Can anybody tell me a similar basic scheme for quantum field theory? For example, in QED, Is there a basic differential equation? What about QCD?
I'll add yet another perspective, since you asked about formulating QFT in terms of differential equations. As mentioned by @DarMM in multiple posts here, the main observables of a QFT are the correlation functions, which are related to experimental quantities. It turns out that the correlation functions of a quantum field theory satisfy a (generally infinite) set of differential equations called the Schwinger-Dyson equations. In principle, one could consider the solution to these equations as giving you the same information as any other method of calculating correlation functions. Whether this is actually the easiest approach depends on the problem, but sometimes numerically solving a truncated subset of them or solving them in some approximation scheme can be useful.