There are two general categories of symmetries in particle physics, internal symmetries and spacetime symmetries. I'm only going to discuss spacetime symmetries here.

Within the category of spacetime symmetries there are continuous symmetries like rotational symmetry (responsible for the conservation of angular momentum), translational symmetry (responsible for regular conservation of momentum), time translation (responsible for conservation of energy), and Lorentz boosts (responsible for Einstein's theory of relativity).

But then there is also another kind of spacetime symmetry--discrete symmetries. There are 2 important discrete spacetime symmetries and they are pretty simple to explain. The first is called time reversal symmetry, usually denoted by the symbol T. As an operator, T represents the operation of flipping the direction of time from forwards to backwards--basically, hitting the rewind button. Parts of physics are symmetric with respect to T and other parts are not. The other important one is P (parity), which flips space instead of time--it's basically what you see when you look in the mirror and left and right are reversed, everything is backwards.

Here is a video of me doing a cartwheel, an every day process which by itself would appear to break both P and T. The animation shows the forward-in-time process first which is a right-handed cartwheel, followed by the time reverse which then looks like a left-handed cartwheel. Because applying T in this case accomplishes exactly the same thing as P (if you ignore the background), this means that this process breaks both P symmetry and T symmetry, but it preserves the combination of the 2, PT:

And now for the front handspring. Unlike the cartwheel, this process respects P symmetry. If you flip left and right, it still looks the same. However, if you time reverse it, it looks like a back handspring instead of a front handspring! So the handspring respects P symmetry but not T symmetry.

Of the 4 fundamental forces of nature--gravity, electromagnetism, the strong force, and the weak force--the first 3 respect time-reversal symmetry while the fourth, the weak force, does not. Because the other 3 are symmetric, it was assumed for a long time (until the 1960's) that all laws of physics had to be symmetric under T. Only in 1964 did the first indirect evidence that the weak force does not respect T symmetry emerge, and more direct proof came in the late 90's and still more interesting examples have piled on within the past decade.

You might think that because there are 3 dimensions, there would be 3 different symmetries like P, where each would reverse a single dimension (say, right and left, up and down, front and back). And that's a good guess, but because you can combine any of these versions of P with a rotation to get any of the others, it doesn't matter how you define it--reversing any single direction in space is the same as reversing either of the other 2, or reversing all 3 at once. The laws of physics are the same no matter how you rotate them in space, because rotation is a continuous symmetry of space(time). This is why when you hold a book up to any mirror it appears as if the writing in it is reversed, as if someone wrote it from right to left. The mirror just reflects light back where it came from, reversing the "into the mirror" and the "out of the mirror" direction, but because intuitively we automatically rotating objects in the left-right direction in our mind (rather than the upside-down direction), we tend to perceive left and right as reversed instead of up and down, or forwards with backwards. All accomplish the same thing--they implement the P operator and flip space backwards, creating a mirror world of backwards images inside the mirror.

The weak force, in addition to violating T, is also the only force which violates P symmetry. Gravity, electromagnetism, and the strong force are all symmetric under P--in other words, if you reverse left and right, gravity still works the same, it just pulls on everything. The fact that the weak force violates parity was discovered in 1954, so for a full decade nearly everyone assumed that it only violated P but not T. Breaking P did not seem like a huge deal, but breaking T was a much bigger surprise.

However, there's one more complication to explain. If the weak force is the only force that violates P, then how can I have a video of an ordinary thing like a cartwheel that also appears to violate P? Does the cartwheel somehow involve the weak force? Couldn't we have guessed that the laws of physics were not symmetric under parity before the 1950's just by noticing that people can do cartwheels?

To answer this, I'll need to say more about the only non symmetry-based rule in physics that I mentioned in the beginning: everything which can happen, will happen. Or as Murray Gell-Mann once said, "everything which is not forbidden is compulsory" (he called this the "totalitarian principle"). What this means is that there are certain symmetries in nature, and to find out what happens in a given experiment, one must sum over all the possible ways in which it could happen, consistent with the symmetries. This is the main difference between classical mechanics and quantum mechanics--instead of just one thing happening at a time, many possibilities happen at once. (And some of those possibilities interfere with each other constructively while others interfere destructively).

So for example, if you look at how a photon (particle of light) travels from point A to point B in space, there are many things that might happen. It might go straight, or it might curve a lot, or it might zig-zag all over the place. But there are also much weirder things it can do, like start at point A, then split into an electron and a positron (the positively charged antimatter version of an electron), each of them wander around for a bit and then come together to annihilate and create a photon again that ends at point B! Once you sum over all of these infinite possibilities, you get the probability for the original photon that starts at A ending up at B.

But here's a question: is an ordinary electron right-handed, left-handed, or is it non-chiral (symmetric between left and right)? The above process I described is an electromagnetic process, so you might guess that the answer is they are non-chiral. But the real answer is "it depends on what you mean by an electron". Before it was discovered that the weak force violates P, it was assumed that electrons weren't right or left handed, since it was obvious that if you reverse left and right you get the same probability of a photon splitting into an electron and a positron and then rejoining. They were called "vector" particles, or "Dirac fermions", or "Dirac spinors". Mathematically, all 3 of those things basically mean a particle that is symmetric between left and right.

But as it turns out, Dirac spinors such as the electron are actually built out of simpler mathematical objects called Weyl spinors. Weyl spinors are chiral in that they are always either left handed or right handed, not both. What happens with the photon during the above process is really that there are two distinct possibilities. One is that the photon splits into a right-handed electron and a left-handed positron, and the other is that it splits into a left-handed electron and a right-handed positron. But because electromagnetism respects P, the probability for each of these to happen is always exactly the same, so whenever you sum over one possibility, the other is there equally as well. So you can never really distinguish the handedness of electrons using electromagnetism--it looks like they are each a single Dirac spinor, rather than two Weyl spinors.

While electromagnetism treats left handed and right handed particles exactly the same, the weak force only acts on left-handed particles and right-handed antiparticles. It is a force that isn't felt at all by right-handed particles or by left-handed antiparticles. So it violates P in a pretty maximal way! Not only is the weak force evidence that the laws of physics are not symmetric under parity, it is also evidence that elementary "vector" particles like the electron are fundamentally built out of simpler "chiral" particles which have a definite handedness. (I should mention if you hadn't guessed already or seen this before, that the word chiral is just a more technically sounding way to say "handedness".)

So finally, now I can answer the big question I mentioned above--about why cartwheels aren't really evidence for parity violation but the weak force is. The reason is that you're only looking at one possible cartwheel I could do. I can do a left-handed cartwheel, but I can also do a right-handed cartwheel. The laws of physics are symmetric in that they don't bias me in favor of doing one or the other, I could do either. (Although as I write this, I realize that it is true that it is easier for most people to do right-handed cartwheels as being right handed is more common. I am not sure what the ultimate explanation is for why more people are right-handed than left-handed, I will have to think about that!) In the quantum analog of this, you would have to sum equally over my left-handed cartwheels and my right-handed cartwheels and you'd get something symmetric as the result, at least if you stuck to gravity, electromagnetism, and the strong force. Breaking it up into one or the other is what makes it look asymmetric.

The only important thing I haven't gotten to here but might in a followup post one day, is charge conjugation symmetry C. It's the action of replacing all particles with antiparticles. I didn't mention it here because it's an internal symmetry rather than a spacetime symmetry, so it's not something that could be demonstrated with gymnastics. But as it turns out, something like the combination of P and T (PT) that is preserved by the cartwheel is always preserved in all of physics. That something is CPT, and there's a theorem called the CPT theorem which shows that it's necessary in order to preserve local Lorentz invariance and hence causality. In other words, if CPT were violated, you really could end up going back in time, killing your grandfather, and thus negating your own existence. As far as we know all evidence suggests that this combined symmetry is never violated.

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