If I understand correctly, the math of wave/particle duality is just that the wavefunctions that we get from solving the Schrödinger equation are specifically standing waves, with quantized properties due to boundary conditions, similarly to how strings in string instruments have discrete wavelengths. At least, that's how it I remember it being explained in the unit on modern physics when I took physics 3 a few years ago. If not for the boundary conditions, there'd be no quantization, right? IIRC, the energy of an electron in free space isn't quantized, but rather the energy of electrons in atoms is quantized only because of how the electron is bound "inside" the atom. And yet, if there's no quantization, does it even make sense to talk about particles at all, as opposed to continuous waves? Or, more specifically, if there were just one field in isolation, any excitations in that field would have to be continuous waves, yes?
I think what I'm really trying to understand here is that, if "everything is fields", as some physicists have said, are the interactions between fields the reason particles are even a thing? Does the "ripple" in a quantum field correspond to the wavefunctions of standard QM? And when those fields interact, does that create wave interference that results in standing waves, and hence the particle nature of things?
