Is there something deeper in soliton-particle resemblence?

Solitons are localized constructions of some field, which:
- generally maintain their structure/properties,
- have some additional rest energy (mass),
- for which there is pair creation/annihilation (there is various number of them),
- their properties usually are integer multiplicities ('quantum numbers' like winding number) for which there appears conservation laws,
- they interact through field deformations they create - causing attraction/repelling of opposite/the same,
- we observe, expect interference of them (e.g. through mathematical decomposition into plane waves),
- ...
This resemblence to particles of our physics or e.g. looking at electric field around electron suggests that maybe particles aren't some additional 'out of the field' entities, but just some its localized constructions - solitons.
So I personally believe that we should search for a field which family of solitons is exactly what we see in our particle physics - I've started discussion about such concrete approach and was suggested by AlphaNumeric to separate it from this general question.

What do you think of solitons?
Are they just abstract concepts? Useful macroscopic models? Separate entities? Or maybe there is something deeper in their resemblance to particles?
Options I gave in the pool don't cover all the possibilities, so please explain and discuss your view on their general status.

 

Log in or Sign up to hide all adverts.

As you wrote, the constituents of solitons are some fields, [I would add, that we can think of solitons as solutions to some nonlinear equations (and here technicalities and definitions follow which are not of concern now) that do not neccessarily have any connection with realistic physical fields]. Fields are composed of particles but fundamental particles are not fields (expl. later). Therefore, If I think about particles, I would rather compare solitons with the particles' pilot wave functions in some situations when solitonic solutions appear, not with the particles as such. For example, the electron is some abstract notion whose structure we identify with a single spinor and with some additional set of labels -- fixed numbers assigned to it, such as the intrinsic spin, electric charge, mass etc. Components of the spinor as solutions to some quantum equation, say Dirac equation supplemented with some additional potential dependent in a nonlinear manner on the psi and maybe its derivatives, and external fields, may in some situation be solitonic in nature, that's all. To me, a particle is some group-theoretical notion, a mathematical entity which transforms in this and that way under some symmetry group action - a fiber in the mathematical language. Only in some representations, say in the position space, we see that the components in some map of that abstract entity (say spinor's components for the electron) are solutions to concrete equations, and the components may occur to be solitons in some situations. I do not think that quatum numbers associated with some symmetry groups (such as the spin or mass of a particle in the case of the Poincare group) are something originating from topological charges. Even the electric charge is not of solitonic nature despite it can be calculated from the Gauss law -- electric charge carries only the zero-energy piece of information, or in other words, the infrared part of the whole electromagnetic field the particle endowed with the charge may produce, the charge it is not an effect of solving a nonlinear equation and finding the associated topological charge, but is the property of space in which the particle 'lives'. Similarly as the particle's spin must be a multiple of 1/2 because of isotropy of the
physical 3dim-space [the group of rotations and various representations] in which the particle 'lives' and not because of a topological property of a solitonic field describing the particle. To me the electric charge (more precisely the fine structure constant e^2/hc which is a pure number like pi) -- its numerical value -- is the property of space in which the particle resides and therefore cannot be something associated with the soliton used to describe the particle.

 

Log in or Sign up to hide all adverts.

I wasn't going to reply as it would bump the thread but... I never said to split the thread, I though you should explain yourself more. 8nstead you spout more nonsense and say nothing. Be SPECIFIC about what problem or equation you consider, not just vague.

 

Log in or Sign up to hide all adverts.

Is 'h' a property of space? Or, for that matter, is 'e' a property of space? When I think of "property of space", I think "Einstein's equations", not "the stress energy tensor that happens to exist in this universe".

 

Constrained, you write "Fields are composed of particles but fundamental particles are not fields", but looking at e.g. electric field around electron, it's very 'singular' there - so electron itself has integrated very nontrivial situation of the field - why do you think it's not enough: that electron need to have also something 'out of the field' in its nature?
You see spin as some abstract spinor, while it brings also magnetic moment (singularity of magnetic field) - so spin can be also seen 'classically' - through spatial structure of EM field around.
You refer to Gauss law - it says divergence of electric field is usually zero (no source), but positions of charged particles - 'standardized' singularities: of charge being integer multiplicities. For topology such standardization is natural - Gauss law can be seen that for boundary of our region, we project electric field directions into 2D sphere and calculate number of times the sphere was covered this way - around a single charged particle it was covered once (ok, it's good description for ellipsoid field, but for electric it's a bit more complicated). While mass correspond just to rest energy of our particle (minimal energy required to glue given quantum numbers/topological constrains).
"Only in some representations, say in the position space(...)" - we can use different representations which are equivalent - soliton models focus on spatial structure, but transforming it into another one, your abstract description could be more practical.

AlphaNumeric, solitons are quite popular today, so if on thread about such simple particle model you were able to discuss everything but the topic, maybe this thread will be more suitable - just explain how do you see status of solitons in physics.
BenTheMan, fundamental constants are just parameters in some fundamental equations - h relates frequency with energy, e relates distance of simplest singularities of electric field with energy of such field configuration...

 

BenTheMan, some numbers are properties of space, for example pi. The fine structure constant involves h, c and e , but it is dimensionless like pi, pure number. You can construct pi in many ways, eg by aplying the Cauchy theorem to any holomorphic function
in some region: take a line integral f(x)dx/(y-x) over a loop lying in that region and encircling a point y and you will get f(y) times 2(pi) i. Here pi appears, always the same independently of f. Here we had a structure on a flat euclidean-like plane and pi appeared as the characteristics of that plane -- try to do that on a cone and the result should be different. In the same way I say that the spin is a property of space -- particles have it because they have some property that can be described by spinors or a set of spinors, which transform under rotations in space in a unique way -- the quantization of spin and its universality irrespectively of the structure of particles comes from the group of rotation not from some solitonic fields that could constitute such particles. The similar kinematical way I think the electric charge is quantized and universal, the fine structure constant or equivalently electic charge e^2/hc=1/137... is dimensionless pure number which is is a property of space -- there must be a kinematical structure involved with electromagnetism that is characteristic of electromagnetic potential -- a fourvector gauge field in spacetime -- which gives rise to the unique number e^2/hc, similarly as there is a kinematical structure associated with
particles living in that space -- the rotation group in space -- which gives rise to intrinsic spin of particles, and not solitons or energy-momentum tensors or other matter fields constituting particles carrying charge or spin. To make it less vague I would say: there are things that cannot be described in the hamiltonian language and which are purely kinematical. To say that a particle is a soliton of some field, is possible only in the language of a Hamiltonian or Lagrangian description -- you must have an action integral for such a field
only then you can construct some currents and the associated charges, energy momentum tensors, conserved quantities or even topological charges. I think that spin is of kinematical origin, just like the group of rotation teaches us, and therefore the quantization of spin and its universality cannot be described at a Hamiltonian level (although the spin can be modeled at the Hamiltonian level -- and this is what solitons offers among other things -- useful models of particles). This is my opinion, and therefore I prefer to say that solitons are useful mthematical concepts that have
much to do with real physics as they enable in a simpler setup to see or mimic some properties of macroscopic or microscopic world or even can be used as effective models
of some complex particles, ensembles of particles and their effective interactions, but cannot be used to explain the spin quantization and to predict the numerical value of the fine structure constant or other coupling constants or the spectrum of masses of fundamental particles.

 

Jarek, I explained what I meant "the out of the field structure" in my reply to BenTheMen. I understand your point with the Coulomb field of an electric charge. It indeed looks like the hedgehog field characteristic of some solitons (like skyrmions) and indeed this structure could be modeled by a soliton with some profile
function that has continously distributed charge and mass or having the charge as some singularity, knot ect and that even the charge quantization could be explained in that way. As I understand you, you would like to
devise a solitonic description of fundmental particles that would be able to explain charge or spin quantization. I also agree that such quantization could be described at solitonic level as topological charges etc and that this description seems natural if not even obvious. However, I think that a fundamental theory of particles should also explain the numerical values of the coupling constants
such as the fine structure constant in the case of particles carrying the electric charge. My point is that these values cannot be explained at the Hamiltonian or Lagrangian level. I'm afraid you will have to put these values to your theory by hands through some eigenvalues of some matrices, but the values will not naturally follow from your model. Something like this is offered by the standard models of paticles, there ar as I remember 19 free parameters
that are fixed by the experiment, by nobody knows why the values are such an not other, and nothing would change in the structure of the standard model if the values were slightly changed -- there is alwys a margin for such values and nothing bad happenes if they are changed within it. I think that from a good model of particles one could expect to obtain unique such values that could not be changed without destroying the structure of the model.
I think based on the group of rotation or the quantum theory of the electric charge that there are some properties of particles that are induced by the
the ambient space in which they live, and which are kinematical in nature and cannot be described at hamiltonian level. I'll write more later in reply to your answer.

 

Charge quantization means that the smallest nonzero charge inside e.g. a ball is the elementary charge - there should be some configuration of the fields inside such ball - e.g. 'hedgehog' of electric field.
Having a tiny ball on which boundary EM field says that there is elementary charge inside, we would like to understand how it is continued inside and finally glued - preferably in continuous way, without infinities. If there are different possibilities of such configurations, physics should prefer the one of lowest energy - while movement, field configuration inside this tiny ball should practically remain the same - it's what being a soliton means.
But what is happening in its exact center?
Extrapolating from external situation, we would say that in its any neighborhood, not only there is achieved any finite value of electric field, but also they cover all possible directions - in the center not only field becomes infinite, but also cannot have defined direction.
Such particle usually has also spin and so magnetic moment - is simultaneously a tiny magnet, so if it's exactly point, there is also infinite magnetic field density going through this point.
Anyway, in close vicinity of this point, the concept of EM field looses its sense ... simultaneously we believe that on extremely close distances weak/strong interaction starts working - so maybe physics doesn't break there, but just in such extreme conditions EM interaction is deformed (GUT) into weak/strong (it's exactly what's happening in ellipsoid field models) ... ?

To summarize - I see building particle soliton model of e.g. electron as just finding spatial continuation of field around it down to the center - which is physical: continuous, without infinities, is electromagnetism on the boundary and has relatively lowest energy.
If we would have such reasonable models, there would be no need for some additional nature of particles - fields on tiny sphere around particle would unequivocally define situation inside it and so define the particle.

Please explain it?
The fact that spin is multiplicity of 1/2 can be naturally seen as topological property.
Spin alignment is described by magnetic field around because of integrated magnetic moment.
Such model might bring some additional nontrivial relations to these 19 parameters, reducing this number ... but it would still probably need some fundamental parameters - using them it should bring numerical values of for example rest energies (masses), expected lifetimes...

 

How exactly do you pass from "your view of a static property of space pi", to a dynamic property of space spin.

Under no other circumstances can you pass from the static to the dynamic without some addition mechanism/force. So, how exactly how do you pass from pi as a property of space to spin as a dynamic property of space without any mechanism other than magic.

 

OK Jarek, if you give the argument that, say, the scalar eigenfunctions of a rotation generator in 3dim must be periodic about, say the rotation axis 'z', then when you require the functions should be periodic in the rotation angle, that is, be proportional to $\exp(im\phi)$ with integer m, then indeed you can interpret the phase \phi as something carrying topological charge of spin and that the spin is something topological in nature: eg: $\phi=ArcTan(y/x)$, hence $d\phi=\frac{-x dy+y dx}{x^2+y^2}$,and then $d d\phi=0$ except for x=0=y which is excluded (here, I mean the exterior derivative). Since $d^2\phi =0$ and the result of taking
a loop integral of $d\phi$ about $z$ gives $2\pi$, there must be in the field $\phi(x,y)$ a singularity located at $x=0=y$ (or smooth field and a hole in the plane) which can be inerpreted as carying the unit angular momentum m=1 and $(\phi(2pi)-\phi(0))/(2pi)$ as the corresponding unit topological charge (in this scalar representation we see that the angular momentum is an integeral multiple of 1). At the moment I do not see a good argument to show in similar way that the spin can be 1/2 , not 1, without referring to spinors. Are you able to construct in similar way a topological charge that equals to 1/2?
Even if yes, this would not prove that spin is a result of the existence of some solitonic field and that spin is a topological charge of that field. The spin could be interpreted in this way, however, it is not necessary for understanding the nature of spin quantization its universality and its quantum. Constructions of the type as above or similar, use the notion of loop space -- eg two loops encircling some point on Euclidean plane are equivalent if they have the same winding number - but this is still a grouptheoretical notion referring to a group of rotations and to the property of Euclidean plane. I surmise this is the naturally seen topological property you meant, isnt it?.
Anyway, why to model spin and its quantization when it already has its purely kinematical explanation (that is such, which is not based on Hamiltonian description and not even touching the issues with constituents of the particles )?
As concerns the value of the fine structure constant, actually there is a closed kinematical scheme which "sees" its numerical value; the principles of the theory can be found in: A. Staruszkiewicz, Ann. Phys. (N.Y.) 190 (1989) 354. I think that if Dirac did not die in 84, seeing the result of the paper, he would call it one of the most interesting observations made in mathematical physics of 20th century. You probably may know the Author and his results

Please Register or Log in to view the hidden image!

.

 

chinclu,
who said that the spin is a dynamical entity? Not me. Have you ever seen a Hamiltonian which explains the spin of Dirac electron? The spin resides in the spinor structure
assigned to the electron and it would be 1/2 irrespectively of which Lagrangian you would propose to derive the equation of motion for the electrons' wave function, as long as you do not choose something different than a single spinor to describe the electron's structure. The spin carried by electron is not something of the same nature as, say, the energy it may happen to have in the hydrogen atom, the energy is dynamical in nature, the electron's spin is not.

 

Constrained, I see the reason for spin quantization condition exactly as you say – ‘z’ axis quantum rotation operator transforms some kind of phase of the field in $\exp(im\phi)$ way, where ‘m’ is spin. Indeed in particle models this ‘m’ isn’t just integer number as for standard phase concept, but integer multiplicity of ½ - for this reason the entity which field we consider, while rotating by half rotation (pi radians), should return to the initial state (like for ellipse/ellipsoid field).
If field has such configuration, there appears problem in the center of such picture – around this point all phases are obtained and so there is no way to assign a phase there in continuous way – differential topology call such situation topological singularity (in ellipse field it's handled by deforming into circle) – this spin is called 2D Conley index.
We can use its 3D version for charge quantization– for that purpose it’s more convenient to use different formulation – take a ball around the singularity (2D/3D) and project local situation (2D/3D 'phase') from its boundary (loop/sphere) into some abstract loop/sphere – Conley index is the integer number of times this loop/sphere is covered this way (homotopy class of this projection) – giving quantization of spin/charge (hedgehog covers it once).
To use it, the ‘phase’ of our field has to be something more general – represent directions not only in 2D, but also in 3D. And if it is so close to spin(magnetic moment)/charge, these rotational modes should correspond to electromagnetic properties (exactly like in ellipsoid field, deformations required to handle singularities correspond to weak/strong interaction).

You feel a need for some separate ‘solitonic field’, but maybe we should rather go another way – not adding succeeding looking independently entities, like additional out of field nature of particles, but just oppositely – try to see the whole physics as emerging from relatively simple model:
- which in vacuum becomes electromagnetism (and gravity), but to handle singularities, near particles it can look like a different interaction (weak/strong),
- which leads to quantization of spin, charge and other quantum numbers, (e.g. on topological level),
- these quantum numbers should fully identify field configuration - particle (even distinguish between long/short living neutral kaons ...),
- they should be usually in the lowest energy state for these given constrains (quantum numbers) – this rest energy is their mass (through Lorentz invariance becomes also inertial mass and should deform gravitational field to became also gravitational mass),
- ... ?
I’d probably should know Staruszkiewicz view, especially he was lecturer of field theory course I've attended to

Please Register or Log in to view the hidden image!

(which occurred to be GRT course) – I think I’ll finally look at his papers as soon as I will have access ... but Dirac have seen EM field as something separate from wavefunction, what leads to that solitons are seen as some separate entities from particles - I disagree and it leads to unobserved monopoles ...

 

Jarek, As to the Dirac monopoles, Staruszkiewicz in his quantum theory of the phase gave a simple counterexample against the existence of magnetic monopoles. He observed that asymptotic
electromagnetic field at spatial infinity that carries the information about charge can be described by two scalar functions which he called the electric and magnetic part on account they enter the total Lagrangian with opposite signs. The electric part which carries the information about the electric charge has the correct sign -- it leads to positively defined scalar products for quantum states, whereas the magnetic part would
lead to negatively definite norms which in turn would lead to ill defined Hilbert space. Therefore the magnetic part must be rejected. And this is the crucial point in the Staruszkiewicz construction, that magnetic monopoles cannot exist in order to have a well
behaved Hilbert space for electrically charged states. One could say based on this standpoint that magnetic monopoles cannot exist, since otherwise there would be an inconsistency in the mathematical structure describing the electric charge. This prediction is consistent with the fact the magnetic monopoles has never been observed.
I do not see any topological argument against the existence of Dirac monopoles, say solitonic reason for the Dirac monopoles' nonexistence, whereas Staruszkiewicz theory does provide an argument which is field-theoretical in nature. In your last sentence you say Dirac regarded EM field as something separate from wavefunctions, I do not understand clearly what you mean, but I can say that Staruszkiewicz construction concerns something very abstract that resides in the electromagnetic field, the so called infrared part of EM fields that carries zero energy and the coresponding quantum states or Hilbert space is quite distinct from wavefunctions of particles. (later Ill answer the other part of your reply)

 

Jarek, if we even assumed that you managed to explain spin as topological charge of your solitons, then what about the isospin of particles? Formally, it is introduced the same way like ordinary spin but the interpretation is different. How would you discriminate between spin and isospin in your model? Is the 'ellipsoid field' sufficient?

 

Constrained, ellipsoid field can be seen as a first step of approach for gluing different interactions without infinities, but I think adding to it a possibility of a similar but different kind of spin would greatly increase its family of solitons ...
Spin correspond to magnetic moment - is defined by EM field around as a magnetic singularity gluing them inside ... while isospin seems to be completely abstract concept(?)
Thinking about the correspondence of solitons of the basic model, I was rather taking into consideration not properties of current models like isospin, but I focused on experimental properties of particles - masses, decay modes, EM properties. Looking at e.g. baryons in this model, there is plenty of ways of creating such singularity - choice of spin curves types, strengeness, charge placement, and most importantly: way of combining spin curve with spin loop - practically each local minimum of such configuration space could be identified as a particle - there would be simulations required to make a better classification ... and there are seen hundreds of such extremely unstable particles ...

Meanwhile I've briefly looked at magnetic monopoles - Wikipedia article says that they require adding q_M(v x E) term to Lorentz force as analogue for standard q(v x B) term ... but thanks of educating Gryzinski's classical models or Aharonov-Cahsher effect (dual to Aharonov-Bohm), I've realized that such term also appears for particles having magnetic moment instead ...
To see it, imagine electron (which have large magnetic moment) traveling classically in electric field of proton - let's change reference frame, so that for infinitesimal time electron stops and proton travels in also magnetic field created by electron's magnetic moment - because of 3rd Newton's law, appeared Lorentz force is also applied to electron ...
So as for magnetic monopoles, electric field should also affect magnetic moments (dipoles) ... ?

 

I had a long post typed yesterday but my crappy internet connection, a netbook with a flat battery and being on a train meant it went to the great Firefox bin in the sky.

My point has been that you want to talk about high level stuff but you don't provide any actual details. The poster Constrained has gone into more details but you haven't risen to the discussion.

As for my view on solitons in mainstream physics there's tons of uses. Solitons arose originally in fluid mechanics, they were witnessed in fluids long before the KdV equation was developed. I happen to know people who research this stuff. Solitons have been in quantum field theory for a long time, particularly in super symmetry (see Nakahara). In string theory they arise even more. Branes are solitonic objects and they are everywhere in string theory. In addition there's the time version of solitons, the instanton, they represent things localised in time rather than space. In quantum field theory they are associated to vacuum transitions, with value \(\langle 0 | 0 \rangle\). In string theory they play important roles in things like Kahler potentials in supergravity theories, something which I can go into detail on if required.

A Lorentz transform can take a static system to a dynamic one, as velocity is frame dependent.

 

Let me preface this with two statements:
1.) I don't know if I disagree with you.
2.) I don't want to make an ass of myself.
Neither one of those has ever stopped me in the past, though

Please Register or Log in to view the hidden image!

I agree that the fact that the fine structure constant, or any of the other coupling constants in the Standard Model, are dimensionless. But these constants most likely stem from some microscopic moduli fixing---that is, there are probably some singlet fields which couple to matter in a certain way, that (when integrated out of the theory) happen to give us a bunch of dimensionless numbers. In this sense, there isn't anything particularly special about the fine structure constant, except that it seems to be finely tuned so that we can exist. Given this, I see the fine structure constant is much more of an "accident" than pi. For example, flat space will _always_ imply a value of pi = 3.14159..., but the value of the fine structure constant will be different in different vacua, if it even exists at all.

Here, you may (rightly) object to my stringy bias, but moduli stabilization is nothing new, and is not inherently a stringy idea. The moduli stabilization problem is manifested most dramatically in String Theory, but exists in _all_ supersymmetric gauge theories, including the MSSM. The SM even has its own moduli stabilization problem, which people usually call the hierarchy problem.

So comparing the fine structure constant with pi is a bit of a mistake, I think.

The kinematical structure must be the photon coupling in the covariant derivative, which (again) is specific to a certain choice of vacuum. This is a microscopic requirement, not a macroscopic one.

 

BenTheMan no I cannot agree, There are some constants that cannot be tuned or be arbitrary. They have to have precise values in order that a given mathematical structure could be consistent. I think that Nature is mathematical and, therefore, the fundamental dimensionless constants such as the fine structure constant or the ratio of the proton mass to that of the electron are not accidental and are results of some unambiguous mathematical structures behind them. For example, somewhere above I gave an example with a loop integral for holomorphic functions that can be regarded as the definition of pi. You cannot change the value of pi without destroying the integral Cauchy theorem, the pi must have precisely fixed value in order that the theorem could be true for all holomorphic functions (the other thing is that the pi is the same as the 'euclidean' pi which is astounding). I think by analogy (see below for justification) that there is a mathematical structure that leads to the fine structure constant and that the value is not accidental but unique.

 

BenTheMan, the discussion is now far from solitons and I do not know if it is all right to continue it here. I do not know the stringtheorists' jargon. But I have also not heard about any single useful prediction of the string theory that could be somehow tested, either it has not gave any solution to unsolved important problems of theoretical physics (at least not yet). I think that the solution to the fine structure constant mystery lies already in the structure of Maxwell fields in Minkowski space, more precisely, in that part of the field which carries the information about the electric charge. One can show that the quantum theory of electric charge is equivalent to a theory of a massless scalar field 'living' on 1+2 dimensional deSitter spacetime + some technicalities. This scalar field in a simple manner is connected with the electromagnetic potential at spatial infinity. There is
a closed dynamical system that can be quantized and which depends in a nontrivial way on the numerical value of that constant. The theory has been discovered in 1989 by Staruszkiewicz who obtained since then important results that some quantum observables change qualitatively if the constant is too large. Among other things the theory distinguishes a spectrum of critical values for the constant, however he has not yet found the observable that would fix the fine structure constant uniquely. Dirac said something that goes like that: important physical theories should be described by beautiful mathematics. Staruszkiewicz construction is indeed beautiful, and leads to beautiful mathematics, touching also the so called pathological representations of the Lorentz group of which Dirac considered to be very important for future physics. The basis of the theory you can find in A. Staruszkiewicz, Ann. Phys. (N.Y.) 190 (1989) 354. Staruszkiewicz seems to me important person at least in the field of theoretical physics, although it is not as 'fasionable' like string-braned-CFT-Multiversed-theorists and other hundredths-of-thousands roundtheworld-ArXivists everysecond spamming the ArXive repository, and therefore not widely known, however, sir Roger Penrose who certainly do is able to distinguish between scientifical fantasy and real science (and this ability is very rare nowadays) in his book "The road to reality" cites Staruszkiewicz paper concerning spinors. There is also a very known and extensively cited Staruszkiewicz paper concerning 1+2 dimensional gravity (although Staruszkiewicz is not very proud of the context in which his paper is cited). I am sure that Staruszkiewicz discovered the right and minimal mathematical scheme that will finally lead us to the precise numerical value of the fine structure constant. Seeing his construction, it is obvious that this value is a geometrical property inscribed into the structure already existing in asymptotic Maxwell fields and Minkowski spacetime.

 

It's worth to add to this thread that there is very interesting topological soliton model of electron by prof. Faber - while we don't observe Dirac magnetic monopoles, he uses dual formulation of EM making them electric monopoles with standard long-range EM interaction (Skyrme's models of mesons and baryons have short-range interaction).
Another view on this model is that it's just reformulation of electromagnetism to have charge quantization deeply written as topological charges - so these quants became separate charge carriers: particles.
His vacuum dynamics is field of unitary vectors, and the simplest topologically nontrivial is hedgehog configuration (v(x) = x/|x|). Curvature of this field drops with the distance - defining electric field directly from this curvature, he gets standard electromagnetic interaction.
Here are some of his papers about this model: http://arxiv.org/abs/hep-th/9910221 , http://iopscience.iop.org/1742-6596/361/1/012022/
Here is my recent FQXi essay with discussion about such search for field configurations behind particles: http://fqxi.org/community/forum/topic/1416