# Soliton particle models?

I see there are mainly discussed here very abstract approaches like string theory. I would like to suggest a general discussion about much less abstract models: to get not exactly beyond, but rather behind the standard model - ask about the internal structure of particles (behind abstract Feynman diagrams) - imagine them as some concrete localized configurations of a field: so called solitons.
There were many physicists suggesting such approaches in the history, like seeing particle as kind of a vortex. The most popular model is probably of Tony Skyrme ( http://en.wikipedia.org/wiki/Skyrmion ), seeing mesons/baryons as topological solitons of some effective field.

There are well known Abrikosov vortices/fluxons - topological solitons in superconductor. We can observe them through a microscope, but it doesn't prevent "quantum" phenomenas for them, like interference ( http://prl.aps.org/abstract/PRL/v71/i14/p2311_1 ).
Recent Couder's droplets are different complex macroscopic objects with quantum properties (thread: https://www.physicsforums.com/showthread.php?t=550729 ) - localized objects conjugated with waves they create, what leads to observed:
- interference in double-slit experiment (particle goes a single way, but "pilot" waves it creates go through all paths - leading to interference): http://prl.aps.org/abstract/PRL/v97/i15/e154101 ,
- tunneling (the field depends on the whole history, making that getting through a barrier is practically random): http://prl.aps.org/abstract/PRL/v102/i24/e240401 ,
- orbit quantization (to find a resonance with the field, the clock has to perform an integer number of periods while enclosing an orbit): http://www.pnas.org/content/107/41/17515 ,
- "classical Zeeman effect" (Coriolis force instead of Lorentz): http://prl.aps.org/abstract/PRL/v108/i26/e264503.

So there is a localized configuration - corpuscle (e.g. charge), conjugated with waves it creates - because of some internal periodic dynamics (so called de Broglie's clock, zitterbewegung).
In the language of fields: localized configuration is a soliton, conservation laws with integer "quantum" numbers naturally appears while using topological solitons (e.g. with topological charge), which often have some internal periodic dynamics, like breathers ( http://en.wikipedia.org/wiki/Breather ).

Prof. Faber has shown that in a field of just unit vectors, defining electric/magnetic field through curvature of the field, the dynamics of topological solitons (hedgehog configurations) is given by Maxwell equations - we get a simple electron model (massive) with standard electrodynamics: http://iopscience.iop.org/1742-6596/361/1/012022/ http://arxiv.org/abs/hep-th/9910221.
Here is my essay with more motivations for using soliton particle models and expansion of Faber's model by a single degree of freedom ("quantum phase"), making that the family of topological solitons grows to resemble our particle menagerie with their dynamics: http://www.fqxi.org/data/essay-contest-files/Duda_elfld_1.pdf.

Are solitons only some abstract mathematical constructs?
Maybe there are also macroscopic solitons like fluxons?
Maybe we can use solitons also for effective models of particles like Skyrme?
... or maybe all particles are just solitons of some fundamental field? What field?

member 11137
I see there are mainly discussed here very abstract approaches like string theory. I would like to suggest a general discussion about much less abstract models: to get not exactly beyond, but rather behind the standard model - ask about the internal structure of particles (behind abstract Feynman diagrams) - imagine them as some concrete localized configurations of a field: so called solitons.
There were many physicists suggesting such approaches in the history, like seeing particle as kind of a vortex. The most popular model is probably of Tony Skyrme ( http://en.wikipedia.org/wiki/Skyrmion ), seeing mesons/baryons as topological solitons of some effective field.

There are well known Abrikosov vortices/fluxons - topological solitons in superconductor. We can observe them through a microscope, but it doesn't prevent "quantum" phenomenas for them, like interference ( http://prl.aps.org/abstract/PRL/v71/i14/p2311_1 ).
Recent Couder's droplets are different complex macroscopic objects with quantum properties (thread: https://www.physicsforums.com/showthread.php?t=550729 ) - localized objects conjugated with waves they create, what leads to observed:
- interference in double-slit experiment (particle goes a single way, but "pilot" waves it creates go through all paths - leading to interference): http://prl.aps.org/abstract/PRL/v97/i15/e154101 ,
- tunneling (the field depends on the whole history, making that getting through a barrier is practically random): http://prl.aps.org/abstract/PRL/v102/i24/e240401 ,
- orbit quantization (to find a resonance with the field, the clock has to perform an integer number of periods while enclosing an orbit): http://www.pnas.org/content/107/41/17515 ,
- "classical Zeeman effect" (Coriolis force instead of Lorentz): http://prl.aps.org/abstract/PRL/v108/i26/e264503.

So there is a localized configuration - corpuscle (e.g. charge), conjugated with waves it creates - because of some internal periodic dynamics (so called de Broglie's clock, zitterbewegung).
In the language of fields: localized configuration is a soliton, conservation laws with integer "quantum" numbers naturally appears while using topological solitons (e.g. with topological charge), which often have some internal periodic dynamics, like breathers ( http://en.wikipedia.org/wiki/Breather ).

Prof. Faber has shown that in a field of just unit vectors, defining electric/magnetic field through curvature of the field, the dynamics of topological solitons (hedgehog configurations) is given by Maxwell equations - we get a simple electron model (massive) with standard electrodynamics: http://iopscience.iop.org/1742-6596/361/1/012022/ http://arxiv.org/abs/hep-th/9910221.
Here is my essay with more motivations for using soliton particle models and expansion of Faber's model by a single degree of freedom ("quantum phase"), making that the family of topological solitons grows to resemble our particle menagerie with their dynamics: http://www.fqxi.org/data/essay-contest-files/Duda_elfld_1.pdf.

Are solitons only some abstract mathematical constructs?
Maybe there are also macroscopic solitons like fluxons?
Maybe we can use solitons also for effective models of particles like Skyrme?
... or maybe all particles are just solitons of some fundamental field? What field?

I didn't had a look at all suggested references because I am on holidays, but to the question "What would I think about that?" I would answer that I would like to confront your approach with another actual discussion on that forum: https://www.physicsforums.com/showthread.php?t=708695. I like to think about the idea that particles are the signature of some topology or some encoded deformation of the topology (... changing the local symmetries...).

Hi Blackforest, I have briefly looked at the thread you have linked, but honestly I don't see much resemblance as it goes further into abstraction ("spacetime emerging from non-geometric theories..."), while soliton particle models are something conceptually extremely simple and intuitive - particles are seen as localized constructs of some classical field theory (which can be finally quantized, but like for Couder's droplets, I am not sure if it is necessary. We can find effective parameters for Feynman diagrams on solitons for QFT on them).

Just imagine an electron - electric field is hedgehog configuration and goes to infinity in the center. Faber's model can be seen as simple expansion of electrodynamics to explain charge quantization here (as topological winding number) and handle the central singularity (smoothed by Higgs-like potential, giving soliton rest energy: mass).
This model can be seen that in vacuum (far from particles) we have field of units vectors ($v:R^4 \to R^3,\ |v|=1$) - there is a direction in every point of the spacetime.
Topologically nontrivial configuration is for example hedgehog (v(x)=x, all directions point outward a point) and generally topology allows only for integer winding number/Conley index for such singularities - we get charge quantization.

Now we need to define dynamics of this field - Faber has done it by curvature of the field (dropping with distance from the center of hedgehog) - such that electron has electric field as it should and magnetic field is defined correspondingly. Now standard electromagnetic Lagrangian for EM fields defined this way leads to standard electrodynamics for these quants of charge - primitive electron models, governed by Maxwell equations.

There have remained the question about behavior inside the singularity - the Higgs-like potential is for this purpose, like $V = (|v|-1)^2$: it is minimal when field is unit vectors (vacuum dynamics), but singularity may enforce getting out of this minimum (near a particle) - giving soliton rest energy: mass.
We know that strong/weak interaction works practically only very near particles - it is natural from soliton model point of view: when the stress near the center of charge is too strong for electrodynamics, the Higgs potential activates weak/strong interaction, which become less energetically costly there than EM interaction.

Here is nice animation of annihilation of 1D solitons - initially they have energy stored in the potential term (mass), which is released while annihilation: http://en.wikipedia.org/wiki/Topological_defect#Images
Here is a simple simulator of analogous 2D pair creation/annihilation: http://demonstrations.wolfram.com/SeparationOfTopologicalSingularities/
For singularities of the same sign, the closer each other they are, the stronger stress of the field - they repel. For opposite signs analogously they attract:
https://dl.dropboxusercontent.com/u/12405967/fig1.jpg [Broken]
The bottom picture is example of de Broglie's clock: enforced by the particle evolution of "quantum phase".

Last edited by a moderator:
member 11137
Hi Blackforest, I have briefly looked at the thread you have linked, but honestly I don't see much resemblance as it goes further into abstraction ("spacetime emerging from non-geometric theories..."), while soliton particle models are something conceptually extremely simple and intuitive - particles are seen as localized constructs of some classical field theory (which can be finally quantized, but like for Couder's droplets, I am not sure if it is necessary. We can find effective parameters for Feynman diagrams on solitons for QFT on them).

So, coming back into the real world... thanks for your answer. I must apologize because my intervention was unclear and not precize. There is effectively no apparent common point between the "soliton approach" and the "pre-geometric" one; and this is exactly the reason why I meant it would perhaps be useful to confront them. The soliton approach is not only a mathematical tool built on a differential equation; it is allready related to experiments and plays a fundamental role in physics (just, e.g., look at the wikipedia article on that topic -and references in it). My impression is that that "behind the standard model" forum here seems to be centred on the research of a modern and new description of what the particles are. The loop quantum gravity (LQG) approach is the prefered one here but the discussion is open enougth to allow "parenthesis". A non-specialist like me get the sensation that the LQG and the spin foam approaches are obligatorily landing on a new territory where graphs and discontinuities play the main role. The Hossenfelder´s essay mentionned in the other discussion (my proposed link for a constructive confrontation) is a progress in that sense that it proposes a link between the discontinuous world of the modern theories (scattering around the topological defects) and our perceptible world. Do we have to model the theories concerning the discontinuous world in such a way that equations of motions characterizing the continuous one are recovered? My intuition is: yes.

Hi Blackfores, soliton models do not require real singularities or discontinuities - they are smoothened thanks to using Higgs-like potential: getting rest energy/mass instead, exactly like in nice animation here.
E.g. potential $V(x)=(|v|^2-1)^2$ has minimum for symmetry breaking situation with unit vectors (generally referred as vacuum), but situation like topological constraint can enforce getting to symmetric: zero vector. So getting closer to the center of hedgehog, finally the stress of the field will cost more than getting out of potential minimum - leading to symmetric $v=0$ situation in the center. Intuitively - electromagnetism would reach infinity there, so there are activated weak/strong interactions to prevent that.
So there is some minimal energy involved with this type of structure: of the potential and stress involved - this rest energy is just a mass of this kind of soliton and can be released while annihilation, like in the animation.

Hi Jarek. I too am very interested in the soliton approach to particles. While the paticle/wave duality has not gone unnoticed, I have not found too much material. As this is not my current field, my "review" of the material will be quite basic, because I didn't go through it
First I stumbled upon the book
https://www.amazon.com/dp/9812562990/?tag=pfamazon01-20
Besides the writing of the book itself, my main criticism is that a great emphasis is put on wave/particle like effects, but little is done on justifying the model equations used, except that they appear in connection with some macroscopic systems. My feeling is that the equations should be justified from first principles.
Then, I found several works on attempts to find pure field solutions of significant particle equations. Finster et al. article on Einstein Dirac Maxwell system solutions
http://arxiv.org/abs/gr-qc/9802012
is the most noticeable I found, but I didn't find anything on multiparticle solutions, or interactions. Mainly proposals for ground state.

This suggests (to me) that solutions to full coupled field equations (and the interaction between localized and distributed solutions) might show several features of the phenomenology attributed to quantum mechanics, as the drop/surface system of Couder (with totally unrelated equations) do. However, I understand a major objection for this line remains: nonlocality of quantum mechanics. Two distant particles may collapse to complementary states "instantanously". However, I feel that might be overcome by suggesting that particles collapsing together might actually not be so far to each other. I think this is what is being attempted recently in
http://arxiv.org/pdf/1306.0533v2.pdf
where a possible relation between entanglement and "wormhole" bridges is set.
All of this is of course an intuitive view, as I have more background in nonlinear dynamics than I have in particle physics.

Last edited by a moderator:
I would like to leave a question arising from the book on nonlinear quantum mechanics referenced above. There, the specific equations chosen have the particularity of being integrable, and having a well defined inverse scattering method (IST). While this is very useful, I don't know if this would be a requirement of the field equations or a coincidence (is the Einstein Dirac Maxwell integrable? if so, is there a systematic way to obtain the IST?).
Moreover, is there a physical interpretation to the associated linear eigenvalue problem? if it is supposed to encode the conserved quantities, it should have a relation to the physical parameters of the particle

ZapperZ
Staff Emeritus