# Thread: Topological solitons of ellipsoid field - our particle menagerie correspondence?

1. Originally Posted by Jarek Duda
I'm mainly asking where to search for eventual differences with our physics - this extremely simple model (combining EM and QM) gives very concrete qualitative picture: particles, properties, decays, interactions ... finding a single essential qualitative discrapency could spare me horrible simulations ...
Deep scattering events show conclusively that quarks, electrons and positrons have a point-like form factor not reflected in your pretty pictures.

Nowhere do you indicate that the EM field is a quantum field.

Nowhere do you present a basis for identifying your pictures with the specific particles you identify. Particles are identified by their mass and how they couple to other fields. (Since your description doesn't have a EM quantum field, at a minimum you need to explain why pions have various charges, i.e. why does the quark model (which also explains deep scattering) also explain the EM charges of the hadrons?

Nowhere do you demonstrate from your axioms that your particles can move.

2. Originally Posted by Jarek Duda
AlphaNumeric, You are constantly saying that I'm using buzzwords - I'm really trying to be clear, but please just make concrete questions and I would gladly explain ...
I have to admit that I'm practically not familiar with algebraic geometry ... [long list of buzzwords]
I don't think you're trying to be clear. The manner in which you 'discuss' things isn't the same as seen in people who understand complicated stuff and who want to discuss it. Anyone comparing your threads to threads started by people like myself, Prometheus, BenTheMan, QuarkHead, Guest and a few others will see that your style doesn't resemble said threads.

If you grasped all these areas and were actively working on a specific problem you'd not be throwing out so many buzzwords, because all that does is overcomplicate things. If you really wanted specific discussion with a specific problem you're up against you'd have said precisely what that is. Instead its "I work on ellipsoid fields, which relate to solitons and topologies and......", not once actually explaining precisely what it is you're working on today.

For instance, if I were to ask Prometheus to explain what precisely he's working on at the moment, pitching the answer at the level of final year undergrads and above, then he'd be specific on the equations and systems he's working with. Likewise with the others I just listed. You just dance around saying all these advanced areas your work touches on without actually saying what it is you're working on.

Are you trying to solve a specific set of equations? If so, what are they? Are you trying to prove some particular result? If so what and how are you approaching it? Are you trying to come up with a specific tool to address a particular problem? If so what problem and what have you tried so far?

One of the things you learn when doing advanced physics (or any science) is how to pitch questions properly. Another thing is developing a feel for how much someone knows of a topic (when said topic is related to your own work) from how they discuss it. I don't get the feeling you understand what you're talking about on a working level and certainly Farsight doesn't. If you disagree then please answer my questions. Farsight, if you disagree with my assessment then step up and show I'm wrong about you. I keep giving you opportunities and you keep avoiding them. Like I said, much of what you've commented on relates directly to my PhD area so if you really wanted to discuss honestly such things you'd be willing to take up my alpha rules discussion offer. Instead you're ignoring any and all corrections I make in regards to your posts, which shows you're dishonest on multiple levels.

3. AlphaNumeric, your posts were rather loosely connected with this topic, so I didn't want discussion to go that way here - I apology if you feel that I've ignored something.
It's generally only an intuitive sketch, bunch of thoughts now - please state concrete questions on the topic and I will really gladly try to explain, clarify how I see/understand it.
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rpenner, really thanks for finally concrete questions I can try to reply to:

Deep inelastic scattering cannot prove that something is exactly point-like, but only limit its size from above.
And generally it's not so simple - there is a difference between electron having spin up and having spin down - so it's rather arrow-like, not a point-like ...
Saying that e.g. electron is exactly point-like means there is infinite energy, charge density in it - in ellipsoid field this density is spread over distance on which weak interaction works, like 10^-18m.
About quarks, in baryon models there are introduced 'stings' connecting quarks (so called quark-gluon strings) and generally there are problems with quark models, for example missing proton's spin - I wouldn't rely too strong on such models.

I don't know what do you mean by the second question?
This is purely classical soliton model and there is general belief that finally it has to be 'quantized' (but thanks of many new arguments, I believe it doesn't have to be necessary).
About why ellipsoid field combines EM field and wavefunction - far from particles it becomes EM, while behave as wavefunction near them - e.g. spin quantum rotation operator make quantum phase rotate spin times while full rotation - this field was constructed for that condition.
Intuitively quantum phase corresponds mainly to rotation in cheapest energetically plane (like in electron spin curve), while modulus of wavefunction corresponds to probability - I see it as effective part of QM and so should be omitted in such trial to construct really fundamental models:
practical quantum mechanics we are able to use always for example neglects some environment, which in fact is essential for system evolution - in such model there is hidden (through wavefunction modulus) statistical ensemble among possible states of environment.

Fermion's spin in this model basically corresponds to outgoing spin curves - while Stern-Gerlach experiment it can connect to external magnetic field in two ways and correspondingly more ways for large spin nucleus.
Charge is here point-like construction on spin curve, so for (not neutral) mesons means that there is such additional construction in Mobius-like loop. Alternative view is that in picture for 'really free electron' (with enclosing spin curves having neutrino-like mass), there is additional internal twist before such spin curves meet.
In baryons there could be charge inside spin loop - it becomes really nasty, but they observe hundreds of possible matastable particles for such states ... but the most essential place for charge is the central spin curve - in such position it can make it easier to combine with spin loop and allow to see beta decay as charge-anticharge creation on this spin curve (and place some of this large energy difference in additionally created spin loop - neutrino).
The difference to standard view is strangeness which now represents the number of additional (half)twists of the spin loop - is not natural, but integer number now - and so e.g. neutral kaons can be constructed in two ways - having the same properties/mass, but differently decaying (no CP violation needed).
... but generally situation becomes really complicated here - there are needed simulations to really understand the situation ...

About particle movement - look at animation here - this model is Lorentz invariant, so using boosts we for example see why soliton's rest energy is proportional to inertial mass ...

4. Originally Posted by Jarek Duda
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.
...
OK, thanks for the post of the information. Before I can accept the topological deformations, are your points in the topology these ellipses?

If so, based on their shapes, your topology is fundamentally dynamic. I have only considered point sets that are static but I am open to "dynamic points". I can't see any reason this cannot be true.

However, my next request is that you must now show me your proof of continuity for your topology.

5. Originally Posted by Jarek Duda
Deep inelastic scattering cannot prove that something is exactly point-like, but only limit its size from above.
I was not making a hand-wavy claim but pointing you to the evidence.

1) The energy levels of the positronium system are those described by the quantum physics of the hydrogen atom with a reduced mass of $\frac{1}{2} m_e$ while hydrogen uses roughly $\frac{240536}{240667} m_e$.

2) This alone is good evidence that the scale of positronium is $a_{e^{+} - e^{-}} = 2 a_0 = \frac{2 \hbar}{m_e c \alpha} = \frac{8 \pi \epsilon_0 \hbar^2}{m_e e^2} \approx 0.1 \, \textrm{nm}$. But this view is also supported by other successful comparisons of the theory with observation of the universe, like the whole of solid state theory, condensed matter physics, quantum chemistry, etc.

3) Research of positronium continues and the Quantum Field Theory version is validated to over 99.99% in even hyperfine spectrum splitting. Also, like hydrogen there is evidence that positronium can form molecules, also with roughly nanometer scales. http://adsabs.harvard.edu/abs/2008PhRvL.100a3401C

4) Contrasted with that is the scattering data that probes electrons (and quarks) to be point-like below the "classical electron radius" of 2.8 fm, below the proton charge radius of about 1 fm, down to a few am.

// Edit. One source ( http://iopscience.iop.org/1402-4896/1988/T22/016/ ) gives 0.0001 am = $10^{-20} \, \textrm{m}$ as the upper limit of the size of the electron, but I have ignored it since I haven't read it myself.

So my point was that the "point-like" nature of electrons was on the scale of (below) 0.000000005 nm while you describe positronium ( which has a scale of 0.1 nm) and ignore the constituents, claiming an extended structure -- but failing to account for a prediction or a comparison with nature.

So you have 8 or 9 orders of magnitude between the scale of electrons and your claims about the structure of positronium. Some very strange results were found in these areas, but nothing like you describe. Your viewpoint seems uniformed, and doesn't begin to lead to a useful discussion of nature because you not only don't tell us what we find in nature, you don't even tell us what to look for or seem to understand that this and this alone controls if your viewpoint is to be considered as scientific.

So if you are going to tell us that the electrons and positrons and electromagnetic fields of nature don't act like described by the mathematics of quantum electrodynamics, you have to tell us how they should act in all the places where quantum electrodynamics has met with success, and then do a better job.

Buzzword bingo is not cutting it. Welcome to the world of psuedo-science. You cannot reap the benefits of science just by pretensions to intellectual chatter, you have to bring the meat and potatoes of useful description of nature.

6. Originally Posted by rpenner
Nowhere do you demonstrate from your axioms that your particles can move.
Originally Posted by Jarek Duda
About particle movement - look at animation here - this model is Lorentz invariant, so using boosts we for example see why soliton's rest energy is proportional to inertial mass ...
OK. Let's have a bet. You explain how you your model is ellipsoidal and Lorentz invariant when the eccentricity of an ellipse is not generally preserved by a Lorentz transform in the plane of the ellipse.

If either of AlphaNumeric or Prometheus or Guest254 signs off on it, I will mail you one of the prize packages leftover from the last physforum.com meet-the-moderator event.

7. chinglu, I don't think I understand your questions, but pictures show discrete sets of points because it's difficult to do otherwise, but this field is intended to be continuous rather - mathematically just tensor field with standard kinetic term, but with Higgs-like potential term.
Ellipsoid representation is only to make it easier to imagine - ellipsoid represents situation of field in given point.
Topological properties don't relate to single ellipsoid, but they emerges on field level.
It is generally happening in 4D spacetime, so both single ellipsoids and topological situation are dynamic - change accordingly to obtained evolution equation.

-----------------
Generally science shouldn't be a dictatorship - if some moderator classify topic as pseudoscience, I believe he should at least explain briefly his decision ... pseudoscience are solitons? tensor fields? topology?
If AlphaNumeric did it - I haven't seen anything concrete on this topic in your posts, but some history based pretensions, calling everybody cranks and everything buzzwords - so I honestly offered to explain what you don't understand, but I still didn't get a single concrete question on topic ...
... while I had concrete question to you about your topology knowledge, but you just ignored it - to suggest some papers generalizing differential topology results to fields of something more complicated than vectors...?

8. rpenner, I don't understand why you see such model contradicting QED?
While in QED we are usually focusing on Fourier transform (momentum space), here we try to figure out spatial structure of particles - how fields glues together to avoid infinites (of electric field, mass density...).
Having such working model - in perfect situation: a field which family of solitons correspond well to our particle menagerie, standard next step for soliton models is some 'quantization' procedures - also for surrounding EM field.
QFT way to construct particles is mathematically extremely complicated, leading to artificially removed infinities - it makes it practically impossible to consider qualitatively more complicated than really simple potentials - it's extremely difficult to make sure that qualitative properties aren't neglected by the way - for example topological ones, like such singularity of electric field directions around electron.
QFT are extremely abstract models - working on Fock space, commutation relations ... soliton models can bring concrete constructions for them, like seeing creation-annihilation pair as concrete field construction - traveling soliton ...

About positronium - I completely agree that it's many orders of magnitude larger than electron - this distance is so large that recent scattering experiments suggests that there is practically no charge screening between these two charges ( http://physicsworld.com/cws/article/news/44265 ).
Especially ortho-positronium: magnetic interaction allow to stabilize it in parallel spin configuration, then they twist into anti-parallel producing one photon and finally annihilate producing additional two.
Electron structure on this picture is just a way to glue surrounding fields to avoid infinities of of electric and gravitational field (charge and mass density).

About Lorentz invariance - ok, I've just assumed that if kinetic term is Lorentz invariant, the whole theory is so (especially that in vacuum we get Lorentz invariant Maxwell's equations) - it's true for scalar fields, but generally might be more complicated for tensor fields.
Preferring some set of eigenvalues can be also made by e.g. potential:

V(M)=(Tr(M)-c_1)^2+(Tr(M^2)-c_2)^2+(Tr(M^3)-c_3)^2+(Tr(M^4)-c_4)^2

where c_k is sum of k-th powers of expected eigenvalues
This way it looks much more Lorentz invariant ... I'll think about it ...

9. Originally Posted by Jarek Duda
Generally science shouldn't be a dictatorship - if some moderator classify topic as pseudoscience, I believe he should at least explain briefly his decision ... pseudoscience are solitons? tensor fields? topology?
But this forum isn't 'science' and we are at the whims of the moderators.

Originally Posted by Jarek Duda
If AlphaNumeric did it
I'm not a moderator. Do you think I'd tolerate such crap as the stuff Farsight comes out with if I were a moderator?

Originally Posted by Jarek Duda
I haven't seen anything concrete on this topic in your posts, but some history based pretensions, calling everybody cranks and everything buzzwords - so I honestly offered to explain what you don't understand, but I still didn't get a single concrete question on topic ...
The issue at hand isn't that I don't understand what you're referring to but that I don't think you do. I asked you to be specific about what it was you are working on in this area, what specific problem, and all you gave was more words and dancing around. If you're unable to even understand what I'm asking you then it only adds weight to my view that you don't understand this sort of science, you just want to be seen to understand it.

Add into that the fact you've started this thread on other forums (I've specifically seen it on PhysForums). If you're in a research group and are studying for a PhD, as you've previously claims, you shouldn't need to rely on forums to spread your work or to get 'informed discussion'. The only people you've engaged in discussion have been Ching and Farsight, neither of which are 'informed' by any stretch of the meaning of 'informed'. Anyone who is informed can see that and so by treating them as such you've given the impression you aren't either.

Originally Posted by Jarek Duda
... while I had concrete question to you about your topology knowledge, but you just ignored it - to suggest some papers generalizing differential topology results to fields of something more complicated than vectors...?
Until I see any reason to think you understand this stuff I see no reason to indulge your delusions. There have been plenty of cranks here who want to 'talk' about complicated stuff but only do it on a superficial level. Posting links to papers so you can copy and paste buzzwords out of the abstract isn't going to be very productive. I asked you to be specific about your work in order to see if you could engage in discussion which involved you showing a working understanding on this stuff.

You failed. Repeatedly.

And then ...

Originally Posted by Jarek Duda
While in QED we are usually focusing on Fourier transform (momentum space)
No, Fourier transforms map between position and momentum spaces, momentum space is not a 'Fourier transform'.

The more you post the more it seems like you're just spewing out terms you don't understand. An over abundance of terminology often is a sign you're trying to make it seem like you're well read. For instance, you put added '(momentum space)' in an attempt to show you know about the relationship between different formulations but anyone familiar with QED would know that anyway and if you were familiar with QED you'd know that too and know adding in the comment about momentum space is pointless. Instead you threw it in and did so incorrectly. My impression is you're trying to cram too many buzzwords in an attempt to compensate for a lack of understanding.

If I'm wrong then say precisely what equations (equations, not words) you've been working on in regards to ellipsoid fields and solitons and what you have done with them. Show you have something more to say than what can be found by using Google.

10. AlphaNumeric, as you can see, I try to deeply answer to concrete questions, comments of others - it's also what I'm asking you for, but instead you produce another leading nowhere long post ...
What is the specific problem I'm asking about? Ok - I'll say it one more time: to find qualitative discrepancies between family of topological singularities of ellipsoid field and our particle menagerie and their dynamics - that's the problem I'm asking about from the beginning of this thread and the rest of participants of this discussion didn't have problem to understand it.
My work? I've recently defended my first PhD and I'm finishing now my PhD in theoretical physics - about new philosophy of stochastic modeling, which finally maximize e.g. entropy and which lead me to this search for soliton particle models ... about which I would gladly discuss face to face, but unfortunately professors I know from studding theoretical mathematics have lacks in physics, while physicists have some deeply imprinted taboo against using solitons not only to model complex particles ... ?
You seem to have a need to test my competence, I really don't like something like that, so for example I've mentioned applying Lyusternik-Schnirelmann and Morse theories on torus, what cannot be just 'googled' (I think?) ... now you are catching me by words about Fourier transform - it's about using integral_p |p><p| form of unity - going into momentum space, or about decomposing into plane waves ...

Please, I really don't want such nonsense argue here - please just state some concrete question/comments about ellipsoid field here and if you don't think I'm competent to reply to them, maybe someone else will be - it's what discussion forums are for ...

11. Originally Posted by Jarek Duda
chinglu, I don't think I understand your questions, but pictures show discrete sets of points because it's difficult to do otherwise, but this field is intended to be continuous rather - mathematically just tensor field with standard kinetic term, but with Higgs-like potential term.
Ellipsoid representation is only to make it easier to imagine - ellipsoid represents situation of field in given point.
Topological properties don't relate to single ellipsoid, but they emerges on field level.
It is generally happening in 4D spacetime, so both single ellipsoids and topological situation are dynamic - change accordingly to obtained evolution equation.

-----------------
Generally science shouldn't be a dictatorship - if some moderator classify topic as pseudoscience, I believe he should at least explain briefly his decision ... pseudoscience are solitons? tensor fields? topology?
If AlphaNumeric did it - I haven't seen anything concrete on this topic in your posts, but some history based pretensions, calling everybody cranks and everything buzzwords - so I honestly offered to explain what you don't understand, but I still didn't get a single concrete question on topic ...
... while I had concrete question to you about your topology knowledge, but you just ignored it - to suggest some papers generalizing differential topology results to fields of something more complicated than vectors...?
Unacceptable.

If you are going to use the term topology and you are going to use continuous deformations, then your point sets must prove continuity.

This is sufficient to refute your theory.

12. Originally Posted by rpenner
I was not making a hand-wavy claim but pointing you to the evidence.

1) The energy levels of the positronium system are those described by the quantum physics of the hydrogen atom with a reduced mass of $\frac{1}{2} m_e$ while hydrogen uses roughly $\frac{240536}{240667} m_e$.

2) This alone is good evidence that the scale of positronium is $a_{e^{+} - e^{-}} = 2 a_0 = \frac{2 \hbar}{m_e c \alpha} = \frac{8 \pi \epsilon_0 \hbar^2}{m_e e^2} \approx 0.1 \, \textrm{nm}$. But this view is also supported by other successful comparisons of the theory with observation of the universe, like the whole of solid state theory, condensed matter physics, quantum chemistry, etc.

3) Research of positronium continues and the Quantum Field Theory version is validated to over 99.99% in even hyperfine spectrum splitting. Also, like hydrogen there is evidence that positronium can form molecules, also with roughly nanometer scales. http://adsabs.harvard.edu/abs/2008PhRvL.100a3401C

4) Contrasted with that is the scattering data that probes electrons (and quarks) to be point-like below the "classical electron radius" of 2.8 fm, below the proton charge radius of about 1 fm, down to a few am.

// Edit. One source ( http://iopscience.iop.org/1402-4896/1988/T22/016/ ) gives 0.0001 am = $10^{-20} \, \textrm{m}$ as the upper limit of the size of the electron, but I have ignored it since I haven't read it myself.

So my point was that the "point-like" nature of electrons was on the scale of (below) 0.000000005 nm while you describe positronium ( which has a scale of 0.1 nm) and ignore the constituents, claiming an extended structure -- but failing to account for a prediction or a comparison with nature.

So you have 8 or 9 orders of magnitude between the scale of electrons and your claims about the structure of positronium. Some very strange results were found in these areas, but nothing like you describe. Your viewpoint seems uniformed, and doesn't begin to lead to a useful discussion of nature because you not only don't tell us what we find in nature, you don't even tell us what to look for or seem to understand that this and this alone controls if your viewpoint is to be considered as scientific.

So if you are going to tell us that the electrons and positrons and electromagnetic fields of nature don't act like described by the mathematics of quantum electrodynamics, you have to tell us how they should act in all the places where quantum electrodynamics has met with success, and then do a better job.

Buzzword bingo is not cutting it. Welcome to the world of psuedo-science. You cannot reap the benefits of science just by pretensions to intellectual chatter, you have to bring the meat and potatoes of useful description of nature.

You spend so much effort and time to refute this theory when it can be done in a few sentences.

13. chinglu, continuity is not something to be proven, but the basic assumption of all field theories - required to use differential equations.
For example evolution of this extremely simple model is governed by Euler-Lagrange equation:

the right term is kind of elasticity force which want to take ellipsoid back to the original shape (set of eigenvalues) - this term allows for deformations required to obtain this family of topological solitons. It practically vanishes far from particles and the remaining wave equation can be written as two sets of Maxwell's equations: for electromagnetism and gravity.
Developed through a few decades string theory cannot be used to reasonable model a single particle, while this equation seem to create all of them with expected dynamics ... and nothing more (no WIMPs, Dirac monopoles, Higgs bosons ...).
Coincidence?

14. Originally Posted by Jarek Duda
chinglu, continuity is not something to be proven, but the basic assumption of all field theories - required to use differential equations.
For example evolution of this extremely simple model is governed by Euler-Lagrange equation:

the right term is kind of elasticity force which want to take ellipsoid back to the original shape (set of eigenvalues) - this term allows for deformations required to obtain this family of topological solitons. It practically vanishes far from particles and the remaining wave equation can be written as two sets of Maxwell's equations: for electromagnetism and gravity.
Developed through a few decades string theory cannot be used to reasonable model a single particle, while this equation seem to create all of them with expected dynamics ... and nothing more (no WIMPs, Dirac monopoles, Higgs bosons ...).
Coincidence?
What are you talking about because it is not topology.

You have a finite set of point sets and claim to emerge them into your continuous deformations.

That means you have a mapping from the finite into the continuous.

Let me see this mapping.

And, if you cannot understand you must prove the continuity of the domain of your topology if you claim a continuous range then you simply are extremely weak at math.

Further, even if you have a continuous domain, that still not not prove a continuous range or deformation in your case.

This all must be proven and you are not able to do it.

Therefore, you have not viable theory.

15. No, I don't have finite number of points, but by assumption - continuous tensor field.
I draw situation of the field on discrete lattice, but use continuous equations to determine these values (here is more continuous presentation).

16. Originally Posted by Jarek Duda
No, I don't have finite number of points, but by assumption - continuous tensor field.
I draw situation of the field on discrete lattice, but use continuous equations to determine these values (here is more continuous presentation).
You cannot go from a continuous tensor field to your ellipsoids and claim these are a continue set for the domain of a topology without proving that.

Then, you must prove that specific continuous domain proves the deformations(range) you claim.

Where is the math.

17. It's not about going from one field to another, but about equivalent ones: about seeing them from a different perspective.
If you have such (stress) tensor field: continuous field of real symmetric matrices (positively defined in our case) - you can think about them in diagonalized form: as given number of eigenvectors and corresponding eigenvalues - which for better intuition can be drawn as ellipsoid with these axes and radii.
For example {{1,0},{0,2}} matrix can be drawn as ellipse which axes are coordinate axes and radii are 1 and 2.
The more eigenvalues are equalized (e.g. by deformation), the more symmetric our ellipsoid is and finally became a ball for identity matrix.
Pictures of ellipsoid field on discrete lattice was made by sampling continuous tensor field on this lattice.

18. Originally Posted by Jarek Duda
If you have such (stress) tensor field: continuous field of real symmetric matrices (positively defined in our case) - you can think about them in diagonalized form: as given number of eigenvectors and corresponding eigenvalues - which for better intuition can be drawn as ellipsoid with these axes and radii.
That only provides a tiny bit of help via some dubious intuition. The notion of drawing a set of axes which changes as you move through space is not new, its built into the notion of a frame bundle. Furthermore, not all tensor fields in physics are real and symmetric. Furthermore, tensors which are not of the form (0,2), (1,1) or (2,0) (which are all related via metric isomorphism if you have a metric) do not lend themselves to such an interpretation as the notion of 'eigenvalue' breaks down. For instance the curvature tensor $R^{a}_{bcd}$ doesn't make vectors to vectors and so can't naively have eigenvalues defined in that regard. Furthermore in the case of frame bundles you can only draw sets of axes which smoothly vary and are never degenerate on manifolds which are 'parallelisable'. The vast majority of interesting manifolds in physics are not parallelisable. The issue of whether or not such frames exist is important in the study of moduli dynamics in string theory and happens to be something I have personal experience with.

19. We generally use different representations because for different purposes given one can be more suitable, give better intuition - for example sometimes it's more convenient to look at evolution of conjugated pendulums through their positions (classical view) and sometimes as unitary evolution: 'superposition of rotations' of its normal modes (quantum view) ...
Real symmetric tensor fields are quite quite popular for example as stress tensor in solids or as famous stress-energy tensors, but I completely agree that there are also different tensor fields considered.
I also completely agree that representing it as ellipsoid field isn't something new - the only thing which seems to be new in this field theory is the Higgs-like potential - preferring only shape and saying nothing about its orientation - with minimum being equivalent to topologically nontrivial reduced space of rotations (rotating 180deg brings to initial state) - far nontrivial differential topology of such undirected reper field (fiber bundle if you prefer) is what I was asking you about ...
These rotations is the essence here - their dynamics becomes EM+gravity in vacuum and brings both spin and charge quantization of its topological singularities - leading to surprisingly good correspondence to our particle menagerie ...

20. Originally Posted by Jarek Duda
Real symmetric tensor fields are quite quite popular for example as stress tensor in solids or as famous stress-energy tensors
I didn't say they weren't 'popular' (though 'common' is more appropriate since popularity has nothing to do with whether or not something has symmetric tensor representations)

Originally Posted by Jarek Duda
I also completely agree that representing it as ellipsoid field isn't something new - the only thing which seems to be new in this field theory is the Higgs-like potential
Which you have derived how?

Scalar potentials are not new. Scalar Higgs-like potentials are not purely confined to Higgs models, symmetry and renormalisability put strict constraints on the sorts of scalar fields typically considered.

Originally Posted by Jarek Duda
preferring only shape and saying nothing about its orientation - with minimum being equivalent to topologically nontrivial reduced space of rotations (rotating 180deg brings to initial state)
The symmetry of the Higgs potential is not found only in the Higgs potential. Usage of SO(N) and SU(M) symmetries are rampant in particle physics. I struggle to see what precisely you've done.

Originally Posted by Jarek Duda
far nontrivial differential topology of such undirected reper field (fiber bundle if you prefer) is what I was asking you about ...
I have never seen the term 'reper field' before and given this thread is on the first page of a Google search for such a term I'm going to give you to slight benefit of the doubt and assume you misspelt something.

Originally Posted by Jarek Duda
These rotations is the essence here - their dynamics becomes EM+gravity in vacuum and brings both spin and charge quantization of its topological singularities - leading to surprisingly good correspondence to our particle menagerie
I haven't seen you justify this and I've asked repeatedly for the details. Show precisely how this occurs. SO(N) symmetry doesn't automatically imply electromagnetism and gravity. Having your fields transform under SO(N) is regularly considered in GUT research and powerful group theoretic tools give the particle make up of such models. For instance SO(10) is considered a lot and it has non-trivial issues with neutrino mass hierarchies and that's not considering gravity. Sure, you could do the naive thing of say "Gravity has an SO(3,1) spin connection and the Standard Model fits into SO(10), therefore I'll consider SO(13,1) ~ SO(14)" but that doesn't mean its renormalisable. You haven't shown how the zeros of the vector field lead to charges or spin, all you've presented so far is "Here's some integers!".

As I said to Farsight, things like the topological non-triviality of compact spaces, the resultant de Rham cohomologies and the moduli fields they define via parameterisations of topology preserving smooth deformations of said spaces is precisely what my PhD was in. BenTheMan also did a PhD in this area, though more on the supergravity field content side, rather than my differential geometry approach, so if you are seriously wanting a discussion on the details you don't need to hold back. Thus far you've skirted around giving details, while claiming to want to talk about advanced stuff. As I've previously said, that sort of behaviour makes me raise an eyebrow. I'm more than willing to admit my initial impressions are wrong, I have no problem with people who do know their stuff. Ben and I worked in similar areas, even crossed paths with the same people from time to time, he's shown he knows that of which he speaks and I would say I've done likewise. If you're seriously doing (or going to be doing) a PhD in this stuff you should have plenty of details already typed up or active in your head, I'm just asking you to provide them.

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