Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas - 275 | Solo: Quantum Fields, Particles, Forces, and Symmetries

Episode Date: May 13, 2024

Publication week! Say hello to Quanta and Fields, the second volume of the planned three-volume series The Biggest Ideas in the Universe. This volume covers quantum physics generally, but focuses especially on the wonders of quantum field theory. To celebrate, this solo podcast talks about some of the big ideas that make QFT so compelling: how quantized fields produce particles, how gauge symmetries lead to forces of nature, and how those forces can manifest in different phases, including Higgs and confinement. Blog post with transcript: https://www.preposterousuniverse.com/podcast/2024/05/13/275-solo-quantum-fields-particles-forces-and-symmetries/ Support Mindscape on Patreon.

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Formulation unique to Shell. Compared to minimum detergent gasoline with continuous use of Shell v. Power Nitro Plus and gasoline direct injection. Actual effects may vary. See shell.us slash more dash protection for more information. Hello everyone. Welcome to the Minescape Podcast. I'm your host, Sean Carroll. And we're very excited these days here at Minescape World International Headquarters for Book Publication Week. As some of you know, on Tuesday, I guess May 14th, here in the United States anyway, we're going to have the publication of Quanta and Fields. This is Volume 2 in the biggest ideas in the universe series. The series will be a three-part series that aims to explain modern physics, the part of modern physics that we know, not the speculative stuff, we're not going to be talking about the multiverse or quantum gravity or anything like that, the kinds that ideas in

modern physics that we think will last for a very, very, very long time. And the gimmick is we're explaining them in a way that involves the equations. So we're going to show you the equations. And that means it's not for everybody, but it's very much for some people. If you've ever read descriptions of physics where you thought that there was something going on, but there's a bunch of metaphors and stories and analogies, and you weren't quite sure why things were working out the way they were working out? These are the books for you. And the biggest ideas you'll really get to see why, for example, the Higgs mechanism gives mass to particles and things like that. You don't need to just wave your hands in any way. And the general theme of the book is quanta and fields.

That's the name of the book, as opposed to the first volume called Space Time and Motion, which was really classical physics. This book is about quantum physics. The third book on complexity and emergence, both is a collection of the things that didn't fit into the first two books, but also is about larger scale things, right? Thermodynamics, complex systems, stuff like that. And it's kind of a appetizer, main course dessert kind of thing. This book, Volume 2, Quanta and Fields, is a book I've been wanting to write for decades. This is a book that I really have thought for a very long time would be super duper useful for a lot of people. Because quantum field theory in particular is a part of modern physics that is absolutely central and gets very, very little airtime in popular discussions.

We talk about quantum mechanics a lot. We talk about entanglement and the measurement problem and all these weird things. Bell inequalities and so forth. And then we skip to speculative stuff about string theory or quantum gravity or the multiverse or inflation. But the actual physicists doing their jobs are thinking about quantum field theory all the time. And you rarely get a very full discussion of that in popular media. We recently did talk to Matt Strassler, who wrote a wonderful book, Waves in an Impossible Sea, that talks about quantum field theory, but it is the opposite level of,

of mathematical explicitness. It's all about trying to explain in words as carefully as you can how quantum fields work, whereas quantum fields, the book that I've written, is a short, action-packed introduction to both quantum mechanics and to its specific implementation in the context of field theory, which is where modern physics lives. That's what we're really thinking about. So we need to understand why in the world you could take a field quantize it and have it look like particles. That's the big mystery of quantum field theory, but that one historically was figured out relatively early on,

and it opens the floodgates. Once you understand that, once you understand that you can quantize fields and get particles out of them without putting particles in explicitly, there's a whole bunch of things that can happen, ways that fields can interact with each other and influence each other,

that leads to interactions, that we measure with Feynman diagrams. I should say that we predict using Feynman diagrams that we measure in experiments, of course, all sorts of conceptual questions with infinities and renormalization, but also new phenomena that can happen, like gauge symmetries, giving rise to forces, breaking those symmetries with the Higgs mechanism, confining things inside nucleons, and violating parity conservation and things like that. So it's a whole wonderful zoo of things that we've spent the whole 20th,

century and beyond, trying to understand. So there's a lot to discuss in a short solo podcast. I'm not going to cover it all. Sorry about that. You can read the book. I did do the audio book version of the book. It's about 10 hours of me talking. If you don't get enough from the podcasts, you can download that. And I just read the book and I try to explain what's in the pictures and the equations and everything like that. So here I'm just going to hit some highlights. I really do want to focus on that idea of why quantum fields, how you get particles out of that and the door that it opens to other things. So it'll be very explicit about where the particles come from, be a little bit more hand-wavy and gestural about where all these other symmetries and how they get involved. But it's

important stuff. It'll give you, I think, a feeling for the kind of thing that we're going to be talking about in the book if you are so inclined to go more deeply. So with that, let's go. I think it's useful to get one thing straight right away. Sometimes you will hear quantum mechanics came first, and then we invented quantum field theory. It's fine to say that because these are words that we make up definitions of, right? You know, you're always allowed to define the words however you want. It's not the best way of thinking about things.

Quantum mechanics is a broad framework that includes quantum field theory as part of it. It is literally the quantum theory of fields. Quantum mechanics applied to fields. That's what it is. And it's a very simple statement, obviously, but I remember literally taking quantum field theory in graduate school, and it took me a while to figure out, you know, oh, it's just quantum mechanics. It's the same quantum mechanics we had before just applied to a different system.

It's a subset of quantum mechanics. The thing that we usually call quantum mechanics is the, quantum mechanics of particles, and quantum field theory is the quantum mechanics of fields. So the very first thing we need to understand to wrap our brains around the idea of quantum field theory is the word of. Now, this isn't some sort of Bill Clinton-esque parsing of language at a very deep level. We're really trying to understand, you know, what do you mean when you say you have a quantum theory of some thing, like particles or fields, or for that matter, strings or loops or whatever. What does it mean to have a quantum theory of some thing? And the answer to that is that

when we do quantum mechanics, whether it's particles or fields or whatever, the usual way we start, not always, but usually, we start with a classical theory, and then we have a cookbook. We have a set of procedures for converting a classical theory into a quantum theory. We call this quantization, surprisingly enough. And there's a lot hidden in that word quantization or to quantize. We don't really have a well-defined map from the space of all classical theories to quantum theories. So think of it as a way to find quantum mechanical theories starting from a classical theory. Okay. And usually we start from particles. That's what we do when we first learn quantum mechanics. And it makes sense when we developed quantum mechanics in the early 1900s, people were more focused on the particles.

The fields were always there. They were thinking about them, but they wanted to get the particles right first. So when you say you have a classical theory of something, you know the vague idea of classical physics, a la Isaac Newton. You have some particles, and they have positions, and they have velocities. Or equally well, if the mass of the particle is a fixed number, you can talk about the momentum of the particle. Momentum is just mass times velocity. For technical reasons that Professor Hamilton figured out, it's actually better to think about momentum than velocity, but it really doesn't matter. They're completely convertible back and forth. And the point is that for any one particle, once you know those two things, position and momentum, you can figure out what's going to happen to it, right?

You can plug into Newton's laws of motion or Hamilton's equations or Lagrange's equations. There's many different mathematically equivalent ways to do classical physics. These are all explained in volume one of the biggest ideas, space, time, and motion. And the thing that is the same in all these different ways of thinking about classical physics is the amount of information you need to say what's going to happen next. If you were Laplace's demon, if you were able to predict exactly what was implied by the laws of physics, the data you need to be given is the position. and momentum of all the particles. And then it's deterministic, right? The wonderful thing about

classical mechanics is it says what's going to happen. It's not ambiguous about it. So quantum mechanics says, okay, think about this one particle, just to start. We have position and momentum. Throw away the momentum. Forget about the momentum, just for a minute, okay? And quantum mechanics says, instead of having a position, we now describe the particle by having a wave function. si of x. If x were the position, si the Greek letter si is the wave function, which assigns a number to every possible position. That's the wave function. And usually that number will be zero

or very, very close to zero almost everywhere. And then in some little blob-like region of space, that number will not be zero. That's the wave function. For any possible place you might see the particle were you to look for it, there's going to be a number, a complex number in particular. And what the wave function tells you is the probability of observing the particle to have that position. So in particular, you calculate the probability by squaring the wave function, or since it's a complex number, you take the modulus squared of the wave function.

And in the new book, Quanta and Fields, we go through quantum mechanics really quickly. First three chapters, we do it. Wave functions, measurement, entanglement, and then we're moving on in devoting most of the book to quantum field theory. because that's a very rich and very complicated set of ideas. So where did momentum go is one obvious question, and the answer is, you know, when I said that the wave function varies with space, how fast it's varying, how fast the wave function is wiggling up and down,

basically tells you the momentum. Or more accurately, you can turn it into a prediction for measuring the momentum to have any particular value in exactly the same way, wave function squared kind of procedure. So the point is, the thing I don't want to get lost, the thing you need to understand for the rest of the discussion is, when you go from classical mechanics to quantum mechanics, you start with some object with a definite state, position and momentum,

and you switch to a wave function that will tell you the probability of getting different observational outcomes. That's the evolution or the transformation from a classical theory, to a quantum theory. And in that case, we were talking about particles. And it's natural to talk about particles because if you think about the development of quantum mechanics, what were they thinking about?

We often attribute the birth of quantum mechanics to Plunk and Einstein. In 1900, Plank understood blackbody radiation in terms of chunks of energy being given off by vibrating electrons and atoms. In 1905, Einstein really said, you know, all electromagnetic radiation can be thought of as coming out in these particle-like quanta, which we now call photons. But subsequent to that, the work by people like Niels Bohr and

Louis DeBroy and Werner Heisenberg, etc., was more focused on particles, in particular on the electrons orbiting atoms. So they were trying to understand the structure of atoms, how atoms would give off radiation, things like that. So it was the particles that were really front and center. And then, therefore, it makes sense when you're an undergraduate physics student, you learn about the quantum mechanics of particles. I should say, vis-a-vis the book that is coming out, the reason why I can pack a whole discussion of both quantum mechanics and quantum field theory into a relatively short book, even though I do include all the equations, is because you're not trying to solve the equations. You're not going to come out of reading this book, being able to

calculate the scattering probability of two particles or the decay rate of the Higgs boson or anything like that, you will understand what those words mean, and you will have an idea what the equations are behind them, but I'm not training you to be a practicing physicist. And that philosophy is remarkably liberating. So, you know, just to be super clear here, a very small percentage of people who get bachelor's degrees in physics, learn quantum field theory. In fact, plenty of people get PhDs in physics without ever learning quantum field theory. Just like in volume one, we did Einstein's equation for general relativity. That's still a classical theory, Einstein's theory of gravity. Many, many people get a physics

education and do not learn general relativity. So part of the reason why I'm writing these books is because I want more people to be able to grasp these ideas, even physics majors, Plenty of physics majors don't get to learn about these things. And they're not that inaccessible. Honestly, you can learn them, I promise. So you'll get that you had a feeling for here in the podcast. Okay. So I'm going to keep referring back to, you know, what we study as physics students, even though I don't expect you to grow up to be necessarily a physics student or a working physicist. That's the idea. Okay. So the first thing we would learn if we were an undergraduate, sometime in your sophomore year, or your junior year maybe. Sometimes the universities will split it up, so you'll sort of get like a little hint, even in your very first year about quantum mechanics, then you'll do it seriously later on in your junior or sophomore years.

And you'll be mostly taught the quantum theory of particles. So you're imagining that the system you're studying is an electron or something like that, okay, orbiting in an atom, and you have this wave function, and you have an equation. that tells you how the wave function behaves, and that's the Schrodinger equation. There's different, again, mathematically equivalent ways of thinking about it, but that's the basic idea. And from that, you can predict things like the spectrum of radiation being absorbed or emitted from the atoms. So the general idea is, as we said, start with particles, make wave functions, and then give you a set of rules for how the way functions behave.

They evolve according to the Schrodinger equation. when you're not measuring them, when you are measuring them, they collapse and you get a probability of observing something. Then you discovered that there's this thing called entanglement. To be very honest, you could possibly take an entire one-semester intro to quantum course as an undergraduate and never hear about entanglement. I know that's weird because if you hear the popular level discussions, they'll talk about entanglement all the time. But you can easily spend a semester just solving the Schrodinger equation for, you know, different kinds of potential energies that the electron could be moving in. So let me tell you, taking that undergraduate course

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hero cheddar biscuit handmade in a Sonoma-based French bakery. Shop now on hero.co. Use code iHeart for 10% off. That's hero.co. Per serving, not a low-calorie foods and products contain aliasi, see nutrition info on hero. com for sodium and sugar content. And all this stuff you learn about the particles, about the wave functions, about the Schrodinger equation, it gives you a wonderfully precise, quantitative, and experimentally testable understanding of what happens in atoms,

of the ways that there are different energy levels, different orbitals, and so forth, and how they show up experimentally. And let me emphasize that bit about different energy levels, because this is absolutely crucial to the essence of quantum mechanics in a way that I think is misunderstood by a lot of people who don't study it professionally, which is the word quantum is a little bit misleading, I would say. What does quantum mean? Before you knew about quantum mechanics, the word quantum was still around.

It meant a discrete amount of something, one chunk of something, right? The idea of a quantum, or many quanta, would be the plural, is rather than having a smooth distribution of things, you have individual pieces, discrete elements, okay? Those are the quanta. So particulate, particle-like, pixelated, or whatever you want to call it. Okay, that's the sort of philosophy behind the word, quanta or quantum. And calling this theory that we're discussing, quantum mechanics or quantum field theory, makes you think, informally, that maybe the world is divided up into chunks, right, into pieces,

into discrete bits, that the world is not fundamentally solidly. smooth, but rather fundamentally discreet. Maybe this inspires you to think that once you include gravity in the game, space and maybe space time will be divided up into chunks and to make a little lattice. It'll be little pixels or something like that. This is a fundamental, deep, terrible, awful, no-good understanding of what quantum mechanics actually says. Because if you think about what I just said, if we take the idea of a particle, replace it by the idea of a wave function, and that wave function obeys the Schrodinger equation, nowhere there is the idea of chunks of anything. Nowhere there is the idea that the world is broken up into discrete bits of matter or

energy. It's as smooth as it could possibly be. Everything is continuous and smooth. The wave function is a function that spreads out all over the place. The Schrodinger equation, smoothly evolves that wave function, right? Where does this idea of quanta come from? And the answer is that, well, it's exactly analogous to the very well-known analogy that we use of plucking a guitar string. So if you have a guitar string that is tied down at two ends and you pluck it, there's a fundamental wavelength at which it vibrates, which leads to a fundamental frequency of sound being given off. But there are also harmonics, right? If you imagine, if you can visualize in your head, I know this is an audio podcast,

but visualize in your head a string being held down at both ends and vibrating sort of uniformly, smoothly, up and down, except at the ends. So in the middle, it's vibrating up and down, tied down at the end. So the amplitude of the vibration kind of goes down to zero at both ends, where it's tied down. That is the fundamental frequency at which is vibrating. And then the first harmonic frequency is it looks kind of like an up and down, wave-like shape. So it starts at one tied-down end, goes up halfway through the string, it comes back down to zero again. And in the other half of the string, it's going down on the right-hand side, where as it was going up on the left-hand side.

And so the wavelength is half of what it was. And therefore, the frequency is twice what it was. And these are two different things that can happen to the same string. And then, of course, you have even higher harmonics where the wave goes up, down, up again, and all the way up for higher and higher frequencies, shorter and shorter wavelengths. The reason why I'm talking about this guitar string analogy, and if you're a talented guitar player, you can actually play the harmonics, get them to come out of your guitar, even without getting the fundamental note to come out of your

your guitar, but that's a different podcast discussion entirely. The point is, the string that you're plucking has nothing discreet about it. It's not that the string is only allowed to do certain things. The string could have any shape that it wants to, as long as it's tied down at both ends, right? But it naturally has a set of ways of vibrating that are sort of easiest to make it vibrate at, that are very natural and direct. In fact, at the technical level, we would say these are the ones that have a definite energy in their vibrations. And so the discreteness comes, not because the string itself is discrete, but because its behavior falls into a discrete set of possibilities, the fundamental frequency, the first harmonic, second harmonic, etc. And you can calculate exactly

what the frequencies are, and so forth. So there's kind of an emergent nature to the discreetness. The discreetness is not built into the system. It is a feature of the solutions to the equations that govern the system in some high-level way, but you can just see it pictorially. You can tell why. You can understand that it makes sense that there is, you know, the lowest note, a little bit higher note, a little bit higher note than that and so forth. That phenomenon is exactly why, in quantum mechanics, we see discrete behavior for things like the energy of radiation being given off by one of these atoms. So instead of thinking of the guitar string being tied down at both ends, think of the wave function of an electron in an atom. Let's make our lives easy. Let's make it a hydrogen atom. So there's only one electron and there's only one particle in the nucleus, a single proton. Okay?

Well, the electron wave function is not tied down. It can extend very gradually out to infinity, right? But the wave function has to go to zero at infinity. As it gets very, very, very far away from the proton, it's going to go to zero because you can only have a finite amount of wave function. The wave function can't spread uniformly over all of space. It has to be sort of concentrated somewhere, and we're imagining that it's concentrated near the proton in the atom.

So even though the electrons wave function is not literally tied down, effectively it is, it goes to zero as you get far away from the atom. And you can play exactly the same game with the electron wave function that you play with the plucked guitar string. There are technical differences because the energy is a little bit different and also because it's a three-dimensional thing, the electron wave function, rather than the single one-dimensional guitar string. but the essence of the issue is the same. There will be a lowest energy state of the electron, which is analogous to the fundamental frequency of the guitar string, and that's the lowest energy that the electron can have in hydrogen atom. And then there is a next highest energy level state

that has some discrete extra amount of energy. It is not the case that the electron's wave function can't do anything at once. It can vibrate in whatever way it wants. But there's a natural set of ways that it can vibrate or ways that it can sort of arrange itself around the proton. And these are the ways that have definite energies. And these are the natural places for the electron to settle down into and what we actually observe. So the discreteness of the energy levels of an electron, which by the way you learn about in chemistry class, we learn about the orbitals of an electron around an atom, that's exactly what's going on.

These are not because you've put in any fundamental discreetness into the nature of the reality that you're talking about. The reality is a smooth function obeying a smooth equation. It's the behavior of the solutions to that equation that has a discrete character. And I'm really sort of dwelling on this and emphasizing it because it's going to become super duper important when we get to quantum field theory. So the lesson here is you start with a completely smooth, continuous, arbitrarily, varying thing, the wave function of the electron, and then you examine its behavior under very certain special circumstances, namely it's sitting there glued to a proton in a hydrogen atom, and you find that the possible places you could find, the possible states you could find the

electron wave function in, come in a discrete set. So again, that discreetness is emergent because it's a property of the different solutions to the equation, in this case, the Schrodinger equation, rather than the equation for a string in a guitar. So that's where Quanta come from. It's not because you pixelated the universe. It's because you've solved an equation, and the solutions come in a discrete set. Okay. And historically, this was all done in the 19-teens and 20s, and we figured this out, quantum mechanics was a great success. Great. But we always knew, even back in the 20s, that the world was not simply made of particles, right?

As we said, the very first glimmers of quantum mechanics came from thinking about fields, not about particles. In the 19th century, we had figured out that electricity and magnetism were two sets of phenomena that were unified in a description that involved two different fields. the electric field and the magnetic field. And in fact, once Einstein and Binkowski came along with special relativity, we realized that the electric field or the magnetic field were literally two different aspects of the same underlying field. So at some point, we just call it the electromagnetic field. And a field is a very different thing than a particle.

A particle classically, let's just think classically now. We haven't quantized anything yet. A particle has a location. As we said, it has a position, it has a velocity, That's what a particle is, basically. Whereas a field is everywhere. Okay, and I need to be very, very careful here, as careful as I can be, to distinguish the different conceptual things that sound very, very similar to each other. In this case, a classical field is very, very different from a quantum wave function.

Okay? They seem similar because a classical field has a value at every point in space. The electric field has a value at every point in space. It's a little vector because the electric field is not just a number. It's a little arrow with a direction and a magnitude at every point in space. You can't see it. It's invisible. It's all around you.

The magnetic field, likewise. You know, you can detect it if you put iron filings on a piece of paper near a magnet. They will line up with the magnetic field, showing the influence of the magnetic field around you. A compass that you can carry around will point. toward the North Pole for exactly that same reason, and you can detect the electric field likewise, but they're there, they're around you, at every point in space around you right now, including inside you, for that matter, and also out in interstellar space, every point in space has a value for the electric field and the magnetic field. That's what a field is. A field is an entity

that takes a value at every point in space. You're very tempted to say, well, what are the fields made of? that would be a bad question to ask. I mean, you're welcome to ask it, but the answer is there isn't anything that the fields are made of. The world is going to be made of the fields. Okay, the fields are the bottom layer of reality, according to quantum field theory. Now, quantum field theory might not be the final answer. I'll even gesture at the very end toward the idea that it's not the final answer, but according to quantum field theory, the rules are that we start with fields and we build everything else out of them. Now, that idea that there is a number or quantity something like that at every point in space time, sounds very similar to the wave function of an electron,

right? I told you that the wave function of electron assigns a complex number to every point in space, and that number squared will give you the probability of observing the electron there. But what if you had two electrons, right? You do know that there is this thing that we talk about in chapter three of the new book called Entanglement. An Entanglement says that when you have two electrons, And let's say if you were actually, even better, let's make it more clear because electrons are identical particles, and that's going to get in the way of being correct sometimes here. Let's say we have an electron and a proton. So two particles, but very distinguishable.

You know which is the electron, which is the proton. And call the position that you might observe the electron at X1, and the position you might observe the proton at X2. You know if you looked for these particles, whether you found the electron or proton. They're very different looking particles. Then the phenomenon of entanglement tells you that the combined state of both particles is not a wave function for the electron and a wave function for the proton. It is a single wave function for the system that is made of the electron and the proton.

In other words, there is only one wave function. Ultimately, it will be called the wave function of the universe, because it actually includes the whole universe, not just these two particles, but for right now, it is a wave function of these two particles, and it is a function of X1 and X2. For every possible position you could find the electron at and position you can find the proton at, there will be a complex number.

And you ask the question, where are these two particles simultaneously? And the wave function tells you a complex number that you square to get the probability of seeing those two particles at that place. So this seems like a little detail that might not be very important, but it's actually super-duper-crucial here, namely that the wave function doesn't depend on space. In this case, in the case of the electron and the proton, the wave-function depends on two copies of space. One copy saying, where is the electron? One copy saying where is the proton.

So in a very real sense, even though it sounds like a technicality, the wave function is not a feel. The wave function of the universe, the wave function in the real world, the wave function that we care about when we do quantum mechanics, is not a field in the same sense that the electric field or the magnetic field is. The magnetic field doesn't care how many particles you have. It still has a value at every point in space. The wave function of a two particle system doesn't have a unique value at some point in space. The thing it depends on is the configuration space of all the particles, and for that matter, as we will see, all the fields in the universe. So it's a different kind of thing. There is sort of an accidental, coincidental resemblance of the wave function of one particle

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Per serving, not a low-calorie foods and products contain alias. See nutrition info on hero.com for sodium and sugar content. But the idea, the conceptual notion of a wave function, is a very different kind of thing than a classical field. Classical fields really, literally, honestly, are things that have values at each point in space. the electric field, the magnetic field, also the gravitational potential field, and other such kinds of fields. Gravity was always thought to be a little bit trickier even before general relativity came along.

So most of the attention in the early 20th century was being focused on the electromagnetic field. So there was this vague idea near the turn of that century that matter was made of particles, electrons, protons, we eventually discovered neutrons. and other things like that, whereas forces were being carried by fields, because we knew about gravity, we knew about electromagnetism, et cetera. So we thought, roughly speaking, even though it was not completely laid down explicitly, that the world was going to consist of these two kinds of things, particles and fields, matter and forces, right? Matter from the particles, forces from the fields. And so once quantum mechanics came along, in its more robust form, in the 1920s, and you understood the quantum mechanics of particles, it was very, very natural

to say, okay, now we will turn to the quantum mechanics of fields. And that's what the word of means. This is where we started this digression. Understanding the word of. The word of means we're starting with a classical theory of something, particles to start now fields, and we're going to construct a quantum theory of it. What does that mean? It means take the classical theory, take the,

thing that plays the role of a position for a particle, right? For a field, fields don't have a position. Fields are everywhere. You can't just say what the position of a field is. But the role of position is basically played by the value of the field at every point in space. And then you're saying, well, okay, what's the role of the momentum or the velocity? Well, it is the rate of change. It's the time derivative of the field at every point in space. But again, happily, we don't need to think about that too deeply, because what happens when you quantize a particle is you temporarily forget about the momentum and just focus on position. When we quantize a field, we're going to start by thinking hard about the space of all possible profiles of the field. That is to say,

at every point in space, we imagine the field has a value, and conceptually we can ask, you know, so what is the space, what is the set of all possible values that can have it all at once? So I'm not asking, you know, what is the value at this point X and this point Y and this point Z, et cetera, et cetera. I'm saying all throughout space, all at once. The field sort of has a shape, right? The field has an individual value at every point. So an infinite number of values, one value at every point. That is a profile of the field, okay? That is a possible answer to the question, if I could imagine measuring the field instantaneously all throughout the universe, what would it look like? That is a profile of the field. And that is the idea that plays the

analogous role to the position of a single particle. So you can see that mathematically, conceptually, this is a little bit more intimidating than particles. For a particle, you just had a position. You set a location. Here it is. I can give you three numbers, X, Y, Z. That's where it is. For the field, you have to tell me what it's doing at every point throughout an infinitely big universe. That seems much harder to do. But fields existed, so we have to try to deal with them. In fact, I should also mention that beyond the existence of the electromagnetic field, there was also the phenomenon of radioactivity. Okay. There was the experimental discovery that particles could turn into other kinds of particles. Particles could decay and give off

particles. Particles could be created and destroyed, right? Even at the level of particles. And the reason why I'm mentioning this is because if you take the Schrodinger equation for a system of one electron or two electrons or three electrons or whatever, the number of electrons never changes, according to the Schrodinger equation. If you just do particle quantum mechanics, you're going to be stuck with the same number of particles that you always had. You need to sort of generalize it a little bit. You can do this. You can sort of say, all right, I'm going to imagine ways that the number of particles can change. But as we'll see, or as will become natural at some point, basically what you're doing is inventing field theory when you do that.

Field theory is a very natural way of accounting for the fact that the number of particles in a system can change over time. Okay, that was just an aside. I want to get back to the electromagnetic field just for a second. So we knew about the electromagnetic field. We knew that we would have to incorporate it into quantum mechanics at some point, and people tried to do it very early on in the development of quantum mechanics. People like Paul Dirac, Heisenberg, and others tried to quantize the electromagnetic field. People like in Rico Fermi wrote down theories where numbers of particles could change, using the idea of fields. So the question became, you know, what does it mean to take something that is already a field,

like the electromagnetic field, and to quantize it, to give it a wave function of some sort. So I can tell you the answer, and it's the correct answer, it's just sort of presented in a way that doesn't seem very helpful. To every possible profile of the field, that is to say to every possible configuration of the field all throughout space, we will attach a number, a complex number. that is the wave function of the field. And were we to imagine observing the field everywhere throughout space simultaneously all at once, we would take that wave function and square it to get the probability we would observe the field to be in that configuration.

So I hope that that made a little bit of sense, but I also hope that it's a little intimidating and a little baffling as to how we would actually make any progress at a practical level with that. It was relatively easy to handle the idea of a wave function of a particle, but the wave function of a field just seems like a completely, you know, impossible to wrestle mathematical beast, right? To every possible field configuration, we assign a number that just seems impractical to write it down. What would that wave function really ever look like? And indeed, were you to take a quantum field theory class at a university somewhere

or pick up a real quantum field theory textbook, they almost never do this. This thing that I just told you, assigning a complex number to every possible field configuration, that is a perfectly valid way of thinking about quantum field theory, but it's not really the practical way. And so that's the reason why I could take a quantum field theory course as a graduate student and have it remain hidden to me for a long time that really what we were doing was just quantum mechanics, because it's presented in a very different way using very different mathematical tools.

But I think that it's conceptually easier if we try to stick as close as possible to the same notion of quantization and converting a classical theory into a quantum theory in the field context that we usually use in the particle context. So the continuity of the ideas remains as explicit as it can be. So what are we supposed to do

with the fact that we're imagining profiles of fields stretching all throughout space and attaching complex numbers to every possible profile, squaring those complex numbers to get the probability that we would observe the field to look like that. The answer is Fourier transforms. That's honestly the answer. If you look online, if you look at the web page that I have for Quanta and Fields, you look at the table of contents that I published there, you will see an appendix. There's like 12 chapters. There's like 12 chapters plus an appendix, and the chapters all have punchy titles like atoms and interactions and effective field theory and matter or whatever. So there's a lot of, you know, basic ideas

that are pretty big, it's pretty obviously big important ideas. And then the appendix is Fourier transforms, which seems like a little bit of a, of an ugly duckling in the collection, right? Like some specific mathy-sounding idea that hopefully I didn't have anything to do it. Indeed, I tell the story that when I was an undergraduate myself and I first learned Fourier transforms, I literally was thinking like this is a quintessential example of a mathematical technique I will never actually use in real life. And then you discover, once you take quantum field theory as well as other parts of physics, the Fourier transforms are the single most important thing in lots of areas of theoretical physics. So what is a Fourier transform? It's just another way of thinking about the value

of a field. Okay. We said the value of the value of field, the profile, the configuration of a field, whatever you want to call it, is basically a function of space or a function of space time. Let's just think space. We don't need to worry about evolution right now, although we will evolve through time at some point. So at every point in space, there's a value of the field. That's the field profile. Okay. So to give you the information about what the field profile is, requires, in principle, an infinite amount of data, right? I have to list every point in space. That's an infinite number of points. And at each point, I tell you what the field is doing. Now, realistically, for fields that we might care about, maybe there's some simple ways of compressing all that information into a

compact notation or whatever, but in principle, that's what I have to give you. The idea of of a Fourier transform is, I can represent that information in a much more convenient way, the information about what is the field doing at every point in space. Think of a simple kind of field configuration, namely a pure wave, a pure sine function or cosine function, a sinusoidal wave. That is just a fancy way of saying a wave that literally just goes up and down with an absolutely fixed wavelength and an absolutely fixed amplitude. Okay? The typical idea of a wave, a plain wave that stretches all throughout space that you might see in a picture, you know, a very wave-like thing, not complicated ripples and going in different directions, just a very smooth,

regular wave stretching all throughout space. That's an example of a simple wave, a wave that a configuration of the field. Waves are just configurations of fields, by the way. Don't, There's almost no difference between the word wave and the word field, except sadly, we have attached the word wave to wave functions, which are different kinds of things. I know. That's very annoying. Sorry about that. I wasn't to blame for that. Anyway, this particular kind of simple wave, a sinusoidal wave, a wave with a definite wave length that remains absolutely constant all throughout infinitely big space. That is a plain wave. It is something we can represent mathematics. using signs and cosines, okay, if you know a little bit about your trigonometry functions. And we call a wave of a definite wavelength a mode of the field. And of course, most field

configurations aren't going to be that simple, right? They are going to be more unpredictable as you go from place to place. But we're choosing just for a moment to focus on very, very simple, very, very regular configurations of the fields, ones that look like absolutely uniform. form sign waves all throughout space. So back in, I don't know, but I think it was the 1800s, Professor Fourier noticed that if I took different waves, that is to say waves of different wavelengths, right? So waves of each wave has a definite wavelength. I'm imagining a mode, is what we call it, a mode is a wave of definite wavelength, so I'm imagining different modes, and he says if I add them together, I can get something that is not a mode, right? Because if I add two waves together

with different wavelengths, I will not get a simple sine wave. I'll get something that looks more complicated than that. And if I add three or a hundred billion waves together, then I can get things that look more complicated than just a simple sine wave. And I don't know what his actual thought process was. It was probably a little bit more systematic than that. But Fourier asked himself, is it possible that if I add an infinite number of waves together, I can get any shape of the field I want? And the answer is yes. Otherwise, we wouldn't be going through this, right? The answer is, I can express any field I want, any configuration of the field, any increase and decrease and change in direction of a field all throughout space. You give that to me. I'm going to express it as a

sum of these nice simple modes, modes with a definite wavelength. I add up different wavelengths together, I get something that doesn't look like it has a wavelength at all. It's an arbitrary mess. Literally any profile of the field can be written as a sum of modes. And the point is, that's the Fourier transform. Rather than giving you the information about what the field value is at every point in space, I give you how much of each mode, contributes to the field. I guess I should, to be super-duper careful here, say not only can you take any profile, any configuration of the field you want and express it as a sum of modes of fixed wavelength, but that expression is unique. That is to say, given a profile, there is one and only one

way of expressing it as a sum of modes of fixed wavelength. That's what the Fourier Transform does. it tells you what that way is, that way of expressing some wildly varying function as a sum of nice sine waves and cosines. Okay. And the information is now been transformed, the information that used to be at every point in space, I tell you the value of the field. Now the information is, I tell you how much of a contribution the field gets from each mode, from each sine wave with the different with a definite wavelength. You know, some of them, some wavelengths might be very important. So there's a big contribution. So there's a big number times the sine wave in the sum of all this infinite number of them. Some might not be that important. You know what's wild? We can video chat

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So in some sense, that's nice, it's cute, right? It's a mathematical fact, but have we really learned anything useful by doing this? The Fourier transform is a way of saying, I can start with whatever field profile I want. I can turn it into an infinite sum over waves of different definite wavelengths. Well, yes, it turns out to be super duper important for the following reasons. If I think about the energy of a field, okay? The very first thing we're going to think about when we quantize our fields is think about what energy the field has, and we're going to look at solutions of definite energy, right?

So what is the essence of the energy of a field? A particle has two kinds of energy. It has kinetic energy, so a particle has an energy that depends on its velocity squared, 1⁄2mv squared in terms of the mass and the velocity of the particle. And it might also have a potential energy. So if the particle is moving in an electric field or something like that, the value of the electric field and charge of the particle will contribute to the potential energy of the field.

Sorry, the potential energy of the particle will get a contribution from the electric field. The field itself has three kinds of energy. It has a kinetic energy, just like a particle does, but it means something different. We call it a kinetic energy, but it's not the field is moving. It's that at each point, the field is changing its value. Okay? The field has a value at each point.

If that field is just constant, then the kinetic energy, constant with time, then the kinetic energy, the field is zero at that point. But if it's sort of vibrating up and down, for example, then it has a non-zero kinetic energy. A field can also have a potential energy, but it means a slightly different thing, because for the particle, the potential energy means how it's sort of moving through space and feeling different values of the electric field or the gravitational field or whatever.

For a field, what we call the potential energy, I know it's very annoying. We use exactly the same words to mean different things, but mathematically they look the same. That's why it's useful to look at the equations. For the field, we think about the field interacting with itself. So an electron gets a potential energy from interacting with the gravitational field or the electric field or the magnetic field. The fields get potential energy from interacting with themselves, and different fields will interact with themselves in different ways. But all that matters for that is the value of the field. So the kinetic energy depends on the derivative, the time rate of change of the field, whereas the potential energy only matters, only depends on the value of the field.

It does not matter how fast things are changing. And you might guess what the third kind of energy is. First kind was kinetic, second time was potential. There's something that is new for fields that doesn't exist for particles, which is the gradient energy. The idea is that not only can the value at each point in space of the field change with time, but as you go to a nearby point in space, there can be differences in the values of the fields.

Like the fields can be stretched a little bit or distorted a little bit from point to point. If the field was exactly constant through space, then there wouldn't be any stretching or distortion energy, what we call the gradient energy. But usually the field is changing from place to place. That will contribute to the energy of the field. Indeed, those of you who have read volume one, space time and motion, know about relativity, and you know that the difference between space and time will change depending on your point of view, and therefore you will not be surprised to learn that for fields, which I didn't say this explicitly yet, but these fields are deeply embedded in the context of relativity. Okay. So there's a relationship between the kinetic energy, the rate of

change with respect to time, and the gradient energy, the rate of change with respect to space, because space and time are related to each other. So we have three kinds of energy, okay? Kinetic energy, gradient energy, potential energy. And to solve the equations for what the field does, you need to know all of these. So you need to know what the field is doing at every point in space, because what it's doing at a nearby point in space will affect what the field does at one point in space. So at just a simple, very basic, dirty, technical level, it's kind of annoying to have these different kinds of energy floating around. Now, remember the question we're trying to answer here. Why do we bother thinking about Fourier transforms? Why do we bother

thinking about the profile of the field as a sum of plane wave modes rather than as just values at each individual point in space. Well, think about those energies. Think about the kinetic energy, the potential energy, the gradient energy. If you write the field as modes, if you take its Fourier transform, it still has a kinetic energy. It's the same kind of thing. It still has a potential energy, the same kind of thing. But the gradient energy is a different. kind of thing now, because by telling you that we're looking, we're focusing in on one particular mode, one particular wave with fixed wavelength stretching throughout all of space, I've told you how the wave changes from place to place, because it has a fixed wavelength, right? I've given you

that information. I don't need to separately give you that information at every point. If I'm just looking at one mode of the field, then I have the gradient energy sussed. I know what it is. I don't need to give you that extra information over and over again. Okay? So it turns out that when we're solving the equations for the behavior of a field, it's much easier to write down the solutions if we look at a mode of the field, or if we look at modes of the field one at a time, rather than looking at positions in space and values of the fields at positions in space. So that may or may not have been a convincing explanation. In this audio podcast, I cannot show you the equations, but the equations make it super duper clear, I promise you.

And it is possible to follow them. Do not get intimidated by the equations. It's just an alien language. You can learn it, and once you learn it, you have a good, warm feeling of accomplishment, so it's worth doing. So the lesson, the upshot of all this digression is, take a particular kind of field configuration, one with an absolutely fixed wavelength. I happen to know that I can write any field configuration as a sum of many such things, but let's keep that in the back of your mind and focus on one of them. pick a mode, pick a wave that has a wavelength, a definite, very, very specific frequency it's vibrating at, ask what it is doing. So in other words, when I quantize this, instead of assigning a complex number,

complex valued wave function to every field configuration, let me focus in on this one mode of the field and assign a complex number to every possible value of the mode. What does it mean value of the mode? The wavelength is fixed, but the amplitude is not. So it might be a wave of a certain wavelength that is very gently vibrating, just very slightly up and down, or it could be vibrating by a lot, right? Those are different ways that this mode could be vibrating. And to each different amplitude, to each different height of the wave, that it goes up and down, I will assign a complex number.

That is the wave function of this particular mode, okay? And you can kind of see how this would be easier to make sense of. You know, modes that are vibrating by a lot, right? Mods that are, again, the wavelength is fixed. So through space, we imagine, you know, from point A to point B, it goes up and down with a certain wavelength, but the amount by which it goes up and down in between where it's zero, one wavelength away,

the amount by which it goes up and down is arbitrary. And there's more energy when it goes up by a lot than when it goes up by just a little bit. And here's where the miracle occurs. You might have thought this was a little bit dry and technical and why are we bothering with this, but a miracle is about to occur, I promise you. We're thinking of the wave function of a single mode of a vibrating quantum field. And the mode itself that we're assigning a wave function to is now, Now that we've told you its wavelength is just characterized by a single number, the height.

Call it H, okay, the height of the wave that is vibrating up and down. So instead of psi of x for a particle being a wave function, we're going to have psi of H, the wave function of the height of the vibrating mode. And the exact thing that happened to the electron in the hydrogen atom, where we said the wave function has to taper off at zero, and you're going to get a set of discrete solutions on the inside that look like orbitals and explain the different energy levels of atoms, that exact same thing is going to happen to our mode of the quantum field. At very, very large height functions, very large amplitudes of the wave shaking up and down,

the wave function is going to have to go to zero because it can only be sort of localized in some particular set of values. And therefore, when we solve the equation, we will get a discrete set of solutions, a discrete set of possible wave functions for this one mode of this one field. There will be a lowest energy wave function, and there will be a simple, separate, next highest energy wave function, and there will be one higher than that, and there'll be a discrete tower of wave functions for this one particular wave with one particular wavelength. Why do you care about that? Okay, I mean, maybe you're with me this far, and you say, okay, I've taken my wave, I've decided to think about my wave as a sum of modes of fixed wavelength, I've focused in on one particular mode, I have assigned a wave function to it, and I've found that the wave function of that mode has a discrete set of solutions. Well, we're going to poke around, and this is something that's going to be hard to, I can't explain it really over words, but you can check out the book.

but I'm going to tell you some facts and you can choose to believe me or not. So look at that lowest energy wave function. It's the lowest energy. We're going to say it's the vacuum, as we call it, okay? There's sort of nothing there.

And then there's the next highest energy wave function. It has an energy. Let's call it M, the letter M. And it turns out that it's not unique. The vacuum, the lowest energy mode, the lowest energy wave function is unique. only one such thing, but there's an infinite collection of next highest energy modes,

which basically correspond to looking at the same kind of behavior but in different reference frames with different velocities, if you like. And you can calculate the energy of all of these different modes, basically the same, sorry, of all of these different wave functions, basically the same kind of wave function just looked at from different reference frames, and they get more and more energy, the more velocity you have with respect to the rest frame. Okay? And then you look at the next highest energy wave functions, and if the first one had a lowest energy of 2M, so you have a vacuum state with energy zero, a next highest one with energy M

or higher depending on if you have velocity with respect to it, and the one after that has an energy 2 times M. So what's going on? There's going to be one with three times M, four times M, et cetera. It's absolutely uniform growth of energies of these different particular wave functions. Well, I'm not going to keep you in suspense much longer. The vacuum state is what we interpret as a quantum state with no particles. It's empty space.

That first excited state, as we call it, with energy M, I should say, yeah, I'm sorry, This is back in my mind, and it certainly will be in your mind if you're reading the book, but I didn't say it out loud. We always in particle physics use units where the speed of light is set equal to one. So C, the speed of light, equals one, to me. And so when I say it has energy M, what I really mean is it has energy MC squared. What is that? Where have we ever heard that before?

The energy of something is MC squared. Ah, that is the energy of a single particle with mass M in its rest frame. If you move to another rest frame, the energy will go up because it will also have a little bit of kinetic energy, as well as the rest energy. What about this wave function that has an energy of two times M? Well, that's really two times mc squared. What is that the energy of? It's the energy of two particles with mass M. and then you have a set of other related wave functions where the particles are moving with respect to each other.

And then you have a vibrational wave function with energy three times MC squared, etc. And they are all related to each other by exactly the ways that you would expect a set of particles and their energies to be related to each other. That was a lot. And in the book, I go through this super explicitly, and I know that the actual manipulations are kind of boring and tedious, right? But the result is so important. It's one of the most important things literally in all of human knowledge, this particular result. Because the result that we got here through talking about four-gate transforms and thinking about quantum fields in terms of modes with different wavelengths and then the different ways that you could get a wave function for that mode. The result we got here is saying that when you take a field, which by itself is continuous and smooth and nothing discrete about it, and you assign a wave

function to the field, which is also by itself continuous and smooth and nothing discrete about it, the solutions to the equations, the different kinds of behavior that that wave function of the field can exhibit come in a discrete set. They look like particles. And you can go much further than this in explicating the fact that they behave like particles. These are what particles are, according to quantum field theory. According to quantum field theory, the world is not made of particles. The world is not made of discrete lumps of energy.

The world is made of fields that have wave functions. If you want to, you can just say the world is made of the wave function of the fields. That's a foundations of physics question that we're not going to get into right now. But the point is there's nothing discreet or lumpy about what the world is made of. The discreetness comes in how the world behaves, and it comes because we have quantized our fields. This is a slightly more mathematically careful and sophisticated way of saying something you may have heard me say before in other contexts. Particles are vibrations in quantum fields. That's what particles are.

When you apply the rules of quantum mechanics to fields, you get a set of solutions that look like particles in exactly the same way that when you solve the Schrodinger equation for an electron in an atom, you get a set of discrete orbitals with discrete energies, or for that matter, when you pluck a guitar string, you get a set of discrete frequencies at which it vibrates. All of the discreetness is emergent. That's where particles come from out of quantum field theory. And you get literally everything that you expect from particles. Many different particles will have different energies going in exactly the same way, etc. Good. So I hope that was convincing. I hope that you see a little bit now when you start with fields.

They can look like particles. And, you know, this means two things. It means that, you know, putting yourself in the mindset of a physicist in 1920s when quantum mechanics was being invented. Okay? One is I can take the electromagnetic field, which I know and love since the time of Maxwell in the 1800s, and I can quantize it. I can do all this. I can treat it in terms of modes, and I can assign wave functions, and I can get lumps of energy, which I interpret as particles, which we will call photons. That's what photons are. The other thing is that these other things we

had lying around, the electrons, the protons, et cetera, the things we thought of as particles, we can now say, well, maybe those are vibrations in quantum fields. So secretly, without even trying very hard, we have stumbled on an enormously powerful unification of different kinds of physics. In the late 1800s, we might have thought that the universe is made of fields and particles. particles being the matter in the universe and fields being the forces. But now we're getting a glimpse of the idea, thanks to quantum mechanics, that it's actually all just fields. And the fields show up, both as big classical fields under the right circumstances, and also as discrete individual particles under other circumstances. And so one challenge that we're faced with right away is, why do electrons and photons

seem to behave so differently from each other. And the answer basically comes down to symmetry. The idea of symmetry, as I say in the book, it's very helpful in classical physics, super helpful in particle-based quantum mechanics, but it becomes absolutely central in quantum field theory. So much so there's a whole chapter in the book devoted to symmetry where I teach you group theory.

You will learn what SU2 and SU3 and all those things are. This is one of the chapters that I actually wrote a lot more than appears in the book. I wrote a lot, and then I'm trying to, you know, even though we're doing the equations, I don't want it to be overwhelming. So I pared it down. There was a lot that I had put in the original version of the chapter that was a little bit more than you need to just do quantum field theory.

But the specific symmetries we're talking about right now are spacetime symmetries. So there's two different kinds of symmetries you can get in. quantum field theory. There are symmetries of space time itself. You know, there is translation invariance. It doesn't matter where you do the experiment. There's time translation invariance. Doesn't matter when you do the experiment. Rotational invariance. Boost invariance. Doesn't matter how fast you're moving in space. But then there's also what are we call, what we call internal symmetries, where you don't change where you are or how you're oriented in space time. you just rotate fields into each other.

And both of those are going to be very important. But for the question at hand right now, the difference between photons and electrons, one of the differences, comes down to how these fields change when you do a rotation in space. We're dealing here with relativistic quantum field theory. That's sort of implicit when we say quantum field theory,

but it's not necessarily that you need to be relativistic, that you need to be thinking about relativity when you do field theory. Relativity, in this case, in this sense, means that everything is relative, including the velocity of two different objects with respect to each other, right? There's no such thing as the velocity of an object. There's just the relative velocity of two different kinds of objects. Now, you can have circumstances where it's appropriate to do non-relativistic field theory. So if you're a condensed matter physicist. So you're not a particle physicist thinking about smashing two particles together in otherwise empty space, but you have a chunk of material here. And maybe there's some

vibrational modes of the material, right? We were talking about modes before, ways that a field can vibrate with a definite wavelength. Well, when you have a crystal or a metal or something like that, there are ways that the material can vibrate, and you can separate that into modes, and you can quantize them. And you're basically dealing with a non-relativistic quantum field theory because there is a rest frame that is well-defined in that case, the rest frame of the material that you're looking at. But anyway, we're not doing that.

We are doing relativistic quantum field theory. But for this question, we really care about rotations in space. How do fields change as you rotate in space? And really we're dealing with the wave function of a field, as we said, and that makes things a wee bit more complicated, but I'm going to, again, try to distill it down to the basic essences. Imagine you have some wave function for some field. In fact, imagine that the state of the field is such that it looks like there's one particle there. Remember, we said that there are quantum states of the field, and an arbitrary

quantum state of a field is basically a superposition of different kinds of. of particles, different numbers of particles. There is a particular wave function in the field that could have representing zero particles and different other ones representing one or two or more. In general, you could be a superposition of all sorts of different kinds. That's quantum mechanics for you, right? That's what let's quantum field theory describe changes between the number of particles, because a single wave function for a single field implicitly includes zero particle states, one particle states, two particle states, and so on upward. So let's zoom in on a wave function representing a single particle. So I want to emphasize this. Even though what is really

going on is a quantum field, there are certain circumstances in which is perfectly sensible to think of a single particle. That is a kind of excitation of a single quantum field. So it's not incompatible. We're not cheating when we say, let's think of one particle within the context of quantum field theory. You can talk about what an electron is doing in the context of quantum field theory. So let's do that. Let's talk about a single electron. And let's ask what happens when you rotate your frame of reference for that wave function of a single electron. In fact, let's not rotate by like 90 degrees. Let's rotate all the way around. 360 degrees. Two pi radians, we scientists, would say. Well, your naive expectation is that if you just do a whole rotation by 360 degrees,

the wave function is going to come back to where it started, right? That's a reflection of the symmetry of rotations. But if you think about it a little bit more carefully, there's another possibility. What if the wave function gets multiplied by minus one when you rotate by 360 degrees? That's interesting because it leaves all of the predictions unchanged, because the predictions in quantum mechanics, quantum field theory included, come from squaring the wave function. Now there's a philosophical question that arises here. Is it true that all we care about are the observational outcomes,

or should we care about other things? Good. Go ahead and contemplate those things. But for these books, for this book series, we're not dwelling on those philosophical questions. We're thinking like hard-nosed physicists. And the physicists will just say, look, I care about the observations. It's just as good if a wave function picks up a minus sign when I rotate by 360 degrees as it is if it remains completely unchanged.

And this, in fact, there's more subtle things you can say. What happens when you do rotate by 180 degrees or by 90 degrees, etc? And long story short, you get all different possible kinds of fields. every individual kind of field will give rise to a single kind of particle that has a definite property when it's rotated in space. And the properties might be that the wave function remains the same under 360 degrees or that it picks up a minus 1. And that corresponds to what we call the spin of the particle. This actually, this fact that the transformation properties of waves, functions under rotations is related to the spin of the particle is the quantum mechanical

version of a statement that we know in classical physics from NERTER's theorem. NERDAZ theorem says that conserved quantities arise from symmetries of the theory, and one such symmetry is rotations in classical mechanics, and the conserved quantity that arises from that is angular momentum, of which spin is one kind. So the quantum version of this is, is that the transformation of the wave functions under rotations tells us the spin of the particle. And basically, again, very long stories being made very short here, the kinds of particles that arise from fields in such a way that their wave function is unchanged under a rotation by 360 degrees have spins that are an integer value, 0, 1, 2, 3,000, etc.

Secretly, just like we set the speed of light equal to 1B4, now we're setting Planck's constant equal to 1. Plonk's constant H-bar is the fundamental unit of quantum mechanics, and it turns out to have the same units as angular momentum. So the spin of a particle is actually measured in units of H-bar. But we said H-bar equal to 1, so rather than saying spin-0, spin-h-bar, we just say spin-0, spin-1, spin-2. Those are all different kinds of particles that come back, their wave function comes back to the same value when you do a 360 degree rotation. If it picks up a minus one, then it turns out that the spin is a half integer value. So it's spin one half, spin three halves, spin five halves, etc.

There's a long mathematically intricate conversation to have about representations of the symmetry groups of space time on quantum fields. And the mathematical object that lets us describe these, spin one-half three-half particles are called spinners. S-P-I-N-O-R-S. It's a coinage that comes from tensors, which are other kinds of objects. So spinners are these spin-one-half spin-3-have fields. So keep that in mind, okay? There are kinds of particles like electrons, like neutrinos, like quarks, lots of different particles that have a wave function that picks up a minus sign when we rotate by 360 degrees. Now think about something that is highly analogous to that but different.

Instead of thinking of one particle and what happens when we rotate it by 360 degrees, think about two particles. In fact, think about two identical particles. As a footnote here, I will mention you might have heard of this story that John Wheeler and Richard Feynman were talking about, why do all electrons have the same mass and the same charge? Maybe because they're all the same electron, because an electron going backward in time is kind of like a positron, and maybe there's only one electron that just bounces forever and ever backward and forward in time. That is false. It did not work. It was a nice idea, and it led Feynman to think about Feynman diagrams and

things like that, but it's just a pretty story. The actual reason why all electrons have the same mass and charge is because they are all vibrations in the same underlying electron field. So what that means is that two electrons are absolutely identical to each other. And this is a technical term in quantum field theory. They are identical particles. Okay. So if I have a state of two electrons and there is a position of one particle of one electron X1 and the position of the other electron X2, again, they're not, they don't have definite positions. positions we could observe them to be at. But there's no difference between x1 and x2. There's a symmetry there. There's no way you can say, oh, it's this electron who I name alice, who is at

position x1, and electron bob is at x2. All you know is that there's one electron at one position and one electron at the other. So we can ask what happens when we take two identical particles, that is to say the wave function corresponding to a quantum state representing two identical particles, and we interchange them. Okay, so we're not rotating. We're not changing our physical coordinate system or anything like that. We're physically taking these two particles

and exchanging one for the other by moving them through space. But again, they're identical particles. So you might think, well, if I exchange one for the other, the wave function doesn't change. But by exactly the same argument, the wave function could also pick up a minus sign overall. And guess what?

Both possibilities are realized in the real world. world. There are particles who, when they get interchanged, identical particles, when they get interchange, their wave function is unchanged, and we call those bosons. And there are other particles whose wave functions, when we interchange them, pick up a minus sign, and we call those fermions. And the physical behavior of these kinds of particles is very different. Think about fermions, because that's the really interesting case here. A fermion set of permium particles have the property that when you take these two particles and you interchange them, then the overall wave function picks up a minus sign.

Okay? So what happens if you try to take two fermions and put them in the same exact quantum state? Right? You can imagine trying to do that. Two electrons with exactly the same wave function just overlapping each other. But then interchanging them doesn't do anything. They are literally already in the same state. You didn't do anything to it when you interchange them.

but we have a rule that says the wave function has to pick up a minus sign. How is it possible for a wave function to pick up a minus sign when you didn't change the wave function at all? And the answer is it's not. That wave function would have to be strictly zero. So these fermions, these particles that pick up a minus sign under interchange of identical particles, have a property that they cannot be in the same quantum state. For any quantum state, there can only be one fermion in it at a time. Or more informally, fermions take up space.

Bosons are the opposite. Bosons, if you have two identical particles in the same quantum state, that's great, because when you interchange them, nothing happens. And indeed, if they're already in the same quantum state, nothing would happen. So physically, this corresponds to very different behavior. The fact that fermions cannot be in the same quantum state is the Pauley exclusion principle, which Pauley actually meant before we really understood fermions and bosons. But it's why atoms are interesting.

It's why matter has the shapes and configurations that it does, because you can't put electrons over and over in the same quantum state orbiting an atom. Electrons have spins, they can be spin up and spin down. So in any one spatially definite wave function, you can put two different electrons in there, because their spins can be oppositely aligned. But then that's all you can do. then you've got to go into more complicated-looking spatial wave functions, and that is the origin of all of chemistry.

So the Pali-exclusion principle, which can be derived in quantum field theory, from the properties of identical particles that get a minus sign under their interchange, is really, really super-duper important. Whereas for bosons, they like to pile up on top of each other, right? They actually prefer it. So bosons, like photons, for example, very naturally pile on top of each other, and that's what gives rise to classical-looking force fields. That's why the electric field, the magnetic field, the gravitational field. These are all fields that we absolutely come into contact with in the ordinary classical macroscopic world. Why? Because they are boson excitations that have piled on top of each other. by a huge amount until they look classical. That's perfectly allowed. Whereas fermions take up space, that's why matter is solid. That's why the table in front of me right now doesn't collapse because the electrons have definite shapes, the wave functions of the electrons.

I won't go into it right now, but one thing I've rant about at a brief period in the book is this idea that atoms are mostly empty space. No, they're not. atoms are mostly wave function, and the wave function takes up space. And that fact is not merely philosophical. It's crucially important. It's why matter is solid. Don't listen to people who tell you that atoms are mostly empty space.

Anyway, you might have noticed, if you hang around on the wrong street corners and know a little bit about particle physics and quantum field theory, that I defined the Fermion boson distinction as what happens under the interchange of two particles. And I defined the spin distinction, the distinction between particles of spin 0, 1, 2, et cetera, versus particles with spin 1 half, 3 halves, et cetera, as what happens under the rotation of a single particle. Now, you might be confused because you might be remembering that people told you that bosons are particles of spin 0, 1⁄2, fermions are particles of spin 1, 1, 3, halves, 5 halves, et cetera. Why did that connection not get made? The answer is that connection exists, but it's not a definition.

It's a theorem that you can prove. You can relate using techniques from relativistic quantum field theory the transformation properties of a single particle wave function under rotation to the transformation properties of a two-particle wave function under interchange. And those minus signs that we talked about in both cases, cases cancel each other out. So there's a theorem, the spin statistics theorem, that says that the particles that pick up a minus sign under a 360 degree rotation are exactly identical, are the same as the particles who pick up a minus sign under the interchange of two particles in the wave

function. So therefore, you can prove that fermions, which are defined as particles that can't be in the same quantum state, particles that take up space, must have half integer spins, one-half, three-haves, et cetera. Electrons are fermions, neutrinos, quarks, et cetera. Bosons, the particles whose wave function are unchanged under a rotation, are exactly, nope, sorry, I said that exactly wrong, didn't I? Bosons, the particles whose wave function are unchanged under the interchange of two identical particles coincide exactly with those particles who pick up whose wave functions are unchanged.

We do a 360 degree rotation. That's the spin statistics theorem. We didn't just prove it right there. In fact, there's like controversy in the literature about who really has proven the spin statistics theorem because there's a lot of hand-wavy arguments that kind of sound like a proof, but really don't rely on relativistic quantum field theory, and therefore they can't be right because the theorem itself actually does rely. and relativistic quantum field theory. But what we're trying to do here is give you a feeling for why.

There is this relationship between spin and statistics, statistics being the question about whether or not the field particle excitations act like fermions or bosons. Okay. So that's an example of how you start from this single unified idea. Everything is fields subject to the rules of quantum mechanics, and you ask yourself, okay, so what kinds of fields could there be? And the answer are, you know, fields whose wave functions change in different ways under rotations, and you invent bosons and fermions. It naturally comes out of the combination of relativity with quantum field theory, which is really quite fun. Now, of course, the real fun comes when you let these particles, or these fields, interact with each other, okay? And that's where you.

you get into Feynman diagrams and renormalization and infinity and effective field theory and the whole bit. If you want to know what a virtual particle really is, read the book. It's a way of talking about the behavior of quantum fields. Virtual particles are not real particles, but they are useful ways of talking about the behavior of quantum fields when other particles scatter off of each other. Now, I will confess, when I was thinking about this solo podcast to advertise the book, I was overwhelmed with the number of things I could possibly talk about. And really, the thing that I think is the most important thing that I talk about in the book is the idea of effective field theory. And effective field theory comes from thinking about the fact that when you scatter particles off of each other and you use Feynman diagrams to calculate the likelihood. of that scattering event happening. Very frequently, the answer is infinity, right? The famous infinities of quantum field theory. And Feynman and Schwinger and Tominaga and others figured out ways to subtract off the infinities and get finite answers, and it all works very, very well.

But it was never very satisfying. So effective field theory is an idea that came along in the 60s and 70s and is the way that modern physicists think, and you were never told this. And this is very frustrating to me that the popular discussions of quantum field theory never mentioned this central organizing principle of the field, the effective field theory paradigm. And basically, Ken Wilson said, look, the reason why you get infinities when you scatter these particles off of each other is because you're summing up contributions from virtual particles with arbitrarily high energies. And in quantum field theory, high energy means short wavelength.

Okay, there's a relationship. Short of the wavelength, the higher the energy. Same thing is true in ordinary light, right? A photon with a short wavelength corresponds to a higher energy particle. And Wilson says, you don't know what's going on at these short distances and high energies. So you should just be honest. You should admit you don't know it. And he figured out a mathematical procedure for putting out a cutoff on your theory and saying, here is the energy below which I think I understand physics. And above, which I don't understand physics. So I can use my quantum field theory knowledge to write down a theory that only is supposed to apply in those regimes with energies below the cutoff. That is to say, only in the regime I understand. I'm ignoring things that go on at higher energies that I don't understand, except for the effects that they have on lower energy, longer wavelength, particles and fields. That's the effective field theory paradigm. It works super well. It's the most important thing, but I will confess. So I did a, you know, I want to give the people what they want. You know me. I'm here to serve. And so I went on to Patreon, where I'm the Mindscape podcast has a Patreon page with wonderful supporters. And so you could be a supporter, by the way. Patreon.com slash Sean M. Carroll. And I asked, I said, like, what are the things in quantum field theory you most want to hear about in the solo podcast?

Effective field theory did not get a lot of votes, I got to say. The votes were for, you know, where do particles come from, symmetries, things like that. So that's what we're here talking about. If you want more on the beauty and elegance and power of effective field theory, read the book. I promise you that it is there. A lot is hiding at short distances and high energies, but we know how to deal with it. So giving the people what they want, let's instead turn to the completing this picture of what kinds of fields are there. I said there are bosons and fermions. These are two different kinds of fields. They're related to the spins of the particles that you get when you look at the quantum wave

functions. Okay. But we know that there's more to it than that. We know that photons and other force-carrying particles have special properties. They're bosons. They're spin-1 bosons, except for the graviton, which is spin-2, but they're a little bit different. There's something going on there. What is going on there? And the answer is, once again a question of symmetry. But now it's an internal symmetry question rather than a space-time symmetry question. So space-time symmetry is, like I said, rotations, translations in space, boosts, changing the velocity, and so forth. But we can also, in quantum field theory, there's a new thing that can happen. We have a set of fields, and rather than rotating in space,

we simply rotate the fields into each other at each point in space. Okay. So imagine I have two fields, phi 1 and phi 2. The Greek letter 5 is the simplest thing that we use for a spin-zero scalar bosonic field in quantum field theory. We always call it phi ever since, I don't know, ever since the day. And so if we have two fields with exactly the same masses and the same interactions, 5-1 and phi-2, we could transform them into each other. We could exchange phi-1 for phi-2, or we could even think of phi-1 and phi-2 as defining a little two-dimensional vector space and rotating them by some arbitrary angle into each other. And if they're truly identical,

if they're truly the same kind of field, we would have no way of knowing that we had done that rotation. It would be a symmetry of the theory. That's really what a symmetry is. A symmetry is just a way of saying there's something we can do that doesn't matter, that has no physical effects. When we were talking about rotations or spatial translations, we said it doesn't doesn't matter how you rotate your experiment. Time translations say it doesn't matter when you do your experiment. Spatial translation say it doesn't matter where you do your experiment. Boasts say it doesn't matter how fast you're moving when you do your experiment, right? So these internal symmetries are saying it doesn't matter how we rotate these fields into each other. So naturally,

this is something that could happen. It turns out it does happen. It's going to happen a lot in quantum field theory. And so we need to think carefully about the mathematics of these symmetry transformations. And so group theory is the mathematical structure for thinking about that. And we talk in the whole chapter on symmetries a lot about group theory. But let's simplify it here down to the bare bones. Let's imagine we have three fields, let's say, three fields that are more or less identical to each other, okay, but we can rotate them into each other. So what is the set of possible ways to rotate three fields into each other. It's just like space. Like space is three-dimensional. We have X, Y, and Z axes. We can rotate those together. And the set of all possible

transformations in a three-dimensional space rotating things and not doing anything else or not stretching them, just rotating them into each other, is called S-O-3, the special orthogonal group in three dimensions. If you had eight fields, that were identical to each other, but there were eight different ones, and you could rotate them into each other, you would say they had an S-O-8 symmetry and so forth. It's just a number of fields that matters. More often, in quantum field theory, you will come across the group SU-3, or SU-2, or S-U-8, or whatever, and that change from S-O- to S-U reflects the fact that very often our quantum fields are complex-valued fields. each field is not a number at each point in space, but a complex number at each point in space.

So this is all to say, don't be intimidated. When you hear people say, the standard model of particle physics has a gauge group, SU3 cross, SU2, cross U1. That's just a way of saying there are three fields that are complex value that get rotated into each other. There's a separate set of two fields that are also complex value that get rotated into each other. That's SU2. And then you can have a single complex valued field that can still, in some sense, rotate into itself because a complex valued field has a real part and an imaginary part.

There is something called the complex plane, where we have a complex number. We write the real value of the number on the horizontal axis, the imaginary value on the vertical axis. Well, you can rotate the complex plane, and that rotation is called, U1. So these are all just fancy ways of talking about rotating fields into each other, okay? And it doesn't matter. It doesn't matter what rotation you do. The same physical predictions

will be made. That's what makes it asymmetry. Indeed, this idea of having three fields that we rotate into each other is a very simple example. They are called quarks. Okay, you may have heard that quarks come in three colors. There are red quarks, green corks, green, quarks and blue quarks. That's not quite right. Like so many things that you've been told, you've been told something that is not quite right, but good enough. And now you're ready for the real stuff. The real stuff is, not that there are three different quarks, but there is a three-dimensional space of quarks. So the subtle difference I'm pointing to here is it's not that there are red quarks, green quarks, and blue quarks. There are quarks that every quark, I should say,

should be thought of as a combination of a bit of red, a bit of green, and a bit of blue. And you can choose, you can rotate your axes, so you define what you mean by red, green, or blue. There's no actual reality to it. It has nothing to do with the real colors. Okay, you're not looking at a blue quark and it looks blue or anything like that. This is just a label. RG and B are completely equivalent to each other, these three-dimensional vector space of quark colors. and the real statement is not there are three different colors of quarks, but that quarks, every quark field is an element at every point in space of this three-dimensional, three complex-dimensional, I should say, vector space. So there is an S-U-3 symmetry rotating the red, green, and blue axes into each other.

Three different kinds of quarks, we conventionally label them by color. There's a symmetry where we can rotate them into each other. S-U-3, very, very nice. And then we'll talk about in the book what that means, SU3, et cetera, et cetera, how to implement it. Great. So what? What difference does that make, really? Well, you instantly know there is a conserved quantity because there is a symmetry, and NIRTHIS theorem tells you there's always a conserved quantity when there's a symmetry.

So that's nice. But it isn't quite the punchline. The punchline comes from this. Let's imagine we have an even better symmetry. So we have these three fields we can rotate into each other, call that rotation SU3. That's nice. But what if I want to rotate the axes of my red, green, blue space in one way at some point and in a completely different way at some other point?

I didn't say that I could do that. I'm just asking whether it would be possible. So there is something called a global symmetry transformation, where we take what we've defined to be red, green, and blue. and we rotate them everywhere all at once in some particular way. So we do a global transformation of what we mean by red, green, and blue. That's fine. No one really is bothered by that.

But if you want, if you have the aspiration of saying, I would like to separately define what I mean by red, green, and blue, independently at every point in space. Can I do that? That would be an enormously bigger symmetry. right? Because I'm separately able to define what I mean by red, green, and blue all throughout space. That's a lot going on. And the answer is you can do that, but you have to sort of figure, you have to make sure that you're allowed to, that it's a well-defined thing. Because whether we call

a quark at any one point in space, a quark field, whether we call it red, green, or blue, or some combination thereof, that's completely arbitrary, right? That makes no physical difference. But if we have a two different points in space. We have the same quark field. Remember, the quark field will take values at every point in space, and let's ask the following question. At these two different points in space, does the quark field have the same color? Or does it have some different color? Is it rotated by 90 degrees or whatever? That should be a well-defined question, right? Is it the same color or a different color? Well, if I'm allowed to rotate my axes separately at every point in space, it seems like I can't answer that question, because I could have some quark field that I

am just simply defining to be red at one point in space by appropriately rotating my little red-green-blue axes. And again, these axes are completely imaginary, like not imaginary in the imaginary number sense, but conceptual. They're not axes of literal space. They're in quark space. Okay. There's three different directions the quark field can be vibrating, the red direction, the green direction, the blue direction. And likewise, okay, so. So I have a red quark at one point in space, and I rotate my axes so that at some other point in space, it's also a red quark. But is that legit? I mean, I could have rotated it, so it was a blue quark.

And there's a physical difference there. So what I'm getting at is, if you want to be able to have this megacymmetry that lets you rotate in red-green blue space independently at every point in space, you also need some way to compare. you need some way to answer the question. What is the value, let's put it this way. How does the value of the color of the quark change as I travel along some path? Okay. I can connect these two points in space with some path, and I can ask how the axes that define red, green, and blue are changing as I move down the path.

So I need some information that lets me answer that question. How do these axes change as I go down the path? how does the definition of red, green, and blue change? It's not a rigid structure like it would have been if I just had that global symmetry. Now I'm able to change what I mean by red, green, and blue independently everywhere. Something needs to absorb the information of how I did that.

What would do that? What would carry that information about how I defined red, green, and blue at every point in space? The answer is a field. I need another kind of field. I already have the quark field. In fact, I have the sort of three different kinds of quark field, the red, green, and blue, three different directions in which the quark can rotate. But now I need an entirely new kind of field to carry the information that tells me how to relate the red, green, and blue axes at different points in space.

And the mathematicians know how to do this. This is called a connection field, and the mathematical subject that studies these things is called fiber bundles. It's a subject within differential geometry. You can read all about it. Again, we're not going to go into all those details. But the point, and this is a big point, and it's another one of those miraculous points that is very, very important,

is that you can do these independent symmetry transformations at different points, but only if you introduce a new field that lets you keep track of the information relating what's going on at different points. So these symmetries that can happen independently at every different point are called gauge symmetries

or simply local symmetries in contrast with the global symmetries that we were talking about to start. Gage symmetries, unlike global symmetries, come along with the new field, the gauge field, aka the connection. Those are the same thing. The gauge field, the connection field,

the field that keeps track of how your axes are changing from point to point in space. Okay? It's much like in space time, if you go back to book one of the biggest ideas, in space time, just to compare the question, what direction is a vector pointing in? You need to be able to parallel transport vectors from one point to another, and you need a connection field to be able to do that.

That connection depends on the metric, in general relativity. Here, in gauge theories of particle physics, there's no metric. There just is the connection. That's a field that shows up as the gauge field. And so what does that field do? What are its dynamics? How does it behave? How does it interact with other fields?

There's a whole bunch of questions. You're going to read the books. You're going to know the answer to these questions. But the punchline is that field, that connection field, you need to implement the gauge symmetry is the force field. It is the electromagnetic field for electromagnetism or the gravitational field for gravity. And it gives rise to particles. photons for electromagnetism, gravitons for gravity,

and there's also the nuclear forces, the strong and weak nuclear forces. Mathematically, they are boson fields, and you can derive that, and there are interactions with other fields. The way that the force-carrying fields interact with the non-force-carrying fields, the electrons and neutrinos and quarks and so forth, is entirely determined by the demands of this symmetry, okay? Given the fact that you want this symmetry to exist, not only do you need this gauge field, but you know how the gauge field interacts with other fields.

And you know more than that. The symmetry is super powerful. This is why physicists sort of start jumping up and down when they talk about symmetry. They get very excited because it really is a very powerful concept. You can derive features of the dynamics of the gauge fields all by itself. So in particular, there's one very, very important feature that can. kind of puzzled people for a long time. Let me back up to just mention. The idea of a gauge field was kind of understood by Maxwell. He understood that his electric field and magnetic field could be

derived from a more fundamental single thing that we would now call the gauge field. It wasn't until the 1950s that Yang and Mills suggested generalizing this idea of the electromagnetic gauge field to more complicated gauge fields, SU2, SU3, etc. And it took a while, it took a while, it took until the 70s, really, for us to figure out how to make that happen. Okay. But it turns out that symmetry, this SU2 or SU3 symmetry or whatever it is, has a very direct implication for the particles that arise from these gauge fields. And the implication is the particles have to have zero mass. And again, this is something where you'll be delighted when you read the book because you will see why that's true, because it comes right out of the equations.

given what you mean by mass, mass is a particular kind of potential energy that the fields can have, and there's no way to have that kind of potential energy and respect the symmetry at the same time. So there's a direct line from saying there's a gauge symmetry to saying, and there's a massless particle. And that's beautiful and lovely because we have the photon, which is a massless particle. We have the graviton, which is a massless particle. I know we haven't detected gravitons yet, and people get fussy about it. but if you believe the basic features of general relativity and quantum mechanics, gravitons will exist. They're just too weakly interacting to ever be detected. I would not have

any skepticism that they really exist. The problem is that those are the only two massless gauge bosons that we know about, the photon and the graviton, right? At least that we knew about back in the day. So when we were thinking about this, back in the 50s and 60s, people were like, wait a minute, this is a problem. We would love it if this beautiful mathematical structure of gauge symmetry and the associated fields could somehow help us understand not only electromagnetism and gravity, but also the nuclear forces. There are two nuclear forces, the weak nuclear force and the strong nuclear force, not the most romantic names, I know, but that's what we're stuck with, okay, the weak force and the strong force. Both of them are short-range forces. That's why

you don't notice them in our everyday lives. In our everyday lives, we really only notice electricity and magnetism or gravity as far as bosonic force fields are concerned. So if Yang and Mills have this idea about gauge symmetries, but gauge symmetries imply that the force-carrying particles are massless, and there's this little mathematical trick you can do to show that massless particles always give rise to long-range forces, which it seems that you can, then we're stuck, right? Because the things that we want to explain experimentally are short-range forces, and it seems like the idea doesn't work.

Of course, physicists are not as easily dissuaded as that. They will keep trying. And they should keep trying. And, you know, this is also true for modern theories where we're thinking about the multiverse or string theory or loop quantum gravity or whatever. You will very often say, okay, if my theory works in a certain way, I predict X. X is not true, right? But you don't give up. You say, well, maybe I didn't think hard enough about what my theory really predicts. That's absolutely the situation they were in in the 50s and 60s, thinking about gauge symmetries. The idea was so incredibly attractive. They wanted to make it work, even though it predicted massless bosons that were not observed. So what is going on?

Well, it turns out there are several different ways to get rid of these massless bosons, and nature uses all of them. Well, there's two very, very straightforward ways, and nature uses both of them. Let's put it that way. So we can think about this in terms of phases of our gauge symmetry. Just like water can be in the liquid, solid, or gas phase, a gauge field can appear in different phases, not really depending on temperature, although that's also true, but it really depends on other fundamental variables, fundamental properties of the underlying physical equations of motion. So one phase is the so-called Kulom phase. Kulom's law is just the electromagnetic version of Newton's inverse square law for gravity. So you could call it the Newton phase if you wanted to.

The Kulom phase is the phase where the gauge bosons are massless and can easily travel throughout the universe. Photons and gravitons both qualify. For that, And in that case, you have long-range forces with an inverse square law. And indeed, for both electricity, magnetism, and gravity, you have long-range forces with an inverse-square law. It's fun and amusing to see the fact that Kulom's law and Newton's Law of Gravity are both inverse square laws coming out of an underlying gauge symmetry for the forces that give rise to them. but there are other phases these gauge symmetries can be in. So one phase happens when you have the symmetry, you have this symmetry, this gauge symmetry that would give you, that does, would and does give rise to gauge bosons, but the symmetry is

broken spontaneously. What does that mean? Spontaneous breaking of a symmetry means when the symmetry is broken not because the fundamental deep down equations of motion violate the symmetry, but because the specific configuration of the world violates the symmetry. For example, rotations in three-dimensional space are a symmetry of the laws of physics, but here on Earth, there's a difference between up and down, right? There's an arrow of space in a very, very direct kind of way, and that is not surprising to us. No one thinks that's because of the fundamental laws of physics. I mean, maybe Aristotle did, but now we know better. We know it's because we have the Earth underneath our feet. The actual

configuration of matter is breaking the symmetry between up and down, even though the laws of physics do not. Now, strictly speaking, that's not a great example of what we're calling spontaneous symmetry breaking, because spontaneous symmetry breaking in quantum field theory happens, again, because the configuration of the world breaks the symmetry, but even in the vacuum. That's the new thing. So the Earth is not the vacuum. The Earth is a big, you know, pile of energy, pile of matter and particles and so forth, so that can break the symmetry. But what if empty space itself could break a symmetry? That would be a whole different kind of thing. And this was studied. I forget whether it's a 50s or 60s where they started studying it, but Yoshiro Nambu, my four colleague at the

University of Chicago, and Jeffrey Goldstone and others, studied how you could. could break a symmetry. And the idea is actually quite simple. You have yet another field, right? And guess what? This is going to grow up into what we now call the Higgs field. So the Higgs field is an example of this kind of field, but we'll get there in a second. If you have another field, not a fermion, not like a quark or an electron or something like that, but a scalar field, a bosaunic field that transforms under the symmetry. Okay. So let's say you have an SU2 symmetry, because you kind of do. You kind of do in the weak interactions, which is what we're going to get to. So you have an SU2 symmetry. So for this scalar field, you have two directions in which the scalar field can vibrate, and there's

an SU2 rotation that rotates them into each other. But if the value of the field in empty space were zero, then not only would the fields, plural, be invariant under this symmetry, but the configuration of these fields in the vacuum would be invariant under this symmetry because the configuration would be centered around zero. And if you rotate a plane around the origin, nothing seems to happen. But what if, in empty space,

the field was at a non-zero value? What if the energy of the field were lower when it had some non-zero value than when it was at zero? When it was at zero. You could do that. You can easily write down equations that make that happen.

If you want to look up the detail, it's the Mexican hat potential that you might have seen under discussions of spontaneous symmetry breaking. And when that happens, the specific value that the field takes in empty space is not invariant under the symmetry. So the equations are invariant, but the value the field has is not invariant. That is what is called spontaneous symmetry breaking. Because it kind of doesn't matter what direction the field is pointing in.

it's pointing in some direction, even in empty space. And so now there's an answer to the question when some other thing, like a quark or an electron, vibrates, you can ask the question, is it vibrating in the same direction as the scalar field that's breaking the symmetry or perpendicularly to it or whatever? The symmetry has been broken in a way that it wasn't before, okay? Sadly, this is not immediately solved the problems because Goldstone proved a theorem, Goldstone's theorem that says that when you have spontaneous symmetry breaking, you will have a new kind of massless particle. The scalar field will turn into a massless particle. And again, those

particles were not observed, and so people were still a little flummoxed. But what they soon realized is that there's a big important difference between the global symmetries that we talked about originally, where the symmetry has to be done everywhere uniformly, versus the gauge symmetries, where you have this connection field that keeps track of what direction you're pointing in. The Goldstone's theorem analysis that predicted the existence of a new kind of massless particle was only for global symmetries. When you do the same thing for gauge symmetries, you do not get a massless particle. Not only is the scalar field not giving you an extra massless particle, but your gauge fields, like the photon or the equivalent of the photon, get heavy.

They go from being massless to being massive. And in every single quantum field theory textbook in the world, what you're told is that the gauge bosons eat the scalar bosons, the Higgs bosons, and they become massive by doing that. So you go from a theory with a lot of massless particles to a theory with no massless particles through spontaneous breaking of a gauge symmetry. And that is the origin in the standard model of particle physics

of the W and Z bosons. You have the Higgs boson, which is a spin-zero boson that is, as we say, charged under the SU2 symmetry. It rotates under the SU2 symmetry. And so do all the other particles of the standard model,

the electrons and the quarks, etc. But that Higgs boson gets an expectation value that is not zero in empty space, and that gives mass to the gauge bosons which we observe at the end of the day as the W and Z bosons,

and they are indeed massive. And it is because those particles are massive, that the force they give rise to is a short-range force rather than a long-range force. So that is one way that we can have our cake needed to, that we can have massless particles

because of the gauge symmetry, but they don't appear to us at the end of the day because they have gained mass because of spontaneous symmetry breaking. There turns out to be a whole thing. another way to hide the massless particles from us, and nature takes advantage of that way in the SU3 part of the standard model. SU3 is the gauge group, the group of symmetries for the quarks, the red, green, and blue quarks, the three dimensions of quark space that get rotated into

each other by this SU3 symmetry. And because those are colors, we call this quantum chromodynamics, or QCD in analogy with QED, coinedage from Murray Gilman. So QCD is a theory of the gauge symmetry in the associated force that comes from invariance under rotations in red, green, blue internal space that the quarks live in. And there, there's no spontaneous symmetry breaking. The SU3 symmetry is unbroken in empty space. So the gluons, which are the gauge boson,

of that force, what we call the strong force, are massless. Gluons are still massless. Why then do they not give rise to a long range 1 over R squared force? The answer is because of the difference between SU3 and U1. And again, some details. We're gliding over. The buzzwords are abelian versus non-abillion. These are different kinds of symmetry groups that you can have.

and the SU3 symmetry group is just more complicated than the U1 symmetry group. Electromagnetism has a very simple symmetry group, U1. It's just rotations in a complex plane. The SU3 symmetry group has more things going on, and at the level of the fields that has a crucially important implication, namely the fields, the gluon fields, interact with each other directly in a way that photons don't interact with each other. Photons are the particles of the U-1 gauge symmetry.

Gluons are the particles of the SU3 gauge symmetry, but photons interact with charged particles like electrons and quarks. They don't directly interact with other photons. Indirectly they can do because of electrons at center, but not directly. Whereas the gluons can interact directly with other gluons because gluons also carry color in a way that photons do not carry electric charge. And what that means is it opens up a possibility that nature, in fact, takes advantage of, which is what we call confinement. You might think that since the gluons

are massless, like electrical fields around an electron, they should spread out in an inverse square kind of potential. But in fact, the gluons keep interacting with each other. They acquire energy by their mutual interaction with other gluons as well as with the quarks. And that energy dramatically changes the way that the gluon field arranges itself around individual quarks. It is not an inverse square law. The force between two gluons or between two quarks mediated by gluons grows with distance rather than decreasing with distance as one over R squared.

So if you take two quarks and they're pulled together by gluons, gluons, if you try to pull them apart, it takes more and more and more energy to get the quarks to pull apart. And so much energy that at some point it makes more sense just to make more quarks or a quark-anti-quark pair to be a little bit more precise. So you just snap the string, if you like, that is connecting, the flux tube, as we call it, that is connecting the two quarks to each other. Because of this, you can never see one quark all by itself. Every quark is connected other quarks held together by these gluons. And that's why the particles we see are protons, neutrons, mesons, things like that, not the individual quarks. As I am recording this, and after I

wrote my book, there has been a claim, an experimental claim, that we have experimentally detected what are called glue balls, which are particle-like excitations made of nothing but gluons, no quarks at all. So that's very exciting. It's too bad for. me, because I said in the book that we haven't detected glue on glue balls yet, but apparently we have. Anyway, this is why the strong force is short-range, not because of spontaneous symmetry-breaking, but because of confinement, because the gluons interact with each other so strongly that you can't separate individual particles from each other, and they completely distort the shape of the gluon field away from being a long-range one-over-r-squared force.

So there's a lot going on. This is why when you take general relativity as a graduate student in physics, you know, different people react differently to different courses. But general relativity is beautiful. You know, when you take that one semester course, you start from, you know, simple geometric postulates and you derive the equation of Einstein, and then you derive all these consequences, black holes and everything. And it's so pristine and logical and lovely.

And then when you take your quantum field theory course, not only is it at least two semesters, often three or four semesters of quantum field theory, but it's not beautiful, really, not in the same way. It's a mess. There's a lot going on with all these different forces, all these different symmetries. And the field itself is just complicated.

You can easily take quantum field theory courses again and again, hopefully from different instructors with different textbooks and learn more and more every time. No one would ever take general relativity more than once if they learned it correctly the first time. There's so much we didn't even get to talk. about parity violation is very, very important in the standard model of particle physics. There are families of particles, right?

There are different ways that neutrinos and electrons hook up with each other versus how quarks hook up with each other. There's, let me just, let me hint at one kind of thing that I haven't been able to tell you about. This gauge group of the strong force, SU3, right? like I said, it's more complex. It's not super duper complex, but it's a little bit more going on there, a little more structure than the U1 gauge symmetry of electromagnetism. The SU2 is much like SU3. They're very similar, but because SU2 is spontaneously broken, it's a different conversation that we have about it. SU3 is not spontaneously broken, but it's confined. And one of the things that can happen, because SU3 is a slightly different gauge group than you won, is that you can have different vacuum configurations of the gauge fields. What do I mean by that? A vacuum configuration of the fields is a configuration as zero energy,

right? But when I say a field configuration, you now know, since you've listened to this podcast, you now know that a field configuration is a specific arrangement of the field at every point in space, But you also know that there is a symmetry, the gauge symmetry. I can rotate what I mean by red, green, and blue of SU3, separately at each point in space. So really what I mean by the vacuum configuration of the field is one particular reference configuration of the field plus any gauge transformation I want to do on it.

A gauge transformation is a symmetry. It doesn't really change the actual field, right? Okay. So that's just a, it sounds like a mathematical detail that I don't need to think too hard. I should just be careful when I say what I mean by a field configuration to say, really what I mean is a field configuration up to a possible symmetry transformation that you might want to do. But it turns out that in SU3, in the strong interactions, there are what we call small gauge transformations and large gauge transformations. And what does that mean? What is small? and large mean, it means that there is a topology to the possible gauge transformations we can do. It's exactly, once again, this is an example of an underlying smooth thing, the gauge fields of SU3 quantum chromodynamics, but having a discrete set of possible arrangements of those fields. So it's kind of like wrapping a circle around another circle. It's actually very analogous to that.

If I have one circle, if I think like a topologist now, and I'm going to map a different circle to my first circle, I can just take the first circle and I can map it the whole thing to a point. Or I could wrap it once around the other circle, or I could wrap it once around the other way, or I could wrap it twice around, et cetera, right? Even though the circles themselves are smooth manifolds, there's a discrete set of ways that I can wrap one circle around another one. These are called the winding numbers of these different maps. Turns out, and again, you don't need to understand this, because this is just like hinting at some fascinating work that happened in quantum field theory in the course of the 1970s, turns out that there are topologically non-trivial gauge transformations

in SU3 that wrap the gauge field around itself, and they're associated with a winding number. Okay, so, you know, once again, you say, all right, it's all empty space. It's just empty space and increasingly complicated configurations because they're all equivalent to each other under the symmetry transformation. But it turns out the field can dynamically flip from one topological configuration to another. And that is not just the vacuum.

That is, it can sort of pop out of the vacuum for a second as it winds around itself and then settle down into a different winding number. So you start with empty space, the vacuum. You end with empty space the vacuum, but through some quantum fluctuation called an instanton, you change dynamically the winding number of the field. And this is a whole conversation that could only be had because we're doing quantum field theory, right? Because the fields are what the universe is made of. They have these mathematical properties.

Things can happen. And the kicker is that these instantons that are quantum transitions that change the wind numbers of the gauge fields have a physical effect. They give rise to masses for different mesons in different ways. And this is called the Ada problem in QCD that is solved by thinking carefully about the structure of the vacuum and the existence of topologically non-trivial gauge transformations. All of which, I hope that everything before the last five minutes was completely crystal clear and you understood everything. The last five minutes was just to make you sort of intrigued. It's not enough detail to possibly be completely understandable. The point is,

there's a lot going on in the standard model of particle physics. It is an amazing structure that I didn't even mention, but of course, the real punchline is it fits the data. It fits so much data. You know, when we turned on the Large Hadron Collider, of course, eventually, you know, we turned it on like 2006, blew up, fixed it, turned it on in 2008, again. By 2012, we discovered the Higgs boson, right? But before we discovered the Higgs boson, we rediscovered the entire rest of the standard model. You know, all the other particles that had been discovered before, all the different quarks and the W and the Z bosons and so forth, the LHC sees them in exactly where they should be doing exactly the things they should do.

There's no more rich and quantitatively accurate theory in the history of physics than the standard model of particle physics. So it's very messy. It's very, it does a little bit of everything. It's not like one beautiful principle just continues on as a line that gives you the entire theory. Many, many ideas come in from seemingly different corners of physics and mathematics, etc. And combine to give us the standard model and it works beautifully. It explains all the data that we have so far. Good news, bad news situation, of course, for reasons that we get into in other podcast conversations. And the final thing to say is, and it's probably not the final answer, right? I mean, the standard model of particle physics, even if you include gravity, you can include

gravity in the standard model as long as the gravitational fields are weak, and we call that the core theory, dubbed by Frank Wilczek. But we don't understand a whole bunch of things. We don't understand the Big Bang or black holes, right? We don't understand conditions where gravity is strong. We don't know what the dark matter is. That's some other kind of energy that is apparently not there in the standard model. And there's things about the standard model that are just puzzling. There's both sort of mildly puzzling things and deeply puzzling things. There are numerical unnatural numbers, right? The cosmological constant, of course, is a famous unnatural number. the mass of the Higgs boson, even though we can measure it, et cetera, it's very different from what we would

expect it to be, according to the logic of effective field theory that we mentioned earlier. So these are apparent fine tunings in the standard model. There's also the fact that that U1 part of the standard model, right? The standard model is SU3 cross U2 cross U1. The SU3 symmetry is quantum chromodynamics. The SU2 and U1 parts are the ElectraWeek unified theory, the part, the theory that theory gives us both electromagnetism and the weak force. But the U1 part is really not well-behaved at high energies.

There's something called a Landau pole that we're not going to get into here. But the U-1 part of the standard model apparently doesn't make sense at arbitrarily high energy scales. It shouldn't. This doesn't bother people because gravity is going to be important at ultra-high energy scales. At low energies where you and I live, gravity is a very weak force, but at high energies it should become important. So there are gaps. There are things the standard model is not up to the task of accounting for, even if they're mostly kind of conceptual things rather than experimental things.

So not only the standard model itself, but probably even quantum field theory as an overall picture, is probably not up to the task of being the final, complete theory of everything in physics. So the reason why I've been wanting to write this book for 20 years, and I'm very excited to finally have done it, is both to impress upon you how wonderful quantum field theory is, how amazing it is,

that it fits all the data in so many intricate and weird and fun ways, but also to prepare you for the fact that we might do better someday, whether it's emergent space time or string theory or whatever it happens to be. Physics isn't done yet. We're still moving on.

We understand a lot. We don't understand everything. We have to accept and celebrate both of those features of our knowledge and try to increase the amount that we understand. Let's get back to work then. Thanks.