[froeb]

It isn't the only way. I would argue that the "sum over paths" is more of a convenient language for categorizing QFTs than an actual statement of physical reality. Path integrals have the benefit that they can be easily connected to correlation functions by a Wick rotation, are naturally invariant under relativity, and work (kinda) well with gauge theories (well better than any other way we know).

However, in practice they are not used directly. Instead, the path integral is either treated perturbatively giving you Feynman diagrams, or by working out the Hamiltonian prescription (probably closest to what you are thinking of), or by using Monte Carlo approaches (there are probably other approaches I don't know of too). All of these approaches are easier to calculate with, but are a less natural language for abstractly describing a QFT.

[evanb]

"Monte Carlo approaches" like lattice QCD do use the path integral directly!

[froeb]

Alright fair enough, I'm not an expert on how those work. I was under the impression that there is quite a bit of work that has to be done to turn a path integral into something amenable to Monte Carlo though.