Are All Gauge Groups in the Standard Model Semisimple for a Physical Reason?

[fresh_42]

Is there a physical reason why all gauge groups considered in SM and especially beyond are always semisimple? [+ U(1)] What would happen if they were solvable?

 

[fzero]

The norm of a quantum state must be positive definite in order that the probability interpretation of quantum mechanics makes sense. For a nonsemisimple group, the Killing form is not definite, so we can't guarantee that there won't be any negative norm states. There are ways to use constraints to remove the negative norm states for certain theories. A simple example is the way that gauge invariance can be used to remove the negative norm states from the $SO(d,1)$ metric on a massless vector field.

I'm not sure about solvable Lie algebras other than the abelian ones. Since the Killing form vanishes on the derived subalgebra, there will be zero norm states that would be considered unphysical. Perhaps one could find suitable constraints to project these out.

 

[fresh_42]

Thank you.