Scalar Higgs potential is not new, but here its essence is taken to tensor field.
Having tensor fields take particular values and then considering their perturbations isn't new either. In fact its the essence of metric linearisation, $$g_{ab}(x) = g_{ab}(x_{0}) + h_{ab}(x-x_{0})$$ where typically $$g_{ab}(x_{0}) = \eta_{ab}$$ for particular coordinates.
that its minimum has nontrivial topology (originally circle, here a subgroup of SO(3) or SO(4))
No, a circle has SO(2) symmetry, it is not a subgroup of SO(3) or SO(4) in and of itself, you're conflating a thing for its structure (or rather the opposite).
it allows for situations in which field topological constrains on boundary of a set, enforce going out of minimum of the potential - allow to glue singularities avoiding infinities and giving them rest energy (mass).
I have repeatedly asked you to show precisely what it is you're doing, to give something more than words. As I just pointed out, your usage of terminology is at best dubious (perhaps due to a language barrier) and perhaps at worst just word salad. If you're really doing this for a PhD then please provide a link to a paper you've got in a reputable journal on this stuff. If you've not published anything yet then can you please send me a link via PM, I will not show it to anyone else but I would like to see you do more than just post words.
About the 'reper' word - I though it is common in differential geometry, GRT, but it seems it is specific for my region - it just means local coordinates: reper field/bundle means there are e.g. 3 or 4 orthogonal distinguishable axes in each point.
Such a thing is a frame bundle on a parallelisable manifold. On non-parallelisable manifolds it is not possible to construct a set of non-degenerate axes at all points in the manifold which vary continuously.
My intention for using it was to simplify situation - from topological point of view, we can forget about ellipsoid radii and their deformations - it's enough to consider field of set of their undirected axes - undirected 'reper field'.
The requirement of parallelisability is an extremely large restriction, the majority of compact spaces are not parallelisable. And speaking from personal experience, intuition is not the way to go about this, you need to be rigorous and work through the mathematics. If all you have is words and some pictures you're not going to get far.
I gave you materials I've written
Which post did you provide them? I haven't seen you provide anything more substantial than what could be got from an afternoon using Google. You claim there's some link between 'topological solitons of ellipsoid fields' and the family of particles in particle physics but if all you've done is all you've posted then I fail to see where you have any justification for any claim you've made. You've talked about spin and charge and symmetry groups and fibre bundles and solitons but if you haven't got anything more than what you've provided in posts I cannot possibly see how you can be seriously planning to get a PhD in this stuff in the not too distant future. If someone had asked me a month ago how much experience I have with solitons I'd have said "Minimal, mostly in the realm of fluid mechanics" but if your experience with them is completely represented by what you've posted you have even less experience with them than me. And you
definitely fall a long way short when it comes to symmetries, topologies, field theory and differential geometry. Sure, those happen to be precisely what I did
my PhD in but that means I know the kind of level of understanding needed to get a PhD in this area and your posts haven't convinced me you even understand them, never mind have contributed enough to physics to be worthy of a PhD.
Are you honestly planning to defend a thesis in this in the next year?
Really?
I can try to answer single concrete questions, but please don't ask me to write everything again.
I'd settle for a few links, either to papers of yours on this stuff or to posts where you give the specifics, which I guess I must have missed. I'm not asking you to post loads of details
again because that would imply you've done it before.
Standard approach to physics is by guessing Lagrangian terms to fit numerical values and ignoring conceptual problems like infinities or that we cannot define electric field in the center of charge ... here I'm trying to go opposite way - start with recreating qualitative picture ...
That is just flat out wrong. The issue of infinities is central to quantum field theory, its what renormalisation is all about. Lagrangians are
massively constrained by renormalisability and symmetries. There is only 1 U(1) renormalisable 4d Lagrangian, that of QED. Its Landau pole shows it should be embedded into a larger model. The next largest one is SU(2), which is the electroweak model. The next one is QCD. Symmetries and renormalisability reduces infinitely many combinations to a handful. And anything which works by point particles is doing to have these issues, even Maxwell's electromagnetism.
here I'm trying to go opposite way - start with recreating qualitative picture ...
You've started with some pictures, throw in a ton of buzzwords and proclaimed you'll be defending a thesis on this stuff soon. Unless you've only shown us 0.001% of the material you've done either you're lying or your thesis isn't worth the $50 it'll cost to print and bind. I've asked repeatedly you provide more than just the qualitative picture and you've failed to provide at every turn.
You can look at it as a trial to avoid undefined infinites - by using Higgs-like tensor potential ... have you seen an attempt to glue electric field around charge without nonphysical situations we had to get used to?
You haven't provided a mechanism to avoid it at all. You've yet to show you can recover anything like the Standard Model. You haven't even shown what you're referring to can be associated to a particle, other than just asserting it.
Or since you are the smart topology guy - just think about it as an interesting topological exercise - classification of topological singularities of not just vector field as usually, but ellipsoid/reper field in 3D or 4D ... looking simple but quite unknown - instead of asking a guy who you see incompetent, just show what you can tell about it
Where to start..... Let's look at this line by line :
"
Or since you are the smart topology guy - just think about it as an interesting topological exercise"
If it was simple enough for someone with
my level of understanding to just knock out in an evening without prior reading then it wouldn't be worth a PhD. It wouldn't be worth a paper. It wouldn't be unknown. Anything worth a PhD is
difficult, that's what makes a PhD worth something. The nut job Terry Giblin made a similar request of me, to solve a problem he claimed was worth a PhD, that it was supposedly something I should be able to do easily. He failed to grasp, as you seem to have failed too, that if it were that easy my criticism it wasn't worth a PhD would be
utterly valid.
"
classification of topological singularities of not just vector field as usually, but ellipsoid/reper field in 3D or 4D"
A number of issues there. Firstly, you don't make it clear whether you mean a 3 or 4 rank tensor field over any base manifold or you mean a general tensor field defined on a base manifold of dimension 3 or 4. Secondly I've already told you what such concepts are called in the literature, they are 'parallelisable manifolds'. Tori are such that its possible to define a non-degenerate frame bundle over them, each point in the torus can have a set of axes associated to it which vary smoothly over the torus and where the axes are never degenerate (ie they form a basis to the fibre vector space). Tori are
very nice manifolds, along with standard Euclidean or Minkowski manifolds, but they are the exception, not the rule. Pick a general compact space and you're not going to have such nicities in general.
I said in a previous post I'd typed something out and it'd gotten eaten by my crappy netbook. In that post I explained this issue in detail. For instance, consider a 6 dimensional compact space which is parallelisable, such as a 6 dimensional torus. It's possible to define a 3 form not unlike the Maxwell 2-form, $$F_{3} = dB_{2}$$ (while for Maxwell you have $$F_{2} = dA_{1}$$) where B is the string 2 form NS-NS field. If the space is parallelisable then you can define the components of the tensor in terms of the frame bundle basis at a particular point $$\eta^{a} = N^{a}_{b}dx^{b}$$ where $$dx^{b}$$ is the usual cotangent bundle basis and N is a transformation matrix. The components are then $$F = F_{abc}\eta^{abc}$$ where [texc]\eta^{abc} = \eta^{a}\wedge \eta^{b} \wedge \eta^{c}[/tex] using usual exterior calculus. This is just a generalisation of the Maxwell tensor components $$F_{ab}$$ where $$F_{0i} = -E_{i}$$ and $$F_{ij} = \epsilon_{ijk}B_{k}$$ in electromagnetism. However, if the space is not parallelisable then $$\eta^{a}$$ are not globally definable, N is not non-singular. Then you have to use the de Rham cohomology for $$H^{3}$$, which is of even dimension and has symplectic structure, $$F_{3} = F_{I}\alpha_{I} - F^{J}\beta^{J}$$ where $$\int_{M}\alpha_{I} \wedge \beta^{J} = \delta_{I}^{J}$$. Via the homology-cohomology duality the dimension of $$H^{3}$$ is $$2h^{1,2}+2$$ where $$h^{1,2}$$ is a Hodge number and related to complex cycles in M and the Betti number, the number of usual real cycles in M. The cycles define the cohomology, which defines the harmonic forms, which defines the massless field content of the model. Thus the extension of usual Maxwell field tensor to higher dimensions is defined.
But wait, if you turn on a field then, via general relativity, you warp the space so in actual fact the massless fields aren't massless and N isn't constant and you can't use harmonic analysis and your exterior derivative picks up a torsion term and
then your torus is no longer a torus, its been 'twisted' and before you know it you're knee deep in twisted generalised Calabi Yaus, generalised geometry and Hitchin functionals.
And how do I know this? Because that's what I spent several years doing. Anyway, back to your post :
"
looking simple but quite unknown"
It only looks simple to you because you clearly haven't got a clue just how deep the rabbit hole is.
"
instead of asking a guy who you see incompetent, just show what you can tell about it"
I've reeled this post off the top of my head and I've given more detail about this stuff in one post than you've given in
all your posts. The fact you challenge me to answer a question, which you admit hasn't been answered, as if I should be able to answer it in a single post without much effort illustrates precisely what I've been getting at. You seem to have no clue as to the level of detail expected and known in the physics community, no understanding of the material yourself and gross naivety as to what constitutes being worth a PhD. You say you've given detail but you haven't. If you think you have given details then you
again show how naive you are about what constitutes 'detail' when you're working at this level in physics.
You posted this stuff on PhysForums too. Why are you doing that? If your *cough* research is guided so much by the views of laypersons and cranks then you should re-evaluate your research. Actually, scrub that, you
should re-evaluate your research regardless, because you are at best very naive and poorly read about this stuff and at worst flat out delusional about your ignorance.
