an excellent answer. This is a crucial feature of quantum field theory
18:45
and one that it would it be very good for everyone to understand a little bit better than we do. So what do I mean by symmetry? What does a physicist or a mathematician mean by a symmetry? Think of it as a transformation of some kind of physical system with the property that, when you do the transformation, the system looks exactly the same after as it did before. In other words, something that you can do that makes no difference. And the simplest example that is very helpful, I find,
19:14
is a triangle. So this figure here that you're looking at is reproduced from the new book. It is probably the most intimidating figure in the book, so I'm just, you know, maxing out how complicated we're getting here. But it's not that complicated. This is a triangle with its vertices labelled A, B, C, and we're asking, what can we do to the triangle to leave the triangle itself looking the same? So the labels A, B, C, don't matter, they just help us keep track of what we're doing.
19:42
So if I rotate that triangle by 120 degrees, that is to say 1/3 of the way around a complete circle, because it's an equilateral triangle, it comes back to looking exactly like it did before. Likewise, if I rotate it the other way, counterclockwise 120 degrees, or if I draw an axis dissecting the triangle through one of its vertices, and reflect it or flip it around that axis, it will also look exactly like it did before. So this is the set of symmetries of the triangle, and that's it.
20:19
We've just listed every possible symmetry of the triangle and you can combine them so you can like do a reflection and then a rotation, et cetera. Mathematicians love this stuff, they make careers out of studying all the different ways we can put these transformations together. We're gonna pump up, we're gonna promote this idea of symmetry to operating on our quantum fields, and what we're gonna find is it's a completely central concept for a very, very good reason.
20:47
It's important in classical mechanics and regular old particle quantum mechanics, but it's absolutely central in quantum field theory. And so let's explain why.
20:58
Think about a quark, okay? Think about a particular kind of quark, like the up quark. I've told you that all particles are really fields, so there is an up quark field filling all of space in front of you, and even inside you, for that matter. And one of the cool things about this that was figured out in the 1960s by Murray Gell-Mann and others is that when I say there is a quark field, technically there are three quark fields for just the up quark.
21:24
Forget about the down and the other kinds of quarks, just the up quark comes in three varieties which we call colours. They're not really colours, they can't look at them and see different colours. These are just fanciful labels to help us keep track of what's going on. But the idea is there is a way for the quark field to vibrate in one direction, which we label red, or it can vibrate in another direction which we label blue, or another one which we label green. These have nothing to do
21:51
with the physical directions of space around us, up down left right. They're just three fields that can sort of mix into each other. And the physical real quark field is not one of these. Again, if you're reading popular science books, you'll often be told, I'm sure I've said it before, there are red quarks, green quarks, and blue quarks. That's not exactly accurate. Really, every quark is a combination of red, green, and blue. The field is vibrating in some direction in this abstract space
22:23
with axes labelled red, green, and blue. So it might be vibrating exactly along one of the axes, and you can say, "Ah, I have a red quark." But more likely it's some combination, it's a little vector with components in the red direction, the green direction and the blue direction. So this is the real world, this is how quarks actually behave. And you might ask, well, what difference does it make how I label red and green and blue? We were just on the previous slide saying
22:51
a symmetry is some transformation you can do that makes no difference. Well, guess what? Rotating what you have labelled red, green, and blue is a symmetry. It's a transformation that makes no difference. Those labels, this direction is red, this direction is green, this direction is blue, we made those up, they were arbitrary for our convenience. So there's a built-in symmetry to this kind of physical situation which says I can rotate the axes that label what I'm labelling red, green, and blue,
23:23
and none of the physical reality underneath it changes. It's purely a convenience for my own sake. So this is a set of rotations in this space of quark colours, and we label those rotations SU3. You don't need to know the details but the 3 is indeed because there are three different colours. And a physical image that you might have of three different directions and just rotating them that does a perfectly good job of capturing what's going on here. So you begin to see
23:53
why symmetry is important in field theory, because you have these fields, the red field, the green field, the blue field are physically identical to each other. The real quark is a combination of them, so there's a built-in redundancy of the description, okay?
24:09
Now, if all that's true, if you bought that, and I said this is the quark field vibrating at one point in space right here at this point right in front of my eyes, it's also true at some other point a little distance away, right? I can also change what I mean by red, green, and blue somewhere else. Now here, there's a question. Can I separately rotate red, green, and blue into each other at different points in different ways? Is that a symmetry? Notice that's an enormously bigger symmetry.
24:41
There's one thing I can imagine doing which is that everywhere in the world I rotate red, green, and blue into each other in some uniform way, and that's a certain pretty big, interesting kind of symmetry. There's a much grander symmetry that says separately, at every point in space and time, I can rotate these different fields into each other and that's a symmetry, okay? That kind of symmetry turns out to be real. The real world has that symmetry, we can do that.
25:10
We can rotate the axes that we call red, green, and blue separately at each point in space. This is called a local symmetry. So it's not global and rigid, it's separate at each point, so it's local or also what we call a gauge symmetry. Now you begin to see the idea of a gauge symmetry coming in. And I'll just repeat myself, a gauge symmetry just says we have the separate ability at each point in space-time to choose what we mean by red, green, and blue quark colours, okay?
25:40
So the reason why we have that ability is 'cause nobody cares. There's no physical meaning to saying, oh, this is a red quark, this is a green quark. There is a physical meaning to comparing. If we have different quarks, we wanna know, are they the same colour or different colours? And indeed, if we have the same quark but at different points in space, we want to be able to say, are they the same colour or different colours, right? That turns out to be important
26:06
to solving the equations for the structure of a proton or something like that. But I just told you that I can separately choose what I call red, green, and blue at different points in space, so how in the world can I know whether two different quark fields at two different points in space are the same colour? The answer is there has to be some way to compare, there has to be a way to take a quark field pointing in some direction at one point in space
26:33
and travel it over its value to some other point and say, are you the same or are you different as what's going on at that point? And if you read the book, you'll not be surprised to learn that the way to do this is to introduce a different kind of field. This is what we do in quantum field theory. If there's a problem, we address it by adding more fields, okay? So what we need is a way to keep track of what it means for the quark field to be the same colour at different points in space,
27:02
and that information is contained in a new field called the connection field, or sometimes just the gauge field. So you see the logic of what's going on. We're doing science, so we're not rigorously proving theorems, we're making hypotheses and testing them. The hypothesis is I can do these separate symmetry transformations all over the place. The implication of that is I need a connection or a gauge field to connect them. Let's see what the implications are of that idea. So here's an analogy.
27:32
Since we're not doing all the equations in all their beautiful glory, here's an analogy I find very helpful. What the connection field is telling us is how to keep the quark field constant as we move from place to place. So imagine you're a waiter working at an outdoor restaurant that is located on hilly terrain, so your different tables full of customers are at different elevations and you have to walk up and down. Your job is to carry the tray with wine glasses and a wine bottle on it
28:00
and keep it so that the wine glasses and wine bottle are pointing vertically. So maybe you can look down at the ground but you're so good at it that you've memorised in your brain how to balance that tray from point to point as you walk. That information, the information about the undulations in the landscape beneath your feet that allows you to keep the tray balanced, is a connection. It is the information that tells you how to connect the orientation of the tray from one point to another
28:32
so as to keep it in the same orientation. And this is a wonderful analogy because you can kind of see the information is basically a topographic map of the landscape you're walking in. And if the landscape were completely flat, you would be able to walk sort of easily from point to point. But if the landscape is rolling, if the connection field has curvature in it, it pushes on you, right? Your poor waiter is being forced to walk up the hill and roll down, et cetera.
29:06
A curvature in the gauge field leads to a force between the particles, okay? When your information about what it means to keep the tray constant is changing from place to place, that field interacts with you and exerts a force on you. And this is just an analogy, it's a good analogy in some ways, a bad analogy in other ways. But overall it works. Here is the miracle. All of the known forces of nature have this property. They're all born out of saying that there is some symmetry,
29:41
saying that it is a gauge symmetry, so it involves separate rotations at different points in space. Introducing a connection field to let us deal with that gauge symmetry, and that connection field provides the forces. The particles that are associated with that field are called the gauge bosons. So electromagnetism, which was known long before quantum field theory, is based on what is called a U1 symmetry. Its gauge bosons are the photons. The strong nuclear force
30:13
that we've already been talking about is based on this SU3 symmetry. Its gauge bosons are called gluons. There's also a weak nuclear force. Different properties, which we'll get into. It is based on an SU2 symmetry. The gauge bosons are the W and Z bosons. And even gravity fits into this paradigm. Gravity is based on an SO3,1 symmetry, even more complicated. And there are gauge bosons associated with that called gravitons. So this is the power of the gauge principle,
30:45
that in fact it leads to all of these different forces that we know and love. But there's a good news, bad news situation. So all of this was put together, you know, maybe in the mid 50s they began to get the idea about how this works, but they instantly hit a problem. The thing about equations is that you can't wriggle out of them. There's no sort of loosey-goosey character. The equation has to be solved exactly, it's supposed to be a law of nature, right?
31:12
So the equations associated with these gauge bosons and their symmetries comes with a very strong consequence. Namely, the gauge bosons should have 0 mass. What do I mean by that? Well, Isaac Newton had a definition of mass. He had an equation, force is mass times acceleration. So basically, the mass of an object is how it resists being accelerated. Einstein comes along, he has a better definition of what you mean by mass, E equals mc squared. The way to read this equation,
31:43
energy is mass times the speed of light squared, is an object which is not moving, an object at rest, which has no kinetic energy, it's just sitting there, it still has some kind of energy, it has rest energy. And that's what we mean by the mass, the mass is the energy of an object at rest. So the quantum field theory version of this is of course, if you have a field, it can have energy because it's changing, because it's vibrating from place to place, right? But what if it's completely uniform?
32:15
That's the equivalent of an object at rest. Is there energy in the field value itself even if the field value is not changing from point to point? In other words, if we uniformly change the field without making it wiggle anywhere, does that cost us some energy? Well, think about our analogy, think about the little waiter walking on the landscape. When we had that discussion of the waiter, we didn't bother to tell you the altitude of the restaurant, right?
32:46
It doesn't matter if the landscape were completely flat, it doesn't matter how high the restaurant is, the waiter can still walk at a completely respectable way without worrying about the curvature of the ground beneath him. What that's telling you is there's no energy cost in the forms of this analogy, there's no energy associated with the value of the gauge field. All of the energy comes from the changes from place to place in the gauge field. In other words,
33:16
and this is actually a halfway respectable translation of what the equations are trying to tell us, but in other words, when you have a field, a connection field or a gauge field that implements a gauge symmetry, the associated particles, the gauge bosons, will always be massless. And as massless particles, they will move at the speed of light. That's good, it's good to have consequences of your actions and of your equations, but it's also bad. Here's why it's both good and bad.
33:48
If you have a force that is meted by massless particles, there's an instant implication of that idea, which is that the force will be long-range, it will stretch out over the cosmos. And indeed we have two very famous examples of this. Gravity and electromagnetism. In both gravity and electromagnetism, there's a force law that is what we call an inverse-square law. The force between two particles depends on their distance as 1 over the distance squared.
34:18
It fades away, but it fades away relatively slowly as you get further and further away. And indeed, you can think of this as happening because the associated gauge bosons, the photon and the graviton, are massless. They're easy to make. Even if the Sun is over there and I'm over here, the gravitons from the Sun can come all the way to affect what I'm doing here on the Earth. So you can even understand this sort of geometrically, think of a lantern giving off light in all directions, right?
34:52
As the lantern gets further away from you, the luminosity, the brightness of the lantern, will appear to diminish. And there's a mathematical rule, it goes down as 1 over r squared, as 1 over the distance squared, an inverse-square law. That's because the light rays from the lantern are spreading out over a broader and broader area as you get further and further away from them. The forces work the same way. If you have massless gauge bosons, lines of force go forever
35:22
and they spread out and dilute away as the area which is r squared. So that happens both for gravity, Newton's inverse-square law of gravity, and for electromagnetism, Coulomb's law of electromagnetism. So, in the 1950s, physicists had this idea that this gauge principle that worked so well for gravity and electromagnetism maybe should work for all of the different forces of nature. The problem was the experiments were not happy with that because, experimentally, the strong nuclear force
35:52
and the weak nuclear force are very, very short-range. They stretch over ranges much, much smaller than the size of an atom. They don't behave like gravity or electromagnetism. So, your 1950s physicist was sort of torn between two facts. One fact is gauge symmetries are very, very beautiful and it would be nice if the world actually worked that way. But on the other hand, they make a prediction that all forces should be long-range, all gauge bosons should be massless. These appear not to be true
36:24
in the case of the strong and weak nuclear forces. So what is going on? I mean, is it a completely different point of view for the strong and weak forces? Or is there something complicated going on? The reason why the strong and weak forces act differently is because the universe loves us. And I mean that not in a sense of, you know, the anthropic principle or theology or anything like that. I mean in the physicist sense that the universe expresses its love
36:51
by making things as difficult as possible. And the reason why that's an expression of love is because when you do figure it out, you feel a true sense of accomplishment. Every physics graduate student, when they finally learn about how the strong and weak nuclear force work, they go, "Oh my goodness, that is amazing." And I'm about to tell you, at least in a very condensed way, what exactly that answer is. So the complication is that the strong and weak nuclear force
37:17
both act at very short range, but for completely different reasons. Interesting reasons in both cases, but different ones. For the strong force, what's going on is what we call confinement. So, photons and gravitons, massless particles that give rise to long-range forces. But there was a little caveat in there in that explanation, which is that they don't themselves interact very strongly. They interact a little bit with each other, but mostly they interact with charged particles
37:49
or massive particles. Charged particles for photons, massive particles or energetic particles for gravitons. But if you shoot two laser beams at each other, they just go right through each other, they don't scatter off in any interesting way. That's crucial to understanding why the force of electromagnetism stretches out over long distances. But the gluons associated with the strong nuclear force, they do interact with each other. In fact, they interact with each other a lot
38:18
very, very strongly, okay? Thus the name the strong force. So this prediction of the equations that gluons are massless is true, gluons are massless, but they also don't escape from the proton. You will typically be told that a proton has three quarks inside. I've already told you that, right? Two up quarks and a down quark. It's a bit of a cheat when you're told that. This image here is closer to the reality of what's going on inside a proton where what you see is many quarks
38:51
and also many antiquarks and many gluons. The strong forces are so strong that a gluon tries to go from one quark to another, it's gonna bump into another gluon. It's gonna create quark, antiquark virtual particles. It's gonna create gluons and antigluons. It's gonna be a mess inside. And indeed, the proton inside is a bit of a mess. But it turns out there's a nice way to think about it. Because the strong force is so strong, if I have two particles that interact with the strong nuclear force,
39:24
let's say an up quark and an anti up quark, if I put them close to each other, the lines of strong force, in other words, the lines of force created by the gluon field, do not stretch out in every direction equally like electromagnetism would. Rather, they bundle up into what is called a flux tube, something that looks almost like a single line of force connecting these two particles. And therefore, when I try to pull them apart, if I imagine doing an experiment where I take tweezers,
39:55
I take one quark and one antiquark and I pull them apart, it's very much like the quarks are connected by a rubber band or a piece of string. You know, if I take an electron and a proton interacting electromagnetically and I pull them apart, it gets easier and easier to pull them apart the further away they get. But if I take two quarks and I try to pull them apart, it gets harder and harder to pull them apart because this string is stretching and it's pulling back on you.
40:24
And eventually what happens is not that you separate the quark from the antiquark, but that the string breaks and it creates another antiquark-quark pair. And so when you started with a quark and an antiquark, by trying to pull them apart, you're left with two pairs of quark-antiquark here and quark-antiquark there. What that means is you can never separate a single quark. Quarks are confined inside these bundles of strongly-interacting quarks and gluons.
40:57
That's why we've never seen a single quark in an experiment. But the hypothesis that they're there fits a wide variety of data. So this is why the strong nuclear force can be based on a gauge symmetry and gluons can be massless but the range of the strong force is very short. It's because the gluons are confined inside the strongly-interacting particles. In the terms of our analogy with the lantern, it's like there's a shutter on the lantern and someone has shut the shutter.
41:28
So it's not that the light beams aren't there, but they're bouncing around the mirrors inside the lantern and they're not escaping to infinity. As promised, for the weak nuclear force, there's another explanation, but it's a completely separate explanation. And the answer has to do with the infamous Higgs boson, okay? So remember, the whole point here is that the forces of nature are based on these gauge symmetries. Once you have a gauge symmetry, you need a connection field.
41:56
That's a whole nother field that gives rise to forces. But the thing about symmetries is they can be broken. There's a phenomenon called spontaneous symmetry breaking. So let me explain what that means. When I say that there's a symmetry in the theory, I mean the dynamical laws, the rules of the game, the equations you would write down are invariant under this symmetry. A symmetry is a transformation that doesn't make a difference. But the solutions to the equations
42:25
might not be invariant under that symmetry. So one thing we can ask when we have a field is, how much energy is there? We were asking this before about the gauge field. How much energy is there when we change the field uniformly through space? That's called the potential energy of the field. So let's think about the Higgs boson. This is not a gauge field. A Higgs boson is a separate kind of thing. It interacts with the gauge bosons of the weak nuclear force.
42:51
So the Higgs field by itself is invariant under a symmetry, an SU2 symmetry, so you can think of there being two different Higgs fields that we can rotate into each other. And the fact that there's a symmetry says it doesn't matter how we label Higgs one and Higgs two, it has to be completely symmetric with respect to that. So the potential energy of the Higgs field has to be completely invariant under these kinds of rotations. On the left-hand side of the graph
43:18
we show one way to make that possible. Imagine the potential energy of the Higgs field is kind of like a parabolic bowl with a minimum at the bottom, right? So I can rotate the bowl, that's the SU2 symmetry, everything is completely invariant, and the Higgs field would naturally settle down into its minimum value which is sitting at the middle, sitting at the bottom of the bowl. And then it too would be invariant when I did this rotation around the centre, right?
43:46
That would be interesting and fine, but it is not the world we live in. There's another possibility, and that other possibility happens to be the one in the real world, which is that, I don't know why but for some reason physicists never learned the word sombrero so they call this the Mexican hat potential. That's what you see on the right-hand side over here where, again, this potential energy is completely invariant under rotations. But there is a bump in the middle,
44:14
a symmetric bump but a bump nonetheless, so that the minimum value of the energy is not at the middle, but rather in the brim of the hat, okay? So you can think about the Higgs field as this little orange ball portrayed in the picture here. It wants to live at the minimum value of energy, that means it wants to live in the brim of the hat, but it has to pick somewhere to live. Any place in the brim is equally good because they're symmetric. But once you've picked one,
44:43
now there's something special about that one particular direction. That's the expectation value of the Higgs field. You have broken the symmetry. So the dynamics, the equations, are completely gauge-invariant, as we say, they're invariant under this SU2 symmetry. But the Higgs field itself that fills all of space is not invariant under the symmetry. So, in yet other words, just to make this come clear, the Higgs field has a very special property that no other known field has,
45:17
which is that its lowest energy value, the value that it takes on in empty space, is not 0. So if you think about all the other fields, so let's reduce space down to two dimensions, so this is location and space on the x and y axis here. And you have all the other fields, so the vertical axis here is the value of the field. And this picture is demonstrating that, for the most part, all the other fields, electromagnetic field, the neutrino field, the quark field, whatever,
45:47
they all sort of sit close to 0, maybe vibrating around 0. You see those vibrations as particles. Whereas the Higgs field in its lowest-energy state rests at a big value nowhere near 0. It's a different value, it's like 246 billion electron volts, if you must know. We can measure what it is. But the point is that's its lowest energy. We've take an enormous amount of energy to move the Higgs field down to 0. And by doing that, it affects the dynamics of everything around it,
46:18
because now all of the other particles of nature are moving through and interacting with that non-zero Higgs field. So the reason, it turns out, the reason why the weak interactions are not long-range even though they're based on a gauge symmetry, is because the lines of force get absorbed by the Higgs field around them. It's as if, in our lantern analogy, it's as if we have a lantern but it's very, very foggy outside. Maybe those of you who live in the UK
46:51
are familiar with this feature of the weather. If you have a lantern on a foggy night, it will fade away more quickly than the inverse-square law would predict because the light is being absorbed by the fog around it, it's being obscured. That's what happens with the weak interactions. The lines of force that would be infinitely ranged if it weren't for the Higgs get absorbed in a very, very short distance. It turns out, you know, you were told, if you've ever talked a little bit to people
47:24
about the Higgs boson, you might be told that the Higgs boson has the job of giving mass to particles like electrons and quarks, et cetera. Well, it's true, but it wasn't why it was invented. This is why they invented the Higgs boson. They were trying to understand how it could be that a gauge force could be short-range. The fact that other particles like electrons and quarks also get their masses from the Higgs field in empty space is just a side bonus.
47:54
People like Higgs himself and Brout and Englert did not anticipate this. This is a feature of the Standard Model that is actually very, very tricky to understand because it comes as a consequence of the Standard Model having so much symmetry. It turns out there is so much symmetry in the Standard Model that without this symmetry breaking, there's no way that particles like electrons or quarks could have any mass. They would all be 0 mass, they would all move at the speed of light.
48:24
It is the breaking of the symmetry by the Higgs boson that allows electrons and quarks to have mass. And you have this picture of it as if, you know, there's the Higgs field in the background and it's kind of like moving through water or maple syrup or something like that. That's a halfway good analogy but the point is, at the end of the day, you've broken the symmetry, you explained where mass comes from for the particles that we know about, and you make predictions,
48:50
you make predictions for what to see in experiments, and you made these predictions in the 1960s. And starting in the 1970s, those predictions started coming true. And in the 80s, we discovered the W and Z bosons. And finally in 2012, we discovered the Higgs boson itself. So, to kind of wrap up here, let me put this in perspective. I wanna emphasise, we are nowhere near finished. You know, forget about what anyone might've thought in 1895. In 2024, it would be foolish to say
49:22
we are close to discovering all of the laws of physics. We are not. There's plenty of questions out there we don't know the answer to. We don't know what goes on when black holes radiate or where the Big Bang came from or what exactly happened near the Big Bang. We don't know why there's more matter than antimatter in the universe. We don't know why there's dark matter or what the nature of the dark matter is. We know there is dark matter, but not exactly what it is.
49:46
So there's plenty of room for more big discoveries in the context of quantum field theory and modern physics more generally. But having said that, I wanna say one other thing. It is simultaneously possible to say we don't know everything and yet we do know some things. And I wanna impress upon you how much we do know. I told you we have a theory that explains electrons and quarks and things like that, and those are the particles that make up you and me, okay?
50:21
I have electrons in me, you have electrons in you. Likewise for quarks and photons and gluons and what have you. What I wanna say is we can claim with very high confidence that there are no particles and forces that we haven't yet discovered that had any role whatsoever in your life. Unless you're a professional particle physicist which, okay, then they have an occupational role in your life. But I mean literally in your brain, in your biology, there's no new particle or field that is affecting
50:53
the firing of neurons in your brain, for example. So how do we know that? It is a feature of quantum field theory, another kind of symmetry, something called crossing symmetry. So in these little pictures you have here, these are the famous Feynman diagrams, right? You should read them as time is going left to right. And they're telling a little story. So the picture on the left tells a story of an ordinary particle in blue and some hypothetical new particle in red.
51:20
Let's imagine the red particle is the consciousness boson. You've hypothesised a theory where there's a new boson that helps account for human consciousness, okay? So, if that's true, according to the laws of quantum field theory, there has to be some interaction where your new boson affects the motion of the ordinary particles in your head, the electrons and the protons and so forth. And then there's a rule of quantum field theory that if that interaction happens,
51:49
if the new particle and the ordinary particle in your head can come together and interact and then go their own way, I can take that diagram and I can rotate it clockwise by 90 degrees and I will get a new diagram, and that new diagram exists just as much as the first one does. What that means is, if this new particle could possibly affect the particles in your brain, then we could make the new particle. Because all we have to do is smash together
52:21
electrons and positrons or quarks and antiquarks, just smash 'em together, see what comes out. And the good news is smashing particles together and seeing what comes out is particle physicists' favourite thing to do. They do it all the time. They've done it very, very accurately. And the answer is we know what comes out. At least we know what comes out within a certain regime of energies and momenta transfers. And those are more than enough to include everything that is happening
52:51
in your brain right now. So we don't know everything. Let me absolutely emphasise we don't know everything. But we do know the laws of physics that govern the stuff out of which you and I are made. That is pretty amazing. That is something that we human beings should be very, very proud of. We understand the laws of physics describing you and me and the rest of the everyday world. This picture here is lacking a certain quantitative precision but it basically includes
53:21
all of the ingredients that you need. The wonder of a human being or a planet or a star is not that they invoke or involve all sorts of weird undiscovered physics. Is they are collective phenomena that you get from many, many, many little particles and atoms acting together, just obeying the individual laws of physics. And so my last slide is an echo of the last slide I gave a year ago when I talked about Einstein's equation. It's the same lesson, the same lesson at the end of the day
53:56
for classical physics or for quantum physics, that the equations are much smarter than we are. When Albert Einstein wrote down his equation in 1915, it's the same equation we use today. It's the equation you will learn in "Space, Time and Motion: Volume 1" or in general relativity textbooks. Einstein's equation was correct when he wrote it down. But he didn't understand everything it predicted. He didn't know about the Big Bang, he didn't know about black holes,
54:22
he didn't know about gravitational waves. Those were all implicit in the equation he wrote down. The same thing is true for the equations of particle physics in quantum field theory. We write down the equations based on certain experimental ideas or data that we have. We get some theoretical ideas, we write down the equations, and then we predict on the basis of those equations new experiments, new phenomena that we haven't seen yet. We finished writing down the equation
54:51
for the Standard Model of particle physics in the 1970s. And we didn't discover the Higgs boson on until 2012, but there it was, clear as day, and now we even have better data for it, and we're gonna continue to do that. So this is sort of the fun and the frustration of physics. The fun is you never know what's gonna come next. You never know exactly what experiment is going to surprise you. The frustration is sometimes your theory does too well.
55:20
You know, if you watch "The 3 Body Problem," the TV series, or if you read the novel, there's a lot of weird physics going on in that show but the weirdest part is when the physicists are frustrated because the experiments don't fit the data. Physicists love it when the experiments don't fit the data, that's what we're always hoping for. But we can still take some solace when the theory is so good. I said the experiments don't fit the data. I mean the data don't fit the theory is what I mean,
55:49
the experiments don't fit the theory. When the theory is so good that it keeps fitting the data over and over again, we might be frustrated that it's hard to discover new things, but we're also proud that we have figured out something real and true about the fundamental way in which nature works. Thank you very much. (audience applauds)
56:11
(applause fades out)
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