Deriving the field operators for Quantum Field theories

I always see the form of the field operators derived by, in the case of a scalar spin 0 particle, imposing the field commutation relations on the classical field solutions of the Klein Gordon equation and interpreting te Fourier coefficients of the field, Fourier expanded in momentum space, as operators which will then turn out to be annihilation and creation operators for particles in momentum eigenstates.

This procedure also works for the photon field and the electron field since both the Wave equation and the Dirac equation, like the Klein Gordon equation, are linear equations that can also be Fourier expanded in momentum space.

When considering interactions the equations are no longer linear but in this case the ingenious "interaction picture" saves the day. The field operators in the hamiltonian operator that is used to calculate the time evolution of the quantum states are just the free-field operators.

I am however wondering if it is possible to derive the exact form of the field operators in a more direct way, that could also work for nonlinear equations that do not have a free field version and a corresponding interaction picture.

Maybe we could use the Heisenberg equation of motion in combination with the commutation relations?

Let’s look at scalar field theory:

The hamiltonian is,

$H=intfrac{1}{2}pi’^{2}+frac{1}{2}(triangledownphi’)^{2}+frac{1}{2}m^{2}phi’^{2}dx’^{3}tag{1}$

Indicating the primed symbols as the ones over which we intgrate, the Heisenberg equation of motion in natural units is:

$frac{mathrm{d} phi }{mathrm{d} t}=ileft[ intfrac{1}{2}pi’^{2}+frac{1}{2}(triangledownphi’)^{2}+frac{1}{2}m^{2}phi’^{2}dx’^{3},phi right ]tag{2}$

Using the equal time commutation relations this finally gives me:

$frac{mathrm{d} phi}{mathrm{d} t}=pitag{3}$

so

$frac{mathrm{d} phi}{mathrm{d} t}=frac{partial phi}{partial t}tag{4}$

It seems to be saying the the field operator doesn’t explicitly depend on $x$, $y$ and $z$.
I don’t know where to go from here though.