What underlies super-selection rule in quantum field theory? Super-selection rule in quantum field theory states that superpositions of two states with different charges do no exist in nature.


*

*What does "charge" mean in this context? 

*Is there a deeper principle from which super-selection rule follows as a mathematical consequence?

 A: A U(1) symmetry (for instance rotation about a given axis, or phase rotation in theories with electric charge) has one-dimensional irreducible representations spanned by the vector $|n\rangle$, where the unitary symmetry operators act as $$U(\theta)|n\rangle=e^{in\theta}|n\rangle$$
The number $n$ distinguishing the representations is called the charge, and it corresponds to the usual definition of charge associated to the symmetry via Noether currents.
Where superselection rules come about is if we require states related by some symmetry to be exactly the same. For instance if we believe that rotating the system by $2\pi$ leads to exactly the same physical state, then it must be the identity up to a phase factor. The phase factor is just $\pm 1$ depending on if it has integer or half integer angular momentum. But notice if we have a superposition of integer and half-integer states, we get a different state after we rotate by $2\pi$. So we say those superpositions don't count as physical states and there is a superselection rule forbidding them.
The same thing happens if we take any $U(1)$ rotation of the phase to correspond to the same physical system. If we have a superposition of states with the same charge, we just get an overall phase factor when we rotate by any $U(\theta)$, so this is allowed. But if there are different charges in the superposition, we do not get the same state up to an overall phase factor due to the relative phase difference. So we say there's a superselection rule forbidding this.
Note this argument all depends on the physical states really being the same after the symmetry operation. If we were to discover a superposition of charges in nature, we could just abandon this and state that a U(1) rotation really is changing something physical.
