isospin

To start off, if this write-up doesn't make much sense, it's not completely my fault. Isospin is a sufficiently abstract concept that most of the people I've met who study its effects for a living weren't able to explain it to me until I'd had three years of college-level physics. (and I'm still not too sure whether I understand it or have just gotten used to hearing about it.)

Here's a try, though. Neutrons and protons are basically a lot more alike than they are different. Roughly the same mass, similar interaction via the weak nuclear force, etc. The major difference is in charge. So at some point, some genius thought to himself (I would add or herself, but it being early 1900s physics it was most likely a he), "What if they're really the same particle in two different states?" When it turned out that several other families of particles like this existed and that treating them this way led to some neat physics, isospin was born.

Thus, isospin is basically just a totally abstract concept to distinguish between nucleons or other sets of particles in a given family. For nucleons, they decided it would be easiest to treat one state as +1/2 and the other as -1/2. Establishing a convention for which was which, however, was a bit tricky. Usually arguments like this go on between physicists and non-conformist chemists; this time, though, it was a sub-disciplinary battle. The particle physicists all thought that naturally, the positive charge should get the positive isospin. In nuclear physics, however, it was more convenient the other way around -- to get the total isospin of a nucleus, it's necessary to add the isospins of all the neutrons to those of all the protons. Since nuclei usually have more neutrons, the total comes out positive if the neutrons get the +1/2. These days, it seems that the particle physicists won.

The major importance of isospin is that it leads to a new symmetry (or conservation law). For reactions that take place primarily by strong force mechanisms, the total isospin of the stuff pre-reaction has to equal the total isospin of the stuff post-reaction. (added in the ever-popular quantum mechanics style, of course, where 1+2 can equal 3,2,or 1)

Some of the neat stuff going on in nuclear physics these days involves examining what happens when isospin is not conserved. (i.e. weak nuclear force reactions)

(note: I'll gladly update this node should I ever reach a deeper insight into the nature of isospin.)

"Isospin" is short for isotopic spin, which comes from the word isotope. Isotopic spin is a misnomer: by definition isotopes differ by the number of nucleons they contain, but isospin rotations do not change the number of nucleons in a nucleus. Isospin rotations “turn protons into neutrons” or vice versa. The term isobaric (meaning “same number of baryons”) spin replaced isospin for a while, but the shorter name isospin is far more common today.

There are actually two kinds of isospin:

• the strong isospin (usually denoted by a capital I) is conserved in strong interactions
• the weak isospin (T) is conserved in weak interactions

The name isospin implies a relation to the (perhaps more familiar) spin - that is only partly true. Isospin does not have anything to do with angular momentum, however it has the same mathematical properties as spin. That is where the peculiar rules for addition come from: Spin is a vector. That means if we have two spins we want to add, the result depends on their relative orientation - if they point in the same direction, we get T1+T2, if they point in opposite directions we get |T1-T2|. Everything in between is also possible, but keep in mind the quantization rules.

Now in particle physics you say there is a thing called a nucleon, which has a strong isospin with an absolute value of 1/2. Which direction is it pointed? Well, that depends. You take an arbitrary direction (usually the z direction in our x-y-z coordinate system) and look at the component of I along it. Because of quantization this quantum number I₃ (which is a scalar, as opposed to the vector I) can only be -1/2 (if I is pointing 'downwards') or +1/2 (if it's pointing 'upwards'). A nucleon with I₃=-1/2 is called a neutron and a nucleon with I₃=+1/2 is called a proton.

But a nucleon consists of quarks! A proton is a combination of two up quarks and one down quark, denoted as uud, and a neutron is udd. It turns out that on a deeper level we can assign a strong isospin to quarks as well - the u quark gets I₃=+1/2 and the d quark gets I₃=-1/2.

We can do the same for leptons. They do not interact strongly, the weak force does affect them though. The weak force is a bit strange, because it differentiates between left and right (see CP violation, or rather parity violation). So we say that the left-handed electron and the left-handed electron neutrino are a weak isospin doublet, ie T=1/2, the electron has T₃=-1/2 and the neutrino T₃=+1/2. Left-handed by the way means that the spin vector is pointing opposite to the momentum vector. There is no right-handed neutrino (which was indeed a puzzling discovery - but since it appears that neutrinos do have some mass after all, it is no longer strictly true, see helicity conservation), and the right-handed electron gets assigned T=0.

So far so good :) Are you still with me? Now it gets even more confusing. In addition to their strong isospin, quarks have weak isospin too. Because the eigenstates of the strong interaction and the weak interaction aren't quite the same (see Cabibbo rotation and CKM mixing) we have to "invent" a new quark, called d'. But the rest stays the same: u and d' are a weak isospin 1/2 doublet with T₃=+1/2 and T₃=-1/2 respectively.

So what's it good for, you ask? After all, it seems more than a little arbitrary! Well, it lets physicists formulate new conservation laws and we can use those to make predictions about experiments, eg which particles may be created in a certain reaction. So far, the predictions of the isospin formalism have been right, and that's what justifies it.