If you take the reaction $$``\ |\text{CH}_4,2 \text O_2 \rangle\ \ \overset{\text {burn}}\longrightarrow \ \ |\text{CO}_2, 2 \text H_2\text O\rangle\ "$$ There are 7 nuclei and 42 electrons. In nonrelativistic quantum mechanics, a state of this system is a function on a ~150-dimensional space. It's essentially impossible to do any calculations on such a function. Even if you just try to write down the function in terms of adjustable parameters, it's too many parameters to store on a computer. Forget about doing integrals etc. You would have to make severe approximations like, oh let's say, using classical physics when possible with semi-heuristic quantum mechanics inserted here and there. So that's exactly what people do.

In relativistic quantum mechanics, you replace a function on a complex 150-dimensional space with something even more complicated, I guess 150 complex linear operators at every point? So the calculation becomes even more impossible in practice. (In principle, of course, you would get the correct answer.)

The fundamental issue is that the transition is a complicated process. First the carbon atom wiggles a little this way, then the valence electrons get distorted in a certain way, then the hydrogen wiggles a little that way, and on and on. Transition path sampling is a powerful technique to figure out this chain of events, and it works within the framework of ordinary classical or semi-classical statistical mechanics. If you tried to use Feynman diagrams, the calculation would be completely intractable.

If you take the reaction $$``\ |\text{CH}_4,2 \text O_2 \rangle\ \ \overset{\text {burn}}\longrightarrow \ \ |\text{CO}_2, 2 \text H_2\text O\rangle\ "$$ There are 7 nuclei and 42 electrons. In nonrelativistic quantum mechanics, a state of this system is a function on a ~150-dimensional space. It's essentially impossible to do any calculations on such a function. Even if you just try to write down the function in terms of adjustable parameters, it's too many parameters to store on a computer. Forget about doing integrals etc. You would have to make severe approximations like, oh let's say, using classical physics when possible with semi-heuristic quantum mechanics inserted here and there. So that's exactly what people do.

In relativistic quantum mechanics, you replace a function on a complex 150-dimensional space with something even more complicated, I guess a bunch of complex linear operators at every point? So the calculation becomes even more impossible in practice. (In principle, of course, you would get the correct answer.)

The fundamental issue is that the transition is a complicated process. First the carbon atom wiggles a little this way, then the valence electrons get distorted in a certain way, then the hydrogen wiggles a little that way, and on and on. Transition path sampling is a powerful technique to figure out this chain of events, and it works within the framework of ordinary classical or semi-classical statistical mechanics. If you tried to use Feynman diagrams, the calculation would be completely intractable.