And now another installment of “Symmetry in Physics.” Recall that in part 1 we introduced the idea of symmetry and mentioned the symmetries of spacetime, such as rotations or translations. These symmetries are all ‘continuous,’ in the sense that you can rotate/translate by any arbitrary amount. Now we’ll introduce some of the *discrete* symmetries of spacetime, meaning that the symmetry only acts by a certain amount. In particular, we’ll focus on symmetries where one flips the sign (‘swaps the polarity’) of an object that can take two values. It turns out that this will be intimately linked to our notion of antimatter.

The spacetime symmetries we discussed in the previous post can be expanded to include three discrete symmetries: parity, charge conjugation, and time-reversal. It turns out (rather surprisingly) that physics chooses *not *to obey these symmetries, and this act of rebellion allowed the universe to develop interesting things like galaxies and life.

**Parity
**

Parity is the symmetry where we reverse all of our space directions. For example, if we draw a coordinate system (*x,y,z*) in space, a parity transformation gives us a new coordinate system *(x’,y’,z’)* drawn below.

What’s the difference between these two coordinates? The first coordinate system obeys the ‘**right hand rule**.’ If you point your right hand in the *x* direction and curl your fingers towards the *y* direction, then your thumb will point in the *z* direction. The parity-transformed coordinate system, on the other hand, does *not* obey this property. It is, in fact, a **left handed** coordinate system. Thus a parity transform essentially swaps left and right.

Does this remind you of anything? One of my favorite puzzles as a child was the question of why a mirror reverses left and right but not up and down. The answer is that the mirror enacts a parity transformation. It reverses the forward-backward direction while maintaining the other two axes. For homework you can convince yourself that this is equivalent to our definition of ‘parity’ above. (For more discussion see this Richard Feynman video clip.)

Parity is a useful quantity when describing spinning particles: the parity transform of a particle spinning in one direction is the particle spinning in the opposite direction.

We might believe that nature obeys parity symmetry, but we’ll see that this is actually not true. Biologists might have already guessed this since the amino acids which make up proteins in cells are all left-handed.

(In fact, when the eminent theorist Wolfgang Pauli heard that Chien-Shiung Wu constructing an experiment to test whether the weak force obeys parity symmetry he scoffed that it was obvious that the answer had to be ‘yes.’ The entire community was shocked to find out that indeed, parity is *not *a good symmetry of nature!)

**Charge conjugation
**

Another symmetry that we might expect nature to respect is **charge conjugation**, i.e. swapping the sign of all charges. We are all familiar with electric charge: electrons (by some poor historical choice) are negatively charged and protons are positively charged by the same amount. If we suddenly swapped the charge of every proton and electron in the universe, we would expect nothing to change since the force between them would be the same.

More generally, charge conjugation would swap the sign of every charged particle. Quarks with electric charge +2/3 would then have charge -2/3 and those with charge -1/3 would then have charge +1/3, and so forth. In fact, charge conjugation applies to *all* charges, including more exotic things like “color charge” in quantum chromodynamics (the theory of the strong nuclear force). Thus a ‘red’ up quark of charge +2/3 is “charge conjugated” into an anti-red charge-conjugate up quark of charge -2/3.

The idea is that as long as all of our “charges” swap sign, all of the forces between them should be the same and nature should look pretty much the same as it would without the charge conjugation. It turns out that this is also not quite true.

**CP, Antimatter, and CP-violation
**

Here’s a tricky question: in the previous section we talked about a “charge-conjugate electron” of charge +1 and a “charge-conjugate up-quark” of color anti-red and charge -2/3. Aren’t these things exactly what we mean by **antimatter**, i.e. matter that is exactly the same, but with the opposite charge?

The answer is surprisingly ‘no,’ and the reason is very subtle.

First of all, we should state that all matter particles (called **fermions**) spin. This is a very weird idea since we think of them as point particles, so it’s not really clear how we’re supposed to picture a point spinning. What I mean by this is that a matter particle carries angular momentum, which we can effectively think of as the particle spinning around on the axis defined by its direction of motion.

Now we can be precise what we mean by antimatter: for a given particle of some spin and some charge, the **antiparticle** has the **opposite spin** and **opposite** **charge**. This is a combination of charge conjugation and parity which we call **CP**. Thus if we have a left-handed electron with charge -1, its antiparticle (called a **positron**) is right-handed with charge +1.

Let’s repeat the lesson one more time to over-emphasize the point: matter and antimatter are related by CP, i.e. swap the sign of the charge and turn left-handed particles into right-handed particles. This leads to an esoteric piece of advice: consider the unlikely situation where you meet a copy of yourself and you’re not sure whether the copy is made of matter or antimatter. If the copy reaches out to shake your hand with his/her left hand (and you are right-handed), *then don’t touch!*

We can ask ourselves if CP is a good symmetry of nature, even if we know that C and P separately are not. You can guess the answer: *no! *This is obvious because all the matter we observe in the universe appears to be matter (otherwise we’d see lots of gamma rays from the annihilation of galactic clumps of antimatter with matter). It seems like all of the antimatter in the universe has annihilated with the matter, leaving behind an excess of matter which now forms everything we see. Clearly matter and antimatter are not symmetric, and this asymmetry allowed matter to be left over rather than having an exact cancellation into ‘uninteresting stuff’ like photons.

Why is CP not conserved? We don’t know! This is one of the outstanding questions in particle physics. While we do know how to build theories that violate CP (this was part of the 2008 Nobel Prize in physics), it turns out to be rather hard to get just the right about of CP violation to account for how much matter is leftover in the universe.

**Extra Credit: Chiral Matter and Antimatter
**

Here’s something that you might have become confused about: why is the “parity” part of CP symmetry so important? If I have left-handed electrons with right-handed antiparticles, surely I must also have right-handed electrons with right-handed antiparticles as well. Thus shouldn’t we have a bunch of left/right handed electrons and a bunch of left/right handed positrons?

The answer is another surprising *no!* This is now a somewhat advanced topic. There’s no fancy math involved, but one does have to sit and think a little bit if this is the first time learning it. If you get lost, feel free to skip this section for a later read, but I hope you come back to this because it’s one of the very elegant pieces of the Standard Model.

Let me start by making three absurd proposals that, nonetheless, are actually the basis for the Standard Model up to some post-facto modifications.

- All electrons are left-handed.
- All electrons start out with no mass.
- Thanks to the Higgs boson, we get right-handed, massive electrons.

First I claim that **all electrons are left-handed**. Your job is to say, “*That can’t be true! If you give me a left-handed electron, I can make it right-handed*.” How would you do this? You could try flipping the electron over so that its angular momentum goes in the opposite direction. This, however, is still a left-handed electron, for the same reason that turning your left hand over doesn’t turn it into a right hand. After a bit of thought, you can say that you could *boost into a frame* where the electron is moving in the opposite direction. By reversing the direction of motion (which defined the axis of spin), you flip the particle from being left-handed to right-handed.

Make sure you understand why. Suppose you’re driving on the highway next to one of those cement-mixing trucks with a rotating barrel. Suppose the barrel is rotating “right-handedly” relative to the direction of the truck. If you’re driving slower than the truck, then you agree that this motion is right-handed. If you speed up so that you’re moving faster than the truck, then you see something different. Relative to you the truck is moving *backwards* so that the rotation is now left-handed. [Homework: think about how this relates to the question of why the mirror changes left-and-right but not up-and-down.]

Ok, so we think we can step on the accelerator and just zoom past an electron to observe it spinning in a right-handed way instead of a left-handed way. Since (as we showed in the previous post) physics is symmetric under change of frame, this right-handed electron must be a valid object in nature. Wrong! Why? This argument only holds if we can move faster than the electron. We know that there’s an ultimate speed limit, however: the speed of light. Thus the argument fails if the electron is traveling at the speed of light.

“*Aha!*” you say, “*But the only things that travel at the speed of light are massless, like photons! Electrons are not massless.*” Now I refer to my second absurd proposal: **electrons start out massless**. If this is true, then it is at least plausible that all electrons could be left-handed because they all travel at the speed of light and we could never boost into a frame where we make them right-handed.

At this point I’m going to add another piece to the tapestry: there are also positrons. All positrons are left-handed, massless and, for the moment, have nothing to do with the electrons. Just like the electrons, you can never boost to a frame where the positrons are made right-handed.

If I’m supposed to get you to believe all this, then I need to also explain why we appear to observe electrons which *do *have mass and which *do *come in right-handed varieties. The reason for all this turns out to be (surprise) the Higgs boson. By now you’ve heard from all sorts of physics outreach sites that the Higgs boson gives particles their mass. What you may not have heard as much is that the Higgs also combines left-handed electrons with right-handed positrons to form these massive left- or right-handed electrons and positrons.

Let’s call the left-handed electrons of charge -1 **e** and the left-handed positrons** **of charge +1** E.** We can also form the CP partner of the left-handed electron, **e***, where we’ve written a * to mean “CP partner.” Thus **e*** is a *right-*handed object with charge +1 and **E***** **is a *right-*handed object with charge -1. Thus we have *four* distinct particles.

What the Higgs boson does is mix together the **e ^{ }**with the

**E***. Recall that both

**e**and

**E***are charge -1, but they now have different spin. Without going into the details of how this mixing occurs, it constructs a

*massive*“electron” of charge -1 that can be right- or left-handed (e.g. one can boost into a frame where it prefers to be one or the other handedness). This mixing is quantum mechanical, so the “electron” actually flickers between being

**e**(which is charge -1, left-handed) and

**E***(which is charge -1, right-handed). This is the object that we call the physical electron with the measured electron mass. The combination of

**e***and

**E**gives the physical positron. The process by which the Higgs mixes these pairs of definite-spin particles into mixed-spin “physical” particles is called

**electroweak symmetry breaking**, which we’ll touch more on later in this series of posts.

We have a good understanding of how electroweak symmetry breaking works in the Standard Model, but it is still an open question *why* nature should undergo this symmetry breaking.

The rabbit hole goes even deeper, and we’ll at least be able to touch on how deep it gets when we start talking about gauge symmetries, but for the moment let me toss out one more complication: even though the construction above seems to treat **e** and **E** rather symmetrically, this too is an oversimplification. In fact, the left-handed charge -1 **e** is also related via a symmetry to the left-handed charge 0 neutrino, while the **E** has no relation to the right-handed charge 0 neutrino. The fact that the left-handed and right-handed particles manifestly behave differently is what is meant when physicists say that the Standard Model is a **chiral theory**.

For “culture,” I’ll note that this chiral structure (treating left-handed things differently from right-handed) is a big road block in building models for new physics. For example, there turns out to be no such thing as a left- or right-handed matter particle in five dimensions (4 space + 1 time) or if your theory exhibits too much “supersymmetry” (which we’ll get to in a later post).

**Time reversal, CPT
**

There’s one more discrete symmetry which is broken by the Standard Model: time-reversal. Consider an electron (left-handed, charge -1) moving away from you. This is a negative charge moving away from you with a left-handed spin. Now consider what happens if we run time backwards (we can’t actually *do* this, of course, but imagine watching a film in reverse). Then we see a negative charge moving toward you with left-handed spin (do you see why it’s still left-handed?).

A negative charge moving toward you is the same thing as a positive charge moving away. So we can call this a positive charge moving away from you with *right*-handed spin (do you see why it’s now right-handed?). Hence we can see that time-reversal is something like CP. In fact, this led many people to think about antiparticles as regular particles moving backwards in time. This can lead to misleading and often plainly incorrect statements like “there is only one electron and it zips backward and forward in time.” This may be poetic, but I don’t find it particularly helpful in thinking about physics.

Just like CP, we can motivate that time-reversal is not a symmetry respected by nature because everything would look *weird* if we reversed the direction of time. Rain would fall upwards, shattered pieces of a broken glass would spontaneously jump together with amazing precision, and the movie *Jaws* becomes much friendlier. In fancy words, entropy would not increase. Clearly there’s something weird and special about time relative to the other dimensions.

Here’s something that’s really surprising the combination CPT (charge conjugation, parity, *and* time-reversal) **is** a good symmetry of nature. Hence the violation of CP is in some sense “equivalent” to the violation of time-reversal symmetry. This is a rather deep theorem that is unfortunately beyond my abilities to explain accessibly.

**Epilogue 2
**

This was a bit of a whirlwind tour of one good (albeit mysterious) discrete symmetry and several broken ones. Along the way we defined the symmetry CP which connects particles and antiparticles. Even though CP symmetry is not a symmetry of nature (since particles and antiparticles must behave differently otherwise how did we end up in a universe with virtually no antimatter?), it sheds light on what we really mean by particle and antiparticle. We’ve learned a bit about spin and chirality, and even mentioned the Higgs boson.

Perhaps the one-line take home message that you should remember for next time is:

Symmetry is interesting. Broken symmetry can be even more interesting.

– Flip