Here is my understanding of the scenario. Please correct me if i go wrong somewhere.
Initially, the perturbative approach to QED (Feynman diagrams) was very successful. But the same approach to QCD proved cumbersome, and hence some sort of 'path integral' approach on vanishing lattice spacetime was developed. This approach was actually non-perturbative.
So, my understanding is that there are two approaches that everyone learns to QFT. The canonical (or perturbative or Feynman diagram approach) and the path integral (or lattice spacetime approach).
Now, having read Feynman and hibb's path integral book a little, i am under impression that the path integral approach works only for Schrodinger's equation, and becomes really difficult when we try to incorporate the relativistic Dirac equation. For the same reason, the same book treats QED as an approximation, using Schrodinger's equation and avoiding Dirac's.
So, i am surprised that in QCD, the path integral approach returns. How did they manage this? How did they overcome the problem of providing the path integral equations for Dirac's equation? Did they?
Please clarify this.