Science

Ok so I think electrons and photons are not actually particles.

Look up phonons

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Don’t electrons and protons carry physical momentum—whereas phonons do not?

Correct me if I’m wrong, but I always took that distinction as the line between particles (protons, electrons and even photons weirdly) and quasiparticles (phonons—aka quantized sound waves)

Genuinely curious—am I missing something?

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Momentum is always conserved. 

Phonons are a bit nutty, but it’s still conserved as long as you are careful about boundary conditions and any defects of the crystalline structure

They carry only "mathematical momentum" in the sense that, within equations of motion, they have a momentum quantity ħK (with K the wave-vector), but that momentum is not always conserved.

Physical momentum, in other hand, is always globally conserved (if system is translation-invariant).