[General] Correspondence between topological structures and charge, spin, baryon number etc.?

[Jarek Duda]

Dear colleagues,

I have recently started and trying to warm up "models-of-particles" 
(e.g. as solitons) Google email group with ~40 physicists and have just 
found this looking related one. We have also a bit more general QM 
foundations online seminar ( http://th.if.uj.edu.pl/~dudaj/QMFNoT ) - 
talks are welcomed, I can also add to the email group.
Let me briefly introduce myself ( http://th.if.uj.edu.pl/~dudaj/ ). I 
have education in Physics (PhD), computer science (PhD) and mathematics 
(MSc). I am interested in physics foundations since ~2006 when I have 
started MERW 
<https://en.wikipedia.org/wiki/Maximal_entropy_random_walk>(alongside 
ANS <https://en.wikipedia.org/wiki/Asymmetric_numeral_systems>) showing 
where disagreement between diffusion and QM comes from (only 
approximating crucial maximal entropy principle), then in 2009 I have 
started working on particle models as topological solitons, which turned 
out expansion of Faber's model of electron (slides: 
https://www.dropbox.com/s/aj6tu93n04rcgra/soliton.pdf ).


Let me start with this general view thered/question, also to get some 
glimpse about your group:

Most of us would like to use topological mechanisms to stabilize field 
configurations into localized models of particle (?)

There is a large variety on both topological and particle sides here, so 
maybe let us try to discuss the most promising correspondences e.g. to 
properties like electric charge, spin, baryon number - to target in such 
models.

Here are some very basic topological structures – what should they 
correspond to?


*1) 2D topological charge* in cross-section e.g. in fluxons/Abrikosov 
vortex in superconductors. It agrees with quantum rotation operator: 
rotating spin ‘s’ particle by phi angle, the phase rotates by phi * s. 
Topological constraint resembles Ampere's law here e.g. for fluxon 
carrying quant of magnetic field, what might be related with spins 
corresponding to magnetic dipole moments (?)

*2) *Such vortices can form *knot**-like structures*: which can be 
relatively stable local minima, but are not global – they could decay 
especially in high temperature.

*3) **3D topological charge* e.g. hedgehog-like localized configuration. 
Gauss-Bonnet theorem acts as Gauss law with built-in charge quantization 
here: integrating field’s curvature over closed surface, we get 3D 
topological charge inside, which has to be integer.



Baryon number doesn’t seem to have Gauss law (?) and usually is believed 
that it can be violated in extreme temperatures – during baryogenesis 
just after Big Bang, or for Hawking radiation inside Black Hole.

So personally I would assign: 2D charge -> spin, 3D charge -> electric 
charge, baryon number as example of knotting.

But e.g. Skyrmion models disagree: assigning 3D charge -> baryon number, 
and usually neglecting electric charge and spin.

So which assignment is the most promising? Why?

Best,

Jarek Duda

ps. Related: is photon number conserved e.g. as topological? If not, why 
single photon does not dissipate? (I would bet on angular momentum)

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