If an electron has no size, then how could a photon scatter an electron?

The traditional answer is that electrons’ influence extends over a space surrounding the electron. This answer is not correct for many reasons. See the references below. The electron, like all quantum “particles”, is not a point particle. In fact, it is not even a particle. It is a field, specifically, a spatially extended bundle of electron-positron energy that fills all space. An electron does not have a “boundary” beyond the point where it can be detected. For more information, please see * my non-technical book Tales of the Quantum, Oxford University Press, 2017.
* “There are no particles; there are only fields,” Am. J. Phys., Vol. 81, p. 211-233 (Mar. 2013).
* “Electrons and field quanta,” Am. J. of Phys. Vol. 73, p. 630 (July 2005).
* “Realistic analysis six controversial quantum issues” in Mario Bunge. A Centenary Festschrift, edited and published by Michael R. Matthews (Springer 2019, Chp. 19, pp 329-348.

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