Continuous symmetries as particles

[arivero]

I am not sure if I recall all the ways for a symmetry to appear as some particle in a Quantum Field Theory.

- The Lagrangian and the vacuum is invariant under the generators of a global symmetry/gauge group. Then the particles in the theory are classified according representations of such group, with all the elements in the same multiplet having equal mass, but... a) Are all the representations expected to appear, and b) is the representation of the generators, the fundamental representation, expected to appear?

- The Lagrangian is invariant under a global gauge group but the vacuum is not. The broken generators then appear as Goldstone bosons, of spin zero. Are they always spin zero? What about the unbrogen generators. No clue of them?

- The Lagrangian is invariant under a local gauge group. Then the generators appear as massless spin 1 bosons, under the fundamental representation of the group.

- The Lagrangian is invariant under a local gauge group and the vacuum is not. The unbroken generators appear as massless spin 1, the broken generators as massive spin 1, all of them join in a fundamental representation of the group that nevertheless has different masses.

 

[ChrisVer]

-I think all the irreducible representations are expected to appear (fundamental,antifundamental etc). I am not sure if there can be representations which don't appear...I don't understand b.

-Yes... as for if the spin is zero, in general I think that's true. Otherwise they would have to be vector bosons, and thus the vacuum which you "break" would have to have a vector field's vev. The unbroken generators/fields still exist afterwards (just like the Higgs boson which exists as a 1 dof after the 4dof of the field are gauged out.

-The Lagrangian invariant under a local gauge group, brings the massless spin-1 generators existing in the adjoint representation. Or at least that's what I have understood from SU(N) theories.

- For the same reason as Higgs... They don't join the fundamental representation though. For example the fundamental representation of SU(2)xU(1) is a (2,1). The gauge bosons exist in the adjoint representation which comes from the tensorial product of the fundamental and antifundamental reprs. As for the masses, I think it depends on the way you are breaking the symmetry.

 

[arivero]

I don't understand b

Sorry, my fault. I always do a mess with the naming. I meant the adjoint. b) Are the unbroken generators expected to appear as particles in the adjoint representation, in a Lagrangian with global unbroken symmetries?

All this stuff can be mathematically consistent. In fact it is. But it is also very confusing when comparing the local gauge with the global case. Nor to speak of approximate global symmetries.

Generically, it seems that in the global case the symmetry itself lives outside of the world, and than it is only in the case of local gauge when the generators also become particles. But then we have the goldstone bosons, breaking this intuition and proving than intuition is not a good guide here.