So if I want to solve the free KG equation, the solution is of the form $$\phi(x) = \int_{p\in \mathbb{R}^4}\frac{1}{2\omega_\vec{p}} \left( a(\vec{p})e^{-ipx} +b(\vec{p})e^{ipx} \right),$$ where the mode coefficients $a$ and $b$ should depend on the initial conditions. Now if I want to implement some momentum distribution $f(\vec{p})$, along which the modes are distributed, how would I go about doing that?
I suppose one would need to choose the correct initial condition $\phi(0,\vec{x})$?
In the special case I am interested in, the situation is even a bit simpler, the field is only evaluated at $x=0$, meaning I have the representation
$$\phi(t) = \int_{p^0} \frac{1}{2\omega_\vec{p}} \left( a(\vec{p})e^{-p^0t} +b(\vec{p})e^{ip^0t} \right),$$
and the initial condition
$$\phi_0 = \int_{p^0} \frac{1}{2\omega_\vec{p}} \left( a(\vec{p})+b(\vec{p}) \right) = \dots$$
Am I correct in assuming that the functions $a,b$ evaluated at $\vec{p}$ should then correspond to the amplitude of the momentum mode $\vec{p}$, given the distribution $f$?
I would imagine this amplitude to be some sort of occupation number, i.e. $\sqrt{f(p)\text{d}p}$.
What would be the correct way to formulate this? Or is there literature that goes more in depth on that?
