# Is the cut-off energy in the calculation of the electron's self-energy associated with an electron size?

I have got two questions on the self energy/point particle concept of the electron:

1. What do physicists exactly mean when they say the elementary particles (e.g. the electron) are point particles? Should I picture the electron like a point with charge $$e$$ and mass $$~0.5 MeV$$? And how does this picture go along with the electron being an excitation in the fermion field. Isn't it better to picture the electron as the excitation in the fermion field (i.e. "a wave packet" with the fwhm as "size") without thinking that the electron is a point like thing "in" this excitation?

I know this question was addressed here 1 and here 2 but I don't think the problems with the point particle were stressed enough. Point particles have infinit mass and charge densities, they should be black wholes according to the Schwarzschild radius. They are smaller than the smallest meaningful distance which is $$2l_{p}$$ according to the generalized uncertainty principle 3

$$\begin{equation} \Delta x\sim \left( \frac { \hbar }{ \Delta p } \right) +{ l }_{ p }^{ 2 }\left( \frac { \Delta p }{ \hbar } \right) \end{equation}$$

It is also senseless to think about an object much smaller than $$l_{p}$$ because it would be unmeasurable also explained here (3 VI.point). So I don't understand why we just say the electron is the excitation in the field and forget about the point particle concept. I also believe that many integrals in QED diverge because we are assuming point particles which leads me to my second questions.

1. The calculation of the self energy of the electron gives the formula (4, p.30):

$$\begin{equation} m{ c }^{ 2 }={ m }_{ 0 }{ c }^{ 2 }\left[ 1+\frac { \alpha }{ 4\pi } \left( 3\ln { \left( \frac { { \Lambda }^{ 2 } }{ { m }^{ 2 } } \right) } +\frac { 3 }{ 2 } \right) \right] \end{equation}$$ When we cut off the energy at a given level arent't we assuming that the electron or the excitation in the field has a minimal size (the size associated with the energy scale of $$\Lambda$$)? Is it right that the bare mass $$m_{0}$$ origins in the coupling of the electron with the higgs field and we therefore don't know how strong this coupling is (because the bare mass depends on $$\Lambda$$)?

• Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. Dec 19, 2019 at 1:11
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– Chris
Dec 19, 2019 at 23:46