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Fundamental particles come with magnetic and electric charge, which makes the particles into a monopole source for the magnetic and electric fields. Of course, the magnetic charge is zero for all particles we know, because magnetic monopoles haven't been demonstrated to exist.

Fundamental particles also come with magnetic and electric spin, which makes the particles into a dipole source for the magnetic and electric fields. Here most particles have a nonzero magnetic dipole moment (the spin), but a zero or very small electric dipole moment.

Is there anything analogous to higher-order moments? Is there like an analog of the spin of a particle but for higher moments? If not, is there a reason why this isn't plausible to consider?

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Your concept does not seem to be entirely consistent with the observed phenomena. Let us proceed step by step.

Fundamental particles come with magnetic and electric charge…. It is true that the two subatomic particles electron and proton have both an electric field and a magnetic dipole.

… which makes the particles into a monopole source for the magnetic and electric fields. This is only correct for the electric field. By separating electrons from the atomic body, electrically negative and electrically positive regions are possible. However, since all subatomic particles are magnetic dipoles, only the alignment of these dipoles in the same direction is possible. This results in macroscopic magnetic dipoles and never monopoles.

Is there anything analogous to higher-order moments? Is there like an analog of the spin of a particle but for higher moments? This was not observed. Such a phenomenon does not seem to exist.

If not, is there a reason why this isn't plausible to consider? Since it has not been observed, there seems to be little inclination to describe such a state theoretically. But perhaps someone will respond to this answer with something that does point to a theoretical background.

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Any quantum object has a wave function. Most often, when we talk about fundamental particles we consider the plane wave approximation to the particle where the wave function of the particle is identified by its linear momentum (and its energy).

A general field configuration of any charged quantum field can carry arbitary angural momentum, and have any higher charge moment that you wish.

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  • $\begingroup$ A general field configuration can carry arbitrary orbital angular momentum. Spin angular momentum is a different creature. See the answers linked under the question. $\endgroup$
    – rob
    Commented Apr 22, 2022 at 3:08

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