# Ellipsoid field solitons - particle menagerie correspondence?

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Looking at electron, there is singularity of electric field in it - its values seem to tend to infinity, but also directions create topological singularity ...

This picture suggests that maybe we don't need some additional (out of field) entities for particles, but this construction of field itself is the electron - that particles are some characteristic localized constructs of the field, maintaining their structures/properties - are solitons.

Skyrme used such constructions to model baryons, they automatically give particles masses (rest energy), allows for various number of particles because of annihilation/creation, there is corresponding attraction/repelling for opposite/the same ones, they have integer 'quantum numbers' ...

For example here is nice animation of soliton/antisoliton annihilation which released energy gathered in them (mass) as analogue of photons: http://en.wikipedia.org/wiki/Topological_defect

Anyway, the perfect situation would be finding a field which family of topological soltions corresponds well to the whole particle menagerie with their properties, decays, dynamics ... and which dynamics became electromagnetism and gravity far from particles (vacuum state).

It occurs that extremely simple field: ellipsoid field surprisingly well qualitatively fulfills these requirements - just a field of real symmetric 3*3 (4*4) matrices, which prefers some set of eigenvalues - it can be seen as stress tensor or as less abstract skyrmion model, but with Higgs-like potential (with topologically nontrivial minimum) or as expansion of ellipse field of light polarization concept (considered by 'singular optics').

Rotating ellipse/ellipsoid by 180deg we get the initial situation, so the simplest constructions of such field have spin 1/2, like in this demonstration allowing also to see attraction/repelling caused by minimizing variousness of the field: http://demonstrations.wolfram.com/SeparationOfTopologicalSingularities/

In ellipsoid field in 3D we can choose these axes in 3 ways - we get 3 families of spin 1/2 constructs. There can be created charge-like construct on it getting 3 families of leptons (topology says that they need also to have spin). Then we get constructions like mesons, baryons which finally can join into something like nucleus. Qualitatively masses, properties, decay modes are practically exactly like in particle physics.

Far from solitons dynamics becomes 2 sets of Maxwell's equations - for electromagnetism and gravity: dynamics of rotations of 3D ellipsoids (no gravity) gives EM and small perturbations of fourth axis of 4D ellipsoids (which has the strongest tendency to align in one direction) gives Lorentz invariant gravity (called gravitomagnetism).

All of it can be basically seen on pictures - they start on 21 page (after motivations for considering solitons) of this presentation.

It is described and derived in 4-5 sections of this paper.

I'm going to make simulations some day, but I would be grateful for any constructive comments now - this model is very 'strict': we cannot just guess and add new Lagrangian terms as in standard approach - it's quite correct or just wrong: a single real qualitative problem would probably take it to trash ...

What do you generally think of soliton particle models?

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Ok, let me take some basic description here ... maybe it will help with discussion ...

Let's start with ellipse field - there is an ellipse in each point of 2D plane, which prefer some shape (2 radii) because of potential.

Mathematically - there is tensor field - real symmetric matrix in each point, which prefers some set of eigenalues being constants of the model (its eigenvectors represent ellipse axis of radius being corresponding eigenvalue).

Now here are two simplest topologically nontrivial situations for such field:

looking at loops around such points, 'phase' make some mulitiplicity not of full rotations like we would expect for vector field, but thanks of ellipse symmetry - some multiplicity of 1/2 rotations - singularities from picture have index/spin +1/2, -1/2.

On such loop, there are achieved all possible angles of ellipse axis - while looking at smaller and smaller loops down to a single point, we see that in some moment these entities have to loose directionality - in this case ellipses have to deform into circle (two eigenvalues equalize).

This enforced by topology deformation means that we get out of potential minimum - soliton chooses minimal energy for this topology, which is nonzero - it has rest energy (mass), which can be released as nontopological excitation (photons) while annihilation with antisoliton.

This mass creation mechanism is based on that potential minimum is topologically nontrivial (circle) - exactly as in Higgs potential: Mexican hat ((|z|^2-1)^2) - if on a circle the field achieves all values from the energy minimum (|z|=1), inside this circle it has to get out of the minimum, giving soliton mass.

Such solitons create/are strong deformations of the field - standard energy density of such field increase with its variousness - taking opposite solitons closer (the same further) make the field less various - give them attraction(repelling) force - it can be see using this demonstration.

Ok, let's go from ellipse field used e.g. by 'singular optics' as representing light polarization to 3D ellispoid field in 3D.

Now singularities as previously create 1D constructs - vortex line/spin curve.

We can make them in three ways - choose one axis along line and remaining two make singularitiy equalizing these 2 eigenvalues.

Now they have mass/energy density per length, which generally should be different in these 3 cases - let's call them electron/muon/tau spin curves correspondingly. By synchronous rotation 90deg of axes along such line, they theoretically can transform one into another.

Loops made of something like this are extremely light (comparing to further excitations), very weakly interacting and generally can transform one into another - we get 3 families of neutrinos.

Now if along such 1D construction, axes rotate toward/outward, we get charge-like singularity on it, transforming spin curve into opposite one, like on this picture:

in such more complicated singularity, now topologically all three axes have to equilibrate in the center, giving it much larger rest energy (mass) - we get three families of leptons.

Alternative view on such singularity is by looking at axis along curve - it's for example targeting the center while such singularity, so looking at perpendicular submanifold which is nearly sphere now, we have to align somehow remaining two axes there - hairy ball theorem says we cannot do it without singularity - or in other words: that electron has to have also spin.

Further excitations is making loop with additional twist along it, like in Mobius strip - in center of something like this appears really nasty topological singularity requiring much larger ellipsoid deformations and so giving these unstable meson-like structures larger mass.

Then there are knots - loop around curve of different type - now on inside curve phase make 1/2 rotation, while on the loop it makes full rotation - enforcing nasty deformations on their contact - we get even heavier constructions: baryon-like. Some integrated irregularity of inside curve could make such combination easier and so proton has smaller mass than neutron.

Now if we have two loops around one line, they generally repels each other, but the energetic income of having charge, make them get closer to share the charge - getting deuteron with centrally placed charge (like on this picture).

Further nucleons can also help holding their structure by creating/reconnecting loops - creating complicated interlacing structures like here:

While deep inelastic scattering, such mesons/baryons seem to be made of 2/3 regions.

Weak interaction here corresponds to spin curve structure, while strong to interaction between two such structures - they work only on specific for these constructions distances (asymptotic freedom).

Far from singularities, ellipsoids have fixed shape and so the only dynamics is through their rotations - it occurs that such spatial rotations can be described using Maxwell's equations - we get electromagnetism and situation around singularities gives them magnetic flux/charge.

To get full spacetime picture, we have to use 4D ellipsoids in 4D instead - fourth axis correspond to local time direction (central axis of light cones) and has energetically strongest tendency to align in one direction - in such case we would get pure EM as previously, but small rotations of this axis gives additionally second set of Maxwell's equations - Lorentz invariant gravity (called gravitomagnetism) - in this picture spacetime is flat and what is curved is space alone - submanifolds orthogonal to time axis.

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ps. If someone is interested, I believe here has finally started a really discussion about ellipsoid field: http://www.sciforums.com/showthread.php?t=106416

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Far from particles, the only working interactions are electromagnetism and gravity - taking a (e.g. 1 fm radius) sphere around a particle, the situation of these two fields fully describe charge, magnetic moment and gravitational mass inside.

We know that charge (and spin) is quantized - directions of electric field around charge are in 'hedgehog configuration', what is called topological singularity, among which there also appears natural quantization of topological charge - how much time the projection from sphere to sphere of directions of field values covers the sphere (Conley index).

The problem with these two interactions is that if we would like to just extrapolate them to the center of sphere, it becomes nonphysical - goes to infinity, and its direction cannot be defined in the center.

So to remain physical, these interactions just have to deform somehow to be able to glue together in a continuous way without infinities ... and we have additional interactions which work only in such regions (weak/strong) - so maybe they are just two faces of a single interaction (GUT) ... ?

We can look at ellipsoid field model of e.g. electron as just a trial of such gluing surrounding EM field in physical way and the required going out of energetically preferred EM interaction, leads to some rest energy prisoned there (mass).

To summarize, we can look at this field as a way to fulfill natural requirements:

- which in vacuum becomes electromagnetism (and gravity), but to glue it without singularities in 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 identify field configuration - particle (even distinguish between long/short living neutral kaons ...),

- field configuration of particle 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)...

Another requirement for such models could be forgotten de Broigle's doctoral thesis concept:

that with particle's energy: E = mc2

comes its internal periodic motion: E = hf

It is remained in very interesting Hestenes paper, in which he also describes recent experimental confirmation of this effect (called e.g. zitterbewegung).

For example while particle moves, the first relativistic correction of its mass is $m\sqrt{1+v^2}\approx m+mv^2/2$

So movement increases frequency, leading to additional phase shift proportional to $\int pvdt = \int p dq$

Requiring that while particle makes a loop, its internal phase returns to initial one, gives Bohr's quantization condition.

Such internal periodic motion creates also periodic wave-like perturbations of surrounding field - giving localized entity also wave nature ...

There are extremely interesting recent papers of Couder, Fort et al. in which they experiment with macroscopic entities having similar wave-particle duality: about oil droplets on vertically vibrating liquid surface - constantly creating periodic waves around - interaction with these waves allows for 'quantum effects': interference, tunneling depending on practically random hidden parameters or orbit quatization condition - that particle has to 'find a resonance' with field perturbations it creates - after one orbit, its internal phase has to return to the initial state.

Returning to ellipsoid field - looking at matrix allowing to represent its vacuum dynamics (rotations) as EM+gravity, electric field corresponds to spatial rotations while time evolution and gravitational field to small rotations of time axis (central axis of light cone) - so its particles have kind of two internal clocks: the first one has frequency proportional to electric charge and the second to gravitational mass.

Edited by Duda Jarek

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I wanted to add that there is conceptually very similar soliton electron model developed for 20 years by prof. Manfried Faber from Vienna: http://arxiv.org/PS_cache/hep-th/pdf/9910/9910221v4.pdf

Instead of three axes of ellipsoids, it has only one - the field in vacuum is from 2D sphere (direction space of 3D), which is equator of abstract 3D sphere, but getting out of the equator costs (Higgs-like) potential energy (instead of deforming in ellipsoid field).

Dynamics of directions in vacuum effectively becomes electromagnetism - in his paper there is very throughout explanation of that (muuuuch better than in mine).

So having hedgehog field configuration for charge, to glue this field to the center, it has to get out of the equator to one of 2 poles of 3D sphere - choosing +- spin of electron and giving it rest energy (becoming also inertial energy thanks of Lorentz invariance).

In ellipsoid field the two additional perpendicular axes give one additional degree of freedom - representing quantum phase required for wave nature - it rotates as soliton's internal clock. Professor invited me and we talk a lot - he doesn't like the internal clock concept and he would like to avoid it.

These additional axes also make the family of solitons grows up to menagerie from our physics ...

Edited by Duda Jarek