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It's always puzzled me how objects with properties like rest mass and charge (and color) could really be geometric points. Is this just a fiction needed for the math to work? Could quantum field theory work with spherical electrons and quarks?

Richardbernstein
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No, it is not fiction.

It is the present conclusion from a large number of data fitted by quantum field theory calculations that hypothesize that the standard model particles are point like. No other models today can describe the totality, or almost totality, of the present data.

anna v
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    You are quite right about the standard model. Beyond the SM the experimental situation is that we know that if the electron has any internal structure then the energy it takes to excite this internal structure is greater than any of the labs have been able to manage. We can easily rule out structures at scales of $\sim 100\ \mathrm{GeV}$, which equates to a distance of roughly $10^{-18}\ \mathrm{m}$. For specific models you can probably get even better limits. The PDG has reviews on this http://pdglive.lbl.gov/Rsummary.brl?nodein=S054&inscript=Y&fsizein=1&clumpin0=A – Michael Jan 18 '13 at 06:33
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    More generally speaking there are two ways (that I know of) to make extended objects in quantum field theory: bound states of multiple point particles (analogous to an atom, or a nucleus), and extended field configurations that are stable due to a topological charge. Examples of these would be solitons, domain walls, vortex tubes, monopoles and skyrmions. There is no evidence any of the known fundamental particles are any of these, although vortices and skyrmions come up in some approximations of QCD, the theory of quarks and gluons. – Michael Jan 18 '13 at 06:39
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    @MichaelBrown yes, you are correct. I am answering for the experimental situation at present. – anna v Jan 18 '13 at 06:50
  • No worries, I just wanted to mention the options and also the sort of size resolution available these days. – Michael Jan 18 '13 at 07:05