Yes, chinglu - you needed only to look at some link I gave ...
Ok - if it's too complicated to look at
21-30 page of presentation of
4-5 section of paper, let me take some basic description here...
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.
...
Questions? Comments?