If we look at canonical quantization approach to QFT, we see that particle numbers are accounted for by means of state vectors, while amplitudes sit in the corresponding field operators. Is it possible to have all of it in one place?
For example, for complex Klein-Gordon field $(\partial^2+m^2)\phi = 0$ is it possible to write something like $$\phi(x) = \sqrt{2} \frac{\exp[-i(px)]}{\sqrt{2E_\vec{p}V}} + \sqrt{1} \frac{\exp[-i(kx)]}{\sqrt{2E_\vec{k}V}}$$ for 2 particles with momentum $\vec{p}$ and one particle with momentum $\vec{k}$?
If we work out energy and momentum for this field configuration, we get:
$$E =\sum_{\vec{q}}E_{\vec{q}}(|a_{\vec{q}}|^2 + |b_{\vec{q}}|^2) = 2E_\vec{p} + E_\vec{k}$$ $$\vec{P} = \sum_{\vec{q}}\vec{q}(|a_{\vec{q}}|^2 + |b_{\vec{q}}|^2) = 2\vec{p} + \vec{k}$$
So, field's energy and momentum are consistent with our expectations. Why then we use creation/annihilation operators to account for chnage in particle numbers during reactions?
