Based on just this primitive analysis, we already know some things about the operation: normal vector quantities like position are inverted under a P transformation, so they have eigenvalues of -1 under P. Other quantities like angular momentum, which is defined as a cross product of two "normal" vectors, are unchanged under parity, because both component vectors change sign. These are called pseudovectors (or axial vectors), and have eigenvalues of +1 under parity transformation. Then, just to make things weirder, there are normal scalars (which are unaffected by parity and thus have eigenvalue +1) and pseudoscalars (like the dot product of a vector and a pseudovector, which is inverted under parity transformation).
Particles can be assigned parities according to their interactions with other particles. This process is somewhat arbitrary, but by convention we choose that quarks have positive parity. Then every strong and electromagnetic interaction conserves parity, which is a multiplicative quantum number. (It turns out that apart from intrinsic parities of particles, the net angular momentum of a multi-particle state also contributes to the state's parity, but we can safely ignore that for now.) This allows us to assign each particle either positive or negative parity.
Feynman described parity symmetry as the indistinguishability of right from left - if we were communicating with some sort of extraterrestrial beings but couldn't actually meet them and parity were conserved, it would be impossible to explain what is right and what is left. Well, luckily for our pen pals, it turns out that strong and electromagnetic interactions preserve parity quite nicely, but weak interactions violate parity in a major way. There are a couple of textbook examples of this:
- Beta decay of cobalt-60: This was the first experimental test of parity conservation in weak interactions. Basically scientists (Wu et al.) polarized the spins of cobalt-60 atoms in a magnetic field. Then they watched the cobalt and measured the direction in which electrons were emitted in beta decay. Astonishingly enough, they observed more electrons emitted in the same direction as the nuclear spin than in the opposite direction. The strange thing about this is that if you look at the experiment in a mirror universe, the nucleus is spinning the opposite way, but the electrons are emitted in the same direction! The mirror universe inherently behaves differently!
- It turns out that all neutrinos are left-handed, and all anti-neutrinos are right-handed. This is a striking violation of parity symmetry: if you look in a mirror, you would see right-handed neutrinos, which don't exist on our side of the mirror! Sheesh!
- The theta-tau puzzle: Once upon a time, particle physicists found two seemingly similar particles, the $\theta^+$ and the $\tau^+$. They were produced in similar reactions, had nearly identical masses, spins, and so on, but their decays were different. The $\theta^+$ decayed into two pions (a $\pi^+$ and a $\pi^0$), while the $\tau^+$ decayed into three (either $\pi^++\pi^++\pi^-$ or $\pi^++\pi^0+\pi^0$). Well, the pions all have negative parity, so by parity conservation, it seems that the $\theta^+$ has a parity of $+1$, while the $\tau^+$ has a parity of $-1$. This baffled physicists for a while, but eventually, when they figured out that parity was violated in weak interactions, they concluded that the $\theta^+$ and $\tau^+$ were actually the same particle, now known as a positive kaon ($K^+$).
Well, that's at least an introduction to parity. I'm still trying to wrap my head around the whole concept, but the bottom line is that parity is squarely not conserved, which led physicists to their next best guess, CP conservation - a topic for another day.
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