Mirror, Mirror

A common brain teaser is “why does a mirror switch left and right, but not up and down?”

Of course, this is a trick question. A mirror on the ceiling or floor absolutely switches up and down: what is “up” for you appears “down” for your mirrored self. A mirror less sketchily oriented on the wall reverses the direction towards that wall. But how does this translate into the reversal of left and right?

It turns out that this property of mirrors, the interchange of “handedness” or “chirality” to use the more impressive terminology, is really important in physics. Thinking about mirrors tells us a great deal about how physics has to work; even more so when you add in some unusual results about how the particles in our Universe work. When you take it far enough, the issues of chirality that thinking about mirrors highlights can bring you all the way to the Higgs boson of recent fame, and perhaps even be related to the mysterious dark matter holding the Milky Way, and all other galaxies, together.

So it might be worthwhile to think about mirrors.

First, we should think about what a mirror does. Mirrors cause rays of light that hit the surface to reflect with the component of their direction of motion perpendicular to the surface reversed. That’s a mouthful, but really very simple. If you have a plane mirror (and I’m going to be sticking with flat mirrors here, sorry for you fans of funhouses), and you shine light straight at the surface, the light reflects: the movement of light rays “into” to mirror becomes “away” from the mirror.

What about light hitting a mirror at an angle? Only the movement of light “into” the mirror gets flipped; the “sideways” motion of light remains the same. Again, though the words might make this sound more complicated than it is, I am willing to bet every person reading this has completely internalized the geometry involved; you use this every time you have ever angled any mirror to be able to see around a corner, or at an angle other than directly back in you face.

As an aside, since it’s totally unrelated to the physics I’m going to go into for the rest of this, I will stop here to mention very briefly how real mirrors “work.” After this, I’m going to go nerd out over what an idealized perfect 100% reflecting mirror does, and never deign to return to real mirrors made of glass or silvered metal or still water.

A mirror is any smooth material that has a different “index of refraction” than the surrounding medium. Unless you are reading this under water, or are in fact an alien, that surrounding medium is air. The “index of refraction” is a measure of how fast light travels in a material. It travels more slowly in water, for example, than air. This is a result of the electromagnetic field that is the very definition of light interacting with the electromagnetic fields of the electrons orbiting in the atoms of the new material, causing them to shake in time with the light wave. The combination of the light wave and the induced oscillation of the electrons results in single new wave moving forward that appears to move more slowly than the light ray would without the atoms around.

It turns out though, that this new slower moving wave of light cannot conserve both energy and momentum on its own. The only way to have both equal energy entering the interface between the air and the new material and leaving that interface is if some of it gets reflected back. Some of it can also be absorbed, of course. Mirrors are just materials that have the right atomic properties to solve this conservation problem via reflection more than via absorption. Glass is one such material; water another. The other thing you need for a mirror is no light coming from the other side of the material. This is how “one-way” windows work: make one room dark and the other light, and people in the well-lit room see more light from their reflection than is being transmitted from the darkened other room. 

So, a mirror reverses one direction. What of it?

First let’s be clear about the terminology. I’m going to imagine a mirror, mirror on the wall. So for me, the mirror is reversing the direction I call “towards” the mirror, which my mirror image would call “away” from the mirror for them. “Up” and “down” are not reversed. Neither are the absolute directions one might call “left” and “right,” but for reasons that hopefully will be obvious soon, I’m going to call these absolute directions “east” and “west” (so I guess this mirror is on the southern wall and I’m facing it).

Now, we all know that when you look in the mirror, your evil mirror twin has flipped left and right (thus why I’m going to use “east” and “west” to identify sideways directions). If I lift my right hand, my mirror image lifts their hand on their right side according to me. If I imagine rotating myself around to stand where my mirror image is, I see that they have lifted their left hand.

How did that happen? Handedness, chirality, left and right, whatever you call it, these concepts involve relative directions: the orientation of one object relative to another. Standing on the surface of Earth, concepts like “up,” “down,” “east,” “north” etc, are absolute directions. Each is a vector pointing in some direction. But “left”? Left is defined relative to two directions. This should be intuitive, once you think about it. If I told you I’m standing somewhere (so you know which way is “up”), you can’t tell me which way I would call “left” until you figure out which way I’m facing (which direction is “forward”). This relative orientation issue is the bane of anyone giving directions to a friend over the phone.

With that in mind, it should be clear why your mirror images lifts their left hand when you lift your right. The mirror reverses one direction, and as left and right are defined as one direction relative to another, what is on my right is on my mirror image’s left. 

Importantly, a “left” hand cannot be rotated into a “right” hand. Obviously, they can be mirrored into each other, but just imagine taking one of your hands and twisting it and moving it until it looks identical to the other. You can’t. This is true of any object that has a “handedness” or “chirality.” It is actually the definition of such objects: something that cannot be rotated or translated into its mirror-image. Incidentally, the term for a pair of chiral objects that are mirror images of each other is “enantiomer.”

Now, if you were comparing your left and right gloves, there is a way to get the left to look like the right: pull the glove inside out. Clearly, this is not recommended for your hand. But if you think about this way of making the left and right hands look the same, you see it is actually just mirroring again: you’ve reversed one direction for the glove (the direction pointing from the cuff to the fingers), while keeping the other directions the same. That’s what a mirror does.

Interestingly, while reversing one spatial direction makes it impossible to rotate a chiral image back into itself, reversing two spatial directions creates an image that can be rotated back into the original. You can see this in the following image. Here I’m showing the three spatial directions: x, y, and z, each oriented 90 degrees away from each other, pointing in specific directions. Then I compare with the mirror image. You see that you cannot rotate the mirrored axes back to the original. Like hands or gloves, the axes are chiral. 

Next, I mirror the axes twice. The result isn’t oriented the same way as the original, but with the right rotations, I can turn the final image back into what I started with. 

Which provides a handy way if you actually want to see yourself as everyone else in the world sees you. Usually when you look at yourself in the mirror, you get the left-right inverted version of yourself, your evil twin as it were. If you look at yourself reflected through two mirrors though, you get back an image with the same handedness as the real thing. The best way to set this up is if you have two mirrors where the mirror goes all the way to the edge, like some mirrored bathroom vanities. Then arrange them edge to edge like pages of a book, at 90 degrees, and look directly at the center. That's what everyone else sees when they look at you.

Ok, so that’s a very pedantic description of mirrors. What does this tell us about physics?

Well, you might think that the laws of physics respect mirroring. After all, throwing a ball when facing to the left is the same as throwing when facing to the right. Electric fields, more concretely, don't care about mirroring: a positive charge will move away from other positive charges and towards negative charges, and that remains true in a mirror as well. This means electric (and magnetic, it turns out) forces respect parity, which is just another way to say they move in the mirror-world the same as they do in the real world. If someone showed you a movie of a charge moving through some electric or magnetic field, you wouldn't be able to tell if you were viewing the real thing, or the mirrored image.

However, not all forces respect parity. Subatomic particles like quarks and electrons are called fermions; they have spin-1/2, which means these particle fundamentally carry a spin-ness in addition to their fundamental mass and electric charges. Any spinning particle can be characterized as either left-handed or right-handed. A right-handed particle, when traveling towards you would be seen to be spinning counter-clockwise. A left-handed particle would be seen as spinning clockwise. The reason for the name is this: take your right hand. Point your thumb in the direction the particle is moving. If you can curve your fingers in the direction that the particle is spinning, it is right-handed. If you can't, it is left-handed, as if you take your left hand and point your thumb in the direction of motion,  you will be able to curve your fingers along the direction of rotation.

Now, it turns out that the weak nuclear force, responsible for some kinds of nuclear decay, cares about this spin of subatomic particles. In particular, the W-bosons that mediate the weak force interact only with left-handed subatomic particles, and not with right-handed particles. But a mirror flips left-handed and right-handed spins. So if someone showed you a video of particles interacting, you'd be able to tell whether you were watching the real world, or the mirror world: look for left-handed particles, and see if W-bosons interact with them.

I didn't say it was an easy way, but the laws of physics don't care about our comfort.

Now, it turns out that while W-bosons don't interact with right-handed particles, they do interact with right-handed antiparticles. So while a left-handed quark can feel the W-boson and a right-handed quark can't, the right-handed antiquark interacts with W's and the left-handed antiquark does not. This interaction respect charge-parity (called CP-symmetry). So you wouldn't be able to tell the difference between watching interactions in the real world and watching interactions in an antimatter universe in a mirror.  

As to why the Universe would bother to set up things like this, I have no idea. In fact, as I discuss in some detail in my post on the Higgs, the fact the weak interaction discriminates between world and mirror-world means that the only way they can have mass is due to a complicated theory we call "The Higgs mechanism," resulting in the famed Higgs boson. If the Universe was built not care about parity, fermions like electrons and quarks could have mass without going through all the bother of the Higgs. Why is it this way? I have no idea. Probably important though.

Before moving on, one last note: there are weak interactions that violate CP-symmetry. However, it is necessary that any law of Nature that can be written using quantum field theory, the best description of how things work we have, must respect charge-parity-time symmetry (CPT). That is, there is no observation on particle interactions you can make that can distinguish between particles in our Universe, and the laws of Nature acting on antiparticles, in a mirror, running backwards in time (this does not include entropy increasing, which is a whole nother thing). That may not be too useful in your day-to-day, but it probably is telling us something important.

As a last thing to think about with mirrors, one really basic fact about our Universe is that we're made of particles: quarks and electrons. Not anti-quarks or anti-electrons. In fact, we don't see many of those antimatter versions around. There are some, flying through space, or being emitted from radioactive decay (potassium from bananas emits antimatter electrons: positrons), but all these sources are creating antimatter in small quantities today. There is no original mass of antimatter around, balancing out all the protons and electrons we see. What happened to the antimatter?

That means that there must have been something in the early moments of the Universe that chose to create matter, not antimatter. That requires charge-parity (CP) violation: something in the laws of Nature "knew" the difference between matter and antimatter, and picked one preferentially (and since we're made of the surviving stuff, we'll call that bit that was picked "matter" and view the other set of particles as that dangerous, evocative "antimatter"). I mentioned that there is a slight CP-violation in the weak interaction, but we now know that there isn't sufficient CP violation to explain all the matter we see today. There is some new source of CP-violation that must have been acting in the early Universe, creating slightly more matter than antimatter, in a process we call baryogenesis.

Finally, some people ask how we know that there really is more matter than antimatter? Couldn't half the galaxies we see be antimatter? Well no: though intergalactic space is empty, it isn't empty enough for this to be a viable theory. Somewhere, halfway between a matter-galaxy and the evil antimatter-galaxy, the wisps of matter and antimatter would collide, and we would be able to see that. To avoid this problem, you'd have to separate matter and antimatter on scales larger than the visible Universe, from the first moments after the Big Bang. And to pick matter and move it here and antimatter there would be a CP-violation in the laws of Nature at least as big as that required to give "normal" baryogenesis.

So, thinking about mirrors (combined with some non-trivial experiments in particle physics) ends up telling us that there is new physics out there, needed to explain the really basic fact that we're made from stuff, and not antistuff.