When treating a quantum field, say the real scalar field, it's totally clear to me how to define a (global) number operator:
ˆN=∫d3pˆa†(p)a(p). This turns out to commute with the hamiltonian and the 3-impulse of the system, therefore the physical interpretation of states with a definite number of particles with a definite total 4-impulse is straightforward. In particular, one could define the vacuum as ˆN|0⟩=0.
Now, consider a field interacting with itself, for example:L=12∂μϕ∂μϕ−12m2ϕ2−λ4!ϕ4.
In this case one still talks (in a sense which is not clear to me) of states with a definite number of particles. In a proof, my professor wrote, for a generic state α:|α⟩=|α⟩0+|α⟩1+|α⟩2+...where the pedices denote the number of particles in each state of the expansion. Now, this equation implicitly says that there is a certain observable ˆN such that ˆN|α⟩n=n|α⟩n.
Question. Is there a theorem which guarantees the existence of such an operator for every physical field theory? Is it possible to construct explicitly ˆN?
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