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When people discuss quantum field theory in a popular context, they say that fundamental particles, such as quarks and electrons, are pointlike, with zero size.

However, I don't think this is what QFT says at all. My impression is that a particle is an excitation of the field, and hence a wavepacket with finite size. Of course, you can have a spike in field value at a single point, like $|x \rangle$ in quantum mechanics, but this state is pathological, and it's definitely not what we mean when we say 'particle'.

Interactions between fields are local and hence only happen at a single point. But to calculate the result of a scattering, you have to integrate this over the positions where the incoming wavepackets overlap. So it doesn't make sense to say that interactions occur at a point either.

If field excitations are not pointlike, and interactions of fields don't occur at points, then what do people mean when they say particles are pointlike?

Edit: I don't think this question answers mine. The top voted answer just reiterates that the Lagrangian is local, but most of the time people seem to be making the much stronger statement that particles are perfectly localized. As argued above, these are not the same thing. I'm wondering if there's any justification for the latter, or if it actually doesn't make sense.

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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/41676/2451 and links therein. $\endgroup$ – Qmechanic Mar 22 '16 at 0:49
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    $\begingroup$ Related: physics.stackexchange.com/q/163691/50583 $\endgroup$ – ACuriousMind Mar 22 '16 at 0:49
  • $\begingroup$ Whenever you hear someone discussing it that way (point particles etc.), you already know that they didn't understand QFT. $\endgroup$ – CuriousOne Mar 22 '16 at 0:52
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    $\begingroup$ Note that from an experimental point of view the size of a particle is the inverse energy scale at which access to the sub-structure begins to effect the interaction cross-section. That is perfectly well defined in a QFT context and for the presumed point-like particle no such scale has been detected. $\endgroup$ – dmckee --- ex-moderator kitten Mar 22 '16 at 1:40
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    $\begingroup$ I always saw this as an historical accident. The properties of elementary and near-elementary fields were first studied in scattering experiments, like Rutherford scattering and Compton scattering (and modern descendants of these at CERN), and these results were (at least initially) interpreted as being like little billiard balls hitting one another. This picture has probably outlasted its value; and can be replaced by the conception that you and the linked questions give. $\endgroup$ – Rococo Mar 22 '16 at 2:34