So to conceptualize the difference between fields with and without restoring forces, I imagine that, for a field that doesn't have a restoring force, the medium itself can move permanently. For example if you have just a bunch of ball bearings lying on the surface of a table, you can cause a wave to go through the balls by hitting one. One bumps into the next, which bumps into the next, etc. There's no restoring force, so the wave is moving through the balls, and the balls are actually moving into a new position and they stay there.
Compare that to a water wave, where gravity is trying to restore the particles to a "flat" position in space. If you cause a wave in water, the medium will return to the space it occupied before through the restoring force, even as the wave travels through it.
Is this really how it works, so that e.g. the EM field itself can move in space, whereas e.g. the electron field cannot move in space, it's "pinned" in some sense by the Higgs field?
First, worth noting that "the EM field" (the thing that shows up in the wave equation) in this case is specifically the EM 4-potential. This doesn't work if you try to treat "the EM field" as the strength of the E and B fields or something - it has to be the 4-potential. I got tripped up by this at one point
Second, this isn't pinning the field in space, it's pinning the magnitude of the field to be close to some value (probably you can call that value 0)
So if the field locally gets "too high" or "too low", there's a restoring force accelerating it back towards the "normal" value, like a spring attached to the normal value.
It's not pinning it in the sense of stopping translation through space or time
In the water wave analogy, we're using the vertical dimension to represent the magnitude of the water wave, but translating that to other contexts, we're not literally talking about a physical height, just the magnitude of the field. (Which, for all I know, maybe you can formulate that as a position in some higher-dimensional space or something)
What trips me up is that we don't think of the field being a real physical thing. But isn't the field really the _true_ physical thing, and the wave is just a concept we overlay on it? Like, water is the real physical thing, and the wave is just an arrangement of the water that we recognize as humans. Isn't it the same with the EM and electron fields etc?
For fields, it’s rather that the wave is the only physical/real thing, and there is no separate “substance” that is waving. “Substance” is a concept that disappears in fundamental physics.
At some point this all kind of drifts apart from ontic science and starts to become a matter of narrative or interpretation, but I would generally agree with that.
Waves are mathematically-friendly possible configurations of the underlying system.
It's mathematically valid to choose the most convenient configurations for analysis because the systems are (pretty) linear, so we can just project any actual state into a sum of wave states, apply our mathematical model, and add it all back to get the new real state.
A lot of physical phenomena are composed of pretty predictable distributions of wave states, so projecting from a realistic state to a sum of wave states is usually straightforward enough.
For example, a moving particle looks like the sum of a bunch of waves all closely grouped around a particular wavelength.
Think of a field as a set of scalar field strength values, one value at every point in space. It's not a "thing" you can grab or see. The field strength values are based on the distances to and the magnitudes of the "particles" have have {charge, mass, color, whatever} (with the complexity that the particles themselves are really just standing waves, thus the scare quotes).
Compare that to a water wave, where gravity is trying to restore the particles to a "flat" position in space. If you cause a wave in water, the medium will return to the space it occupied before through the restoring force, even as the wave travels through it.
Is this really how it works, so that e.g. the EM field itself can move in space, whereas e.g. the electron field cannot move in space, it's "pinned" in some sense by the Higgs field?