What is Weak isospin in Laymans terms? I apologise if this is a bit of a basic question but I have read the Wikipedia page and all it says is that it is quantum number relating to weak interaction. I'm just looking for a basic, intuitive explanation of what it is and how it fits into QFT.


Actually, in order to understand the Isospin, the history of this notion helps. The isospin was first considered for the strong interaction, the interaction that governs the nucleus. In 1932 Heisenberg considered the proton and neutron as the same particle "nucleon" being in 2 different states. In 1937 Wigner coined the word "Isospin" for it since the nucleon seems to be in 2 different states as the spin of an electron. As a proton and neutron are indeed very similar this seems still rather intuitive, although the charge of the proton makes it nevertheless rather different from the neutron. As well as the spin the isospin is conserved in nuclear reactions, only flips (or doesn't flip at all) between between one isospin state $I_z=1/2$ to $I_z=-1/2$ and back. The pair (neutron, proton) is grouped together in a isospin multiplett with isospin $I=1/2$ (similar to an electron with spin $s=1/2$ and its 2 spin states $s_z=\pm 1/2$). Later pions were discovered. There 3 of them, and one could group them into another isospin multiplett with $I=1$ and 3 individual isospin states $I_z=(1,0,-1)$ for $(\pi^+,\pi^0, \pi^-)$. That made sense as it was observed that in nuclear reactions where nucleons and pions interacted with each other, the isospin was conserved. It had well the function of a conserved quantum number.

Weak interaction (WI): Its understanding was for several decades a mystery. Things changed when theory of massless particles was considered. The actors there are massless electrons and neutrinos $\nu_L$ (higher generations in the standard model will be neglected). The weird thing about WI is that left-handed electrons $e_L$ interact differently than right-handed electrons $e_R$. So in fact the left-handed electron and left-handed neutrino were grouped together in a multiplett of "weak isospin $T=1/2$". This makes sense: you can shoot an left-handed electron on a protron, the electron can emit a $W^-$ boson and transform in a left-handed neutrino. The $W^-$ boson hits the proton and converts it into a neutron for instance. The weak isospin of the $(\nu_L,e_L)$ is conserved, only the z-component can flip. And the (p,n) turns not only being a strong isospin multiplet, but also a weak isospin multiplett (In fact, once the quarks were discovered, the isospin (strong and weak) was attributed to the corresponding quarks). What happens with the right-handed electron ? It was identified as weak isospin singlet $e_R$ $T=0,\,\, T_z=0$. Actually, one could think that an $e_L$ is very different from a $\nu_L$, but they are a priori not more different as a protron is different from a neutron. The proton has charge, the neutron not, the electron has charge, the neutrino not. Finally quark pairs $(u,d)$ can be grouped into a weak isospin multiplett, and similar grouping can be done in the higher generation of the standard model. As in the spin space rotations can even be considered in the isospin space. And these (rather abstract) rotations can be considered as new symmetries similar to the well-known 3d rotational symmetry.

I could also explain about interaction of the weak isospin with the weak hypercharge etc. etc. but that will be definitively too long.

Anyway, most of the stuff can be found certainly in more technical terms on wikipedia.org.


There is no "intuitive" picture of what weak isospin is because it is the charge associated to a force - the weak force - that is invisible at classical scales, and our (untrained) intuition is almost exclusively classical. However, one might draw an analogy to "ordinary" spin:

Spin also is pretty unintuitive - it's "angular momentum-like", but it does not refer to anything spinning in the sense of classical motion, see e.g. this question. However, if you accept that spin is simply something that defines the intrinsic behaviour of a particle under the group of rotations, then weak isospin is exactly the same, only for a group of "rotations" $\mathrm{SU}(2)$ that has nothing to do with spatial rotations - it is an internal symmetry group that purely exists in the mathematics of the model and does not correspond to any transformation in spacetime.

However, this does not stop it from being physically significant. Statements like Noether's theorem or indeed the whole of mechanics - be it classical or quantum - do not really care about whether the variables and transformations in question correspond directly to spacetime or not. (If you feel uneasy about this, just recall that electromagnetic charge is no different - it's the charge belonging to an internal $\mathrm{U}(1)$ symmetry.) So weak isospin is really nothing but an additional type of charge, relating to a different kind of "force", about which you can't really think like electromagnetism because it disappears from the model in the classical, macroscopic limit.


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