A Primer on Particle Physics, Part 2: Fun with Quantum Field Theory

Link to part 1 in case you missed it: https://the-avocado.org/2020/07/07/a-primer-on-particle-physics-part-1-what-is-a-particle-the-answer-may-surprise-you/

Last time, I talked in a very abstract way about fields (and particularly quantum fields), and I claimed that particles are fundamentally excitations of these fields. This time, we’ll get just a little bit more concrete, and introduce a few different features of Quantum Field Theory and the particles that arise in it. As I was writing this, though, I realized that one topic that I wanted to cover here really wanted to expand into an article of its own, so I’ve broken off the section on Feynman diagrams and will post that as part 3 next week.

The picture, again, is one of infinitesimal springs (in the language of physics, “harmonic oscillators”) filling all of space, all attached to one another, so that if one spring starts bouncing up and down, it will start its neighbors going, and the excitation will spread out. These springs are only allowed to bounce up and down in discrete steps, which means that there is a smallest “unit” (in physics-ese, a “quantum”) of excitation that is allowed. If a spring has just one unit of excitation, that little bit of bounciness will move from spring to spring in a probabilistic way that ends up looking just like the motion of a particle in quantum mechanics. So, I claimed, that smallest excitation is a particle.

That’s the framework of QFT. To get a specific theory, we need to specify two things:

– What fields exist in this theory, and what their properties (mass, charge, spin, etc.).

– What couplings exist between these fields.

These features are specified in a mathematical expression called the Lagrangian of a field theory. The Lagrangian, in turn, plays a role in the equations, or laws, that tell us how the fields behave.

To make a very rough analogy, if you’re familiar with Newtonian physics, you know that the motion of objects in that theory is governed by Newton’s second law, F = ma. This relates the force (F) on an object of a certain mass (m) to its acceleration (a). To actually do anything with this, though, you need the theory to give you rules specifying how to calculate the force on the object. In field theory, instead of a force, you have a Lagrangian; instead of a law relating force to acceleration, you have a law relating that Lagrangian to the oscillations of the “springs” in the field; and instead of rules telling you what forces exist, and how to calculate them, you have rules telling you what fields and couplings are included in the Lagrangian.

Just for fun, let’s actually look at the expression for Lagrangian of Quantum Electrodynamics, the first “real” QFT that was developed. This theory includes just two fields: one for the photon and one for the electron. There is obviously a lot of math here that we’re not going to touch, so please don’t let this frighten you away:

QED Lagrangian
Wow, OK, so that looks complicated. But let me quickly explain what these various symbols mean. First of all, you have some that are just mathematical operators – the \partial symbols, and the \gamma (with their \mu and \nu subscripts and superscripts). Then there are the physical constants \hbar and c, which are basically just unit conversions. The m and e are constants of a different sort – they are the fundamental mass and charge for the electron field – which is to say, the mass and charge of an electron. That just leaves \psi, which is the electron field itself (in other words, it’s a function giving the value of that field at every point in space) and A (with its various subscripts and superscripts), which, similarly, is the electromagnetic field (i.e. the photon field) itself.

But the main reason I wanted to show you this is to make the following point. Notice that this Lagrangian is a sum of three terms, which I’ll highlight with different colors:

QED Lagrangian color
The first term, in red, contains only the electron field \psi. This term governs the behavior of the electron field by itself; it describes, for example an electron propagating freely through space. The second term, in blue, similarly contains only the electromagnetic field A, and it governs the behavior of the that field by itself – that is, the behavior of free photons. The third term, in green, contains both \psi and A. This term governs the coupling between the two fields; in other words, it describes the interaction between photons and electrons. And that’s roughly how these Lagrangians always work; every term that contains two different fields gives rise to a coupling between those fields – it means that those two fields are “connected” so that oscillations in one can be transferred to the other. In other words, a particle in one of those fields can create a particle in the other; the particles can interact.

From the Lagrangian for a quantum field theory, we can derive everything about how the fields and particles behave and interact. The Lagrangian itself, however, can rapidly become unwieldy and difficult to work with. To give you an idea of what I mean, here is the full Lagrangian for the Standard Model – the most complete and successful quantum field theory we’ve developed so far:

Spoiler

SM Lagrangian

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Spin, fermions, and bosons

It turns out that fundamental particles can be usefully divided into two types: fermions and bosons.

The technical difference between these two classes of particle has to do with a property all particles have called spin. The textbook definition of spin is the “intrinsic angular momentum” of a particle. What does that mean? Well, just for a moment, forget everything I’ve told you about how particles are really fields; pretend they’re actually hard, tiny spheres. Imagine this sphere is rotating. We would classically assign it a quantity called “angular momentum”; basically, the faster it’s spinning, the higher it’s angular momentum. If it’s not spinning at all, its angular momentum is zero.

We know that particles aren’t really little spheres; they’re a phenomenon that arises from quantum fields. But it turns out that they still possess angular momentum. That angular momentum doesn’t correspond to any physical thing that is actually rotating, which is why it’s called “intrinsic” – it’s just a property of the field. But if you want to picture it as an infinitesimal particle rotating, you won’t be far off.

Now, a classical object could have any angular momentum, but in quantum physics angular momentum is quantized. Just as the “springs” of our quantum fields can only move up and down in discrete steps, angular momentum, or spin, always comes in discrete units. We talk about spin in terms of the physical constant \hbar, and we find that it only comes in units of 1/2 \hbar. Further, a given type of particle will always have the same spin – just as all electrons have the same mass and the same charge, they also all have the same spin.

Since all spins come with units of \hbar, we generally drop that unit when we talk about spin. So, some particles have a spin of 0, some have a spin of 1/2, some have a spin of 1, and so on. And there turns out to be a pretty big difference between particles that have a integer spins (0, 1, 2, etc.) and those that have half-integer spins (1/2, 3/2, 5/2, etc.). We call the former bosons and the latter fermions.

At first glance, this might seem a somewhat esoteric distinction. Why should a particle’s angular momentum be of such fundamental importance, and why should, say, a particle with spin 0 have more in common with one with spin 1 than one with spin 1/2? There’s no easy non-mathematical way to see why this should be so, so I’ll just give you the upshot and you’ll have to take it on faith. It turns out that particles with half-integer spins (fermions) form different statistical distributions from those with integer spins (bosons). In particular, no two fermions can occupy the same quantum state. If you’ve studied any chemistry, you’ve run into one manifestation of this in the Pauli Exclusion Principle – this is why you can only have so many electrons in a given orbital. The electrons in an atom all “want” to be in the lowest-energy state they can, but if those low energy states are already filled, any further electrons have to occupy higher energy states. Bosons, on the other hand, can occupy the same state, and even tend to be in the same state.1 Among other things, this is what makes a laser possible – a laser is essentially a lot of photons that are all in the same state.

This somewhat subtle difference ends up having tremendous implications for the way particles behave. The fundamental fermions tend to be what we think of as “matter”, and the bosons largely act as “force carriers” that mediate interactions between those fermions. So, for example, in QED, we have the electron (a fermion) and the photon (a boson), and the usual way to think about it is that the electrons are the “stuff” and the photons are sort of “messenger particles” that carry the electromagnetic force.

Symmetries

A concept that comes up again and again in physics is that of symmetry or invariance. There is way, way more to this topic than I will even touch on here, but I thought it would be worthwhile to introduce it, if for no other reason than to be able to talk about some surprising asymmetries that come up in quantum field theories.

Loosely speaking, a symmetry is something you can change in a physical system (a “transformation”, in physics lingo) without affecting the way it behaves. An easy example is translational symmetry. Imagine a universe identical to this one, except that literally everything is shifted left by five centimeters – obviously, that universe will behave exactly like ours. The laws of physics don’t care about absolute position, only about the relative positions of things, and a universe where everything is shifted five centimeters in the same direction would be empirically indistinguishable from our universe.2 In physics-speak, we’d say that the laws of physics are invariant under a translation. Mathematically, this means that if you shifted the coordinate system of your equations (e.g. replace every x coordinate with x + 5), all the outputs you get from your equations will be the same, except with that coordinate shift.

There are three transformations that were for a long time assumed to be symmetries: parity, charge conjugation, and time reversal. A parity transformation amounts to flipping everything along one axis, like a mirror image. In a parity-flipped universe, for instance, I would be left-handed instead of right-handed, English would be printed from right to left, and the Earth would rotate clockwise, instead of counterclockwise, when viewed from the North Pole. It certainly seems, intuitively, like this ought to be a symmetry of the laws of physics; in this mirror universe3 everything would behave the same, except backwards. In other words, if you took our universe, flipped its parity, allowed it to evolve for some amount of time, and then flipped it back, you’d expect the result to be exactly the same as in our non-flipped universe, after the same amount of time.

Parity
Parity symmetry

It was, then, something of a shock to physicists when they discovered in the 1950s that parity is not a symmetry of the laws of physics. The Quantum Field Theory that describes the weak interaction (which we’ll talk about in more detail later) actually cares about the “handedness” of particles. To put this concretely, take an electron. I’ve told you that all electrons have a spin of 1/2. That spin could be oriented in any direction, though. So imagine that you’re looking at an electron and, relative to you, its spin is clockwise. When I say that parity is not a symmetry of the universe, I mean that the laws of physics apply differently to that clockwise-spinning electron than they would if it were instead spinning counterclockwise. So, that’s pretty weird. This is called parity-violation, or P-violation.

Then there’s charge conjugation. This transformation means taking changing all positive charges to negative, and vice versa4. So, again, if this were a symmetry, then if you flipped the charge of every particle in the universe, you’d expect the universe to behave exactly the same, except of course that the charges of everything will the opposite of those in our universe.

Charge
Charge conjugation symmetry

But, wouldn’t you know it, that also turns out to be false. The laws of physics (and, again, it’s the weak interaction that is the culprit here) apply differently to a system of particles than they do to an otherwise identical but oppositely-charged system.

But here’s an idea: What if we combine these two transformations, charge (C) and parity (P) into a charge-parity (CP) transformation? Soon after discovering C-violation and P-violation, physicists noticed that if you do perform both of these transformations, the deviations that you get from each one seem to cancel each other out. If that were to hold true, then we could declare at least that CP is a symmetry of the universe; the laws of physics would, so to speak, treat a positively-charged, left-handed particle the same as a negatively-charged, right-handed particle.

Charge parity
Charge-Parity symmetry

This seemed to work for a while, but then in the ’60s, more sensitive experiments started to find clear evidence of CP-violation. In other words, although the combined CP transformation is closer to a symmetry than either C or P by itself, it still isn’t one. The laws of physics still treat the universe differently from an otherwise identical but mirror-flipped, opposite-charge universe.

What about the time reversal symmetry I mentioned? Well, it was long assumed that the laws of physics were time-symmetric. Let’s picture this in terms of classical physics first, to see what I mean. Suppose you have a bunch of billiard balls colliding and bouncing around on a pool table. Let’s say that you know the position and velocity of every ball at some time that we’ll arbitrarily call the start time. Now, let them bounce around and collide until some other time, which we’ll call the end time – now they all have some different positions and velocities. OK, now let’s imagine that we’ve got another pool table, and on table 2, it just so happens that at the start time, all the balls have the exact same positions and speeds that they had at the end time on table 1, except that the balls are all moving in the opposite direction. What can we say about where the balls will be on that second table at the end time? 5 Well, in classical physics, we’d find that on our table 2, the balls all end up in exactly the same positions as the ones where they started on table 1. In other words, the laws of (classical) physics don’t care which direction in time is “forward” and which is “backward”. If you have some billiard balls colliding and you then “reverse time” (in other words, reverse all their velocities), the laws of physics will, so to speak, play the movie in reverse for you.6

Time
Time reversal symmetry

But, as I’m sure you’ve guessed that I was going to say, in particle physics time reversal turns out not to be a symmetry either. The movie does not play the same backward as it does forward. The deviations are small, but they are experimentally very well established.

However! Imagine we perform a parity-flip, a charge conjugation, and a time-reversal. This would be a CPT transformation. And this (finally) looks like it is a symmetry of the universe. Those deviations that you get with a CP-transformation look to exactly cancel out with the deviations you get from a T-transformation. A hypothetical universe where everything is backwards, opposite charged, and time-reversed would otherwise behave exactly like ours.

Charge parity time
Charge-Parity-Time symmetry

Here’s another way to think of this. Since the deviations due to a CP-transformation cancel out with those due to a T-transformation, a movie of a world in which charge and parity are both flipped would look exactly like a movie of the unflipped world played backwards. And this is why you’ll sometimes here people say, rather misleadingly, I think, that an antiparticle is like a normal particle moving backward in time. Which brings us to . . .

Antiparticles

One of the more famous surprises to come out of Quantum Field Theory is that every type of particle has a corresponding “antiparticle” with the same mass and spin but opposite charge.

As with a lot of this stuff, to show you how exactly QFT leads to the notion of antiparticles would involve actually doing the math, so you’ll have to take this on faith. Take what I’ve called the “electron field” – our infinite array of springs connected by string. I told you that if you have a single “unit” (i.e. quantum) of oscillation in one of those springs, you can work out from QFT that it will move around in that field just like an electron moving through space – to the point where we can say that an electron moving through space really is a quantum of oscillation in one of these fields. But it also turns out that there are two kinds of oscillations that QFT allows to exist in this field, a positive one and a negative one. To get a very loose image of what that means, you might imagine one spring bouncing up and down, and another bouncing, so to speak, down and up. In other words, this second spring is oscillating at the same frequency as the first one (i.e. it takes the same amount of time to complete one oscillation), but in the opposite direction – every time the first spring is at the top of its oscillation, the second spring is at the bottom. Both of these oscillations can move around from spring to spring – both, in other words, are particles of the field. But what would happen if the two of them ran into each other? If both of these oscillations moved to the same spring, they’d cancel each other out.

Now, I’m really straining the springs analogy here, because there’s a lot of stuff going on mathematically that can’t be captured in the simple picture of bouncing springs. But, granting that the analogy is flawed, that “upside-down” oscillation is what we call an antiparticle. In the case of the electron field, the right-side-up oscillation is an electron, and the upside-down one is an anti-electron, also known as a positron.

This “flipping” of the oscillation that we’ve imagine turns out, mathematically, to amount to a CP-transformation. In other words, the antiparticle has the opposite charge and “handedness” of the original particle (don’t worry too much about what “handedness” means; it’ll only really be relevant when we talk about the weak interaction, which we’ll come to later). So the antiparticle is a kind of negative version of its corresponding particle. Because the electron has a negative charge, the positron has an equal positive charge.7

Antiparticles can annihilate with their corresponding particle. You can think of this sort of as the two opposite oscillations of the field cancelling each other out, though as I said, the analogy here is far from perfect. Energy has to be conserved, though, so the energy of the annihilating particle/antiparticle pair has to go somewhere – it has to be transferred to another field. In other words, this “annihilation” is really more just a conversion into some other kind of particle (typically a photon).

Every charged particle has an antiparticle with opposite charge. But what about neutral particles? Well, it turns out that some neutral particles have distinct antiparticles, while in other cases, a neutral particle might, in a sense, be “its own antiparticle”. To get into the reasons for this would require diving way more deeply into the formalism of QFT than we can do here, so you’ll have to take my word for it that, for instance, an antiphoton is a photon.

Because antiparticles behave much the same as their counterparts (with the exception of deviations due to CP-violation), there’s no reason one couldn’t have antimatter atoms, molecules, and larger structures.  Instead of a world made of protons, neutrons, and electrons, one could imagine one made of antiprotons, antineutrons, and positrons. And yet, the universe we see is filled only with matter, not antimatter. We can produce antiparticles in particle colliders, and there are natural processes like radioactive decay that produce them, but since there is so much normal matter around, these antiparticles will typically exist for only a very short time finding a partner to annihilate with.

In fact, the prevalence of normal matter, as opposed to antimatter, in the universe is something of a puzzle. If CP were a symmetry of the universe – that is, if the laws of physics treated particles and antiparticles the same way – then we’d expect equal amounts of matter and antimatter to have been produced in the Big Bang. And if that had happened, then pretty much all the matter and antimatter should have long ago annihilated with each other, and the universe should be empty of pretty much anything but photons. That’s obviously not the case; somehow significantly more matter than antimatter was produced in the first seconds of the universe. Now, I did tell you that CP turns out not to be an exact symmetry of the universe; but so far, we’ve observed only very small deviations from that symmetry, not nearly large enough to account for the large imbalance between matter and antimatter that we have. This is one of the biggest mysteries in physics right now, though there is a somewhat promising lead that I hope to discuss when we talk about neutrinos later.

This article turned out to be a bit of a hodgepodge, but as I mentioned above, I’ve decided to break off the section on Feynman diagrams into its own article, and I think that one may help to put some of this stuff into a clearer context. That part should appear next Tuesday. As always, if you’ve made it this far, thanks for reading, and if you have any questions, I’d be happy to (try to) answer them in the comments!