How are intertwinners related to conservation laws?

[naima]

In "The Algebra of Grand Unified Theories" http://arxiv.org/abs/0904.1556" john Baez (see page nb 9) shows how the conservation of eigenvalues (T3 isospin)
appears if there is an intertwiner between the representation spaces of the symmetry group.
I discovered intertwiners for the first time in LQG (in this forum).
Can one avoid lagrangians to explain conservation laws with them?

 

[naima]

I think it will be easier if i quote Baez:

Quite generally, symmetries give rise to conserved quantities. In quantum me-
chanics this works as follows. Suppose that G is a Lie group with a unitary rep-
resentation on the finite-dimensional Hilbert spaces V and W . Then V and W
automatically become representations of g, the Lie algebra of G, and any intertwin-
ing operator F : V → W respects the action of g. In other words,
F (T ψ) = T F (ψ)
for every ψ ∈ V and T ∈ g. Next suppose that ψ ∈ V is an eigenvector of T :
T ψ = iλψ
for some real number λ. Then it is easy to check F (ψ) is again an eigenvector of T
with the same eigenvalue:
T F (ψ) = iλF (ψ).
So, the number λ is ‘conserved’ by the operator F .
The element T ∈ g will act as a skew-adjoint operator on any unitary representa-
tion of G. Physicists prefer to work with self-adjoint operators since these have real
eigenvalues. In quantum mechanics, self-adjoint operators are called ‘observables’.
We can get an observable by dividing T by i.
In Casson and Condon’s isospin theory of the strong interaction, the symmetry
group G is SU(2). Here isospin, or more precisely I3 , arises as above: it is just the
eigenvalue of a certain element of su(2), divided by i to get a real number. Because
any physical process caused by the strong force is described by an intertwining
operator, isospin is conserved.

I was surprised to see a conservation law without reference to Noether nor lagrangians.

 

[naima]

Hi Marcus

When you opened the thread 'intuitive content of LQG' intertwiners appeared in the first posts
meteor said:

However, in various documents on Arxiv, I've found that the edges of the graph are labeled with group representations of Lie groups, and the vertices with intertwining operators(damned if i now what's an intertwining operator!)

and Self Adjoint gave a good answer:

The intertwiner functions are like black boxes - deterministically relating spin reps into spin reps out. These again are physics, somewhat like Heisenbeg's S-matrix relating momenta into momenta out.

and that closed the exchange on that point
8 years later i come back with a well known text of John Baez beginning with the notion of intertwiners.
And i got no answer about its physical importance. (250 readers)
Do you think that during these 8 years such things became so intuitive, so basic
that it deserves no comment?