robousy said:
If you can expose more of the 'new' isospin idea that would be good.
Perhaps it would be good to remember that the fundamental particles are not actually little indivisible things that cannot be divided. A better description of them is to think of the wave functions one dealt with in quantum mechanics.
With wave functions, you can take the sum of two wave functions and, because the equations are linear, the result is another wave function with properties somehow sort of midway between the other two wave functions. For example:
\psi = \psi_A + \psi_B
where the two wave functions on the right satisfy an operator equation like:
\mathcal{O} \psi_A = A \psi_A and same for B.
The sum of the A and B wave functions is a valid wave function but is not likely to be an eigenfunction for the operator like A and B were. So if you define "particle" as the things that are eigenfunctions for that operator, then the sum is not a "particle".
In the case of the elementary particles, we consider charge, Q, to be one of the operators that define what are the elementary particles. For example, if "e" is the electron, and \psi_e is an electron wave function, we have the operator equation:
\mathcal{Q} \psi_e = -e \psi_e
since the charge of the electron is -e. Similarly,
\mathcal{Q} \psi_\nu = 0
since the charge of the neutrino is zero.
So we classify the elementary particles in a way that makes them eigenfunctions of (electric) charge and mass (and parity or whatever).
But just because we classify elementary particles in this way does not mean that we cannot reclasify them in the same we can reclassify wave functions by taking linear combinations of them. And some of the alternative methods of classifying them would make more sense in certain circumstances.
An example is the Cabibbo angle. The up and down quarks (of the "electron family") are eigenstates of the electric charge operator and are eigenstates of mass, but they are not eigenstates of the "weak charge" (when I was in grad school it was called "neutral charge") operator in the sense that when you change an up quark into a down quark by emitting a W-, you don't actually get a pure down quark. Instead, you have to mix in some of the other families of quarks, the "muon family" and "tau family". W+ and W- interactions for quarks involve changing from a +2/3 to a -1/3 (or -2/3 to a +1/3), so you can fix the weak charges by either mixing the (u,c,t), or by mixing the (d,s,b). Nowadays, it is done by mixing the (down,strange,bottom) and it is called the "CKM" matrix.
As an alternative, when you have a pure down quark and arrange for it to emit a W-, it doesn't become a pure up quark but instead ends up with a mixture of top and charm. Thus you could instead define a CKM type matrix as mixing the (u,t,c).
With the leptons, there is also a mixing between the neutral leptons (neutrinos) as compared to the charged leptons. As with the quarks, the mixing appears when you define the particles according to their masses (and therefore into the families or "flavors"). The electron, muon and tau are the mass eigenstates of the charged leptons. When one of these particles emits a W-, the charged lepton changes to a mixture of neutrinos.
Now with the quarks, the effect of the emission of a W+ or W- is a relatively small probability of a change in the family. Therefore we collect the quarks into pairs, (up,down), etc. But with the neutrinos, the mixtures are more democratic so it's hard to say which of the neutrino eigenstates corresponds to the electron and which to the muon and tau. So the neutrinos are generally defined according to what you get when you take a W+/- out of the corresponding lepton mass eigenstate. That is, we talk about an electron neutrino, a muon neutrino and a tau neutrino. This means that the lepton analog of the quark CKM matrix is defined in sort of reverse, an extra opportunity for confusion. It is my belief that the quark mixing is more pure than the lepton mixing arises naturally from the way they are produced from subparticles but that's another story; in the standard model, these are all fairly arbitrary parameters.
If you want to always define "particle" as things that are eigenstates of electric charge and mass, then you will have the weak interactions mixing particle types. But you could instead define "particle" as things that are eigenstates of weak charge and mass, and that would leave you with electric interactions that mixed particle types.
It is also possible to define particles as the things that are eigenstates of both electric and weak interactions. If you do this, then you will have particles that are of mixed mass eigenstates. I suspect that things will be simplest in this basis. In any of these three cases, it is important to note that the mixing is over corresponding particles in the different families, and that the families differ only in their masses.
Now what does this have to do with weak isospin and weak hypercharge? Weak isospin and hypercharge, together, give electric charge and weak charge. Weak isospin is the SU(2) type symmetry between the two objects that a weak force W+/- converts between. Once you've defined weak isospin and electric charge, the difference between these is also defined and (twice the difference) is called weak hypercharge.
When the electric and weak interactions are combined into an electroweak interaction, the appropriate currents (i.e. moving charges) are weak isospin (which is a vector) and weak hypercharge (a scalar). These two are mixed by the Weinberg angle so that instead of interacting directly with gauge bosons according to weak isospin and weak hypercharge, they instead use weak charge and electric charge. Thus the photon is a mixture of a weak isospin and weak hypercharge interaction.
It's late. I hope I haven't made too many serious mistakes.
Carl
