# What is the difference between isospin, weak isospin, hypercharge, weak hypercharge and the isospin third component $I_{3}$?

These terms crop up all the time in particle physics but I am confused as to what the difference is. I have read that hypercharge is the third component of isospin. This wikipedia page differentiates between hypercharge and weak hypercharge: https://en.wikipedia.org/wiki/Hypercharge. Please could someone just provide a breakdown of all these terms and where they are the same thing and where they are not? Thanks!

• Do you know what the term "internal spaces" refers to?
– user226006
Mar 24, 2022 at 15:46
• No, but I may be familiar with them just not by that name. From a Google search it is if the transformation is occuring in spacetime. Mar 24, 2022 at 16:07
• A very good book on this is Deep Down Things by Bruce Schumm, it does a great job of explaining exactly what you are asking. "From a Google search it is if the transformation is occuring in spacetime". No, it's pretty much the exact opposite. Spacetime is external (ordinary) spaces, internal spaces are imaginary and allow us to view different groups of particles as rotations of one particle inside internal abstract spaces. Reading the book I recommend will make this clearer.
– user226006
Mar 24, 2022 at 16:52

Your particle physics book should precisely specify all these terms, and WP is only meant to summarize what you already know: it is not a tutorial. Google can only expose you to sources, but not organize them for you.

• In the strong interactions, isospin represents a vectorlike SU(2) flavor symmetry, explicitly broken by quark masses. (Vectorlike means it acts the same for left or right quarks chiralities.) SU(2) has three (almost) conserved generators, and one of them, $$I_3$$, is instrumental in the Gell-Mann-Nishijima formula, $$Q=I_3+Y/2$$, yielding the charge of the multiplet in terms of it and Y, the

• Hypercharge, a different vectorlike conserved quantity, which depends on other charges, like baryon number and strangeness.

This pattern/arrangement is replicated in a skewed fun-house-mirror manner in the Weak Interactions, which, however, are chiral: they treat right and left fermions very differently. This is the heart of the Standard Model. The symmetries now are exact (not explicitly broken), but they are now realized in a peculiar nonlinear (shift) mode, spontaneous symmetry breaking, the mode of Nambu and Goldstone.

• As a result, weak isospin sits in the left-chiral group SU(2) underlying the gauge theory of the electroweak interactions, and the weak hypercharge is the corresponding "correction" in the Gell-Mann-Nishijima formula giving the correct electric charges for all fermions, again. So a right- and a left-chiral fermion have identical electric charge, even though it is put together very differently from very different weak $$I_3$$s and

• weak hypercharges. Since right-chiral particles have absolutely no weak isospin,$$I=0$$, $$I_3=0$$), their electric charge is due to just weak hypercharge.

You find those tabulated in WP, and the first duty of the student is to put them together properly and see why they must be what they are.