Isospin ad conservation of J

Yes. The total angular momentum conservation is twin to energy and linear momentum conservation, and that has never been observed to be broken.

 

I'm sorry I did not notice that you include isospin in your angular momentum. I'm not sure what you are doing here. Possibly G-symmetry...

Usually, J=S+L is the total angular momentum. Isospin is analogous to spin but acts in internal space, unlike spin and angular momentum.

 

No, isospin isn't angular momentum in any sense. Yes, it obeys an algebra that is a carbon copy of that used for angular momentum. But a nucleon that is "isospin up" is a proton, and a nucleon that is "isospin down" is a neutron. This has got nothing to do with the z-component of its angular momentum.

I think you're getting 2 different things confused here. You quote the total angular momentum as J=L+S+I, and then call I the isospin. But most textbooks refer to I as the nuclear spin, which is an angular momentum. Isospin is usually represented by [itex]\tau[/itex].

 

He does !?
That only defines a symmetry group, not to what the symmetry is applied (as Tom Mattson said).
I am quite bugged.

So, what you are doing has no link with G-symmetry ? You were mentionning weak interaction. It maximally violates parity, so combining parity and isospin reverse, you often get (almost) conserved quatities...

 

The J=1/2 is wrong. Ispin has nothing to do with angular momentum.
Just the math is similar.
The two pions have L=0, a symmetrical state.

 

what is the previous sentence... or what was he talking about

As Meir Achuz said, the pions are in a L=0 state, which must be symmetrical.

 

OK, fine: Isospin doesn't satisfy that algebra, if only because there is no [itex]\hbar[/itex] in there. The isospin matrices satisfy:

[tex][\tau_i,\tau_j]=i\epsilon_{ijk}\tau_k[/tex]

There are no angular momentum units anywhere in there.

 

Actually, I've been thinking about my last post and I'm not satisfied with it. In natural units (where [itex]\hbar=1[/itex]), the algebras are identical. So that's not why isospin is not an angular momentum.

Angular momentum is the generator of rotations in the normal 3-space in which we all live. It is conserved in physical systems that are invariant under rotations in that space. Isospin, on the other hand, is the generator of rotations in a completely different space altogether, called isospin space. Isospinors are not elements of the eigenspace of [itex]J[/itex], and neither are spinors elements of isospin space. And there is no reason that conservation of [itex]J[/itex] should imply anything about conservation of [itex]\tau[/itex], and vice versa.