I'm going to have to butcher it a bit to make this fit in a comment :) Basically the thing you need to think about is the energy associated with the field itse

I'm going to have to butcher it a bit to make this fit in a comment :) Basically the thing you need to think about is the energy associated with the field itself. For the electromagnetic fields, the energy in the field is like (E^2 + B^2), or in terms of the electromagnetic field tensor F (see [0] if this isn't a familiar concept), it's F^2. You can write F in terms of a potential A, basically F = dA, where d means gradient [1]. So the energy looks like (dA)^2, which is a kinetic energy for the field A, because it tells you that an oscillation in the field costs energy. Integrating over the possible values of the A field is a lot like the integral you would do for the matter fields [2], in particular the math looks basically the same as what you would do for electrons for example [3]. And just like with electrons, the local propagating degrees of freedom are the ones we call particles. So this is how you make dynamics happen in QFT--you integrate over all the possible field values, and the local propagating degrees of freedom are called particles. And for EM and other gauge theories, the math is basically the same as for electrons, and it should be interpreted the same. The same story carries over for gravity, too, where the Einstein-Hilbert Lagrangian (R) is like the EM field strength (F^2), and the metric g enters into it in a similar way as A does for EM (basically, R = (dg)^2). [0] https://en.wikipedia.org/wiki/Electromagnetic_tensor https://en.wikipedia.org/wiki/Electromagnetic_tensor [1] This isn't quite right, you have to make sure everything is gauge invariant, but that's not important for understanding what's going on at a high level. [2] See [1]. [3] If this isn't a story that you understand yet, take some time to study free scalar field theory, and make sure to understand both the path integral method and the canonical method. By seeing how these translate between each other, you can build intuition.