1. In quantum mechanics spin can be described as that while rotating around the spin axis, the phase rotates "spin" times – in mathematics it’s called Conley (or Morse) index of topological singularity, it’s conservation can be also seen in argument principle in complex analysis.
So particles are at least topological singularities. I'll try to convince that this underestimated property can lead to explanations from that fermions are extremely common particles up to the 'coincidence' that the number of lepton/quark generations is ... the number of spatial dimensions.

I've made a simple demonstration which shows qualitative behavior of the phase while separation of topological singularities, like in particle decay or spontaneous creation of particle-antiparticle: http://demonstrations.wolfram.com/Se...Singularities/
The other reason to imagine particles as topological singularity or a combination of a few of them is very strong property of spin/charge conservation. Generally for these conservation properties it’s important that some ‘phase’ is well defined almost everywhere – for example when two atoms are getting closer, phases of their wavefunctions should have to synchronize before.
Looking form this perspective, phases can be imagined as a continuous field of nonzero length vectors – there is some nonzero vector in every point.
The problem is in the center of a singularity – the phase cannot be continuous there. A solution is that the length of vectors decreases to zero in such critical points. To explain it form physical point of view we can look at Higg’s mechanism – that energy is minimal not for zero vectors, but for vectors for example of given length.
So finally fields required to construct such topological singularities can be field of vectors with almost the same length everywhere but some neighborhoods of the singularities where they vanishes in continuous way.
These necessary out of energetic minimum vectors could explain (sum up to?) the mass of the particle.

Topological singularity for charge doesn’t have something like ‘spin axis’ – they can be ‘pointlike’ (like blurred Planck's scale balls).
Spins are much more complicated – they are kind of two-dimensional – singularity is ‘inside’ 2D plane orthogonal to the spin axis. Like the middle of a tornado – it’s rather ‘curvelike’.

The first ‘problem’ is the construction of 1/2 spin particles – after rotating around the singularity, the spin makes only half rotation – vector becomes opposite one.
So if we forget about arrows of vectors – use field of directions – spin 1/2 particles are allowed as in the demonstration – in fact they are the simplest ‘topological excitations’ of such fields … and most of our fundamental particles have 1/2 spin …
How directions – ‘vectors without arrows’ can be physical?
For example imagine stress tensor – symmetric matrix in each point –
we can diagonize it and imagine as an ellipsoid in each point – longest axis (dominant eigenvector) doesn’t choose ‘arrow’ – direction fields can be also natural in physics … and they naturally produce fermions …
It's emphasized axis - eigenvector for the smallest or largest or negative eigenvalue would have the strongest energetic preference to align in the same direction - it would create local time dimension and its rotation toward energy - creating gravity and GR related effects.
One of other three axises could create one type of singularity, and there still would remain enough degrees of freedom to create additional singularity - to combine spin and charge singularity in one particle - it could explain why there is 3*3 leptons/quarks types of particles.

Another ‘problem’ about spins is behavior while moving the plane in ‘spin axis’ direction – like looking on tornado restricted to higher and higher 2D horizontal planes - the field should change continuously, so the critical point should so. We see that conservation doesn’t allow it to just vanish – to do it, it has to meet with opposite spin.
This problem occurs also in standard quantum mechanics – for example there are e^(i phi) like terms in basic solutions for hydrogen atom – what happens with them ‘outside the atom’?
It strongly suggest that against intuition, spin is not ‘pointlike’ but rather curve-like – it’s a ‘curve along it’s spin axis’.
For example a couple of electrons could look like: a curve for spin up with the charge singularity somewhere in the middle, the same for spin down - connected in ending points, creating kind of loop.
Without the charges which somehow energetically ‘likes’ to connect with spin, the loop would annihilate and it’s momentums should create two photon-like excitations.
Two ‘spin curves’ could reconnect exchanging its parts, creating some complicated, dynamical structure of spin curves.
Maybe it’s why electrons like to pair in orbits of atoms, or as a stable Cooper pairs (reconnections should create viscosity…)

Bolzman distribution among trajectories gives something similar to QM, but without Wick’s rotation
http://www.thescienceforum.com/viewtopic.php?p=158008
In some way this model corresponds better to reality – in standard QM all energy levels of a well like made by a nucleus are stable, but in the real physics they want to get to the ground state (producing a photon). Without Wick’s rotation eigenfunctions are still stable, but the smallest fluctuation make them drop to the ground state. What this model misses is interference, but it can be added by some internal rotation of particles.
Anyway this simple model shows that there is no problem with connecting deterministic physics with squares appearing in QM. It suggests that maybe a classical field theory would be sufficient … when we understand what creation/annihilation operators really do – what particles are … the strongest conservation principle – of spin and charge suggests that they are just made of topological singularities… ?

What do you think about it?
I was said that this kind of ideas are considered, but I couldn’t find any concrete papers?

There started some discussion here:

2.

3. Originally Posted by Jarek Duda
In quantum mechanics spin can be described as that while rotating around the spin axis, the phase rotates "spin" times – in mathematics it’s called Conley (or Morse) index of topological singularity, it’s conservation can be also seen in argument principle in complex analysis.
So particles are at least topological singularities. I'll try to convince that this underestimated property can lead to explanations from that fermions are extremely common particles up to the 'coincidence' that the number of lepton/quark generations is ... the number of spatial dimensions.

I've made a simple demonstration which shows qualitative behavior of the phase while separation of topological singularities, like in particle decay or spontaneous creation of particle-antiparticle: http://demonstrations.wolfram.com/Se...Singularities/
The other reason to imagine particles as topological singularity or a combination of a few of them is very strong property of spin/charge conservation. Generally for these conservation properties it’s important that some ‘phase’ is well defined almost everywhere – for example when two atoms are getting closer, phases of their wavefunctions should have to synchronize before.
Looking form this perspective, phases can be imagined as a continuous field of nonzero length vectors – there is some nonzero vector in every point.
The problem is in the center of a singularity – the phase cannot be continuous there. A solution is that the length of vectors decreases to zero in such critical points. To explain it form physical point of view we can look at Higg’s mechanism – that energy is minimal not for zero vectors, but for vectors for example of given length.
So finally fields required to construct such topological singularities can be field of vectors with almost the same length everywhere but some neighborhoods of the singularities where they vanishes in continuous way.
These necessary out of energetic minimum vectors could explain (sum up to?) the mass of the particle.

Topological singularity for charge doesn’t have something like ‘spin axis’ – they can be ‘pointlike’ (like blurred Planck's scale balls).
Spins are much more complicated – they are kind of two-dimensional – singularity is ‘inside’ 2D plane orthogonal to the spin axis. Like the middle of a tornado – it’s rather ‘curvelike’.

The first ‘problem’ is the construction of 1/2 spin particles – after rotating around the singularity, the spin makes only half rotation – vector becomes opposite one.
So if we forget about arrows of vectors – use field of directions – spin 1/2 particles are allowed as in the demonstration – in fact they are the simplest ‘topological excitations’ of such fields … and most of our fundamental particles have 1/2 spin …
How directions – ‘vectors without arrows’ can be physical?
For example imagine stress tensor – symmetric matrix in each point –
we can diagonize it and imagine as an ellipsoid in each point – longest axis (dominant eigenvector) doesn’t choose ‘arrow’ – direction fields can be also natural in physics … and they naturally produce fermions …
It's emphasized axis - eigenvector for the smallest or largest or negative eigenvalue would have the strongest energetic preference to align in the same direction - it would create local time dimension and its rotation toward energy - creating gravity and GR related effects.
One of other three axises could create one type of singularity, and there still would remain enough degrees of freedom to create additional singularity - to combine spin and charge singularity in one particle - it could explain why there is 3*3 leptons/quarks types of particles.

Another ‘problem’ about spins is behavior while moving the plane in ‘spin axis’ direction – like looking on tornado restricted to higher and higher 2D horizontal planes - the field should change continuously, so the critical point should so. We see that conservation doesn’t allow it to just vanish – to do it, it has to meet with opposite spin.
This problem occurs also in standard quantum mechanics – for example there are e^(i phi) like terms in basic solutions for hydrogen atom – what happens with them ‘outside the atom’?
It strongly suggest that against intuition, spin is not ‘pointlike’ but rather curve-like – it’s a ‘curve along it’s spin axis’.
For example a couple of electrons could look like: a curve for spin up with the charge singularity somewhere in the middle, the same for spin down - connected in ending points, creating kind of loop.
Without the charges which somehow energetically ‘likes’ to connect with spin, the loop would annihilate and it’s momentums should create two photon-like excitations.
Two ‘spin curves’ could reconnect exchanging its parts, creating some complicated, dynamical structure of spin curves.
Maybe it’s why electrons like to pair in orbits of atoms, or as a stable Cooper pairs (reconnections should create viscosity…)

Bolzman distribution among trajectories gives something similar to QM, but without Wick’s rotation
http://www.thescienceforum.com/viewtopic.php?p=158008
In some way this model corresponds better to reality – in standard QM all energy levels of a well like made by a nucleus are stable, but in the real physics they want to get to the ground state (producing a photon). Without Wick’s rotation eigenfunctions are still stable, but the smallest fluctuation make them drop to the ground state. What this model misses is interference, but it can be added by some internal rotation of particles.
Anyway this simple model shows that there is no problem with connecting deterministic physics with squares appearing in QM. It suggests that maybe a classical field theory would be sufficient … when we understand what creation/annihilation operators really do – what particles are … the strongest conservation principle – of spin and charge suggests that they are just made of topological singularities… ?

What do you think about it?
I was said that this kind of ideas are considered, but I couldn’t find any concrete papers?

There started some discussion here:
What do you mean by a "topological singularity" ? Singularities generally refer to the differentible structure of a manifold or the behavior of a differentiable function.

4. Conley index for nonzero length vector field in N dimensions means that we take a (N-1 dimensional) sphere around the critical point and look how many times the field 'wraps around' the sphere (homotopy group).
In 2 dimensions it's exactly how many rotations makes the phase while making a loop around a critical point - it's equivalent to spin in physics or to http://en.wikipedia.org/wiki/Argument_principle
in complex analysis.
In 3 dimensions (charge) it's a bit more complicated, but still there are known from physics conservational properties.

Or please just look at the demonstration I've linked - if you choose 0 spin for one singularity, You will see only single one.

5. It sounds like what you are attempting to describe are instantons--these are positive energy solutions to the classical field equations which are confined to remain in the higher energy state because of topological constraints.

For example, the lowest energy state of a vortex might require it to have winding number = 0, so a vortex with winding number >= 1 would not be able to continuously collapse into the vacuum state.

As instantons tend to be localized in space-time, I can see how it would be easy to confuse them for particles in the standard model sense of the word. However, the localized properties of instantons arise at the classical level, not the quantum level. So they are, in some sense, classical particles (or strings, or something else, depending on the situation), not quantum particles.

One can, of course, develop a quantum theory of instantons--this why the study of the moduli space of instantons (i.e., Donaldson theory and Seiberg-Witten theory) is so relevant to physics.

But, as I understand it, the particles that appear in the standard model are pertubations around the vacuum solutions to the classical field equations.

Again, I don't blame you for confusing these two topics--the math/physics community is not exactly doing a stellar job of making this distinction clear. Perhaps if you're interested I can write more on this later.

6. I see that there the nomenclature is completely mixed I remember that on QM lectures when we were solving potential with two minimals using Feynman path integrals, it was called instantons - that they stay a long time in one well and jumps between them practically instantly ...
Physicists call turbulence or vortex something 'spiral-like' - of spin or Conley index 1.
But as in the demonstration - there is much more qualitatively different possibilities...

About that I'm talking about classical field theory - I completely agree - but my search started a year before, when I get some analytic result which made me realize that it's enough - extremely simple model: take Bolzman distribution among trajectories, gives something very similar to Schrodinger's equations
Bolzman distribution among trajectories gives something similar to QM, but without Wick’s rotation
http://www.thescienceforum.com/viewtopic.php?p=158008
In some way this model corresponds better to reality – in standard QM all energy levels of a well like made by a nucleus are stable, but in the real physics they want to get to the ground state (producing a photon). Without Wick’s rotation eigenfunctions are still stable, but the smallest fluctuation make them drop to the ground state. What this model misses is interference, but it can be added by some internal rotation of particles.
We can extend this model by using a classical field theory in which particles are some special solutions and they have some 'internal rotation' to be able to create interference. Now particles can interact through the field, and they can also be created/annihilated
QFT creates some universal math on creation/annihilation operators in some abstract (and nonphysical) Fock space - I believe that we can place these operators in some classical field theory and for example automatically get required cutoffs...

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I'll try to explain it from the other side:
Some time ago I've considered some simple model - take a graph and consider a space of all paths on it. Assumption that all of them are equally probable, leads to some new random walk on graph, which maximize entropy globally (MERW). It can be also defined that for given two vertices, all paths of given length between them are equally probable.
Standard random walk - for all vertices, all edges are equally probable - maximize uncertainty only locally - usually gives smaller entropy.
http://scitation.aip.org/getabs/serv...cvips&gifs=yes
This model can be generalized that paths are not equally probable, but there is Bolzman distribution among them for some given potential.
Now if we cover R^3 with lattice and take continuous limit, we get that Bolzman distribution among paths gives near QM behavior - more precisely we get something like Schrodinger equation, but without Wick rotation - stationary state probability density is the square of the dominant eigenfunction of Hamiltonian - like in QM. Derivations are in the second section of
http://arxiv.org/abs/0710.3861
In some way this model is better than QM - for example in physics excited electrons aren't stable like in Schrodinger's equations, but should drop to the ground state producing photon, like wihout Wick's rotation.
Anyway in this MERW based model, electron would make some concrete trajectory around nucleus, which would average to probability distribution as in QM.
This simple model shows that there is no problem with 'squares', which are believed to lead to contradictions for deterministic physics (Bell's inequalities) - they are result of 4D nature of our world - the square is because there have to meet trajectories from past and future.

This simple model - Bolzman's distribution among paths is near real physics, but misses a few things:
- there is no interference,
- there is no energy conservation,
- is stochastic not deterministic,
- there is single particle in potential.
But it can be expanded - in some classical field theory, in which particles are some special solutions (like topological singularities suggested by e.g. strong spin/charge conservation).
To add interference they have to make some rotation of its internal degree of freedom. If it's based on some Hamiltonian, we get energy conservation, determinism and potentials (used for Bolzman distribution in the previous model).
To handle with many particles, there are some creation/annihilation operators which creates particle path between some two points in spacetime and interacts somehow (like in Feynman's diagrams) - and so creates behavior known from quantum field theories, but this time everything is happening not in some abstract and clearly nonphysical Fock space, but these operator really makes something in the classical field.

The basic particles creating our world are spin 1/2 - while making a loop, phase makes 1/2 rotation - changes vector to the opposite one. So if we identify vectors with opposite ones - use field of directions instead, fermions can naturally appear - as in the demonstration - in fact they are the simplest and so the most probable topological excitations for such field - and so in our world.
A simple and physical way to create directional field is a field of symmetric matrices - which after diagonalisation can be imagined as ellipsoids. To create topological singularities they should have distinguishable axises (different eigenvalues) - it should be energetically optimal. In critical points (like the middle of tornado), they have to make some axises indistinguishable at cost of energy - creating ground energy of topological singularity - particle's mass.
Now one (of 3+1) axis has the strongest energetic tendency to align in one direction - creating local time arrows, which somehow rotates toward energy gradient to create gravity/GR like behaviors.
The other three axises creates singularities - one of them creates one singularity, the other has enough degrees of freedom to create additional one - to connect spin+charge in one particle - giving family of solution similar to known from physics - with characteristic 3 for the number of generations of leptons/quarks. With time everything rotates, but not exactly around some eigenvector, giving neutrino oscillations.

7. There is nice animation for topological defects in 1D here:
http://en.wikipedia.org/wiki/Topological_defect
thanks of potential, going from 1 to -1 contains some energy - these nontrivial and localized solutions are called (anti)solitons and this energy is their mass. Such pair can annihilate and this energy is released as 'waves' (photons/nontopological excitations).

My point is that in analogous way in 3D, starting from what spin is, our physics occurs naturally.
I think I see how mesons and baryons appears as kind of the simplest topological excitations in picture I've presented - in each point there is ellipsoid (symmetric matrix) which energetically prefers to have all radiuses (eigenvalues) different (distinguishable).

First of all singularity for spin requires making 2 dimensions indistinguishable, for charge requires 3 - it should explain why 'charges are heavier than spins'. We will see that mass gradation: neutrino - electron - meson - baryon is also natural. Spins as the simplest and so the most stable should be somehow fundamental.
As I've written in the first post - from topological reasons two spins 'likes' to pair and normally would annihilate, but are usually stabilized by additional property which has to be conserved - charge.
And so electron (muon,tau) would be a simple charge+spin combination - imagine a sphere such that one axis of ellipsoids is always aiming the center (charge singularity). Now the other two axises can make two spin type singularities on this sphere. And similarly for other spheres with the same center and finally in the middle all three axises have to be indistinguishable. The choice of axis chooses lepton.
Now mesons - for now I think that it's simple spin loop (up+down spin) ... but while making the loop phases make half rotation (like in Mobius strip) - it tries to annihilate itself but it cannot - and so creates some complicated and not too stable singularity in the middle. Zero charge pions are extremely unstable (like 10^-18 s), but charge can stabilize them for a bit longer.
The hardest ones are baryons - three spins creating some complicated pattern and so have to be difficult to decay - the solution could be that two of them makes spin loop and the third goes through its middle preventing from collapse and creating large and 'heavy' singularity. Spin curves are directed, so there are two possibilities (neutron isn't antineutron). We believe we see up and down quarks because two creating the loop are different form the third one.

8. It occurs that rotational modes of the simplest energy density in such ellipsoid field already creates electromagnetic and gravitational interaction between such topological excitations.