Writing down a field theory is seems to be writing out a path integral of some sort: \int Dx Exp[S[x]] Where \int Dx is like a sum, x is a field (a thing that

Writing down a field theory is seems to be writing out a path integral of some sort: \int Dx Exp[S[x]] Where \int Dx is like a sum, x is a field (a thing that assigns numbers or an object to a point in space), Exp[S[x]] is a weighting. You sum over all possible field configurations weighted by S[x], which is called the action. You can feed in other fields to this sum and calculate expectations of events happening, a bit like a Fourier transform or Gaussian integral from statistics. Classical configurations (classical paths/trajectories) this is the biggest contribution, then all the quantum stuff gets weighted less. Formalising this notion is complicated for the sorts of objects and spaces needed in physics. This is also problematic when it comes to quantising general relativity, the gravitational field in a sense is not just a field on space, it's more like space itself, although you can try to describe that as a field. People get around this by summing over geometry. In a sense you can test what the geometry is by seeing what it does to objects, maybe a bit like forcing particles to interact with each other and seeing what gets produced says something about internal structure. It is interesting of course that we try and use classical information plus some choices to build a field theory. In my opinion it is sort of unnatural to quantise, you are making choices. Broadly quantisation is sticking some commutative theory in a non commutative theory. But since we know the universe actually runs on those quantum rules, seems wrong to define it as like the embedding of a classical theory, it should go the other way, classical information should emerge naturally instead. Perhaps the other issue is that field theories are effective in a sense. Much like fluid dynamics doesn't say anything about the internal molecular composition of water, fields are an effective theory. Perhaps string theory tries to get around this by having large scale dynamics and then saying what the micro local structure is, but that is even less of a well defined notion than a QFT.