Expansion of the solution if interacting KG equation

[ShayanJ]

The exponentials $\phi_p(x)=e^{ipx}$(where $px=p_\mu x^\mu$), are solutions of the free KG equation $(\partial_\mu \partial^\mu+m^2) \phi =0$. I can expand the solutions of the interacting KG equation $(\partial_\mu \partial^\mu+m^2)\psi=V\psi$ in terms of solutions of the free KG equation $\psi(x)=\int e^{ipx} \pi(p) d^4p$.
But because there is an interaction, it seems to me that the evolution of $\psi(x)$ should be different from a free field and so I think the function $\pi$ should also be a function of time. But because we want to think covariantly and also because it seems a natural generalization to let the field have different expansion coefficients in different events of space-time, I think we should let $\pi$ depend on all four coordinates, i.e. we should write $\psi(x)=\int e^{ipx} \pi(p,x) d^4p$
Is this the right way of thinking about it?
Thanks

 

[gianeshwar]

It seems an advanced level question requiring tensors.
Please make the question simple if possible!