When you look into your mirror in the morning, the face looking back at you isn’t exactly your own. Your mirror image is flipped: left-handed if you’re right-handed, and right-handed if you’re left-handed. Your body is not symmetric in the mirror: we say it does not respect parity symmetry. Zoom in, and many of the molecules in your body also have a “handedness” to them: biology is not the same when flipped in a mirror.

What about physics? At first, you might expect the laws of physics themselves to respect parity symmetry. Newton’s laws are the same when reflected in a mirror, and so are Maxwell’s. But one part of physics breaks this rule: the weak nuclear force, the force that causes nuclear beta decay. The weak nuclear force interacts differently with “right-handed” and “left-handed” particles (shorthand for particles that spin counterclockwise or clockwise with respect to their motion). This came as a surprise to most physicists, but it was predicted by Tsung-Dao Lee and Chen-Ning Yang and demonstrated in 1956 by Chien-Shiung Wu, known in her day as the “Queen of Nuclear Research”. The world really does look different when flipped in a mirror.

I gave a lecture on the weak force for the pedagogy course I took a few weeks back. One piece of feedback I got was that the topic wasn’t very relatable. People wanted to know why they should care about the handedness of the weak force, they wanted to hear about “real-life” applications. Once scientists learned that the weak force didn’t respect parity, what did that let us do?

Thinking about this, I realized this is actually a pretty tricky story to tell. With enough time and background, I could explain that the “handedness” of the Standard Model is a major constraint on attempts to unify physics, ruling out a lot of the simpler options. That’s hard to fit in a short lecture though, and it still isn’t especially close to “real life”.

Then I realized I don’t need to talk about “real life” to give a “real-life example”. People explaining relativity get away with science fiction scenarios, spaceships on voyages to black holes. The key isn’t to be familiar, just relatable. If I can tell a story (with people in it), then maybe I can make this work.

All I need, then, is a person who cares a lot about the world behind a mirror.

When Alice goes through the looking glass in the novel of that name, she enters a world flipped left-to-right, a world with its parity inverted. Following Alice, we have a natural opportunity to explore such a world. Others have used this to explore parity symmetry in biology: for example, a side-plot in Alan Moore’s League of Extraordinary Gentlemen sees Alice come back flipped, and starve when she can’t process mirror-reversed nutrients. I haven’t seen it explored for physics, though.

In order to make this story work, we have to get Alice to care about the weak nuclear force. The most familiar thing the weak force does is cause beta decay. And the most familiar thing that undergoes beta decay is a banana. Bananas contain radioactive potassium, which can transform to calcium by emitting an electron and an anti-electron-neutrino.

The radioactive potassium from a banana doesn’t stay in the body very long, only a few hours at most. But if Alice was especially paranoid about radioactivity, maybe she would want to avoid eating bananas. (We shouldn’t tell her that other foods contain potassium too.) If so, she might view the looking glass as a golden opportunity, a chance to eat as many bananas as she likes without worrying about radiation.

Does this work?

A first problem: can Alice even eat mirror-reversed bananas? I told you many biological molecules have handedness, which led Alan Moore’s version of Alice to starve. If we assume, unlike Moore, that Alice comes back in her original configuration and survives, we should still ask if she gets any benefit out of the bananas in the looking glass.

Researching this, I found that the main thing that makes bananas taste “banana-ish”, isoamyl acetate, does not have handedness: mirror bananas will still taste like bananas. Fructose, a sugar in bananas, does have handedness however: it isn’t the same when flipped in a mirror. Chatting with a chemist, the impression I got was that this isn’t a total loss: often, flipping a sugar results in another, different sugar. A mirror banana might still taste sweet, but less so. Overall, it may still be worth eating.

The next problem is a tougher one: flipping a potassium atom doesn’t actually make it immune to the weak force. The weak force only interacts with left-handed particles and right-handed antiparticles: in beta decay, it transforms a left-handed down quark to a left-handed up quark, producing a left-handed electron and a right-handed anti-neutrino.

Alice would have been fine if all of the quarks in potassium were left-handed, but they aren’t: an equal amount are right-handed, so the mirror weak force will still act on them, and they will still undergo beta decay. Actually, it’s worse than that: quarks, and massive particles in general, don’t actually have a definite handedness. If you speed up enough to catch up to a quark and pass it, then from your perspective it’s now going in the opposite direction, and its handedness is flipped. The only particles with definite handedness are massless particles: those go at the speed of light, so you can never catch up to them. Another way to think about this is that quarks get their mass from the Higgs field, and this happens because the Higgs lets left- and right-handed quarks interact. What we call the quark’s mass is in some sense just left- and right-handed quarks constantly mixing back and forth.

Alice does have the opportunity to do something interesting here, if she can somehow capture the anti-neutrinos from those bananas. Our world appears to only have left-handed neutrinos and right-handed anti-neutrinos. This seemed reasonable when we thought neutrinos were massless, but now we know neutrinos have a (very small) mass. As a result, the hunt is on for right-handed neutrinos or left-handed anti-neutrinos: if we can measure them, we could fix one of the lingering mysteries of the Standard Model. With this in mind, Alice has the potential to really confuse some particle physicists, giving them some left-handed anti-neutrinos from beyond the looking-glass.

It turns out there’s a problem with even this scheme, though. The problem is a much wider one: the whole story is physically inconsistent.

I’d been acting like Alice can pass back and forth through the mirror, carrying all her particles with her. But what are “her particles”? If she carries a banana through the mirror, you might imagine the quarks in the potassium atoms carry over. But those quarks are constantly exchanging other quarks and gluons, as part of the strong force holding them together. They’re also exchanging photons with electrons via the electromagnetic force, and they’re also exchanging W bosons via beta decay. In quantum field theory, all of this is in some sense happening at once, an infinite sum over all possible exchanges. It doesn’t make sense to just carve out one set of particles and plug them in to different fields somewhere else.

If we actually wanted to describe a mirror like Alice’s looking glass in physics, we’d want to do it consistently. This is similar to how physicists think of time travel: you can’t go back in time and murder your grandparents because your whole path in space-time has to stay consistent. You can only go back and do things you “already did”. We treat space in a similar way to time. A mirror like Alice’s imposes a condition, that fields on one side are equal to their mirror image on the other side. Conditions like these get used in string theory on occasion, and they have broad implications for physics on the whole of space-time, not just near the boundary. The upshot is that a world with a mirror like Alice’s in it would be totally different from a world without the looking glass: the weak force as we know it would not exist.

So unfortunately, I still don’t have a good “real life” story for a class about parity symmetry. It’s fun trying to follow Alice through a parity transformation, but there are a few too many problems for the tale to make any real sense. Feel free to suggest improvements!