I've had the same question for ages. Shouldn't there be an equation in "Schroedinger form" with some relativistic Hamiltonian?
Writing it this way goes against the basic idea of QFT, which is that, in a relativistic context, quantum systems can no longer be described as "wave functions evolving in time", which is what the Schrodinger/Hamiltonian formulation describes.
If you're using the non-relativistic approximation, most of the Standard Model is irrelevant since you're limited to interaction energies much less than the rest mass of the lightest particle involved. You're basically looking at the low energy regime of QED, or straightforward non-relativistic QM with an appropriate potential in the Hamiltonian and no pretense of even trying to derive things from an underlying QFT model.