General analysis of internal symmetries in QFT I am trying to understand as much as I can about internal symmetries in QFT, without using a Lagrangian or the canonical formalism (nor perturbation theory), but I am having a hard time to find good references.
For example, if we consider a free theory with a $U(1)$ symmetry it is easy to construct the number operator and to show that its eigenvalues are integers. 


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*When we add interactions, does this remain true? It is clearly true for asymptotic states, but what about bound states? must these also have integer charge?


As an example of what I have in mind: one can use the algebra of the angular-momentum operators $[J_i,J_j]=i\epsilon_{ijk}J_k$ to show that the eigenvalues of $J_i$ are discrete:


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*Is it possible to do the same with, say, the charge operator? In other words, is it possible to use the algebra of internal symmetries to show that internal quantum numbers must be discrete? 

 A: I am not sure to have understood well your question. However, something general can be said regarding the spectrum, just looking at the abstract group. If you represent a compact Lie group as $SU(n)$ or $SO(n)$ by means of a unitary strongly continuous representation, you are sure that the spectrum of each generator and of the Casimir operators is discrete, barring rare situations I describe below, as a consequence of the celebrated Peter-Weyl theorem. 
This result establishes that, under the afore-mentioned hypotheses, the representation is the direct orthogonal sum (and not a direct integral as it happens in the general case) of irreducible unitary representation with finite dimension. 
In each invariant finite-dimensional subspace of every irreducible representation everything reduces to a matrix representation and thus the spectrum of each self-adjoint generator (which is a Hermitian matrix) is a finite and discrete set of points and every eigenvector is a proper eigenvector. 
Summing all irreducible subrepresentations of the initial representation, every generator is the sum of all corresponding generators of the sub representations. (There are some mathematical details on the used domains and the topology used in the sum but they are quite irrelevant here).
The Casimir operators are the sum of the corresponding (constant) Casimir operators in each irreducible sub representation. The spectrum is the closure of the union of the spectra. The possible added points to the simple union of discrete points are the only possible points the continuous spectrum of the overall representation and are just the accumulation points (if any) of the said union.
