[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.

[evanb]

Well it depends on what you mean by "quite a bit of work"! Wick rotate, focus on a finite volume of spacetime, and discretize the spacetime. That's all. The first step changes a difficult-to-sample distribution exp( i S ) into an easy-to-sample exp( – S ). The other two steps keep you from needing infinite RAM.

The discretization and finite-volume approximations may be removed by doing calculations with different spacings and volumes and extrapolating. The Wick rotation is more subtle, and typically means only some observables are accessible. To study the path integral with real time causes a very difficult sign problem [essentially: you need an exponentially large number of samples in order to get cancellations under control], which is why real-time quantum dynamics is an exciting potential application of quantum computers.