# Can massive particles be seen as soliton solutions?

I wonder if the common relativistic wave equations contain a sort of soliton solutions, which might be considered as particle localisations.

• That's not a bad question but if you try this for an electron you get a "size" that is much larger by several orders of magnitude than any reasonable values for the radius of the electron. This is the one example I recall where this has been tried; I don't recall lots of other details as that particular attempt spectacularly failed. The electron is a good test case for any such calculation. – ZeroTheHero Feb 23 '17 at 21:57
• Something like this can happen in gauge theories. One can have electrically charged excitations which are fundamental and magnetically charged ones that look more like solitons (or monopoles). Then there can be a dual description where the electric and magnetic "particles" are swapped. The "size" of the particle has something to do with the coupling strength. As the coupling gets tuned up, one type of particle gets bigger and the other smaller. – user2309840 Feb 23 '17 at 23:11

There are popular Skyrme-like topological soliton models especially of baryons and nuclei, interpreting topological charge as baryon number. Here are some nuclei diagrams from https://www.sciencenews.org/article/nuclear-knots-skyrmions-could-unravel-mysteries-atoms :

Interpreting topological charge as electric charge instead, we can derive charge quantization. Gauss-Bonnet theorem can be reformulated that integral of field's curvature over a closed surface gives topological charge inside this surface. Hence interpreting curvature of some vector field as EM field, Gauss-Bonnet theorem becomes Gauss law, which finally allows only for integer charges like in nature - leading to Maxwell's equations with built in charge quantization, see e.g. Faber's model of electron, slides.

However, not to exceed 511keV electron mass with $$E\propto r^{-2}$$ electric field alone, such infinite energy field of perfect point charge requires deformation in femtometer scale. In contrast, there is a belief that electron is much smaller. However, asking for experimental evidence for this belief, requiring to exceed 511keV mass by energy of electric field alone, turns out it is based only on fitting parabola to two points in 1988: Experimental boundaries for size of electron?