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With my pop-sci level of understanding, it seems to me that quantum fields exhibit particle-like properties only when interacting with a different quantum field - i.e. electromagnetic field interacts with electron field to produce an excitation of an electron and this interaction is localized so both fields appear as particles to us (photon / electron respectively). A field seems to interact only in a wave-like fashion with itself, i.e. constructive or destructive interference.

Is this a valid impression? Are there examples of particle-like interactions of the same field or wave-like interactions of different types of fields?

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On the question: "Is this a valid impression". The answer is "meh". It's not great, it's not terrible. A photon interacting with an electron has a quantum field description (e.g. Feynman diagrams), or a Fock state description in quantum optics. Both of those are technical.

Any general statement about "it's a particle" or "it's a wave" is, well, too general, to be of any pedagogical value.

An example of particle behavior that is wavelike is the Pomeron in diffractive (wave-like) electro-rho (particle like) production. Ofc, that example is useless at a pop-sci level of understanding.

A general statement you can make is: classical analogies of quantum phenomenon with out actual quantum mechanics will lead you astray if you take them too seriously. The reason is: while classical physics is the $\hbar \rightarrow 0$ limit of quantum mechanics, you can't go the other way. A classical analogy will break.

When you replace quantum mechanics with quantum field theory--it only gets worse. We don't even solve real problems exactly; rather, we compute corrections to approximate states, or transition probabilities between approximate initial and final states.

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