Number operator in interacting quantum field theory

When treating a quantum field, say the real scalar field, it's totally clear to me how to define a (global) number operator:

$$\hat N = \intop \text d ^3 \mathbf p \hat a ^\dagger (\mathbf p )a(\mathbf p ).$$ 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 $\hat N \rvert 0 \rangle =0$.

Now, consider a field interacting with itself, for example:$$\mathscr L =\frac{1}{2}\partial _\mu \phi \partial ^\mu \phi -\frac{1}{2}m^2 \phi ^2 -\frac{\lambda}{4!} \phi ^4.$$

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 $\alpha$:$$\lvert \alpha \rangle = \lvert \alpha \rangle _0 + \lvert \alpha \rangle _1 + \lvert \alpha \rangle _2 + ...$$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 $\hat N $ such that $\hat N \lvert \alpha \rangle _n = n\lvert \alpha \rangle _n$.

Question. Is there a theorem which guarantees the existence of such an operator for every physical field theory? Is it possible to construct explicitly $\hat N $?