When I was in school I remember the math class made sure, for no apparent reason, that we knew what "COMMUTATIVE" meant. A+B=B+A That seems pretty banal. Of course A-B isn't B-A, but that difference didn't seem worth making up a new word for.
That was the fault of the textbooks. Easy exercise. Take 2 dollar bills and spread them out beside each other so George is looking at you. You will do the same 2 rotations on each one, but in a different order. Rotate the first clockwise by 90 degrees. This rotates around a vertical axis. Then pick up the far end and bring it towards you, turning the bill over in the process. This rotates around an axis that goes left-to-right.
Now take the other bill. Pick up the far end and bring it towards you, turning the bill over in the process. Then rotate it clockwise by 90 degrees.
A lot of things in life are more like those rotations than like grade-school addition--you have to get the order right. You can't put on shoes and socks and expect the same result as if you put on socks and shoes.
Most operations you can do to an object don't commute.
Suppose we have a function that describes the state of a brick. (Humor me) You can look at that function and say: "The brick is in this spot." Now suppose we have a "do-something-to-the-function" thing-a-mabob that we call an operator. The "operator" that I am thinking of right now changes the function so that the position of the brick is "over there." In the real world we call that "shifting the brick." The mathematical description involves an "operator."
That seems a little odd at first glance, but it is very handy for describing elementary particles. (You know that you can't tell one electron from another one, right? That has profound implications for descriptions.)
The model reflects the real world: operators do not always "commute." The order matters.
Some kinds of operators turn the simple function into a more complicated one: sort of like an operator that turns a function about the state of a brick into a function about the state of two half-bricks. I leave the physical interpretation of this as an exercise for the reader.
You can think of a measurement as a kind of operator that multiplies the function by some number characteristic of the thing it is describing--like the mass. Or the color. (Humor me.)
Quantum mechanics was designed to account for the observed fact that some measurements interfere with each other. You can't measure both the x-position and the x-momentum of a particle perfectly simultaneously. (Heisenberg's Uncertainty Principle) Quantum mechanics reflects this: the operators that measure those things do not commute. The descriptions match--I won't go into the math.(*). Quantum mechanics also relies on the observation that particles are waves.
Different forces involve different operators. So does identification; operators which measure "What kind of neutrino is this?" for example.
We've 3 kinds of free elementary charged particles: electrons, muons, and taus. (4 if you count the W-boson, which I'm not going to worry about here.) The only things that seem to be fundamentally different between them are their masses and their "identity." "Identity" is conserved, btw. A muon and antimuon together add up to 0 muon-ness. You may ask what happened to protons and the rest of the zoo--they're composed of quarks, which can't move around freely like electrons.
Neutrinos seem to match 1:1 with those 3 particle types: electron neutrino, muon neutrino, tau neutrino. They partake of the same "identity" as the 3 also. An electron and an anti-electron-neutrino together have 0 electron-ness.
Here's the weird bit. You can't measure the mass and the "electron-ness" of a neutrino perfectly at the same time. With an electron, you can. With a neutrino, no.
The neutrino is created from some interaction, perhaps radioactive decay of a nucleus. That would give us an unambiguous electron neutrino. But that electron neutrino is a combination of three possible different mass states (alternatively, each mass state is a combination of electron, muon, and tau neutrino), and different masses with the same energy don't move at quite the same rates.
So what started out with unambiguous "electron-ness" will, over time, wind up having "muon-ness" and "tau-ness" too. The operators for measuring the mass don't commute with the operators for measuring the identity.
This results in what's called neutrino oscillation.
The title of The 'True' Neutrino is therefore a bit of a misnomer. Either way of looking at the neutrinos is perfectly fine--in its place. The neutrinos we interact with interact using an identity, and so will always be a blend of the mass states. But we'd really like to know what those masses are. They're small... And you'll notice that I lied up above when I said that "identity is conserved." Because the neutrinos can change, "lepton" number can change too. Just not during an interaction.
The report linked above explains a recent IceCube result in which it seems as though those unobserved mass states have their masses lining up in the expected order--though it's too soon to be sure. The community has had bounds on their size of their mass differences, but that doesn't tell you which is bigger.
(*) Lubos Motl is quite a bit smarter than I, but he keeps mistaking the map for the territory and describing quantum mechanics operators as fundamental. The observations are fundamental and quantum mechanics is the simplest description.
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