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In many pop physics books I have read that an electron has no internal structure.

How do we know that and how can we rigorously prove that it has no such structure at all?

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We study it using deep inelastic scattering. It was this type of experiment that first revealed the proton had an internal structure, and if the electron has an internal structure it will be this type of experiment that reveals it.

No experiment has yet discovered evidence for an internal structure in the electron, however that doesn't prove the electron has no internal structure. It proves only that any structure is too small scale for us to probe at the energies available. So contrary to what you say in your question, we haven't rigorously proved that the electron has no structure.

There is a long standing idea that the apparently fundamental particles like electrons and quarks might be bound states of fundamental particles called preons, though I must emphasise that there is no experimental evidence for this idea and there is some theoretical evidence against the idea.

We know the electron size must be less than about $10^{-18}$m because we've already probed that length scale in collider experiments. That means any preons have to be confined within $10^{-18}$m and the uncertainty principle implies a correspondingly large uncertainty in momentum. This would suggest the preon bound state has a high energy and therefore a high mass - certainly higher than the electron mass. There are ways around this so it isn't proof that the electron isn't a bound state. However the ways to avoid an excessively high mass are a bit contrived and will get more so as we probe to smaller and smaller length scales.

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  • $\begingroup$ A free electron cannot absorb a photon, while the combined system of an electron and a proton can absorb a photon. Can we then use spectroscopy to probe if a particle has internal structure? It seems that a bound state, due to boundary conditions, produce discrete energy levels, while a free state such as a free electron has a continuous energy level. Can these be used to probe the internal structure of particles? $\endgroup$
    – MOON
    Commented Apr 10, 2021 at 10:03
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You can't "prove" the electron has or has not internal structure. All you can do are collision experiments at the energies available to you. Along these lines the proposed ILC (International Linear Collider) - if build - will provide an unprecedented new look at the electron (and positron).

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The electron has a magnetic dipole moment and an intrinsic spin. Both phenomenon need an extend and a structure and hence the electron has to have an internal structure.

That until know we haven't found this structure has to do with our capabilities to build the right instrument.

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    $\begingroup$ "Both phenomenon need an extend and a structure and hence the electron has to have an internal structure." [citation needed] Standard quantum field theory has no issues with having point-like particles with intrinsic spin ("intrinsic" precisely means that this has *nothing to do with its extent!). $\endgroup$
    – ACuriousMind ♦
    Commented Sep 10, 2016 at 14:46
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    $\begingroup$ The two dimensional hilbert space that is needed to describe spin does not require some kind of spatial extent to be there. It is fine for something with no internal structure to have a state space of many dimensions $\endgroup$
    – Secret
    Commented Sep 10, 2016 at 14:50
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    $\begingroup$ see Where do the quantum fields encode the spin information?. Spin does not require a particle to have an extended structure. I'm afraid your answer is plain wrong. $\endgroup$
    – John Rennie
    Commented Sep 10, 2016 at 15:24

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